FTCE Elementary Education K-6 Math

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

Inverse Property Additive/ Multiplicative

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

Inverse

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

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

Iteration

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

Components of math fluency

Accuracy, Automaticity, Rate, Flexibility.

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=__

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.

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.

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.

1. Recalls information

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

Whole Numbers

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

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.

Prime numbers

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

Norm referenced

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

Rhombus

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

Major operations on rational numbers

addition subtraction multiplication division

Authentic assessment

alternative assessment that incorporates real-life functions and applications.

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

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

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.

Accuracy

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

Attribute blocks

classifying, geomtry, logical reasoning

Achievement test battery

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

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.

Natural numbers

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

Concrete

hands on manipulatives

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)

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.

Automaticity

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

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.

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

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.

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.

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.

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.

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=__

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

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.

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

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=__

Separate

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

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.

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.

Alternative assessment

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

Multiples

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

Accuracy

Getting the correct answer

Abstract model

Using numbers only to perform operations

The Multiplicative Identity

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

$197.95

What is 25% off of $264?

Integers

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

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

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

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)

Parallelogram

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

Quadrilateral

4 sides sum of angles =360

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

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.

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

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.

Algorithms

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

Ratio

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

Flexibility

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

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.

Types of problems for Multiplication/Division

Equal groups or repeated addition Area and array Combination Multiplicative Comparison

Rate

How quickly computations are made

Rate

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

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?

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

Array

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

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.

Pictorial

Picture or mental image. Visualize

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)

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.

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

Natural Numbers

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

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

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

Semi abstract model

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

Concrete Model

Use objects to demonstrate operations

Semi Concrete Model

Use pictures (instead of actual objects) to demonstrate operations

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

1. ex. 190^0 = 1.

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

4. Extended thinking and complex reasoning

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

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

Addition and Subtraction: 4 types of problems

join part-part-whole separate compare

Whole numbers

natural numbers and zero

Quadrilaterals

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

Abstract

using numbers or numerals