Math Final (Ch 16, 18, 19, 21)

what happens if the teacher tries to teach at a level of thought that is above a student's level

- not understand - memorize the material (seem to be understanding) - easily forget material - unable to apply it

Explain how the area formula of a circle is related to the area of a "bumpy" parallelogram

- start with circumference (distance around circle) (2 pi R) - cut it into 8 wedges - configure them to look like a near parallelogram - student should notice that the height of the parallelogram is the radius of the circle - the base of the parallelogram = half of the edges of the circle on the top, half at the bottom, so it is half of the circumference (pi R) - all of this equals piR squared which is the area of a parallelogram

explicitation

- students describe what they have learned about the topic in their own words - teacher introduces relevant math terms

guided orientation

- students explore the objects of instruction in carefully structured tasks such as folding, measuring, or constructing - ensures that students explore specific concepts

5 levels of the van Hiele model of geometric thought

0. recognition or visualization 1. analysis 2. Ordering or Informal Deductive 3. Deduction or Formal Deductive 4. Rigor

five phases of learning

1. information 2. guided orientation 3. explicitation 4. free orientation 5. integration

What is a key idea in connecting whole number place value and decimal fraction place​ value?

10 to 1 multiplicative relationship between values of any two adjacent positions

diagonals in a square

2

the diagonals of a rhombus

2 diagonals of reflectional symmetry

rhombus sides & characteristics

2 pairs of parallel sides

The process of​ concrete-semi-concrete-abstract (CSA) described earlier in this text can refer to the phases of teaching probability. Which of the following would match with​ semi-concrete?

Representation

Following steps to set up a successful simulation involve three of the following. Which one would not be​ useful?

Select an expensive method to manipulate the real situation

Which of the following statements is not true about​ simulations?

Simulations are important in middle school because they provide an engaging way in which to explore probability and connect to the abstract and difficult concepts related to compound events.

What activity described below would guide students understanding of​ halves, thirds,​ fourths, and eighths as decimal​ fractions?

Students shade in a 10 x 10 grid to illustrate a familiar fraction

The following statements are true about early experiences in probability except​:

Students should begin in middle school when they are developmentally ready.

Which of the following statements is true​?

Teach experiments and theoretical probability​ together, focusing on the number of trials needed for experiments to reflect the theoretical probability.

When students explore how shapes fit together to form larger​ shapes, it is​ called:

composing shapes.

When a figure can be reflected over a line and rotated about a point this combination of transformations is​ called:

composition

A good teaching option for developing a full understanding of computation with decimals is to focus​ on:

concrete​ models, drawings, place value​ knowledge, and estimation.

The following activities support students learning about​ two-dimensional shapes in different orientations except​:

constructing figures with centimeter cubes.

The role of the decimal point in a number is​ to:

designate the units position.

The following are​ "place learning" words learned as a position description in kindergarten except​:

direction.

t/f: all parallelograms have congruent diagonals

false

t/f: all pyramids have square bases

false

t/f: if it has exactly two lines of symmetry, it must be a quadrilateral

false

The most important factor in moving students up the van Hiele levels​ is:

geometric experiences that teachers provide to the students.

The following are suggested strategy for estimating except​:

guessing

Spatial sense includes all the following except​:

identifying hierarchy of geometric properties.

A good task to use with students to assess their understanding of the use of a standard ruler​ is:

measuring with a​ "broken ruler."

All of the materials below can be used to represent an area model of decimal fractions​ except:

meter stick

angles of rhombus

opposite angles are congruent (same size)

Students should explore the area of a triangle after they have a conceptual understanding of the area of​ a:

parallelogram

A collection of uniform objects with the same mass can serve as nonstandard weight units except​:

plastic toys.

Probability is about how likely an event is. A good place to begin is​ with:

possible and not possible

Exploring properties of quadrilaterals is a rich investigation for students. The following are important concepts that emerge from these investigations except​:

prisms are special cylinders

The following are money ideas and skills typically required in primary grades except​:

reading and writing number and word money amounts.

When a shape can be folded on a​ line—so that the two halves​ match—that fold line is also a line​ of:

reflection

symmetry of rhombus

reflection symmetry of 180

symmetry of parallelogram

rotational symmetry of 180

When you are measuring an object using a tool and choosing the attribute to be​ measured, then you​ must:

select a tool with the same attribute to measure with

The four major content goals in geometry for all grade levels​ are:

shapes and​ properties, transformation,​ location, and visualization.

One of the basic ideas of length measurement is that when the unit is​ longer, the measure​ is:

smaller

Rotational symmetry is described​ as:

smallest angle required to have shape match its footprint

The value of a collection of coins is best learned by having​ students:

sort the coins starting with the highest value and skip counting.

free orientation

students apply relationships they are learning to solve problems and investigate more open-ended tasks

integration

students summarize and integrate what they have learned, developing a new network of objects and relations

An area model that demonstrates how figures can have the same area composed of different shapes​ is:

tangrams

Area representations have all of the following features except​:

they are readily adaptable to situations with three events.

information

through discussion, teacher identifies what students already know about a topic and the students become oriented to the new topic

Coordinate grids are often used in geometry to​ explore:

transformations.

t/f: all squares are rectangles

true

t/f: if it is a square, then it is a rhombus

true

t/f: some parallelograms are rectangles

true

Angles are measured​ by:

using a smaller angle to fill or cover the spread of the rays.

5 skill areas of geometry

visual, verbal, drawing, logical, applied skills

Physical models provide the main link between​ fractions, decimals, and percents. Identify the one model that is suitable for all three because they all represent the same idea.

​Base-ten models

What are the standard units used to measure​ capacity?

​Milliliters, centiliters, and liters

According to your​ textbook, which of the following trios of​ real-world situations represent common uses for estimating​ percentages?

​Tips, taxes, and discounts

The following are examples of independent probability events except​:

Drawing a blue cube from a bag of six different colored cubes

Measurement​ _______________________ is the process of using mental and visual information to measure or make comparisons without the use of measuring instruments.

Estimation

​Base-ten models, the rational number wheel with 100 markings around the​ edge, and a​ 10-by-10 grid are all models for linking which three concepts​ together?

Fractions, decimals, and percents

In the​ real-world decimal fractions are rarely those with exact equivalents to common fractions. Students need to wrestle with the magnitude of decimal fractions. Identify the activity below that addresses magnitude.

Identify what 7.396 is close to​ 7 or 7 1/2 or 8

Which of the following is true about the van Hiele​ levels?

They are a progression of ways in which students understand geometric ideas.

Students need to be acquainted with various visual models to help them think flexibly of quantities in terms of tenths and hundredths. Which example below would help students understand the decimal fraction 65/100 in terms of place​ value?

This decimal fraction could be thought of as 6 tenths and 5 hundredths

To answer the question​ "What is the chance of having triplets being all​ girls?" the best random device for a simulation would​ be:

Two color spinner

One of the main goals of the visualization strand is to be able to identify and draw which of the​ following?

Two-dimensional images of​ three-dimensional shapes

Three of the four important principles of iterating units of length are listed below. Identify the one that is actually a misconception.

Units are measured by the ending point of a ruler

It is important for students to experience and explore percent relationships in realistic contexts. Three of the statements below are guidelines to follow for presenting percents. Identify the one that does not support best practices.

Use the following sentences in their solutions​ "____ is​ ____ percent of​ _____"

Students need to learn that time is something that can be measured. All the activities listed help them think in terms of​ seconds, minutes, and hours except​:

a.m. and p.m.

Symmetries of a Square

all sides are symmetrical

For students to have a conceptual understanding developing formulas for perimeter and​ area, they should do all the following except​:

be told the formula.

diagonals of parallelogram

bisect each other

How can one obtain an accurate measure of the volume of a rectangular prism when given a set of the​ same-sized cubes?

Layer the cubes on the bottom of the box to fit the dimensions and then see how many layers are needed.

Which of the following is one of the best approaches for teaching elapsed​ time?

An empty number line

Which van Hiele level is it when students are considering classes of shapes and focusing on properties of​ shapes?

Analysis

When using​ base-ten materials in developing decimal concepts what is an important idea to be​ realized?

Any piece could be effectively chosen as the ones piece

Purposes of conducting experiments​ (doing simulations) are important for all of the following reasons except​:

Assess whether students have a probability sense

What type of scale measures mass from small to​ large?

Beam or balance scale

The natural progression for teaching students to understand and read analog clocks includes starting with which of the following​ steps?

Begin with a​ one-handed clock that can be read with reasonable accuracy.

What tool would allow students to make an axonometric drawing where scale is​ preserved?

Centimeter isometric dot paper

What is the cognitive skill that helps students recognize and group shapes according to their attributes and​ properties?

Classification

The​ "place learning" words lay the foundation for students to identify points on what​ system?

Coordinate plane

Money skip counting and using the hundreds chart to money count support what mental mathematics​ strategy?

Counting on

Which term refers to an event whose results depend on the results of the first​ event?

Dependent event

Which of the following transformations is a nonrigid​ transformation?

Dilation

What description below describes a visualization​ activity?

Draw and recognize objects from different viewpoints

What would be a prerequisite to being successful in measuring​ angles?

Mental images of angle size

Multiplication of decimals is poorly understood for many reasons. Identify the misunderstanding that relates to whole number multiplication.

Multiplying makes the product larger

Which of the following would result in an unequal likelihood versus and equal​ likelihood?

Number cube with sides​ 4,4,4, 5,5,5,​ 6,6,6

Angles in a Parallelogram

Opposite Angles are Congruent

sides of parallelogram

Opposite sides of a parallelogram are congruent

Which of the following is a nonstandard unit of​ measure?

Paper clip

To compare the weights of two​ objects, which of the following is the best​ approach?

Place the two objects in the two pans of a balance.

What mathematical​ tool(s) would provide students with the opportunity to make a conjecture about how likely an event​ is?

Probability number line 0 impossible to 1 certain

Angle measurement can be a challenge for some students for the following reason.

Protractors are used to measure angles

Comparison activities guide students understanding of volume and capacity. Identify the activity that would not use volume comparison.

Provide students with grid paper and rulers to construct different sized rectangular prisms

Estimation is particularly important for students who have learned the rules of computation but cannot decide​ about?

Reasonable answers

Which of the following is shown through research to be a common error or misconception when students are comparing or ordering​ decimals?

The decimal that is the shortest is the largest.

What event listed below would be an example of known sample​ space?

What is the probability of drawing a red cube from a bag of six different colored​ cubes?