Teaching Elementary Math 1

Pataasin ang iyong marka sa homework at exams ngayon gamit ang Quizwiz!

Can be represented in two ways; measurement and partition.

Division

Line segment

Part of a line with two endpoints.

Return (RET)

An output instruction that causes the scan to return to the main program. It can be a conditional, or unconditional output depending on the desired operation of the subroutine.

What does the geometry standard call for students to do?

Analyze characteristics of geometric shapes and make math arguments about their relationship and use it to visualize, and for spatial reasoning and geometric modeling for problems.

Number multiplied by the sum of two other numbers can be handed out, or distributed, to both numbers, multiplied by each of them separately, products added together. Ex: a(b+c) = (a 3 b) + (a 3 c)

Distributive

reflecting on the design

8. reflect and refine

Natural numbers

counting numbers (1,2,3,4,5)

Which of the following skills would student develop first?

Counting objects

3-dimensional solid figure. Ex: 6 faces, 12 edges, + 8 vertices. F + V = E + 2

Cube

Associative Property

+/x are _________:; the order that numbers are grouped in +/x does not affect the result. ex: 2x(10x6)=(2x10)x6=120->a(bc)=(ab)c

What are the three facets involved in a math lesson?

Content Level Concept Development Class Coherence

what should be assessed

conceptual understanding, procedural fluency, strategic competence, adaptive reasoning

objective=

concrete

Teach _____ to _____

concrete, abstract

Isoceles Triangle

Has at least two sides that are exactly the same length. This forces two of their angles to also be acute angles of exactly the same size.

Range

The difference between the highest and lowest values in a data set.

When is a student's understanding of math deeper and more lasting?

When they connect mathematical ideas and see the coherent whole.

1. ex. 190^0 = 1.

When you have a number that is being raised to zero it always equals ______.

What is classification?

Where a child can name and identify sets of objects according to appearance, size, or other characteristic.

What is decentering?

Where a child can take into account multiple aspects of a problem to solve it.

What is elimination of egocentrism?

Where a child can view things from another person's perspective.

What is seriation?

Where the child is able to arrange objects in order according to size, shape, or any other attribute.

Spatial temporal knowledge

How children think about space and time, is another area that can be strengthened using puzzles such as working with a Möbius strip

Rate

How quickly computations are made

In the number 1984 the number 9 would be in what place value?

Hundred: 900

TYPES OF REASONING -

INDUCTIVE VS. DEDUCTIVE

What is reversibility?

Where the child understands that objects can be changed and returned to their original state.

common fraction

a number in the form a/b, such that a and b are whole numbers

Prime numbers

a number with exactly two whole- number factors (1 and the number itself) 2,3,5,7,11,13

Commutative property of addition and multiplication

+/x are commutative: switching the order of two numbers beging added or multiplied does not change result. ex: 100+8=8+100->a+b=b+a 100x8=8x100->ab=ba

zero property of multiplication

a number x 0 = 0

multiplicative identity element

a number x 1 = the number

Assessment types and applications ( error patterns, task analysis, student verbalizations )

-

Connecting an objective to standards and research based norms

-

RIDE ( Intervention )

- Read the problem correctly - identify the relevant information - -

a cost

a price

guess and check

a problem-solving strategy in which a student identifies the best solution they can and then checks its accuracy

Major operations on rational numbers

addition subtraction multiplication division

Scatterplot

A graph showing paired data values. A scatter plot can suggest various kinds of correlations between variables with a certain confidence interval

Histogram

A graph showing the results of tabulating the number of items found in defined categories and shown using vertical bars; often referred to as a bar chart.

Polygon

A simple closed figure with any number of sides (square, triangle, hexagon etc.).

Protractor

A square, circular or semicircular tool, typically in transparent perspex, for measuring angles. The units of measurement utilized are usually degrees

Algorithm

A step by step procedure or formula for solving a problem

Common Factor

A number that is a factor of two or more numbers. A common factor of 9 and 6 is 3.

Principle

Generalizations, develop rules

Equity

High expectations and strong support for all students

Concrete

Manipulatives --algebra tiles

Distance around the object. Ex: Around fence

Perimeter

Common multiple

A number that is a multiple of two or more numbers. A common multiple of 2 and 3 is 6.

Addend

A number that is added. in 5 + 8 = 13, the addends are 5 and 8

Dividend

A number that is divided by another number. For example, in 36/4 = 9, 36 is the dividend.

Association

Words or symbols

pictorial models

a visual representation which models the essential characteristic of a mathematical concept

Multiplicand/multiplier

A number that is multiplied by another number. In 7 x 4 =28 the multiplicand is 7 and the multiplier is 4.

proportion

equation in which a fraction is set equal to another

confidence interval

estimated range of values, likely to include a particular population parameter

Concrete

hands on manipulatives

irrational number

have a decimal, aren't whole

parallel postulates

if 2 lines make perfect right angles, they are parallel

india was primary source of developments

in 6th-12th century

What measure could be used to report the distance traveled in walking around a track?

kilometers

Modeling the Operations: Student matches the elements of a given group with abstract numbers. Ex: 3 rabbits eating 4 carrots daily would be set up like: 3 x 4

Abstract method

slope

rate of change

Grades K-1 Manipulatives

Beads, straws, tiles, macaroni, buttons, paper clips, toothpicks, legos, board games, pictographs

Teach _____ rather than just facts.

Concepts

Math should be taught _____

Conceptually

Modeling the Operations: This method the teacher allows the students to use real objects.

Concreate method

Sequence of instructional tools

Concrete, pictorial, abstract

Operations are what?

systematic

Inverse operations

Operations that are the opposite of each other and cancel each other out. Addition and subtractions are inverse operations, as are multiplication and division

Whole numbers

natural numbers and zero

Learning Logs

1. Communication of student understanding 2. For writing and drawing about concepts and solutions 3. Also for reflective responses on divergent questions --drawing solutions

Write down lesson plans how long in advance?

a week before

strategic competence

ability to formulate, represent, and solve mathematical problems

Alternative assessment

non traditional options to assess students' learning. portfolios, journals, notebooks, projects, and presentations.

3 common probability distributions

normal, binomial, geometric

*study skill 17.2

page 176

Exponent

A number that tells how many times the base is to be used as a factor or to be multiplied by itself. in 2 to the 3rd power, 2 is the base and 3 is the exponent meaning 2x2x2

Divergent Questions

1. Extending students' thinking beyond basic facts (predicting)(More than 1 answer/solution) Benefits: -Instigates discovery -Probes for deeper understanding -Uses upper levels of Bloom's -Encourages creative thinking and reflecting

Think pair share

1. For peer discussion 2. Limited to brief periods (15-30 seconds) 3. Good for less verbal students 4. Partner talk or shoulder talk

Wait time

1. Pause time (at least 3-5 seconds) between question and response Benefits: -Student responses increase dramatically over time -Confidence is strengthened -Struggling students contribute more -More reflective thinking is exercised

Factor

A number to be multiplied or a number that divides evenly into a given second number is a factor of that number. In 2x3 =6, 2 and 3 are factors of 6

To perform Prime Factorization you can use?

A number tree

Irrational Number

A number whose decimal form is nonterminating and nonrepeating. Irrational numbers can't be expressed as fractions.

Ordered Pair

A pair of numbers used to locate a point on a coordinate plane is called an ordered pair. An ordered pair is written in the form (x, y) where x is the x-coordinate and y is the y-coordinate.

Rhombus

A parallelogram with four congruent sides.

Convergent Questions

1. Questions that focus on basic knowledge and comprehension (factual answer) Benefits: -Directs attention to parts of an activity -Recalls order of info or procedures -Recalls needed facts -Ensure understanding ex. 7x8 -- only one answer

worthwhile tasks

1. the students have no prescribed or memorized rules or methods to sole 2. there is not a perception that there is one correct solution

How much time is spent teaching a new concept?

1/3

What is pre operational thought?

A pattern of thinking that is egocentric, centered, irreversibly and non transformational and dominates thinking and reasoning of children.

Collaboration

A philosophy about hot to relate to others--how to learn and work.

Cognitive guided instruction (instructional practises )

- involves ____________________ of process, activation of prior knowledge, and explicit teaching - helps students become more efficient at selecting and using effective strategies - sequence: read math problems, paraphrase, draw a diagram, establish a plan, predict correct answer, solve the problem, evaluate for correctness

Formal Operational Stage

(11-adult) The stage during which the individual can think hypothetically, can consider future possibilities, can use deductive logic, and can think abstractly

Preoperational Stage

(2-7) The stage during which a child learns to use language but does not yet comprehend the mental operations of concrete logic Responses are based on their own perception and not on logical reasoning

associative property of addition

(4+2) + 3 = 4 + (2+3)

associatie property of multiplication

(6x8) x 2= 6 x (8x2)

Concrete Operational Stage

(7-11) The stage during which children gain the mental operations that enable them to think logically about concrete events

Sensorimotor Stage

(Birth-2 years) Infants are busy discovering relationships between their bodies and the environment

Jump (JMP)

) instruction is used to bypass portions of the ladder program the jump out will leave every output in their state

factors that contribute to difficulties with problem solving

- difficulty in reading makes understanding the math problem almost impossible - difficulty with logical reasoning, which is the basis of many story problems - insufficient instruction in math - mathematics education has focused on operations and not on understanding the reasons for operations or even a thorough understanding of numbers that are involved in operations

Purposes of assessment

- instructional design. - student placement. - monitoring student progress. - summative evaluation of student. - accountability. - validating student achievement. - true/false - worked out problems. - essays. - fill in the blank. - matching. - multiple choice. - program evaluation.

Research

- it supports conceptually based teaching or teaching concepts rather than just memorizing facts. Ex: students should learn the concept of multiplication before they learning multiplication itself. - it supports the use of communication, problem solving, and working in cooperative groups to develop a broader base of understanding. - it shows that teachers should promote the idea that math is used as a thinking tool for answering important questions, not just a set of rules and procedures to memorize.

Effective use of Formative Assessment in Math includes

- learning progressions, learning goals, descriptive feedback, self-peer assessment, collaboration

Guidelines for implementing and scoring a math CBM

- multiple examples of a variety of math problems - standardized directions, 2 minutes of time

Concrete, semi concrete, and abstract( instructional practises )

- process for teaching mathematical computational and problem solving skills - sequence: hands on, tactile elements, illustrations or drawings - complete the problem without tactile elements or illstrations

Peer tutors ( intervention )

- the use of tutors is beneficial both in math and other subject areas - the use of __________ ___________ is necessary in effective peer tutoring programs

Goal setting ( intervention )

- used to address poor attitudes in math - self regulation and goal setting were linked to higher levels of achievement in math

At what degree does the F and C tempertures meet?

-40 degrees

Number Sequence

-Each number being raised to an exponent: 1,3,9,27...(3*0,3*1,3*2,3*3) -Each number may be a prime number in order (2,3,5,7...) or skipping every other prime (2,5,11,13...) -Each number might be the sum of that number plus the number before it (2,4,6,10,16)

Minuend

...

standard numerals

...

Piaget stages of development

1- Birth to 2 Years of Age: (Sensorimotor Stage) First stage of child's mental development which mainly involves sensation and motor skills such as hearing, seeing, feeling, tasting, moving, manipulating, biting, chewing, etc. In this stage the child does not know that physical objects remain in existence when out of sight. 2- 2 to 7 years of age (preoperational Stage) In this stage children use their mental ability to represent events and objects in various ways like using symbols, gestures and communication ...they are not yet able to conceptualize abstractly and need concrete physical situations to help with understanding. 3- 7-11 years of age (Concrete-Operational Stage) At this stage the child starts to conceptualize, creating logical structures that explain physical experiences. Abstract problem solving is also possible at this stage. Math problems can be solved with numbers and not just objects. 4- 11 years to adulthood (Formal-Operational Stage) Children become more systematic and reasonable; they reason tangibly and are also capable of reasoning and thinking in more abstract, hypothetical and idealistic terms.

Types of learning

1- association: words or symbols Example: students associate the word triangle by its symbol. 2- concept: relational or concrete attributes Example: the corresponding angles are equal and the ratios of corresponding sides are equal. 3- principle: generalization, developed rules Example: the area of trapezoid is developed from the concept of a trapezoid and the area of triangles, rectangles and/or parallelograms. 4- problem solving: putting together concepts and principles to solve a problem new to the learner Example: given a composite figure the student determines the area using the areas of triangles and rectangles.

Types of Assessment

1- formal vs informal. 2- diagnostic / formative / summative. 3- standardized vs criterion referenced.

Types of reasoning

1- inductive (informal): This reasoning goes from the specific to general. It uses observations and patterns to infer a generalization. 2- deductive (formal): This reasoning process reaches conclusions based on accepted truths and logical reasoning. And it goes from general to specific.

Piaget stages of development

1- sensorimotor (birth-2 years of age) First stage of child's mental development which mainly involves sensations and motor skills, such as: hearing, seeing, feeling, tasting, moving, manipulating, biting, chewing, etc. In this stage the child does not know that physical objects remain in existence when out of sight. 2- pre-operational stage (2-7 years of age) Chicken use their mental ability to represent events and objects in various ways like using symbols, gestures and communication. 3- concrete-operational stage (7-11 years of age) The child start to conceptualize, creating logical structure that explain physical activities. Abstract problem solving is also possible at this stage. Math problems can be solved with numbers not just objects. 4- formal-operational stage (11 years +) Children become more systematic and reasonabl; they reason tangibly and are also capable of reasoning and thinking in more abstract, hypothetical and idealistic terms.

A general problem solving method that can be applied to many types of problems

1- understand. 2- plan. 3- solve. 4- check.

Performance based assessments

1. Authentic- real use of skill concept, student generated. (Students show what they know) 2. Students demonstrate while teachers assess 3. Assessment is semi-concrete/concrete 4. Individual or group

commutative property of multiplication

7 X 9 = 9 x 7

What are the five types of questions teachers should ask to elicit justification and reasoning?

1. Questions that help students learn how to make math meaningful. 2. Questions that help students become self-relient and determine if something is mathematically correct? 3. Questions that help students learn to reason mathematically. 4. Questions that help students learn to solve problems. 5. Questions that relate to helping students connect mathematics, it's ideas, and application.

Group Think Lab

1. Teacher designs problems for students to solve. 2. Students are put into groups of 2 to 4 (solving problems with the use of materials) 3. Students via math reasoning and use of appropriate math manipulatives solve the problem and share results with whole group.

5 Developmental math goals

1. The students will become math problem solvers 2. The students will learn to communicate mathematically 3. The students will learn to reason mathematically 4. The students will learn to value mathematics 5. The students will become confident in one's ability to do/use math

Multi-sensory approach

1. Using more than one sense for learning a skill/concept (based on brain research of left/right hemisphere use) 2. Using a visual/auditory/kin-esthetic process for teaching/learning.

Math Word Wall

1. Word 2. Definition 3. Drawing 4. Personal Connection

the before phase of a lesson

1. activate prior knowledge 2. be sure the task is understood 3. establish clear expectations

5 sources of numbers that may be useful for teaching place value to older students

1. an odometer 2. numbers from students science or s.s. text 3. numbers from population of school 4. population data from town, country, state, or county 5. financial data page from newspaper

three phases of learning

1. before: getting ready 2. during: students work 3. after: Class discussion

what are several numeration concepts

1. cardinality 2. grouping pattern 3. place value 4. place value (base 10) 5. one digit per place 6. places- linear/ordered 7. decimal point 8. place relation/regrouping 9.implied zeros 10. face times place 11. implied addition 12. order 13. verbal names (0-9) 14. verbal names with places 15. periods and names 16. naming in the ones period 17. naming multi digit numbers 18. decimal places and their verbal names

nrc five interwoven strands that compose proficiency and definitions

1. conceptual understanding - refers to understanding mathematics concepts and operations 2. procedural fluency - being able to accurately and efficiently conduct operations and math practices 3. strategic competence - ability to formulate and conduct math problems 4. adaptive reasoning - refers to thinking about, explaining and justifying mathematical work 5. productive disposition - appreciating the useful and positive influences of understanding mathematics and how ones disposition toward mathematics influences success

preparing a lesson 3 brackets

1. content and task decisions 2. lesson plan 3. reflecting on the design

tiered lessons considerations

1. degree of assistance 2. how structured the task is 3. complexity of the task given 4. complexity of process

content and task decisions

1. determine the learning goals 2. consider your students' needs 3. select, design, or adapt a worthwhile task 4. design lesson assessments

during phase of lesson

1. let go: don't give too much guidance, embrace struggle 2. note students mathematical thinking; what ideas they are using, approaches, interactions 3. provide appropriate support: support student thinking without taking away their responsibility, solving in a way that makes sense 4. provide worthwhile extensions: preparation for students who finish quickly

conversion of units

1. multiply by conversion factor 2. cancel the miles units 3. solve

how is math performance assessed and progress monitored

1. need to see if teacher has time to use an individually administered assessment or to use a group administered measure 2. needs to determine whether measure is designed for students in age range of students he is teaching progress monitoring - use CBM (curriculum based measurement) - documenting the extent to which the student is learning the critical elements in the curriculum that you have targeted

prenumber skills students need to progress in math

1. one-to-one correspondence-matching one object with another 2. classification - ability to group or sort objects on the basis of one or more common properties 3. Seriation - depends on recognition of common attributes or properties of objects, ordering depends on the degree to which the object possesses the attribute 4. algebraic principles

maximize test

1. permit students to use calculators 2. use manipulatives and drawings 3. include opportunities for explanations 4. use open ended questions

create a classroom environment for doing mathematics

1. persistance, effort, and concentration are important in learning math 2. students share their ideas 3. errors or strategies that didn't work are opportunities for learning 4. students listen to each other 5. students look for and discuss connections

after phase of lesson

1. promote a mathematical community of learners: productive discussion, helping students learn together 2. listen actively without evaluation: find out how they approve the problem 3. summarize main ideas and identify future problems: make connections, lay groundwork for future tasks

factors that influence math ability

1. psychological factors - intelligence/cognitive ability, distractibility, and cognitive learning strategies 2. education factors- quality and amount of instructional intervention across the range of areas of math 3. personality factors- persistence, self-concept, and attitudes toward mathematics 4. neuropsychological factors- perception and neurological trauma

how can RTI be used in math

1. screening - to determine if they have math problems in numeracy, math calculations, and problem solving 2. evidence-based math - instruction based on best research available 3. interventions - additional instruction through short-term interventions when students have difficulties that aren't addressed in math program 4. progress monitoring - students progress is documented to make sure they are staying on track and meeting benchmarks

All of the following are measurements of obtuse angles

110 degrees, 135 degrees, 91 degrees

The mass of a cookie is closest to:

15 grams

divisibility rule

1: divisible by 2 if even 2: divisble by there if sum of numbers is divided by 3: divisible by 4 if last 2 digits are divisible by four

Parallelogram

2 parings of parallel lines, 2 pares of congruent angles, opposite angles are congruent , diagonals bisect

Distributive Property

2(4 + 3) = 2 x 4 + 2 x 3 = 8 + 6 = 14

net

2-dimensional figure that can be cut out and folded up to make a 3-dimensional solid

Review is what part of the lesson?

2/3

What number comes next in this pattern? 3, 8, 13, 18

23

Learning Environment - Teaching Math

4 categories: 1. task - projects, questions, problems that students engage 2. discourse - manner of representing, thinking, talking, agreeing, disagreeing that teachers and students use to engage 3. environment - setting for learning 4. analysis - systematic reflection in which teachers engage

Quadrilateral

4 sides sum of angles =360

four point rubric

4. excellent; full accomplishment; met consistent, minor errors 3. proficient; substantial accomplishment; errors minor, needs minimal feedback 2. marginal; partial accomplishment; lack of evidence of understanding, further teaching is required 1. unsatisfactory; little accomplishment; fragments are accomplished, little or no success

The typical lesson lasts how long?

45 min

commutative property of addition

5 + 3 = 3 + 5

Which is a good example to explain the idea of "doubles plus one" to students?

5 + 6 = 11

polyhedra

5 regular solids- cube, tetrahedron, octahedron, icosahedron, dodecahedron

lesson plan

5. plan the before phase of the lesson 6. plan the during phase of the lesson 7. plan the after phase of the lesson

At what age do children tend to fix their attention on a single aspect of a relationship?

6 years old/pre operational stage.

What is the smallest multiple off 12,15 and 20?

60 Start with the number 20 and go out by 10s.

Dividend, divisor and quotient? 80/8=10

80=dividend 8=divisor 10= quotient

Plane

A 2-sided surface

Geometry

A branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Considered to have its formal beginning in about 300 BC, when the Greek mathematician Euclid gathered what was known at the time, added original work of his own, and arranged 465 propositions into 13 books, called 'Elements'.

What is centration?

A child focuses only on one aspect of a situation or problem.

Guided participation

A child's performance, with guidance and support, of an activity in the adult world.

Common Denominator

A common multiple of two or more denominators for 1/6 and 5/8, it is 24.

Ratio

A comparison of two or more values( 1/2, 4/6 or 1:2, 4:6) or ( 1 is to 2; 4 is to 6).

Iteration

A computational process in which a cycle of actions or operations is repeated, generally to get closer to a final answer.

Iteration

A computational process in which the same steps are repeated until the final answer is found.

Factor tree

A diagram showing how a composite number breaks down into its prime factors.

Factor

A factor of a number divides the number evenly. This means the quotient is a whole number and the remainder is 0. ex: 24->2,3,4,6,8,12

Face

A flat surface of a solid figure

Fractions

A fraction is part of an entire object. ex: 1/5,3/7,12/50 1=Numerator _ 5=Denominator parts of integers and therefore fit between then, when comparing size. In order to compare fractions, their DENOMINATORS must be equal.

Polyhedron

A geometric figure solid with flat faces and straight edges.

line graph

A graph in which a line shows changes in data, often over time

Coordinate system

A graph with a horizontal number line (x-axis) and a vertical number line (y-axis) that are perpendicular to each other. The point of intersection is called the origin and labeled 0 on the graph. An ordered pair (x,y) is used to name a point on a coordinate system.

Performance based assessment ex.

A group project that designs a seating arrangement for a certain amount of people

Expression

A mathematical phrase that combines numbers and/or variables using mathematical operations. An expression is a representation of a value; for example, variables and/or numerals that appear alone or in combination with operators are expressions. Examples: 5 + 6 - (3 + 2)/18 a + b - c

Mass

A measure of how much matter is in an object. Commonly measured by how much something weighs. But weight can change depending on where you are (such as on the moon) while this stays the same.

Inductive Reasoning

A method of drawing a probable conclusion from an emerging configuration of data. It occurs by analyzing observations and discovering common patterns. When patterns repeat for an extended period of time, an analyst can logically predict that those patterns will continue to repeat.

Multiples

A multiple of a number is the product of that number and any whole number. ex: 3->6,9,12,15...

Pi

A name given to the ratio of the circumference of a circle to the diameter. That means, for any circle, you can divide the circumference (the distance around the circle) by the diameter and always get exactly the same number. It doesn't matter how big or small the circle is, Pi remains the same. While Pi has a decimal that never ends, it is usually estimated as 3.14.

Inverse Relationship x/division & +/-

A number fact is made up of 3 numbers. -These 3 numbers can be used to make up other number facts. 3,4,7 Addition->3+4=7, 4+3=7 Subtraction->7-3=4, 7-4=3 Multiplication/Division->Fact families 4x4, 5x5, 6x6

Power (of a number)

A number found by multiplying the number by itself one or more times.

Rules of Divisibility

A number is divisible by another number if the quotient is a counting number and the remainder is 0. 2: The number is even. 3: The sum of the digits is divisible by 3. 5: The last digit is 0 or 5. 6: The number is even and divisible by 3. 9: The sum of the digits is divisible by 9.

Average

A number obtained by dividing the sum of tow or more addends by the number of addends (2+4+6 = 12/3 = 4

Ordered pair

A number pair, such as (2,3), in which the 2 (x-axis) is the first number and the 3 (y-axis) is the second number.

Prime Number

A number that can be divided evenly only by 1 or itself. And it must be a whole number greater than 1.

Divisor

A number that divides another number. In the example 36/4 = 9, the 4 is the divisor.

Understanding Word Problems

A problem solving process can help students make sense of the problems. They can do this by reading the problem more than once, annotating words and numbers, visualizing the situation etc.

Mathematical Proofs

A proof is a rhetorical device for convincing someone else that a mathematical statement is true or valid.

Parallelogram

A quadrilateral (any four-sided polygon) with opposite sides parallel and congruent.

Trapezoid

A quadrilateral with one pair of parallel sides. These sides are called the upper and lower bases.

Function

A relation that uniquely associates members of one set with members of another set. There will always be three main parts: The input, The relationship, The output. For instance, 4(input) X 2(function) = 8(output)

Circle Graph

A round graph that uses different-sized wedges to show how portions of a set of data compare with the whole set.

Heterogeneous ability grouping

A strategy that groups students of varied ability instead of by grade/age level.

Array

A systematic arrangement of objects or numbers, generally in rows and columns

Acute triangle

A triangle that contains acute angles (<0 and > 90 degrees

Scalene Triangle

A triangle where all three sides are different in length.

Equilateral Triangle

A triangle with all three sides the same length. All equilateral triangles are also isoceles triangles. All three internal angles are also congruent to each other and are each 60°.

Pictograph

A visual representation used to make comparisons. A key always appears at the bottom of a pictograph or picture graph showing how many each object represents

Visual Representations

A way for students to access abstract math ideas. Drawing a situation, graphing lists of data, or placing numbers on the number line. This strategy gives students tools and ways of thinking that they can use as they advance their learning.

What is mathematical communication?

A way of sharing ideas and clarifying understanding.

Algorithms

A way to solve problems without visual models. Algorithms are standard step by step procedures for solving mathematical problems.

Set

A well-defined collection of objects or numbers.

Composite number

A whole number greater than 1 that is not a prime number (e.g 4, 6, 9, 10, 12 etc).

2 pairs of angles of 2 triangles are = in measurement + pair of corresponding sides are equal in length = congruent

AAS (Angle-Angle-Side)

2 triangles are = measurement, included sides = then = congruent

ASA (Angle-Side Angle)

What are numbers and operations curriculum for sixth graders?

Able to write prime factorizations using exponents, compares and orders non-negative rational numbers.

Number's distance from zero on the number line. Ignores + or - sign of a number. |x| used on calculator. Ex: |-5| = 5

Absolute Value

Components of math fluency

Accuracy, Automaticity, Rate, Flexibility.

Learning

Actively build new knowledge from experience and previous knowledge

Service learning

Activity that promotes learning and development through participation in a meaningful community service project.

The Zero Property of Addition/Multiplication

Adding 0 to a number leaves it unchanged. We call 0 the additive identity ex:88+0=88->a+0=a Multiplying any number by 0 gives 0 ex: 88x0=0->0x1003=0

Associative Property

Addition: a(b+c)= (a+b)+c Multiplication: a(bc)= (ab)c

Commutative Property

Addition: a+b= b+a Multiplication: ab=ba

Identity

Additive : a + 0= a Multiplicative: a x1=a

The number 0 added to any natural number yields a sum that is the same as the other natural number is?

Additive identity, or identity element of addition

Inverse Property Additive/ Multiplicative

Additive-> a+(-a)=0 Multiplicative->ax(1/a)=1

Inverse

Additive: a+ (-a)= 0 Multiplicative: a x (1/a)=1

Requires knowledge of the concepts of a variable, function, and equation.

Algebraic Expressions

A statement that is written using one or more variables and constants that shows a greater than or less than relationship. Ex: 2x+ 8 > 24

Algebraic Inequality

fenced zone,

All rungs that exist between the two MCR instructions are part of a

Interior Angles

An angle inside a shape. When you add up the Interior Angle and Exterior Angle you get a straight line, 180°. The sum of the measures of the interior angles of a triangle is 180 degrees. All the interior angles of a square are right angles -- that means that they are all 90 degrees.

Obtuse angle

An angle that measures greater than 90 degrees and less than 180 degrees.

Acute Angle

An angle that measures greater that 0 degrees and less than 90 degrees

Peer Tutoring

An instructional grouping practice where pairs of students work on assigned skills.

Manipulatives

An object which is designed so that a learner can perceive some mathematical concept by manipulating it. The use of manipulatives provides a way for children to learn concepts in a developmentally appropriate, hands-on and an experiencing way. Mathematical manipulatives are used in the first step of teaching mathematical concepts, that of concrete representation.

What opportunities do digital videos provide for students?

An opportunity to evaluate their own explanation of problem solutions.

Jump to Subroutine

An output instruction that causes the scan to jump to a specified subroutine file (U3 through U255) when input conditions are true.

Ratio

Another way to write a fraction -if the ratio is 2:3, it means two out of 3 or 2/3

Some/Sum

Answer to addition problem

Rational number

Any number that can be expressed as a fraction a/b where a and b are integers and b does not equal 0, such as 3, 3/1, 1/4, .34 and 56 percent.

Math Scene Investigator

Application used to solve mathematics word problems. Involves inspecting and finding clues, plan and solve, and retrace.

Expository instruction

Approach to instruction in which information is presented in more or less the same form in which students are expected to learn.

Peer tutoring

Approach to instruction in which one student provides instruction to help another student master a classroom topic.

Discovery learning

Approach to instruction in which students develop an understanding of a topic through firsthand interaction with the environment.

Learner-directed instruction

Approach to instruction in which students have considerable say in the issues they address and how to address them.

Convergent thinking

Approach to instruction in which students work with a small group of peers to achieve a common goal and help one another learn.

Cooperative learning

Approach to instruction in which students work with a small group of peers to achieve a common goal and help one another learn.

Teacher-directed instruction

Approach to instruction in which the teacher is largely in control of the content and course of the lesson.

Authentic activity

Approach to instruction similar to one students might encounter in the outside world.

Learner-centered instruction

Approach to teaching in which instructional strategies are chosen largely on the basis of students' existing abilities, predispositions, and needs.

Reciprocal teaching

Approach to teaching reading and listening comprehension in which students take turns asking teacher-like questions of classmates.

What are numbers and operations curriculum for Pre-K

Count by ones to 10 or higher, by fives or higher, and combine, separate, and name concrete objects. Begin to recognize the concept of zero, identify first and last in a series, and compare concrete objects using same, equal, more than,and less than.

Squaring the measure of the side of the square. Rep as: A = s2

Area of a square

Found by multiplying the measure of the length by width of the rectangle. Rep as: A = 2xw

Aread of a rectangle

Logical-mathematical knowledge/skills to work on

Arithmetic, classifying, ordering, number concdepts

Can model a multiplication problem. First number is vertical and second number is horizontal.

Array

How is algebra best learned?

As a set of concepts and techniques tied to the representation of quantitative relations and as a style of mathematical thinking for formalizing patterns, function, and generalizations.

At what age do children develop a basic understanding of numbers?

As early as two.

Types of learning

Association, concept, principle, problem solving

Property: grouping three or more addends or factors in a different way does not change the sum or product. Ex: (3+7)+5 results the same sum as 3 + (7+5)

Associative

Grade 6 Manipulatives

Attribute logic blocks, algebra tiles, video games, math software games, iPods, balanced metric eight scale, computer/internet, smart boards, digital cameras, computer templates

Research Basis:

Ausabel, Bloom, Brownell, Gagne, Piaget, Vygotsky, etc. Keep in mind that it is not so much the name, but what their research tells us. The Competency does not specify the researcher.

Mean

Average of a set of values

Grade 3 Manipulatives

Base-ten blocks, tangrams, patterns, scales: customary and metric, spinners, dice, games, measuring tape, 0-99 charts, bar graphs

Flexibility

Being able to solve problems in more than one way and selecting the most appropriate method.

"A job worth doing, is worth doing right."

Ben Franklin

Grade 2 Manipulatives

Blocks, cubes, chips, measuring cups, money, number lines, dominos, unified cubes, ruler, balance scales, linear graphs

Abstract is _____

Brains only

"Every pole has to be laid in the right place..."

C.T. Studd

Square Root (SQR)

Calculates the square root of the source and stores the value in a specified destination.

Logical mathematical knowledge

Can be developed by using arithmetic, classifying and ordering and using number concepts activities.

Tiling

Can be used to relate to calculating the area of rectangles wherein a rectangle is divided into unit squares and counted to find the area.

Discrete Models

Counters

NCTM standards proposed 5 categories of questions that teachers should ask to elicit justification and reasoning

Category 1: questions that hep students learn how to make learning math meaningful "Do you agree (disagree) with the correct answer?" Category 2: questions that help students become self-reliant and determine whether something is mathematically correct "Does that make sense?" Category 3: questions that help students learn to reason mathematically "How could you explain this is your own words?" Category 4: questions that help students learn to solve problems "What would happen if (if not)....?" Category 5: questions that relate to helping students connect mathematics, its ideas, and its applications "Can you write another problem like this one?"

Negate (NEG):

Changes the sign of the source and stores the value in a specified destination.

Physical knowledge

Children get by observing their physical environment, seems basic. However, good listening and observing skills must be sharpened through teaching these skills directly and practicing them.

Problem-based learning

Classroom activity in which students acquire new knowledge and skills while working on a complex problem similar to those in the outside world.

Project-based learning

Classroom activity in which students acquire new knowledge and skills while working on a complex, multifaceted project that yields a concrete end project.

Ausabel

Cognitive frameworks

What must the curriculum be like in mathematics instruction?

Coherent, focused on important mathematics, and well-articulated concepts across the grades.

Inductive reasoning

Collecting data to draw a conclusion that may or may not be true.

A whole number that is a multiple of two or more given numbers. Ex: 2, 3, and 4 are 12, 24, 36, 48....

Common Multiple

Property: Order of adding addends or multiplying factors does not determine the sum or product. Ex: 6x9 gives the same product as 9x6. Division and subtraction can not use this.

Commutative

Inner Limit:

Compares a test value to an upper and lower limit. If the test value is between upper and lower limit, the instruction is true.

Outer Limit

Compares a test value to an upper and lower limit. If the test value is outside the two limits, the instruction is true

Focal points: 6th grade

Compares and order nonnegative rational numbers, generates equivalent forms of rational numbers, including who numbers, fractions, and decimals Uses integers to represent real-life situations Is able to write prime factorizations using exponents; identifies facts of a positive integer, coachman facts, and the GCF of a set of positive integers

Less Than or Equal:

Compares two words, or one word to a preset value. The instruction is true if A <= B.

Greater Than or Equal

Compares two words, or one word to a preset value. The instruction is true if A >= B.

Not Equal:

Compares two words, or one word to a preset value. The instruction is true if A != B.

Less Than

Compares two words, or one word to a preset value. The instruction is true if A < B.

Greater Than

Compares two words, or one word to a preset value. The instruction is true if A > B.

Equal:

Compares two words, or one word to a preset value. The instruction is true if A = B.

Working memory

Component of memory that holds and actively thinks about and processes a limited amount of information.

Long-term memory

Component of memory that holds knowledge and skills for a relatively long time.

Number greater than zero which is divisible by at least one other number besides one (1) and itself resulting in an integer (has at least 3 factors). Ex: 9....1,3, and 9 are all?

Composite Number

Other counting numbers, composed of several counting number factors. Number 1 is neither prime nor composite, has only one counting number factor: 1.

Composite numbers

COMPREHENSION LEVEL

Comprehension questions have students represent information in their own words or in a different way = What are some ways to express a sum of 10? = What other problems could you write that show how 7, 3, and 10 are related to each other? Key words to aid in determining whether questions and activities are at this level include: = Tell in your own words = Describe in your own words = interpret in your own words = compare and contrast = explain in your own words = what does it mean.

Children in Preoperational stage experience problems with at least these 2 perceptual concepts

Concentration and Conservation

Grade 5 Manipulatives

Cuisenaire rods, virtual manipulative, graduated cylinders, timers, metric beaker for volume, metric trundle wheel, fraction tower cubes, connecting cubes, board games, reaction tiles, pie charts and graphs

What are the process in the concrete operational stage?

Decentering, reversibility, conservation, serialization, classification, and elimination of egocentrism.

Characteristics of children in the Concrete Operational Stage

Decentering: child can form a conclusion based on reason rather that perception Reversibility: child understands that objects can be changed and then returned to their original state Conservation: child understands that size of an object is unrelated to the arrangement or appearance of it Serialization: able to arrange objects in the order of size or shape Classification: child can name and identify sets of objects according ro appearance, size or there characteristics Elimination of Egocentrism: child is able to view things from another person's perspective

Decimals

Decimals are a method of writing fractional numbers without writing a fraction having a numerator and denominator. The fraction 7/10 could be written as the decimal 0.7. The period or decimal point indicates that this is a decimal.

Deductive Reasoning

Deductive reasoning is one of the two basic forms of valid reasoning. It begins with a general hypothesis or known fact and creates a specific conclusion from that generalization. The basic idea of deductive reasoning is that if something is true of a class of things in general, this truth applies to all members of that class.

What are numbers and operations curriculum for kindergarten?

Demonstrate part of and whole with real objects, uses patterns, and model and create addition and subtraction problems, uses whole number concepts to describe how many objects are in a set up to 20.

Modeling

Demonstrating a behavior for another; also, observing and imitating another's behavior.

2- diagnostic/formative/summative

Diagnostic - Used to pre-determine stu- dents' knowledge. Example: Pre-test before instruction on a unit has begun. • Formative - Used to determine students' ongoing learning and retention, whether or not additional instruction is needed. Example: Giving a quiz within a unit. • Summative - Used to determine mastery of material. Example: A semester exam or a project incorporating all objectives from a unit.

Diagnostic/ formative / summative

Diagnostic: used to pre-determine students knowledge. Example: pr-test before instruction on a unit has begun. Formative: used to determine students ongoing learning and retention, whether or not additional instruction is needed. Example: giving a quiz within a unit. Summative: used to determine mastery of material. Example: a semester exam or a project incorporating all objectives from a unit.

Is a line segment where two faces of a three-dimensional figure meet.

Edge

Types of problems for Multiplication/Division

Equal groups or repeated addition Area and array Combination Multiplicative Comparison

The NCTM identified principles that should guide mathematics instruction

Equity Curriculum Teaching Learning Assessment Technology

What are the six principles that should guide mathematics instruction?

Equity, curriculum, teaching, learning, assessment and technology.

Technology

Essential in teaching and learning math; it influences the teaching of math while enhancing and facilitating students' learning

Rounding numbers to the nearest decimal place for accuracy.

Estimating

Compute (CPT):

Evaluates an expression, and then stores the value in a specified destination. This instruction takes more processing power/scan time than those previously discussed.

Equivalent expression without parentheses. Ex: 263= 200 + 60 + 3

Expanded Form

Showing place value by multiplying each digit in a number by the appropriate power of 10.

Expanded Notation

Focal points: Pre-K

Explore concrete models and materials Counts to 10 or higher Names "how many" concrete objects Begins to recognize the concept of zero "same", "equal", "one more", "less than"

Symbolic way of showing how many times a number or variable is used as a factor. Exponent expressed out: 5x5x5=125

Exponential Notation

Expanded form

Expressing a number as factors [325 = (3x100) + (2x10) + (5x1)]

Rounding

Expressing a number to the nearest thousandth, hundredth, tenth, one, ten, hundred, thousand, and so on as directed.

Each of the plain regions of a geometric body is a face.

Faces

Any of the numbers, or symbols in mathematics that, when multiplied together, form a product.

Factors

1. Recalls information

Facts definitions, terms, properties, rules, procedures, algorithm, rote responses.

Estimating

Finding a number that is close enough to the right answer. You are not trying to get the exact right answer. What you want is something that is close enough. Also, involves the concept of predicting, or making an educated guess

Prime Factorization

Finding which prime numbers multiply together to make the original number.

Standard System of Measurement

Fluid Ounces Cups Pints Quarts Gallons Ounces Pounds Tons Inches Feet Yards Miles

Concentration

Focusing on only one aspect of a situation of problem

One is considered the what of multiplication?

Identity element

1- formal vs informal

Formal - An end-product testing. Example: A unit test, a final exam • Informal-The teacher observes, listens, and questions students in order to gather information regarding student learning. Example: When conducting guided prac- tice, the teacher observed that three stu- dents were "counting" using their fingers. • Formal assessments are often the familiar paper-and-pencil tests such as teacher- made tests or quizzes, but can also be such tests as diagnostic screenings, stan- dardized tests, spelling tests or final exams. Informal assessments may take the form of teacher observation, graded daily work, or student answers to oral questions. Teachers, who often need to know before the final grading period if their students are having difficulty, should use both.

Right

Forming a 90 degree angle

What must students do in data analysis and probability standard?

Formulate questions and collect, organize, and display relevant data.

Equivalent Fractions

Fractions that may look different, but are equal to each other. Two equivalent fractions may have a different numerator and a different denominator. For instance, The fractions 2/3 and 4/6 are equivalent. (A fraction is also equivalent to itself. In this case, the numerator and denominator would be the same.)

Congruent Triangles

If two triangles are congruent they will have exactly the same three sides and exactly the same three angles. The equal sides and angles may not be in the same position (if there is a turn or a flip), but they will be there.

Venn Diagram

Illustrates the similarities and differences between two sets of concepts

Grade 4 Manipulatives

Geometric solids, geoboards, calculators, protractors, tangrams, pentominoes, graphing paper, fraction kit, platform scale, graphic boards, scatter plot graphs

four step problem solving

George polya, 1. understand the problem (What is the problem?) 2. Devising a plan (Think about how to solve it) 3. carrying out the plan (implementation) 4. looking back (reflection, does it make sense, is it understood)

Accuracy

Getting the correct answer

What do tangrams do?

Give tactile experience with triangles.

Focal points: 1st grade

Has ability to create sets of 10 Reads and writes numbers up to 99 Separates a whole into two, three, or four equal parts and uses appropriate language Medels and creata addition and subtraction problems Identifies individual coins by name and value

Diagonal

In a polygon, a segment that connects one vertex to another vertex but is not a side of the polygon.

The largest positive integer that divides into the numbers without producing a remainder.

Greatest Common Divisor (GCD)

70%

Group discussion

NTCM 6 principles

Guide math instruction 1. Equity - high expectation and strong support for all students 2. curriculum - coherent, focused on important and well articulated math concepts across grades 3. teaching - understanding what students know and need to learn while challenging/supporting students 4. learning - learn math with understanding, actively building new knowledge from experience and previous knowledge 5. assessment - support the learning of math concepts & give useful information to both teachers and students 6. technology - influences the teaching of math while enhancing and facilitating student learning

A fourth-grade student with a specific learning disability in mathematics has just failed a mathematics quiz. When the student shows the quiz to his special education teacher, she sees that he has made the same error repeatedly, as illustrated by the following example: 239 +446=6715. The special education teacher arranges to meet later that afternoon with the student's fourth-grade teacher. The special education teacher's best recommendation regarding an instructional accommodation for the student would be to suggest that the fourth-grade teacher:

Have the student solve problems that reinforce the concept of place value

Which of the following experiences demonstrates a meaningful way to show math integrated with another subject area?

Having students read a short story and record how many times they come across a given set of sight words, then working with the class to construct a bar graph representing the data

How does the use of manipulatives help in mathemastics instruction?

Helps teachers move students from concrete through abstract stages of reasoning necessary for learning higher-level concepts.

ANALYSIS LEVEL

Here is where outcomes are = thought through, = analyzed, and = alternatives are considered. Jake now has more than $12. He had $5 at the beginning of the day and completed several jobs in his neighborhood for which he was paid. = what is the minimum amount Jake could have earned working today? = How did you arrive at your answer? = Are other answers possible? Keywords in questions and activities might include: = reason = why? = what are the causes? = what are the consequences? = what are the steps in the process? = how did you start? = what are some examples? = list all solutions possible? = what problems might arise?

What is equity in mathematics instruction?

High expectations and strong support for all students.

Cognitive apprenticeship

Mentorship in which a teacher and a student work together on a challenging task and the teacher gives guidance about how to think about the task.

Pythagorean Theorem

In any right triangle, the are of the square whose side in the the hypotenuse (longest side and opposite from the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the sides that meet at the right angle). Where c is the length of the hypotenuse and a and b are the lengths of the two sides, this may be expressed as a squared + b squared = c squared.

Place Value

In our decimal number system, the value of a digit depends on its place, or position, in the number. Each place has a value of 10 times the place to its right.

Rate

In tracking how many exercises were correctly done in a fixed amount of time.

Explicit and systematic instruction

Includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review.

Computer-based instruction (CBI)

Instruction provided via computer technology.

Development of Learning:

Instructional Moves: Linked between and among the following using modeling, describing and recording by both teacher and student. • Concrete—manipulatives, models, hands-on • Pictorial—pictures, diagrams, graphs, technology • Abstract—symbols, words Properties and Concepts: • Learner centered • Mathematically correct • Developmentally appropriate

Development of learning

Instructional moves, properties and concepts

Whole number that includes all positive and negative numbers, including zero.

Integers

Learning strategy

Intentional use of one or more cognitive processes for a particular learning task.

What are the three main segments of thematic units?

Introduction to unit, presentation of content, and closing activity.

Combination

Involve different combinations that can be made from two sets. ex: shirts and pants Multiplication->Oscar 17apples/Tom 5 times as many 5x17=__ apples as Oscar, how many does Tim have? Division-> Tom has 85 apples, 5 times as many as Oscar. 85 divided by 5=__

A number that cannot be represented as an exact ratio or two integers.

Irrational Numbers

Concrete instruction

Is connected to students real experiences, and uses activities that students can see, hear, taste and touch.

Basic strategy for inequalities and equations is to?

Isolate x on one side

What are the challenges students face with math nomenclature?

It abounds assumptions concerning students' prior knowledge of special terms. Terms that have a meaning in one subject can assume a different meaning in the vocabulary of math, and the vocabulary tends to encompass a variety of homophones.

Individual education plan (IEP)

It's a plan with legal requirements for assisting students with physical or learning disabilities

constructivism

Jean Piagets work, developed in the 1930s, notion that learners are not blank slates but rather creators or constructors of their own learning. integrated networks, cognitive schemas

All basic mathematics skills are introduced in grades?

K-2ed

Bloom's Taxonomy?

Knowledge Comprehension Application Analysis Synthesis Evaluation

Properties and concepts must be

Learner centered, correct, developmentally appropriate

What is the deductive approach to teaching?

Learning step by step.

Inductive teaching

Learning through examples or Learning step by step

What is inductive teaching?

Learning through examples.

sociocultural theory

Lev Vygotsky, mental processes exist between and among people in social learning settings and from this social settings the learner moves ideas into his or her own psychological realm information is accessible if the learner has support from peers or more knowledgeable others

Gagne

Levels or learning, each requires different types of instruction

When graph of pairs of numbers results in points lying on a straight line, a function is said to be?

Linear

Parallel lines

Lines in the same plane that do not intersect.

Perpendicular lines

Lines that intersect at right angles.

First find the greatest common denominator. Start with the biggest common denominator listed. Thats 10. Not all denominators fit into 10. So go up by 10's until you find a number that fits. 40. 5x8=40, 3x8= 24 giving you 24/40. Do the same to the rest. 20/40, 24/40, 30/40, 36/40 aka 2/4, 3/5, 6/8, 9/10.

List 3/5, 9/10, 6/8, and 2/4 in least to greatest order.

Physical knowledge/skills to work on

Listening, observation

Manipulatives

Manipulatives are any objects that can be touched and moved (manipulated) to assist understanding. Manipulatives can take the form of counters such as small toys or buttons used to help a kindergarten student add, 100 noodles glued onto construction paper to show place value, or a circle folded and cut into eighths to assist with understanding fractions.

What trains the intellect?

Math

MOVE FROM CONCRETE TO ABSTRACT

Mathematics should be taught conceptually. Children should learn the "big ideas" behind what they do before they learn to do it. Concrete instruction is connected to students' real experiences, and uses activities that students can see, hear, taste and touch. Abstract ideas require children to use their imaginations or their brains only, without help from pictures or real objects. With each new idea taught, good teachers start with easier, more reality-based thinking and move to more imaginary ideas.

Quadrilaterals

Means "four sides. "Any four-sided shape is a Quadrilateral. But the sides have to be straight, and it has to be 2-dimensional.

Students know how many in each group (set) but do not know how many sets.

Measurement division

Mnemonic

Memory aid or trick designed to help students learn and remember a specific piece of information.

Metric System of Measurement

Milliliters Liters Grams Kilograms Tonnes Millimeters Centimeters Meters Kilometers

PEMDAS

Mnemonic code for students to remember the orders of operations.

Of any counting number are the results of multiplying that continues number by all the counting numbers. Ex: 7: 7,14, 21,28....

Multiples

When a factor is multiplied by 0, then the product is 0.

Multiplication property of 0

Multiplication and Division of Fractions

Multiplication->Denominators do not have to be the same. Multiply across both n/d. ex: 3/7x3/5=9/35 or 4/6 x5/8=20/48 Division-> Denominators do not need to be the same BUT need to KCF=keep change flip aka: INVERSE

Combination

Multiplication: How many combinations of shirts and pants can be made out of 5 shirts and 17 pants? Division: If you have 5 shirts, how many pants are needed to make 85 combinations of pants and shirts?

Multiplicative Comparison

Multiplication: Oscar has 17 apples and Tom has 5 times as many apples as oscar does? How Many apples Does Tom have? Division: Tom has 85 apples. This is 5 times as many as what Oscar has. How many apples does Oscar have?

Equal Groups or repeated addition

Multiplication: Oscar has 5 bags of apples with 17 apples in each bag. How many apples does Oscar have altogether. Partition or sharing division: Oscar has 85 apples. He Arranges the apples into 5 bags with the same amount of apples in each bag. How many apples are in each bag? Measurement or subtractive division: Oscar has 85 apples. He arranges the apples into bags of 17 apples each. How many bags of apples did he make?

Area and Array

Multiplication: Oscar has a farm of apple trees planted in 5 rows of 17 apple trees in each row. How many apple trees does he have on his farm? Division: Oscar planted 85 palm trees on his farm. He want's to plant the trees in 5 equal rows of palm trees. How many palm trees will he need to plant in each row?

Any number multiplied by 1 remains the same... 34 x 1 = 34.

Multiplicative identity property of 1.

Scientific Notation

Multiply out the scientific notation. -Multiplied by 10(to a positive number such as 2), the decimal is moved right. 6.89x10=689. -Multiplied by 10( to a negative number such as -2, you move the decimal place to the left ex: 5367.x10=53.67 ex: 3.4567x10=3456.7 (BEFORE COMPARING), 26543X10*-3=26.543

Curriculum

Must be coherent, focused on important and well-articulated math concepts across the grades

Positive integer or non-negative integer

Natural Numbers

Equivalent to the set of counting numbers.

Natural numbers

Parities Division

Needed when students know how many groups there needs to be but not how many objects will be in each group.

Measurement Division

Needed when students know how many objects are in each group but do not know how many groups there are.

Creativity (creative thinking)

New and original behavior that yields a productive and culturally appropriate result.

When graph of pair numbers is not on a line the function is?

Non-Linear

What are the five strands of content students should learn in mathematics?

Number and operations, algebra, geometry, measurement, and data analysis and probability.

Continuous Models

Number lines that children might choose for solving problems

Name three types of conservation in math?

Numbers Mass/ Energy Volume Time Length Area

Even

Numbers divisible by 2

Three Biblical Considerations When Teaching Math:

Numbers reveal God's order Math embodies truth Math builds character

Symbolic references

Objects or experiences

Jean Piaget

Observed and recorded the intellectual abilities of infants, children, and adolescents Stages of intellectual development are related to brain growth and led him to conclude that the thinking and reasoning of children were dominated by preparation thought (i.e., a pattern of thinking that is egocentric, centered, irreversible, and non transformational)

Array

One way to model multiplication visually. One factor is shown vertically and the other is horizontally. 2x4 **** ****

With two points this is exactly one straight line, straight lines are considered?

One-dimensional

In the number 1984 the number 4 would be in what place value?

One: 4

Organizing

Organizing information in a problem and help students better understand the patterns needed to solve the problem. Organize info in the problem.

Bivariate data

Pairs of linked numerical observations. Ex. a list of heights and weights for each player on a football team. Box plot. A method of visually displaying a distribution of data values by using the median, quartiles, and extremes of the data set. A box shows the middle 50% of the data.

Students know the number of groups (sets) but they do not know the number of objects in each set.

Partitive division

A sequence governed by a rule that can be expressed in words or symbols.

Patterns

P = 22 + 2w

Perimeter

P = 4s

Perimeter

By adding twice the length of the rectangle to twice the width. Ex: l=10m w=5m P = 2(10m) + 2(5m) = 30m

Perimeter of a rectangle

Multiplying four times the measure of a side of the square.

Perimeter of a square

Found by adding the measures of 3 sides of the triangle. Rep as: P = s1 + s2 + s3

Perimeter of a triangle

Model

Person who demonstrates a behavior for someone else.

Significant Number (or Figures)

The digits in a value that are known with some degree of confidence. As the number of these increases, the more certain the measurement. They are especially important in rounding.

Who came up with the stages of development?

Piaget

Pictorial

Picture or mental image. Visualize

Basic Foundation for understanding mathematic computation.

Place Values

Flat surfaces without edges are?

Planes

A third grade student with dyscalculia is likely to have the most difficulty with which of the following tasks?

Playing a game involving quantity and place value

Apply the order of operations

Please excuse my dear aunt sally P: Parenthese E: Exponents(work from left to right) M/D: Multiplication/Division (work from left to right) A/D: Addition/Subtraction (work from left to right)

Is a specific location, taking up no space, having no area, no dimensions and frequently represented by a dot.

Point

A special education teacher and a general education teacher co-teach a kindergarten class that includes several students with disabilities. The teachers are designing a mathematics learning center to give students practice with shapes and patterns. The general education teacher feels may not be very accessible to the students with disabilities. Which of the following strategies would likely be most effective for the special education teacher to use to promote communication and collaboration between the teachers?

Pointing out specific aspects of the materials that may be challenging for the students with disabilities and suggesting possible alternative materials.

Differentiated instruction

Practice of individualizing instructional methods, and possibly also individualizing specific content and instructional goals, to align with each student's existing knowledge, skills, and needs.

What does PANIC stand for?

Preparation Attention Need Information Closing

Methods of Mathematics and Engagement?

Presentation Organization Presenter

Every composite number can be written as a product of prime numbers and is called?

Prime Factorization

Number with only two counting number factors - 1 and the number itself. Ex: 2,3,5,7,11,13, and 17.

Prime number

Does not matter which factor pair you start with in prime factorization as long as you continue factoring until you only have?

Prime numbers

Probability

Probability (or likelihood) is a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a value between 0 (0% chance or will not happen) and 1 (100% chance or will happen). The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen.

Deductive reasoning

Process of drawing a logical inference about something that must be true, given other information that has already been presented as true.

Critical thinking

Process of evaluating the accuracy and worth of information and lines of reasoning.

Divergent thinking

Process of moving mentally in a variety of directions from a single idea.

Modeling

Promote mathematical thinking and facilitate and understanding of key concepts and mathematical structure. Students engage their senses to better understand and reason with abstract concepts.

Manipulatives

Provide "hands-on-learning" Excellent way for students to develop self-verbalization learning strategies

What do geoboards do?

Provide the tactile experience of creating right triangles.

The Readiness Rule

Provide young students with readiness of math abstractions one to two years (time frame)before the proficient use of the math is performed in paper-and-pencil or mental math formats. Move from concrete to semi-concrete to abstract math performance

A third grade student with a physical disability has difficulty manipulating objects. The student's teacher often has students work in small groups, using manipulatives to solve mathematics problems. Which of the following instructional strategies would most effectively promote the student's participation in these learning activities?

Providing the student and his group with adapted versions of the manipulatives used by the other groups in the class.

Problem Solving

Putting together concepts and principles to solve a problem new to the learner

Volume

Quantity of liquid

KNOWLEDGE LEVEL

Questions and activities at this level are simple memory and recall questions. = How much is 2 + 3? = How many sides does a triangle have? Activities and questions ask the students to tell list describe state define identify ask who what when where

Zone of proximal development (ZPD)

Range of tasks that child can perform with the help and guidance of others but cannot yet perform independently.

Can be expressed as a ratio or quotient of two non-zero integers.

Rational Numbers

Any number that is positive, negative, or zero, and is used to measure continuous quantities.

Real Numbers

Brownell

Real-life connections

Inductive Reasoning

Reasoning in which conclusions are based on observation.

Deductive Reasoning

Reasoning in which conclusions are based on the logical synthesis of prior knowledge of facts and truths.

Mathematical Reasoning

Reasoning refers to students ability to hypothesize, test their theories, and draw conclusions. Three main types: Inductive, Deductive, Adaptive.

Reciprocals

Reciprocals are two numbers which multiply together and make 1. They are also called multiplicative inverses of each other. For example: 3 and 1/3 are reciprocals because 3 × 1/3 = 1 5/6 and 6/5 are reciprocals because 5/6 × 6/5 = 1 -0.2 and -5 are reciprocals because -0.2 × -5 = 1

Indicates the renaming of a number from one place value to another.

Regrouping

Concept

Relational or concrete attributes

What promotes diligence?

Repetition

Forms of assessment

Reports, applications, models, lab investigations, projects, always/sometimes/never.

Abstract ideas

Require children to use their imaginations or their brain only, without help from pictures or real objects.

Teaching

Requires understanding what students know and need to learn while challenging and supporting students to learn it well

Two pairs of sides of two triangles are equal in length + included angles are equal in a measurement = congruent

SAS (Side-Angle Side)

3 pairs of sides of 2 triangles equal in length = congruent

SSS (side-side-side)

Norm referenced

a standardized test that focuses on a comparison of a students score to average of a norm group.

Form of writing a number as the product of a power of 10 and decimal number greater or equal to 1 and less than 10. Ex: 2,400,000 = 2.4x 10 to the 6 power.

Scientific Notation

What is the process for planning a theme?

Select a broad theme, use state curriculum to find main principles or generalizations and key objectives to involve each content area, gather materials.

The basic steps to planning and organizing thematic units

Selecting a theme Designing the integrated curriculum Gathering materials for the unit Arranging thematic activities

Automaticity

Selecting problem solving methods and performing computations without requiring much time to think the process through

Modeling the Operations: This method students work with visual representations (pictures) instead of actual objects.

Semi concrete method

Modeling the Operations: Students work with one symbol (tally marks, x's, y's, etc) to represent objects instead of actual objects, pictures, or abstract (numerical) representations. Students use one symbol.

Semiabstract method

What are the 4 general stages of development?

Sensorimotor (0-2) Pre-Operational (3-7) Concrete Operational (8-11) Formal Operational (12-15)

What are the four main developmental stages that children acquire information?

Sensorimotor stage (birth - 2 years) Preoperational stage ( Years 2-7) Concrete operational stage (years 7-11) Formal operational stage (years 11 - adult)

Subset

Set A is a _______ of B if and only if each element of A is also an element of B.

Proper Subset

Set A is a ________ _______ of set B if and only if A is a subset of B, and A is not equal to B. (C)

Equal Sets

Sets with exactly the SAME elements.

Collective self-efficacy

Shared beleif of members of a group that they can be successful when they work together on a task.

Assessment

Should support the learning of important math concepts, and furnish useful information to both teachers and students

What should teachers do to help try and get kids interested in math?

Show them real world professions that use math.

Hierarchy Diagram

Shows all the functions of a particular concept. Which is broken down to subordinate relationship and super-ordinate concepts.

Why should talking about their work be encouraged?

So students learn to use mathematical vocabulary.

Spatial-temporal knowledge

Space and time thinking

Mobius Strip good for

Spatial-temporal reasoning

3- standardized vs criterion referenced

Standardized-written to general content, and the performance on the test is based on a comparison to other similar students who took the test. Example: SAT, ACT, GRE, National Assessment of Educational Progress. • Criterion Referenced-written to measure specific content and the criteria for passing the test is pre-specified. Examples: Well- designed teacher unit test, STAAR, Texas Examination of Educator Standards

Standardized vs criterion referenced

Standardized: written to general content, and the performance on the test is based on a comparison to other similar students who took the test Ex: SAT, GRE, ACT. Criterion referenced: written to measure specific content and the criteria for passing the test is pre-specified. Ex: well designed teacher unit test, STAAR, TEXES

Commutative Property

States that changing the order of addends does not change the sum. That is, a + b = b + a. Commutative Property of Multiplication: It states that changing the order of factors does not change the product. That is, a × b = b × a.

Zero - Product Property

States that if the product of two factors is zero, then at least one of the factors must be zero

Distributive Property

States that the product of a number and a sum is equal to the sum of the individual products of addends and the number. That is: a(b + c) = ab + ac.

80%

Student activity-experience by doing

EVALUATION LEVEL

Students are asked to = make judgments based on set criteria and to = defend or support their thinking Evaluation is the highest level of thinking in Bloom's Taxonomy In the purest sense, the Evaluation stage is rarely reached at the elementary level simply because of the nature of cognitive development present until later However, students can be involved in many activities where they are asked to make choices and support their choices. Activities and questions where students are asked to explain their thinking, give students the opportunity to support choices they have made. keywords include = which is the best choice and why = what do you think about = rate from good to poor = what is the problem = are all of the solutions the same = will all the solutions work = decide which = justify your choice

Math curriculum for grades 3-5

Students continue developing number concepts to include multiplication, division, fraction and decimal representation, geometric principles, and algebraic reasoning

Math curriculum for grades 1-2

Students continue exploring number concepts and begin learning basic computation skills

Which of the following best promotes understanding of math operations?

Students verbalizing their reasoning and solutions to problems

Addition and Subtraction of Fractions

The denominators must be equal the add or subtract and leave the denominator the same. -5/7-(-3/7)=5-3/7 -2/7 or 4/5+1/5=4+1/5=5/5 Unlike Denominator->You have to look at the denominators and find the lowest common denominator both share.

Supplementary Angles

Supplementary angles are two angles that add up to give a straight angle, 180°.

Scaffolding

Support mechanism that helps a learner successfully perform a task within his or her zone of proximal development.

Face

Surface of a geometric solid

What processes are included in the pre operational stage of development?

Symbolic functioning, centration, intuitive thought, egocentrism, and inability to conserve.

The Professional Standards for Teaching Mathematics (NCTM) presents standards for the teaching of math, organized under four categories

Tasks Discourse Environment Analysis *These strands are integrated and interdependent, and are crucial in shaping what does on in math classrooms

Presenter?

Teacher Student

50%

Teacher modeling

What are advantages of collaborative or small-group learning?

Teachers reduce their workloads and the amount of time spent on helping students and planning lessons

Peer Interaction

Teachers should pair students carefully, model effective ways to interact, provide students with relevant tools, and offer specific advice. Struggling students can benefit from peer explanations, clarify a process and ask and answer questions.

Unfamiliar Structures for ELLs

Technical terms, problem solving verses computation and notation, math symbols, multiple words indicating the same operation, ...."if"....."then" statements, etc.

Nomenclature of Math

Technical vocabulary

In the number 1984 the number 8 would be in what place value?

Ten: 80

What must young children understand about counting objects?

That the idea of an amount is inclusive of all perviously counted things, not just the same as the last block.

label (LBL)

The JMP instruction is always paired with a

Subitizing

The ability to instantly "see" the number of objects in a small set without having to count them.

Adaptive Reasoning

The ability to think logically about the relationships between concepts and to adapt when problems and situations change.

What are characteristics of the formal operational stage?

The ability to use symbols and think abstractly.

Absolute Value

The absolute value of a real number is equal to the numeric value of the number without regard to its sign (e.g -3 is 3) Absolute value is often thought of as the distance a number is from zero on the number line

Volume

The amount of 3-dimensional space an object occupies. For a rectangle the formula would be length times width time depth or height. Since there are three dimensions it is expressed as cubics (9 cubic inches).

Area

The amount of space inside the boundary of a flat (2-dimensional) object such as a triangle, rectangle or circle. Different objects have different formulas to determine area. For a rectangle it is length X width. Area of a triangle is 1/2 of the base times the height. The area would be expressed as a square (10 square feet, etc.)

Mean

The average of a set of numbers; the sum of the numbers divided by how many number there are; 2 + 5 +5 =12, then 12/3 = an average/mean of 4

Percents

The best way to compare a percent to other number expression is either to _convert it to a decimal and leave it as that. _convert it to a fraction (depending on how the other numbers are expressed) ex: 76%->move 2 decimal places to the left .76. ex: .76-> 76/100->can be simplified to 16/25

Associative Property

The change in grouping of three or more addends or factors does not change their sum or product. holds good for both addition and multiplication, but not for subtraction and division. Addition: (2 + 3) + 5 = 2 + (3 + 5) Multiplication: (4 X 5) X 10 = 4 X (5 X 10)

Vertex

The common endpoints of two rays that form an angle, or the point of intersection of two sides of a polygon of polyhedron

Natural Numbers

The counting numbers 1,2,3,4,5...

Diameter

The distance across a circle through its center point. It is twice the radius of the circle.

Circumference

The distance around a circle (C). C=pi times the diameter or d X pi. Or, 2radius X pi. (Pi - 3.14).

Perimeter

The distance around a two-dimensional shape, such as a triangle or rectangle.

Girth

The distance around something; the circumference.

Radius

The distance from the center of a circle to the edge of the circle. It is also half the diameter of a circle.

Distributive Property

The distributive property of x over +multiplucation may be distributed over addition ex:10x(50+3)=(10x50)+(10x3) 3x(12+99)=(3x12)+(3x99( a(b+c)=ab+ac

Subroutine (SBR)

The first input condition entered in the subroutine. It serves to identify the file as a subroutine. It is always true, and is actually optional, though recommended.

Greatest Common Factor (GFC)

The greatest number that is a factor of each of two or more given numbers. Examples: The greatest common factor of 24 and 15 is 3. The greatest common factor of 40, 50, and 25 is 5.

Translation

The image of a figure that has been "slid" to a new position without flipping of turning.

Least Common Multiple (LCM)

The least number that is a common multiple of two or more numbers. Find the LCM of 30 and 20. 30: 30,60,90,120 20: 20,40,60,80,100,120

Discourse (strands)

The manner of representing, thinking, talking, agreeing, and disagreeing that teachers and students use to engage in these tasks

Probability

The measure of how likely an event is. It is usually determined by dividing the number of ways something can happen by the total number of outcomes.

Median

The middle number of a set of numbers after they have been placed in numerical order. In the set [2,3,4], 3 is the median. If there are an even number of numbers, the median is the average of the two middle numbers.

Frequency

The number of times a score appears in a list of data.

Mode

The number that occurs the most frequently in a set of data. In the set [2,4,4,3,5] 4 is the mode.

Additive Inverse

The opposite of the number. A number and its opposite add up to give zero. They are called inverse additives of each other.

Thematic instruction

The organization of curriculum content based on themes or tops Integrates basic disciplines (e.g., reading, math, music, art, and science) with the exploration of broad subjects

Endpoint

The point at the end of a line segment

CONNECTIONS

The process of making connections should be modeled and reinforced. Students should understand how mathematics can be part of everything they see, do and know. They should be guided to an awareness of how their knowledge of mathematics can help them with other school subjects, such as using a population graph in social studies. They should also appreciate how knowing mathematics helps lead to better jobs and future opportunities. People who know arithmetic can make good decisions about how to spend allowance money, or whether or not they can afford a new bike or a vacation to Mexico.

Tasks (strands)

The projects, questions, problems, constructions, applications, and exercises in which students engage

Fact Families

The related number sentences for addition and subtraction or multiplication and division that contains all the same numbers (e.g. 2+3 =5; 3+2 =5; 5-3 =2; and 5-2 =3)

Environment (strands)

The setting for learning

A fourth grade student receives special education services due to a specific learning disability in mathematics. The student's special education teacher is scaffolding instruction to help her complete several addition problems involving decimals. First, the teacher recites aloud each step for completing an addition problem while the student listens and watches the teacher complete the problem. Next, the student recites the same steps aloud as she and the teacher complete a second addition problem together. Which of the following approaches would be most appropriate for the student and the teacher to use to complete a third addition problem?

The student quietly says the steps to herself as she completes the problem, and the teacher helps as needed.

What is the importance of a placing patterns activity?

The students are practicing patterns, which will help build a foundation for algebraic thinking

Perimeter

The sum of the lengths of the sides of a polygon (p = 2l x 2w where l =length and w = width).

Analysis (strands)

The systematic reflection in which teachers engage

Competency

The teacher understands how students learn mathematical skills and uses that knowledge to plan, organize and implement instruction and assess learning.

Objectives

The teacher understands how students learn mathematical skills and uses that knowledge to plan, organize, and implement instruction and assess learning.

Rote counting

The verbal repetition of numbers

Integers

The whole numbers and their negatives (e.g. -2, -1, 0, 1, 2).

Whole Numbers

The whole numbers are the counting numbers and 0. The whole numbers are 0,1,2,3,4,5... Cannot be negative

What are the mathematical process standards and what do they do?

They highlight ways of acquiring and applying content knowledge. Problem solving, reasoning and proof, communication, connections, and representations.

Think Pair Share ex.

Think: Students think independently about the question that has been posed, forming ideas of their own. Pair: Students are grouped in pairs to discuss their thoughts. This step allows students to articulate their ideas and to consider those of others. Share: Student pairs share their ideas with a larger group, such as the whole class. Often, students are more comfortable presenting ideas to a group with the support of a partner. In addition, students' ideas have become more refined through this three-step process.

APPLICATION LEVEL

This is where most problem-solving will occur in the elementary classroom. If Jake had $7 and his father gave him an additional $3 how much money does he have now? Keywords in questions and activities at this level include: = demonstrate = solve = what are the next steps = how can this be used = contrast = separate into parts = use = change = apply = estimate = determine

SYNTHESIS LEVEL

This level is where = divergent thinking, = originality, and = imagination occur At this level, concepts that have been learned = are taken apart and thought about in a totally different way. In upper-level mathematics, synthesis is a major component of algebra, geometry, trigonometry, and calculus Although synthesis can certainly occur at the elementary level, it is less frequent Keywords include = create = design = how many hypotheses can you suggest = how many different ways = how else = what would happen if = how many ways are possible = compose = develop = suppose

1. Inductive (Informal):

This reasoning goes from the specific to the general. It uses observations and patterns to infer a generalization.

2. Deductive (Formal):

This reasoning process reaches conclusions based on accepted truths and logical reasoning. This reasoning goes from the general to the specific, and uses a general rule or statement to draw a reasonable conclusion.

In the number 1984 the number 1 would be in what place value?

Thousand: 1000

How should a teacher of math create a learning environment that fosters development of each student's ability?

Through structuring time necessary to explore math and grapple with problems, using physical space and materials, providing context, and respecting students' ideas and ways of thinking.

doing mathematics

engaging in the science of pattern and order, posing worthwhile tasks and then creating an environment where students take risks, share, and defend mathematical ideas

Symmetry

To discover what symmetry is, take a piece of paper, fold it, and cut out a shape along the fold. Unfold the shape that you cut out. This figure is symmetric. That means it is exactly the same on both sides of the crease. The simplest symmetry is Reflection Symmetry (sometimes called Line Symmetry or Mirror Symmetry). It is easy to recognise, because one half is the reflection of the other half.

Bisect

To divide into two congruent parts

What does thematic planning provide an opportunity of for students?

To hear similar information in various instructional segments and from a variety of sources.

The NCTM standards includes 5 major shifts in the environment of mathematics classrooms from current practice to teaching for the empowerment of students

Treating classrooms s math communities rather than simply a collection of individuals Students using logic and math evidence as verification rather than students relying solely on the teacher for correct answers Students employing math reasoning rather than early memorizing answer-finding Emphasizing problem solving rather than an emphasis on mechanistic answer-finding Students connecting math, its ideas, and its applications rather than viewing math as a body of isolated concepts and procedures

Complementary Angles

Two Angles that add up to 90 degrees (a Right Angle). They don't have to be next to each other, just so long as the total is 90 degrees.

Congruent angles

Two angles that have the same degree of measurement.

Complementary angle

Two angles whose sum is equal to 90 degrees.

Adjacent angles

Two angles with a common vertex, a common ray, and not common interior points

Additive Inverses

Two numbers whose sum is 0 are additive inverses of one another. Ex. 3/4 & -3/4 are additive inverses of one another because 3/4 + (-3/4) = (-3/4) + 3/4 = 0.

Planes are infinite length and breadth but no depth are considered to be?

Two-dimensional

Cultural differences in coherence?

USA 1.9 topics Germany 1.6 topics Japan 1.3 topics

What were the cultural differences in grade level between the USA, Germany, and Japan?

USA 7.4 Germany 8.7 Japan 9.1

Cultural differences in development in the US, Germany, and Japan.

USA 80% stated Germany 25% stated Japan 20% stated

Which schools are interrupted more then any other country?

USA schools

Problem-solving method UPSC

Understand, plan, solve, check

What does effective mathematics teaching require?

Understanding of what students know and need to learn and then challenging and supporting students to learn it well.

Conservation

Understanding that quantity, length, or number of items is unrelated to the arrangement or appearance of the object or items

What is conservation?

Understanding that quantity, length, or number of items is unrelated to the arrangement or appearance of the object or items.

What does the measurement standard include?

Understanding unit systems and process of measurement and applying techniques and formulas to determine measurements.

Semi abstract model

Use a single symbol (such as an x or a tally mark) to represent numbers of objects while performing operations.

What are numbers and operations curriculum for second grade.

Use concrete models of hundreds, tens, and ones to represent a given whole number. Use place values to read, write, and describe numbers, uses concrete models to represent and name fractional parts of a whole object, models addition and subtraction of two-digit numbers with objects, pictures, and words, can recall and apply basic addition and subtraction facts, determine value of collection of coins up to a dollar.

What are number and operations curriculum for first grade.

Use concrete models to represent and name fractional parts of a whole object, model addition and subtraction of two-digit numbers with objects, separates a whole into two, three, and four equal parts, identify individual coins by name and value and describe relationship, create sets of tens and ones using concrete objects and reads and writes numbers up to 99.

Concrete Model

Use objects to demonstrate operations

Semi Concrete Model

Use pictures (instead of actual objects) to demonstrate operations

What are numbers and operations curriculum for third grade?

Use place values to describe numbers through 9,999, uses fraction names to describe fractional parts of whole objects and compares parts of whole objects in a problem situation, Uses operations to solve problems involving whole numbers up to 999, applies multiplication facts through 12 by using concrete models and objects.

Focal points: 2nd grade

Uses concrete models of hundreds, tens, and ones Read and write the value of whole numbers up to 999 Uses concrete models to represent and name fractional parts Models addition and subtraction of 2-digit numbers with objects, pictures, words, and numbers Determines the value of a collection of coins up to one dollar

Focal points: 3rd grade

Uses place value to read, write, and describe the value of whole numbers, compares and orders who numbers through 9,999 Uses fraction names and symbols to describe fractional parts Selects and uses addition or subtraction to solve problems involving whole numbers through 999 Applies multiplication facts through 12 using concrete models and objects Uses models to solve division problems

Focal points: 4th grade

Uses place value to read, write, compare, and order whole numbers through 999,999 Uses concrete objects and pictorial models to generate equivalent fractions Uses multiplication and division to solve problems (no more than 2 digits times 2 digits) Uses strategies, including rounding and compatible number, to estimate solution to +,-,x,/

Focal points: 5th grade

Uses place value to read, write, compare, and order whole numbers through 999,999,999,999 and decimals through the 1000th place Identifies common factor of a set of whole numbers; uses multiplication and division to solve problems (no more than 3 digits times 3 digits)

What are numbers and operations curriculum for fifth graders?

Uses place value to read, write, compare, and order whole numbers through 999,999,999,999 and decimals through thousandths, identifies common factors of a set of whole numbers, uses multiplication up to three digits x two digits.

What are numbers and operations curriculum for fourth graders?

Uses place values to read, write, compare, and order numbers through 999,999,999 and decimals involving tenths and hundredths, can use concrete objects to generate equivalent fractions, uses rounding and compatible numbers to estimate solutions to problems, multiplication of 2 digits by 2 digits.

Focal points: K

Uses whole number concepts Uses sets of concrete objects to represent quantities given in verbal and symbolic descriptions Begins to demonstrate the concept of "part of" and "whole" Sorts to explore number; uses patterns

Abstract model

Using numbers only to perform operations

Union of two segments or point of intersection of two sides of a polygon.

Vertex

Learning Modalities:

Visual, Auditory, Kinesthetic

Learning modalities

Visual, auditory and kinesthetic.

Mapping Diagrams

Visually illustrates relationship between two or more concepts and are linked by a term.

Which of the following are basic assumptions of the criteria for teaching problem solving in mathematics?

a teacher can expect students to approach problems with varied strategies, problem solving can be applied to subject areas other than math, and students can help each other investigate possible problem-solving strategies

How much space is inside of a 3-dimensional closed containers.

Volume

The Multiplicative Identity

We call 1 the multiplicative identity multiplying any number by 1 leaves the number unchanged ex: 88x1=88-> ax1=a

Read

We remember 10%

Hear

We remember 20%

See

We remember 30%

See and Hear

We remember 50%

Discuss with others

We remember 70% --discussion with peers

Experience by doing

We remember 80%

Teach

We remember 95%

$197.95

What is 25% off of $264?

PROBLEM SOLVING

When encountering different kinds of prob- lems, students should be able to choose the best problem-solving method from several that they know. A general problem-solving method that can be applied to many types of problems is: 1. Understand 2. Plan 3. Solve 4. Check For any word or logic problem, students must read and understand the problem care- fully (including removing any unnecessary information), plan how to find the answer, solve the problem, and check their work. Students may also choose to use other strategy tools such as drawing a picture, making a list, chart or table, estimating, guessing and checking, working backwards, or using objects.

When does math because more meaningful and easier to internalize?

When it is used in real-life situations.

Efficiency

When students do not get caught up in too many steps or get confused with the logic of the problem or strategy or conceptual meaning. An efficient algorithm is carried out easily with out confusion

Thinking Aloud

When students verbalize what they know, it helps them reflect upon and clarify the problem and focus one step at a time. "Thinking Aloud require talking through the details, decisions, and reasoning behind those decisions. Gives time to fully comprehend the problem.

Order of operations

When there is more than one operation and parentheses are used, first do what is inside the parentheses, then the exponents. Next, multiply or divide from left to right. Then add of subtract from left to right (PEMDAS of Please Excuse My Dear Aunt Sally).

How should addition problems be posed when moving from quantification to computation?

With concrete objects

How must student's learn mathematics?

With understanding, actively building new knowledge from experience and previous knowledge.

differentiating instruction

a teachers lesson plan includes strategies to support the range of different academic backgrounds found in classrooms that are academically, culturally, and linguistically diverse what content, process, product can be diverse physical environment

MATH INSTRUCTION

Young children learn mathematics best when they DO mathematics. Conceptual development, rather than reliance on teaching a set of step by step procedures, or algorithms, is the goal. Constance Kamii, a specialist in early childhood math education, tells of observing a class of first graders. When presented with a math problem similar to 3 + 5 would write: 3 + 5 = 5 To determine what the problem was, she had them model the problem with counters. She began to watch what they were doing and discovered that they would put out three counters to model the 3 then place two more counters with the three to make five counters in all. So their answer to 3 + 5 = 5 made sense to them. Kamii used this example to point out the importance of not rushing too quickly to the abstract representation of a concept, in this case the concept of addition. Addition is much more complex than one might think initially.

Integers

______ are called whole numbers, negative whole numbers, and zero.

Which is an example of the additive identity?

a + 0 = a

Which type of graphs would best be used to represent the number of students who like red, green, or yellow best?

a bar graph or pictograph

Numbers provide what?

a complete continuum

Geoboards

a mathematical manipulative used to explore basic concepts in plane geometry such as perimeter, area and the characteristics of triangles and other polygons. It consists of a physical board with a certain number of nails half driven in, around which are wrapped rubber bands.

ratio notation

a method of notating fractions for the purpose of comparison

additive identity element

a number + 0 = the number

Arithmetic Sequence

a sequence in which each term is obtained by adding a constant value to the preceding term. The constant value is called the common difference (d).

Geometric Sequence

a sequence in which the ratio of each term to its predecessor is the same for all terms. or each term is obtained by multiplying the preceding term by a constant value called the common ratio (r).

Empty Set

a set containing no elements

Concrete operational stage (7-11 years)

ability to think logically about concrete objects or relationships. Decentering - form a conclusion based on reason rather than perception reversibility - objects can be changed and then returned to their original state ex) 4+4 = 8 . . . 8-4 = 4 conservation - quantity, length, number of items is unrelated to the arrangement or appearance of the object. serialization - arrange objects in an order according to size, shape or any other attribute classification - name/identify sets of objects according to appearance, size, or other characteristics elimination of egocentrism - view things from another's perspective

Three things that need to progress in higher math levels?

abstract thought complexity problem length

Which angle would measure less than 90 degrees?

acute

Diagnostic tests

are used with the diagnostic-prescriptive teaching of mathematics. This process is an instructional model that consists of diagnosis, prescription, instructions, and ongoing assessment. can be used to help identify specific problem areas. can be teacher made or commercially developed

What types of manipulatives should be used in grade 6

algebra tiles, video games, ipods, math games software, smart boards, digital cameras.

real number

all numbers, rational or irrational

regular polygon

all sides and angles are equal

engage students in productive struggle

aloo students to have disequilibrium in learning, it is part of the process and developing concepts, must have tools and prior knowledge

Authentic assessment

alternative assessment that incorporates real-life functions and applications.

A "shape set" is:

assortment of triangles, rectangles, squares, and circles in various colors and sizes

Numerical values are always ________________

ambiguous

dispersion

amount of spread

Using scissors to cut a piece of paper represents which of the following spatial concepts?

an angle intersecting a place

Know your class.....

and teach to the middle

the same problem will always give you the same __________

answer

Help them to develop what?

answers

tool

any object, picture, or drawing that can be used to explore a concept. includes calculators and manipulatives

expository instruction

approach to instruction in which info is presented in essentially the same form in which students are expected to learn it

learner-directed instruction

approach to instruction in which students have considerable control regarding the issues they address and the ways they address them

teacher-directed instruction

approach to instruction in which the teacher is largely in control of the content and course of the lesson

backward design

approach to instructional planning in which a teacher first determines the desired end result (i.e.. what skills a students should acquire), then identifies appropriates assessments, and finally determines appropriate instructional strategies

Integers

are also called whole numbers negative whole numbers and zero. ex: 560,-35,2,0,-197744

Number Properties

are another important aspect of number theory.

Composite

are numbers composed of several whole number factors. 30 is composed of several whole-numbers

manipulatives

are physical objects that students and teachers can use to illustrate and discover mathematical concepts, whether made specifically for mathematics or for other purposes. Choices abound from common objects such as lima beans to commercially produced objects

Subroutines

are short programs that are used by the main program to perform a specific function. are stored in program files numbered LAD 3 through LAD 255 subroutines can be nest

Base Ten Blocks

area, classification, comparing, computations (whole numbers and decimals), decimal fractional- percent equivalences, metric measurement, number concepts, ordering, percent, perimeter, place value, polynomials, sorting, square, and cubic numbers.

Which represents shape set activities?

arrange pieces in repeating pattern, sorting the pieces in the set by shape, and using the set's pieces to make a model of a real-life object

tessellation

arrangement of closed shapes that completely covers a plane without overlapping or leaving gaps

deduction

arrive at conclusion based on statements known to be true

assessment

as a way to collect evidence about students content knowledge, flexibility in applying that knowledge, and disposition or attitudes toward mathematics

inquiry students

ask questions, determine solutions, use math tools, make conjectures, seek patterns, communicate, reflect, make connections

rephrasing

asking students to restate someone else's ideas in their own words will ensure that ideas are stated in a variety of ways

reasoning

asks the student what they think of the idea proposed by another student

formative assessment

assessment conducted before or during instruction to facilitate instructional planning and enhance student's learning

you can find a child's aptitude in math from this?

assessment test

variance

average sqaured distance from each value to mean

when doing a diagnostic interview

avoid revealing whether a student is right or wrong, wait silently for an answer, shouldn't interject clues or teach, do not interrupt the student

symbolic representation

basic language of math

be sure the task is understood

be able to explain what the task is asking them to do, not explain how

Open-ended questions, portfolios, and writing activities can:

be used as assessment of mathematical skill and knowledge

What are types of manipulatives to be used in grades K - 1

beads, strings, sewing cards, color tiles, macaroni, paper clips, legos, board games, pictographs, buttons.

start and jump numbers

begin with a number(start) and add(jump) a fixed amount. for example start with 3 and then jump by 5s.

What types of manipulatives should be used in grade 2

blocks, cubes, chips, measuring cups, money models, number lines, dominoes, clock faces, ruler, yard stick, balance scales, linear graphs.

implications for teaching mathematics

build new knowledge from prior knowledge provide opportunities to communicate about mathematics create opportunities for reflective thought encourage multiple approaches engage students in productive struggle treat errors as an opportunity for learning scaffold new content honor diversity create a classroom environment for doing mathematics

technology principle

calculators, computers, and other emerging technology are essential for learning mathematics

What types of manipulatives should be used in grade 5

calculators, timers, metric beaker sets ,connecting cubes, board games, fraction tiles, pie charts and graphs

adaptive reasoning

capacity for logical thought, reflection, explanation, and justification

Manipulatives

enhance student understanding, enable students to have conversations that are grounded in a common model, help students recognize and correct any misconceptions Games - children learn best when they can manipulate materials to check their understanding and link what they are learning to real life situations

Accuracy

careful recording of the computational algorithm, memorizing basic facts, knowing number relationships and place value, and checking reasonableness or correctness of the results

scaffold new content

carefully structured content, new concepts require more structure or assistance

when did math start

cave paintings prior to 20,000 bce in africa and france

central tendency

center of data set

Equal groups or repeated addition

involves making a certain numbers of equal sized groups repeated 3x6=6+6+6 3 numbers involved: numbers of groups (factor), size of the group (factor), total number of the objects (product) ex:3x6 3 groups of 6 objects

Presentation Organization?

class work seat work

Attribute blocks

classifying, geomtry, logical reasoning

authentic activity

classroom activity similar to an activity that students are apt to encounter in the outside world

how can math interventions used to improve math performance

cognitive approaches behavioral approaches alternative instructional delivery systems which includes cooperative learning, computer assisted instruction and interactive video games

verbs that engage mathematics

collaborate, describe, justify, predict, compare, create, invent, explain, formulate, use, and verify, these lead to higher level thinking

Number Sequence

collection of numbers, called terms arranged in order. Sequences are arranged from left to right, the numbers are separated by commas.

Set

collection of objects

stanine "standard nine"

combine the understandability of percentages with the properties of normal curve of probability

scoring

compares students work by established criteria set in advance, many times collected by a rubric

Achievement test battery

composed of subtests of math concepts and skills and usually includes technical aspects of math.

conceptual understanding

comprehension of mathematical concepts, operations, and relations flexible web of connections and relationships within and between ideas, interpretations and images of mathematical concepts, will allow students to connect what they know

Math helps a child develop ___________?

concepts

Concrete instruction is _____

connected, tactile

learning progressions

contain clearly articulate subgoals of the ultimate learning goal

Finite Set

contains zero elements or a number of elements that can be stated as a specific natural number.

Standardized test

content areas and provide useful information about students' math skills. their validity on three basic assumptions: students have been equally exposed to test content, students know the languages of the the directions and responses, and students just like those taking the test have been included in the standardization samples to establish norms and make infrequence.

Compare

involves no action, but involve comparision between 2 different sets- how much or how much less is one/than another. Variation: Difference unknown: 56+__=85 or 85-56=__ Larger unknown: 56+29=__ Smaller unknown: 29+__=85 or 85-29=__

continued --- Concrete Stage (7 - 11) Brings the ability to think logically and classify based on multiple attributes other than simply visual ones. children at this stage are = able to think and reason in 2 and 3 dimensions = can understand problems and problem solving approaches in ways other than their own. = They can see and understand different points of view and perspectives. This addition of different perspectives opens up the idea that there is often more than one way to approach and solve a problem

continued ---- Formal Stage (11 - 16 - adult) Marks a movement to a better understanding of abstract mathematics. It should be noted here that not all children arrive at this stage at the same age (this is also true of the other stages as well) and this is one reason why not all children are ready for upper-level mathematics at age 12 or 13. Developmental age is very important in determining readiness for abstract mathematics. Frustration will ensue if students are forced into abstract thinking before they are ready Students at this stage are capable of = abstract thinking such as limits, = areas under a curve, = infinity, = a formal geometric proof or = complex numbers

continued ------- This is also a good time to discuss the importance of pre-assessment Pre-assessment is the process of determining where the students are with respect to their understanding of the prerequisites necessary to move into the new unit. Example: = It would not be wise to move into the exploration of multiplication if students do not have a firm understanding of addition They mastery of addition is a prerequisite to the study of multiplication

continued ----- Regardless of the grade level you teach, it is important to know the state outlined curriculum for your grade level as well as the ones for the grade level below and above your grade level These curriculum guidelines will give you a better idea of the depth of understanding required for the next grade. For instance, = A 6th grade teacher may be very frustrated that her 6th grade students have no understanding of how to convert a fraction to a decimal. When she looks at the 5th grade curriculum, it becomes evident that converting fractions to decimals in NOT a 5th grade curriculum. In the 5th grade students are required to convert a decimal representation to a fraction with denominators of 10, 100, and 1000

continued ---- Developmentally, children begin by counting 1, 2, 3, 4, 5...10. They have no understanding of numbers or one-to-one correspondence or relative size. They simply count. Much like they recite their ABCs before they actually recognize letters or sounds or words. Before children can begin putting numbers together to add, they must develop an understanding of one-to-one correspondence. As they count the objects with their finger, the finger moves from object to object, increasing the count by one with each object. The problem 3 + 5 can then be understood to mean: "If one set has three objects and another set has five objects, how many objects are there when the two sets are put together? The child will build a three counter set and a five counter set, put the sets together, and count the total in the combined set.

continued ----- Piaget studied the stages of cognitive development in children. His research determined that there were 4 general stages of development 1. Sensorimotor 2. Preoperational 3. Concrete 4. Formal

continued ------ One final area to address as units are being planned is the level of thinking that the teacher wants the students to achieve. One way to analyze the level of the activities you are planning is to use Bloom's Taxonomy. Bloom's Taxonomy consists of 6 levels of understanding, all of which are necessary for long-term retention.

continued ----- 6 levels Knowledge Comprehension Application Analysis Synthesis Evaluation

continued ------- Students do not learn in isolation It is important to incorporate time for = individual work, = pair work, = small group work and = whole class work If children can explain their thinking to others, they have truly mastered the concept, and often children, even young ones, have a way of explaining concepts to their peers that enhances understanding

continued ------ In elementary school, thematic units are often used. Thematic planning lends itself to the integration of = music = science = social studies = reading = writing = and mathematics into a cohesive, interdependent unit

continued ------- Lesson planning should begin with the desired end or outcome in mind The teacher must first understand what end result is desired The she should clearly decide how mastery is to be measured and develop the assessment to be used first This is similar to planning a vacation. You first choose the destination then plan how to get there. Otherwise, you could wander aimlessly, never to reach the destination, or not even realize when you arrive The same is true for planning a unit of study So lets suppose the teacher wants to plan a unit using this 1st grade math state outlined curriculum (3) Number, operation, and quantitative reasoning. The student recognizes and solves problems in addition and subtraction situations. The students are expected to: (A) model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences; and (B) use concrete and pictorial models to apply basic addition and subtraction facts { up to 9 + 9 = 18 and 18 - 9 = 9 } { up to sums of 9 + 1 = 10, teacher decision }

continued ------- As the teacher considers these state outlined curriculum that are to be taught to mastery at 1st grade, they must first decide what the final assessment will be. Example: Let's suppose that this is the first time in the school year these areas in the curriculum have been addressed. The teacher might first decide that her goal is for students to understand through pictures and model sums of ten. The assessment could be pictorial. It could have pictures of objects where students must circle a number that represents the total, and all the sums will be ten or less. There will be a section on the final assessment where students will use objects to build addition problems and draw their representations in spaces provided where sums will be ten or less.

continued ---- Sensorimotor Stage (birth-2yrs.) Children experience their world through their own senses = touch = taste = smell = sight = hearing Children in this stage of development are = egocentric and can only experience the world from their own perspective With respects to math, children in this stage begin to develop the idea on = one to one correspondence Children who cannot count to ten and match that number to the counting of ten objects, are not yet ready to move to the next stage of development and will need additional practice in the development of counting and one to one correspondence

continued ------- Preoperational Stage (2 - 7) Children begin to use their understanding of one to one correspondence to develop a concrete (hands on/Manipulative based) understanding of addition and subtraction They can recognize = patterns = give the next 2 or 3 members of a simple sequence Children at this stage generally operate in only = one dimension Children at this stage are unable to understand = two dimensional mathematics If two objects look different they are assumed actually to be different. This means that while the rectangles are actually identical because they have different orientations, they will be perceived as being different Students at this stage are still limited to what they can experience in their world. Therefore concrete, manipulative tools are an absolute necessity. It is also a great idea to incorporate objects from the students everyday world to demonstrate mathematical concepts

continued ------- Notice that the assessment described in the scenario preceding the discussion about Bloom's Taxonomy only addresses a portion of the state outlined curriculum: = addition = pictorial representations = concrete models = sums of ten Now begins the process of developing carefully crafted lessons that will guide the students to mastery of this portion of the state outlined curriculum delineated above One of the biggest issues that beginning teachers face is = Pacing = choosing enough material to challenge the students but not frustrate them. These portions of the curriculum will be addressed multiple times in the school year with each new segment building on what has already been mastered. As you plan, be creative and incorporate as many higher level questions and activities as possible

continued -------- Assessments fall into 2 general categories: = formative = summative Formative: = are ongoing and can be formal or informal Formal Formative = would be a = quiz = a paper = tangible student work that a teacher would grade Informal Formative = might be = teacher observation = questioning the students orally and checking for understanding The purpose for Formative assessments is the = gathering of data in order to inform instruction Informed instruction is = the ability to analyze where students are in their understanding of the learning, = what misconceptions they might have, and = adjusting the instruction accordingly Summative assessments = may take many forms but are in essence, = the final assessment for a unit of study

continued ------- Elementary students generally fall into the = Pre-operational Stage and = Concrete Operations Stage Mathematical planning for these students should involve the use of a wide variety of manipulative tools = pattern blocks = base ten blocks = tangrams = geoboards = counters = cm cubes = fraction bars = fraction circles = rulers = meter sticks = scales and weights = cylinders for liquid measure in metric and standard units In general, commercial manipulatives are either = proportional or = non - proportional Proportional materials consist of objects that are proportional to each other with respect to shape and size Example of Proportional manipulative materials: = base ten blocks = Cuisenaire rods = tangrams = fraction tiles = fraction circles Non - Proportional materials are materials like color counters where the color is the only thing that distinguishes one from another

continued -------- Measurement can be either formal or informal. Formal measurement = uses traditional measurement tools. = rulers = tape measures = meter sticks = yardsticks = cups = gallons = pounds = grams Informal measurement = uses readily available objects at hand = footsteps = arm lengths = book lengths = pitcher-fills and so on Informal measurement precedes formal measurement and is necessary and useful in the conceptual development of measurement

assessment principle

continuously gathering data and looking at a variety of techniques

discrete models

counters

formative assessment (alternative assessment)

create response rather than select answer

learning goals

criteria for success are clearly identified and communicated to students

summative assessments

cumulative evaluations that might generate a single score, such as an end of unit test or standardized test that is used in your state or school district

Separate

involves removing elements 3quanities/ variations result unknown->85-29=__?__ change unknown->85-__?__=29 start unknown->__?__-56=29

simple event

describes a single outcome

instructional goals

desired long-term outcome of instruction

instructional objectives

desired outcome of a lesson or unit

National Council of Teachers of Math (NCTM)

development and improvement of math education 6 principles // 10 standards that children K-12 should master

Rhombus

diagonals are right angles, all sides are congruent, diagnols bisect

Multiplicative Comparison

involves the comparison of 2 quantities manipulatively, involves finding "how many times as much" of 1 quantity is compared in another quantity, or "stretching" the original by a certain quantity.

apothem

distance from center of polygon to one side

distributive properties

distribute numbers outside of parenthesis

Multiplication of the reciprocal

division of fractions

instrumental understanding

doing something without understanding introduced by Richard Skemp in 1978, a student who only knows the procedure for simplifying a fraction for example

honor diversity

each learner is unique, with a different collection of prior knowledge and cultural experiences. lessons begin with eliciting prior experiences and understandings and contexts for the lessons selected based on students knowledge and experiences

Recursive Pattern

each successive term of the sequence is obtained from the previous term(s), at least after the first few. to reveal a pattern, the sequence should contain a minimum of 3 terms.

fractal

endlessly repeating pattern that varies according to a set formula

family in mathematics

engage them, talk about their learning, describe their role, use cooperative groups, advocate technology, provide support

provide opportunities to communicate about mathematics

engagement with other learners, reflective thinking, expanding on different ideas and networks

equation solves to y=any number

equation is horizontal line, slope is 0

equation solves to x= any number

equation is vertical line, slope is undefined

six principles

equity, curriculum, teaching, learning, assessment, technology

One-to-One-Correspondence

exactly one element for each element of the other set.

The equity principle

excellence in mathematics education requires high expectations and strong support for all students

communication

explains ideas in writing using words, pictures, numbers/equations, graphs, tables analyzes the thinking of others uses precise language, units, and labeling to clearly communicate ideas

Line of Symmetry

f you can reflect (or flip) a figure over a line and the figure appears unchanged, then the figure has reflection symmetry or line symmetry. The line that you reflect over is called the line of symmetry. A line of symmetry divides a figure into two mirror-image halves. The dashed lines below are lines of symmetry.

problem translation

factual knowledge - one meter equals 100 centimeters linguistic knowledge- floor tiles and tiles refer to the same thing

induction

find a pattern from group of examples

drill limitations

focus on a singular method and an exclusion of flexible alternatives -false appearance of student understanding -a rule oriented or procedural view of mathematics

problem based tasks

focus on important mathematics concepts, stimulate connection of students previous knowledge, allow multiple solutions methods, offer opportunities for correction, encourage students to explain, create opportunities to observe, and generate data

procedures WITH connection tasks

focus students attentions on the use of procedures for the purpose of developing deeper understanding, general procedures, are usually represented in multiple ways, require engagement

cartesian (rectangular coordinate system)

formed by 2 perpendicular axes (x and y)

decimal numbers

fractions written in respect to base 10

standards

general statements regarding the knowledge and skills that students should gain and the characteristics that their accomplishments should reflect

Types of learning: Principle -

generalizations, developed rules Example: The area of a trapezoid is developed from the concept of a trapezoid and the area of triangles, rectangles, and/or parallelograms.

What types of manipulatives should be used in grade 4

geometric solids, geoboards, calculators, protracters, pentominoes, graphing paper, fraction kit, weight set, scatter plot graphs

translation task

give a computational task, ask to write a word problem, ask for an illustration, and explain their process of arriving at an answer or the meaning of the operation

productive disposition

habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy

Part-Part_Whole

involve no action or change over time as happens with join and separate problems. Focus on the relationship between a set and its two subsets (or a whole and 2 parts) Variations involve situations the whole is unknown-> 29+56=__?__ part of the whole is unknown -> 29+__?__=85

Denominator

he bottom number of a fraction, telling in how many parts the whole is divided. in 1/3 the 3 is the denominator.

Least Common Multiple

he multiples of a number are what you get when you multiply it by other numbers. When you list the multiples of two (or more) numbers, and find the same value in both lists, then that is a common multiple of those numbers. The "Least Common Multiple" is simply the smallest of the common multiples.

Equity principle

high expectations and intentional ways to support all students. all need the opportunity, support, to learn no matter what

classroom discussion

how did you decide what to do? what did you do that helped you make sense of the problem? did you find any numbers or information you did not need? did you try something that didn't work?

Exponents

how many times a base number is used. to compare number with exponents it is best to multiply them out or estimate what they might be. ex: 2*2=4, if you had 2*-2 the answer would be 1/4 or (1 over 2*2) ex: when you have a number that is being raised to zero it always equals 1. (195*0=1)

Naturalistic assessment

involves evaluations that is based on the natural setting of the classroom. It involves the observations of students' performance and behavior in an informal context.

Equivalent Set

if and only if there exists a one-to-one correspondence between the sets. (A~B)

inverse

if not p, then not q

contrapostive

if not q, then not p

conditional

if p, then q

converse

if q, then p

Where can a teacher expect students to find items representing the shape of a rectangle?

in the classroom, in their room at home, in a picture book

establish clear expectations

includes both how they will be working and what they what students to demonstrate with their learning

procedural knowledge

includes information about how to perform specific tasks includes basic action sequences as well as more complex knowledge needed to carry out complex cognitive processing of information

4. Extended thinking and complex reasoning

incorporate demands from other content ares in the development and support of real world mathematical arguments.

Master Control Reset

instruction belongs to a group of instructions referred to as program control instructions. These instructions are used to enable/disable blocks of program logic, or move the program scan from one location to another.

Fill (FIL)

instruction can be used when working with a word-to-file data transfer.

File Arithmetic and Logic (FAL)

instruction is more complex than the previous to file-level data transfer instructions.

bit distribute (BTD)

instruction is similar to the move instruction. It differs in that it can be used to move data within a single word.

evidence-based practice

instructional method or other classroom strategy that research has consistently shown to bring about significant gains in student's development and/or academic achievement

Data Compare

instructions are used to compare the numerical data stored in specified words. They do not move, or change the data stored in the specified words.

logarithms

inverse of exponential

Join

involve adding or joining elements to a set. 3 quantities involved: the starting amount__?__ +56=85 the changing amount 29+__?__=85 the resulting amount 29+56=__?__ variations of the join problem include situations when the result is unknown, the change is unknown, or the starting amount is unknown.

Area and Array

involve finding the area of a rectangular area or arrangement. 5 rows of 17 apple trees ex:Multiplication->5x17=__ 85 palm trees/5rows Division->85 divided by 5=__

Variable

is a letter or other symbol that stands for any number within a specified set of numbers.

Expression

is a meaningful string of numbers or variables, or both and operation symbols, and possibly also grouping symbols.

Equation

is a statement of the equality of mathematical expressions; it is a sentence in which the verb "is" represents = to solve an equation, or find the solutions for an equation, means to find all replacements for the variables that make the equation true.

Prime Numbers

is a whole number greater than 1 that has exactly two factors, 1 and itself. 2-the only even prime number, 3,5,7,11,13,17,19,23,31,41,43,47,53,59,61,71,73,79,83,89,97 1 IS NOT PRIME

Composite Number

is a whole number greater than 1 that has more than two factors 4,6,8,9 1 IS NOT COMPOSITE

1

is not considered prime or composite

drill and practice

is present to at least some degree

Copy (COP) instruction

is used to duplicate the data in file, to a different file specified by the user program.

Scale (SCL):

is used to enlarge or reduce very small or very large numbers by the rate value. This is utilized for converting values generated by a 4-20mA

What is true of multiplication?

it represents repeated addition, is commutative and associative, can be modeled with area diagrams

A pictograph of results is appropriate for young children because:

it shows them how math is used in their own lives, it gives them a chance to participate directly in the lesson, and it provides the opportunity to quantify results and compare values

Addition and Subtraction: 4 types of problems

join part-part-whole separate compare

reasoning

justifies solution methods and results recognizes and uses counterexamples makes conjectures and/or constructs logical progressions of statements based on reasoning

relational understanding

knowing what to do and why, introduced by Richard Skemp in 1978, a child that can draw diagrams, give examples, find equivalencies, and tell the approximate size

recalling simple facts?

knowledge

ELL and Math

math is not universal use Arabic numbers does not present problems for ELL's can't comprehend the explanation of the process in a language that ELL's have not mastered. Inductive teaching - learning through examples deductive teaching - learning step by step

law of large numbers

larger the sample size, closer sample mean will be to population mean

greatest common factor

largest number that a factor of all numbers given in a problem

learning principle

learning mathematics with understanding is essential, it not only requires computational skills but also the ability to think and reason and solve problems

metric system

length: meter weight: gram volume: liter

problem solving develops

mathematical processes, student confidence, provides a context to help students, allows extensions and elaborations, engages students so that there were fewer problems

theorums

mathematical statements that can be proven by geometry

problem solving strategies

look for patterns, predict and check, justify claims, create a list, create a table, create a chart, simplify the problem, write an equation

questioning technique

make questions to clarify problem, eliminate solutions, and simplify the problem-solving process

standards for mathematical practice

make sense of problems and persevere in solving them reason abstractly and quantitatively construct viable arguments and critique the reasoning of others model with mathematics use appropriate tools strategically attend to precision look for and make use of structure look for and express regularity in repeated reasoning

manipulatives (concrete)

materials students can physically handle and move

1. Nomenclature of Math for ELL

math classrooms assume students know specific terms such as denominator, subtraction, minuend, divisor, subtrahend based off of prior knowledge. ELL's do not have this prior knowledge

z- score

measure of distance in standard deviation of a sample from the mean

low level of cognitive demand

memorization tasks, procedures without connection tasks

create opportunities for reflective thought

mentally engaged, see how concepts are connected to each other, interconnected rich web of interrelated ideas

early math started...

mesopotamia, egypt, greece, rome

generative learning

must provide the students with an opportunity to mentally 'play with' information to create a personal understanding of the subject to be learned

continuous model

number lines that students might choose for solving problems

central limit theorem

number of sample increases, distribution sample means approaches normal distribution

integers

numbers preceded by either a + or -

Tessellation

ometimes referred to as tiling of the plane. A tessellation is a collection of plane figures that fills the plane with no overlaps or gaps. Tiles on a kitchen floor can be thought of as a simple form of tessellation.

permutation

one of a number of possible selections of items, without repetition, where order of selection is important

combination

one of a numer of possible selections, without repetition, where order of selection is not important

Student passes out 15 napkins for 18 students and determines he needs more by

one-to-one correspondence and pre-subtraction skills

function

only one x corresponds to one y

linear programming

optimization of linear quantity that is subject to constraints expressed as linear equations or inequalities

What is the progression of a lesson?

oral review written review new concept review of new concept

Thematic instruction

organization of curriculum content based on themes or topics integrates basic disciples (reading, math, music, art, science) with the exploration of broad subjects provides opportunity for students to hear similar information in various instructional segments and form a variety of sources.

Thematic activity

organized into 3 main segments 1. introduction to the unit 2. presentation of the content 3. closing activity

Abstract

paper and pencil/mental math

alternate interior angle theorem

parellel lines cut by transversal, alternate interior angles are congruent (corresponding angles are equal

semiotic meditation

part of sociocultural theory, refers to the use of language and other tools, such as diagrams, pictures, and actions they are exchanged between and among people when this occurs

frieze

pattern that repeats in one direction; seven possible patterns

Use __________ to ___________ influence

peer to peer

study us system

pg 186

fundamental counting principles

pg 201

4 basic tessellations

translation, rotation, reflection, glide reflection

Quadrilaterals

trapezoids, kites, parallelograms, rectangles, rhombuses, and squares.

differentiated instruction

practice of individualizing instructional methods- and possibly also individualizing specific content and instructional goals- to align with each student's existing knowledge, skills, and needs

lesson plan

predetermined guide for a lesson that identifies instructional goals or objectives, necessary materials, instructional strategies, and one or more assessment methods

constructivism

prior knowledge greatly influences learning math

Students use a balance and cubes to determine which toy is heavier, they are using:

problem solving and number sense

five process standards

problem solving, reasoning and proof, communication, connections, representation

Solving 20-15 with p + 15 = 20 shows that:

problems can be solved in multiple ways, addition and subtraction are inverse operations, and a letter can stand in the place of an unknown number when solving a problem

Open-ended problems

problems for which students are asked to find more than one answer ours more than one method. Students may also be asked to design an extension to the problem.

solution execution

procedural knowledge - 7.2 x 5.4 = 38.88 0.3x 0.3 = .09 38.88 x .09 =432 432 x $0.72 = $311.04

high level cognitive demand

procedures with connection tasks, doing mathematics tasks

task analysis

process of identifying the specific behaviors, knowledge, or cognitive processes necessary to master a particular topic or skill

Deductive Reasoning

process of reaching a necessary conclusion solely from a set of facts or hypotheses. These facts are called the assumptions or premises.

Preoperational Stage (2-7 years)

process of symbolic functioning, centration, intuitive thought, egocentrism, inability to conserve. child experience problems with two perceptual concepts (centration and conservation) 1. centration - focusing on only one aspect of a situation or problem 2. understanding that quantity, length, or number of items is unrelated to the arrangement or appearance of the object or item.

Formative Assessment

process used by teachers and students during instruction that provides feedback to adjust on going teaching and learning to improve students achievement of intended instructional outcomes

memorization tasks

producing already learned facts, are routine, have no connection to related concepts

When introducing students to a new math topic, a teacher should:

provide opportunities for students to investigate and explore the new material in small groups and individually

self and peer assessment

providing student an opportunity to think meta cognitively about their learning

to simplify

put in simplest form

Types of learning: Problem Solving—

putting together concepts and principles to solve a problem new to the learner Example: Given a composite figure the student determines the area using the areas of triangles and rectangles.

Without math science is what?

qualitative

subjective and relative?

qualitative

Objective and absolute?

quantitative

With math science is what?

quantitative

waiting

quiet time should not be uncomfortable, but should feel like thinking time

treat errors as opportunities for learning

rarely give random answers, so their wrong answers are an insight to what they may have not understood correctly

scale factor

ratio of any 2 corresponding measurements of similar solids

3. Strategic thinking and complex reasoning

reasoning,planning, using evidence, and a higher level of thinking than the previous two levels; making conjectures is also at the this level.

Is 1 divided by that number.

reciprocal

Product of any number (except 0) multiplied by its __________ is 1.

reciprocal

practice

refers to different tasks or experiences, spread over numerous class periods, each addressing the same basic ideas increased opportunity to develop conceptual ideas and useful connections to develop alternative and flexible strategies a greater chance for all students to understand a clear message that math is about figuring things out

drill

refers to repetitive exercises designed to improve skills or procedures already acquired -an increased facility with a procedure -review of facts or procedures so they are not forgotten

Reliability

refers to the consistency of a measure. A test is considered reliable if we get the same result repeatedly. For example, if a person is administered the same test repeatedly his/her results on the test should be approximately the same each time, if the test is reliable.

number sense

refers to whether a students understanding of a number and of its use and meaning is flexible and fully developed

Types of learning: Concept—

relational or concrete attributes. Example: Similar figures have relational attributes. The corresponding angles are equal and the ratios of corresponding sides are equal.

compensation

replacing numbers when guessing and checking

models

represent mathematicals concepts by relating concepts to real-world situations

elaborating

request for students to challenge, add on, or give an example. intended to get more participation, deepen understanding

doing mathematical tasks

require complex and non algorithmic thinking, exploration, demand self monitoring, require cognitive effort, access to relevant knowledge, and analyzation of the tasks

2.Basic application of concepts and skills

requires engagement of some mental processing beyond a habitual response, and making some decisions as to how to approach the problem or activity, following a defined series of steps.

anchored instruction

requires putting the students in the context of a problem-based story. The students "play" an authentic role while investigating the problem, identifying gaps to their knowledge, researching the information needed to solve the problem, and developing solutions. For example, the students play the role of a pilot to learn about aeronautics subject matter such as gravity, airflow, weather concepts, and basic flight dynamics. The teacher facilitates and coaches the students through the process. Learning and teaching activities should be designed around an "anchor" which is based on a contextualized case study or problem situation. Curriculum materials should allow exploration by the learner (e.g., interactive sites) to allow active manipulation, questioning, and involvement in the situation.

quantity discrimination

requires students to name which of two numbers is larger or smaller

Performance assessment

requires that completion of a task, project, or investigations; communicates information ; or constructs a response that demonstrates knowledge or understanding of a skill or concept.

revoicing

restating the statement as a question in order to clarify, apply appropriate language, and involve more students

one up one down

results in an answer that is one less than the original problem. 7x7=49 8x6=48

congruent figure

same size, same shape

postulates

sas (side angle side) sss (side side side) asa (angle side angle)

problem integration

schematic knowledge - the tile problem requires the formula area = length x width

Math is the language of what?

science

Mathematics is the language of _________?

science

geometric sequence

series of numbers in which a common ratio can be multipled by a term to yield the next term =

Domain

set

Intersection

set of all elements common to both sets A and B.

Universal Set

set of all the numbers being considered

Coordinates

set of numbers called an ordered pair (x,y) indicating the horizontal and vertical location of a point in space.

arithmetic sequence

set of numbers with specific difference between terms = a + (n-1)d

common core state standards

set of standards that most U.S states have adopted to guide instruction and assessment in English-language arts and math

y>or= 2x + 2

shaded top left

curriculum principle

should be coherent and built around 'big ideas' in the curriculum and in daily classroom instruction

tiered lessons

similar problems focused on the same mathematical goals, but adapted to meet the strange of learners, with different groups of students working on different tasks

polygon

simple closed figure composed of line segments

Range

single element in a set.

teachable moment

situation or event (often unplanned) in which students might be especially predisposed to acquire particular knowledge or skills

procedural fluency

skill in carrying out procedures flexibly, accurately, efficiently, and appropriately includes flexibility and ability to choose an appropriate strategy

Using a number line is appropriate to introduce

skip counting and counting backward

least common multipler

smallest number that all given numbers will divide into

Which are behaviors demonstrating concrete understanding of mathematics?

sorting objects into equally sized groups, giving one pencil to each student in the class, and counting the number of books on each shelf

activate prior knowledge

specific mathematical learning goals, what has been previously learned

procedures without connections tasks

specifically call for use of procedure, are straightforward, have no connection to related concepts, require no explanations, are focused on producing correct answers

standard deviation

square root of variance

solution planning and monitoring

strategic knowledge - step by step procedure

Traditional math workouts ______________ the mind?

strengthen

Reading a story out loud about two children playing hide-and-seek would

strengthen the children's understanding of spatial concepts such as behind, under, and inside

Alternative assessment in math may include:

student explanation of reasoning behind the answer, analysis of data, and multimedia

build new knowledge from prior

students apply their knowledge, test ideas, make connections, compare and make conjectures. The more students see the connections among problems and among mathematical concepts, the more deeply they understand mathematics

missing number

students are provided with a string of numbers and are asked to identify which number is missing

computation

students asked to complete computations that are representative of their grade level - have 2 minutes to complete as many problems as possible

number writing

students asked to write number when given a number orally between 1 and 20

Math Language

students must build a shared understanding of mathematical terms to successfully share and refine their ideas. Differentiated instruction when learning terms and symbols.

number identification

students must orally identify numbers between 0 and 20 when presented randomly on a piece of paper

descriptive feedback

students receive evidence based feedback linked to the intended instructional outcomes and criteria for success

Learning Log ex.

students record the process they go through in learning something new. This allows students to make connections to what they have learned, set goals, and reflect upon their learning process.

grading

summarizing a students performance through the accumulation of a variety of scores and data about their understanding of important skills and concepts

What types of manipulatives should be used in grade 3

tangrams, pattern blocks, playing cards, scales, magnetic numbers, chalk boards, spinners, dice, calendar, games, measuring tape, charts, bar graphs.

bloom's taxonomy

taxonomy of 6 cognitive processes, varying in complexity, that lessons might be designed to foster

collaboration

teachers and students are partners in learning.

Quality Assessment

teachers gather student work and materials in order to gauge and advance student learning

teaching principle

teachers must understand deeply the mathematics content, understand how students learn it, and select meaningful instructional tasks and generalizable strategies that will enhance learning

Progress Monitoring

teachers should be monitoring students learning progress by asking these questions before, during and after math instructions: - is the student making progress? - what does the student need to learn next? - how solid is the students understanding? - does the student need more work with a specific content? -etc.

2. Nomenclature of Math for ELL

terms have one meaning in one subject and another meaning in math which can confuse ELL's. quarter, column, product, rational, even, table.

summative assesment

test what has been learned; like a test or final project

Objective assessment

testing that requires the selections of one item from a list of choices provided with the question. This type of assessment includes true false responses, yes- no answers and questions with multiple- choice answers.

Complement

the __________ of set A is the set of all elements in the universal set U that are not in A

Greatest Common Factor (GCF)

the greatest factor that two or more numbers have in common. ex: 6 is the greatest common factor of 18 and 30. Use the factor tree, include 1

associative property

the grouping of numbers does not matter

Elements

the objects in a set

communiatvie properties

the order of numbers does not matter

Inductive Reasoning

the process of making a generalization based on a limited number of observations or examples.

Ability grouping

the process of placing students of similar abilities into groups and attempting to match instruction to the needs of these groups.

Natural Numbers

the set composed of the natural numbers and zero

whole numbers

the set containing number 0, 1, 2, 3, ...

counting numbers

the set containing numbers 1, 2, 3, ....

Union

the set of all elements in set A or B or both.

word names

the specific names given to the concept of a specific quantitative unit

operations

the standard process through which a qualitative entity is manipulated

Jean Piaget

thinking and reasoning of children were dominated by preoperational thought (patter of thinking that is egocentric, centered, irreversible, and nontransformational). 4 schemes of children: sensorimotor stage, preoperational stage, concrete operational stage, formal operational stage

to multiply

times

plus

to add

to increase by

to add

Which of the following is an effective math teaching strategy?

to be clear about instructional goals, to progress experiences and understanding from the concrete to the abstract stage, and to communicate to students what is expected of them and why

to solve

to find the answer

to solve for "x"

to find what the value of "x" is

A good way to create a concrete learning experience for students to learn how to count money is:

to give students an assortment of coins to sort and count

to round a number

to make a number simpler

to enlarge

to make bigger

to reduce

to make smaller

to label

to mark

times

to multiply

to double

to multiply by 2

to triple

to multiply by 3

to square a number

to multiply the number by itself

Why are manipulative, models, and technology used by math teachers?

to promote interest, to address diverse learning needs, to give hands-on math experience

to box answer

to put a square around answer

to explain

to put into words

to add

to put together

What is the main purpose of having K students count by twos?

to recognize patterns in numbers

to cost

to sell for

to complete

to solve

to divide

to split into equal parts

minus

to subtract

to decrease by

to subtract

to subtract

to take away

probability strategies

tree diagrams, grids

Correct steps yield to correct answers

true

Math provides a way to measure

true

conditional probability

two events that are not independent are dependent

compound event

two or more simple events

Competency 001-Math Instruction

understand how students learn math

Infinite Set

unlimited number of elements.

solve problem with overlapping lists

use a venn diagram

encourage multiple approaches

use of strategies that make sense to students, allow them to use their own strategy to find the answer or solution

formative assessment

used to check status of students development during instructional activities, to pre assess, or to attempt to identify students naive understandings or misconceptions 1. identify where learners are 2. identify goals for learners 3. identify path to reach the goal

visualize

using manipulatives, acting it out, drawings, helps understandings

Abstract

using numbers or numerals

cooperative learning

usually small groups of students (3-5) can be used to have students work together to solve problems

rote counting

verbal repetition of numbers begins around the age of 2-3 years foundation for building and understanding of the number concepts of combining, separating, naming amounts

Semi-concrete

visuals, no manipulatives --Using pictures to add

3. Nomenclature of Math for ELL

vocab tends to encompass tons of homophones (same pronunciation but different meanings) which can confuse ELL's. Even, face, plane, mean, right, some/sum, volume

Disjoint

when A and B sets = O

assimilation

when a new concept fits with prior knowledge and the new information expands an existing network

accommodation

when the new concept does not fit with the existing network causing a disequilibrium

Flexibility

when the students are able to understand more than one computational algorithm for a particular exercise. The students are able to choose the most appropriate approach for a given exercise.

rational number

whole numbers, integers, and fractions (with no decimals)

Types of learning: Association—

words or symbols Example: Even very young children associate the word "triangle" with the ∆, without knowing the attributes and properties of a triangle.

problem solving

works to make sense of and fully understand problems before beginning perseveres to demonstrate a variety of strategies assesses the reasonableness of answers

National Council of Teachers of Mathematics

worlds largest mathematics education organization. has an emphasis on what is best for students and learners

3rd grade students are recording the length of the hallway. What unit should they use?

yards


Kaugnay na mga set ng pag-aaral

Global Business: Practice exam #1

View Set

Course 3 Module 10: Formula Investing and Investment Strategies

View Set

BOC Chemistry Practice Questions

View Set

Basic Insurance Concepts and Principles

View Set