Try It Out! Measuring Up to the TEKS

Try It Out! Sample Pack | Math | Grade 7 | Lesson 9

Measuring Up to the TEKS

The Try It Out! sample pack features:

• 1 full student lesson with complete Teacher Edition lesson• 1 full Table of Contents for your grade level• Lesson Correlations

Developed to meet the rigor of the TEKS, Measuring Up employs support for using and applying critical thinking skills with direct standards instruction that elevate and engage student thinking.

TEKS-based lessons featureintroductions that set students up for success with:

aAcademic Vocabulary

aStep-by-Step Problem Solving

aDemonstrate Higher-OrderThinking Skills

aMulti-Step and Dual-CodedQuestions

Guided Instruction and IndependentLearning strengthen learning with:

aDeep thinking prompts

aCollaborative learning

aSelf-evaluation

aDemonstration of problem-solving logic

aApplication of higher-order thinking

Flexible design meets the needs ofwhole- or small-group instruction.Use for:

aIntroducing TEKS

aReinforcement

aIntervention

aSaturday Program

aBefore or After School

Extend learning with online digital resources!Measuring Up Live 2.0 blends instructional print resources with online, dynamic assessment andpractice. Meet the needs of all students for standards mastery with resources that pinpoint student needs with customized practice.

MasteryEducation.com | 800-822-1080 | Fax: 201-712-0045

ADAPTIVE, DIFFERENTIATED

PRACTICE

+TEKS-BASED

ASSESSMENTS

+TARGETED

INSTRUCTION

Mathematics • Level G Copying is illegal. Measuring Up to the Texas Essential Knowledge and Skills52

Understand the TEKSYou can use proportions to solve problems involving similar figures.

Two figures that are similar have the same shape, but not necessarily the same size. Corresponding dimensions or angles in similar figures are the pair that in the same relative positions in the figure. In similar figures, corresponding angles have equal measures and the measures of corresponding dimensions are proportional.

If �ABC is similar to �DEF, then the pairs of corresponding angles with equal measures are: � A and � D, � B and � E, and � C and � F. The corresponding sides are

___ AB and

___ DE ,

___ BC and

___ EF ,

and ___

CA and ___

FD . Since their measures are proportional, ratios with corresponding sides are equal. So, for �ABC and �DEF: AB ___ DE � BC ___ EF and AB ___ BC � DE ___ EF .

To solve for unknown side lengths of similar figures, set up a proportion, use a variable for the missing length, and then cross multiply to solve the proportion.

Guided Instruction

Two books are similar in size. The larger book is 8 inches wide and 11 inches tall. The smaller book is 6 inches wide. Will the smaller book fit on a shelf that is 9 inches tall?

Use the fact that corresponding sides of similar fi gures are proportional.

Step 1 Draw a diagram to model the situation.

Mark the given information on the diagram.

Let h � the height of the smaller book.

Step 2 Identify corresponding sides of the similar figures to write a proportion.

__ � 11 ___ h

Step 3 Solve the proportion to find the value of h.

8h � 6 �

8h �

Words to Knowsimilarcorrespondingproportional

8 in.

11 in.

6 in.

h

Problem 1

S 7.5(A) Generalize the critical attributes of similarity, including ratios within and between similar shapes.R 7.5(C) Solve mathematical and real-world problems involving similar shape and scale drawings.

Similar FiguresLesson9

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Mastery Education • MasteryEducation.com Copying is illegal. Chapter 2 • Proportional Relationships 53

Similar Figures Lesson 9

h �

The smaller book is inches tall.

Will the smaller book fit on a shelf that is 9 inches tall?

The triangles shown are similar triangles. How does the ratio of LM to MN compare to the ratio of RS to ST? How does the ratio of LM to RS compare to the ratio of MN to ST?

M N20

6

8

15

L

T S

R

Step 1 Write each ratio in fractional form.

LM ___ MN � ___

RS ___ ST � __

LM ___ RS � ___

MN ___ ST � ___

Step 2 Simplify each ratio.

LM ___ MN � __

RS ___ ST � __

LM ___ RS � __

MN ___ ST � __

Step 3 Compare the ratios.

LM ___ MN RS ___ ST

LM ___ RS MN ___ ST

Solution

Problem 2



Lesson 9 Similar Figures

Note that because �L and � R are marked as congruent, similarity is such that �LMN is similar to �RST. Therefore, the ratios defined in this problem relate corresponding sides:

LM and RS as well as MN and ST.

How does the ratio of LM to MN compare to the ratio of RS to ST?

How does the ratio of LM to RS compare to the ratio of MN to ST?

Another Example

At the park, there are two similar fountains. The smaller fountain has a diameter of 12 feet and a height of 4 feet. The larger fountain has a height of 8 1 __ 4 feet. Estimate the diameter of the larger fountain.

Since the fountains are similar, we know that their dimensions are proportional.

4 ft

d

8 ft1 4

12 ft

Since the larger fountain’s height is approximately twice the height of the smaller fountain, its diameter is approximately twice the diameter of the smaller fountain.

12 � �

The larger fountain’s diameter is about feet.

Solution




Critical ThinkingSolve each problem.

1. In triangle MAY, MA � 4 feet, AY � 8 feet, and MY � 10 feet. It is similar to triangle ROW. Jackie said that RO � 8 feet, OW � 4 feet, and RW � 20 feet. Find the error in her reasoning.

2. Work with a partner to create similar rectangles and analyze the areas. One partner cuts out a small rectangle and record the measurements. The other partner cuts out a rectangle with dimensions that are doubled. Use observation to record the number of small rectangles that would fit inside the area of the large rectangle. Record the areas of both rectangles. Now, repeat the activity - this time creating a large rectangle with dimensions that are tripled. Is the ratio of corresponding dimensions ratios the same as the ratio of areas? Record your observations in your math journal.

3. You can find the height of an object without measuring it if you can measure its shadow and another object’s height and shadow. Work with a partner. Find an object that is too tall to be measured directly, but that is casting a shadow with a length you can measure (e.g., school building, flagpole, etc.). Then, measure the height of one of you and your shadow. Use similar triangles to find the height of the immeasurable object. Make a poster of your findings and discuss your results with the class. Did you all come up with the same ratio to use in your proportion?

4. Work in groups. Open a spreadsheet file and set up 4 columns with the following headings: Furniture Item, D, W, H. Using the standard measurements for furniture below, set up a formula to find the sizes of similar figures for miniature replicas. Use a 1/12 (.083) ratio. You will need two rows for each piece of furniture: full size and miniature size. Make sure to place each measurement in a separate cell so you can apply the formula.

Standard measurements: Couch: 36" D � 84" W � 36" H; Loveseat: 36" D � 60" W � 36" HArmchair: 36" D � 36" W � 36" H; Coffee table: 30" W � 48" L � 18" H

Elevate

CollaborativeLearning

Discuss



Lesson 9 Similar Figures

★ Practice

DIRECTIONS Read each question. Then circle the letter for the correct answer.

1 Two similar triangular pastures meet at a vertex. The dimensions are given in yards. How much fencing would be needed to enclose the larger pasture?

11

57

13

y

x

A 28.2 yards

B 42.7 yards

C 47 yards

D 59.8 yards

2 The ratio of the corresponding dimensions of two similar rectangles is 3 : 2. The ratio of the length to width for each is 4 : 3. If the width of the smaller rectangle is 120, what is the length of the larger rectangle?

F 135

G 160

H 180

J 240

3 A tree casts a shadow 20 feet long. At the same time, a boy 5.5 feet tall casts a shadow 8 feet long. Which proportion CANNOT be used to find the height of the tree?

A 5 . 5 ___ 8 � 20 __ x

B 5 . 5 ___ x � 8 __ 20

C 8 ___ 5 . 5 � 20 __ x

D 20 ___ 5 . 5 � 8 _ x

4 Ellen has created a design using two parallelograms. Determine whether the parallelograms in the design are similar or not.

D

A B

C

3

1577

D

A B

C

5

24

103

F No, because the angles do not have equal measures

G No, because 15 __ 24 ≠ 3 _ 5

H No, because 5 � 3 ≠ 24 � 15

J Yes, because 3 : 15 is equivalent to 5 : 24




1 A surveyor found a distance across a pond by measuring similar triangles ACD and BCE on land.

E

D

AB

C

�ACD is similar to �BCE

If AD � 10 yards, AC � 45 yards, and BE � 60 yards, what is the distance across the pond?

A 13.33 yards

B 95 yards

C 270 yards

D 360 yards

2 A man, who is t feet tall, leans against a building by putting his hand h inches up the side of the building. In a photo of the man, he is p feet tall. How many inches up the building is the man’s hand in the photo if h � 9t?

F 9

G 9p

H 9tp

J p __ 9

3 The two right triangles shown are similar.

6 ft10 ft

12 ft x ft

What is the value of x?

A 18

B 7.2

C 30

D 8

4 The two rectangles shown are similar.

9 cm

x

24 cm

16 cm

Which relationship could be used to find x?

F 24 __ 9 � x __ 16

G 16x � 24 � 9

H x _ 9 � 9 __ 24

J 24x � 9 � 16

★ Assessment

DIRECTIONS Read each question. Then circle the letter for the correct answer.


Teacher Edition

MasteryEducation.com | 800-822-1080 | Fax: 201-712-0045

i

Lesson Correlation to the Grade 7 Texas Essential Knowledge and Skills . . . . . . . . . . . . . . . . . . . . iv

Letter to Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Letter to Parents and Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

What’s Ahead in Measuring Up® to the TEKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

What’s Inside: A Lesson Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Chapter 1 Rational Numbers

TEKS Lesson

S 7.2(A) 1 Sets of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

S 7.3(A), R 7.3(B)

2 Add and Subtract Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

S 7.3(A), R 7.3(B)

3 Solve Problems Involving Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

★ Building Stamina®: Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 2 Proportional Relationships

TEKS Lesson

R 7.4(A), S 7.4(B), S 7.4(C)

4 Unit Rates and Constants of Proportionality in Tables and Graphs . . . . . . . . . . . . . 24

R 7.4(A), S 7.4(B), S 7.4(C), S 7.5(B)

5 Unit Rates and Constants of Proportionality in Equations, Verbal, and Pictorial Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

R 7.4(D) 6 Solve Ratio and Rate Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

R 7.4(D) 7 Solve Percent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

S 7.4(E) 8 Convert Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

S 7.5(A), R 7.5(C)

9 Similar Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

R 7.5(C) 10 Scale Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

★ Building Stamina: Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Contents

Note: Eligible standards are written in boldface. R = Readiness standard S = Supporting standard

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Chapter 3 Probability and Statistics

TEKS Lesson

S 7.6(A), R 7.6(H)

11 Sample Spaces and Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

S 7.6(E), R 7.6(I)

12 Theoretical Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

R 7.6(I), 7.6(B)

13 Experimental Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

R 7.6(I) 14 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

R 7.6(I) 15 Dependent Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

S 7.6(C), S 7.6(D)

16 Make Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

R 7.6(G), 7.6(F)

17 Solve Problems About Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

R 7.12(A), S 7.12(B), S 7.12(C)

18 Solve Problems About Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

★ Building Stamina: Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Chapter 4 Expressions, Equations, and Relationships

TEKS Lesson

R 7.7(A) 19 Make Tables and Graphs for Linear Relationships . . . . . . . . . . . . . . . . . . . . . . . . . 120

R 7.7(A) 20 Write Linear Equations and Verbal Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

S 7.10(B), R 7.11(A)

21 Solve Two-Step Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

S 7.10(A), S 7.10(C), R 7.11(A)

22 Write and Solve Two-Step Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . 136

S 7.11(B), S 7.11(C)

23 Write and Solve Geometry Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142


Chapter 5 Geometric Figures

TEKS Lesson

R 7.9(A), 7.8(A), 7.8(B)

24 Volume of Prisms and Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

R 7.9(B), R 7.9(C), 7.8(C)

25 Area and Circumference of Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

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Chapter 5 Geometric Figures (continued)

TEKS Lesson

R 7.9(C) 26 Area of Composite Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

S 7.9(D) 27 Surface Area of Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

S 7.9(D) 28 Surface Area of Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179


Chapter 6 Personal Finance

TEKS Lesson

S 7.13(A) 29 Sales Tax and Income Tax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

S 7.13(B), S 7.13(D)

30 Budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

S 7.13(C) 31 Net Worth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

S 7.13(F) 32 Sales, Rebates, and Coupons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

S 7.13(E) 33 Simple Interest and Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215


Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Problem-Solving Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

Texas College Readiness Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

Mathematics Reference Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Copy Masters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

Measuring Up Supplements Practice TestsThese assessments, written to match the STAAR® blueprints, will help students prepare for the rigor of the STAAR® and are included as blackline masters in the Teacher Edition. They are also available in Measuring Up Insight®.

Measuring Up InsightThis Web-based formative assessment program allows teachers to administer pre-made tests (including the STAAR®-emulating Practice Tests), and create and assign custom tests. Analytic reports help monitor student results and customize instruction, review, and remediation.

Measuring Up MyQuest®

Student-centered, standards-based Web-based drill with integrated games makes mastering the TEKS fun. Optional linking to Insight makes practice purposeful.

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iv

Lesson Correlation to the Grade 7 Texas Essential Knowledge and Skills

This worktext is customized to the Texas Essential Knowledge and Skills and will help you prepare for the State of Texas Assessments of Academic Readiness (STAAR®) in Mathematics for Grade 7.

Mathematical process standards are not listed under separate lessons. Because application of mathematical process standards is part of each knowledge statement, these standards are incorporated into instruction and practice throughout the lessons.

Texas Essential Knowledge and Skills Measuring Up Lessons

TEKS 7.2 Number and operations. The student applies mathematical process standards to represent and use rational numbers in a variety of forms.

(A) extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of rational numbers

1

TEKS 7.3 Number and operations. The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions.

(A) add, subtract, multiply, and divide rational numbers fl uently 2, 3

(B) apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers

2, 3

TEKS 7.4 Proportionality. The student applies mathematical process standards to represent and solve problems involving proportional relationships.

(A) represent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt

4, 5

(B) calculate unit rates from rates in mathematical and real-world problems 4, 5

(C) determine the constant of proportionality (k = y _ x ) within mathematical and real-world problems 4, 5

(D) solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and fi nancial literacy problems

6, 7

(E) convert between measurement systems, including the use of proportions and the use of unit rates 8

TEKS 7.5 Proportionality. The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships.

(A) generalize the critical attributes of similarity, including ratios within and between similar shapes 9

(B) describe π as the ratio of the circumference of a circle to its diameter 5

(C) solve mathematical and real-world problems involving similar shape and scale drawings 9, 10

TEKS 7.6 Proportionality. The student applies mathematical process standards to use probability and statistics to describe or solve problems involving proportional relationships.

(A) represent sample spaces for simple and compound events using lists and tree diagrams 11

(B) select and use different simulations to represent simple and compound events with and without technology

13

(C) make predictions and determine solutions using experimental data for simple and compound events 16

(D) make predictions and determine solutions using theoretical probability for simple and compound events 16

(E) fi nd the probabilities of a simple event and its complement and describe the relationship between the two 12

(F) use data from a random sample to make inferences about a population 17

(G) solve problems using data represented in bar graphs, dot plots, and circle graphs, including part-to-whole and part-to-part comparisons and equivalents

17

(H) solve problems using qualitative and quantitative predictions and comparisons from simple experiments 11

(I) determine experimental and theoretical probabilities related to simple and compound events using data and sample spaces

12, 13, 14, 15

TEKS 7.7 Expressions, equations, and relationships. The student applies mathematical process standards to represent linear relationships using multiple representations.

(A) represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b

19, 20

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v

Texas Essential Knowledge and Skills Measuring Up Lessons

TEKS 7.8 Expressions, equations, and relationships. The student applies mathematical process standards to develop geometric relationships with volume.

(A) model the relationship between the volume of a rectangular prism and a rectangular pyramid having both congruent bases and heights and connect that relationship to the formulas

24

(B) explain verbally and symbolically the relationship between the volume of a triangular prism and a triangular pyramid having both congruent bases and heights and connect that relationship to the formulas

24

(C) use models to determine the approximate formulas for the circumference and area of a circle and connect the models to the actual formulas

25

TEKS 7.9 Expressions, equations, and relationships. The student applies mathematical process standards to solve geometric problems.

(A) solve problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids

24

(B) determine the circumference and area of circles 25

(C) determine the area of composite fi gures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles

25, 26

(D) solve problems involving the lateral and total surface area of a rectangular prism, rectangular pyramid, triangular prism, and triangular pyramid by determining the area of the shape’s net

27, 28

TEKS 7.10 Expressions, equations, and relationships. The student applies mathematical process standards to use one-variable equations and inequalities to represent situations.

(A) write one-variable, two-step equations and inequalities to represent constraints or conditions within problems

22

(B) represent solutions for one-variable, two-step equations and inequalities on number lines 21

(C) write a corresponding real-world problem given a one-variable, two-step equation or inequality 22

TEKS 7.11 Expressions, equations, and relationships. The student applies mathematical process standards to solve one-variable equations and inequalities.

(A) model and solve one-variable, two-step equations and inequalities 21, 22

(B) determine if the given value(s) make(s) one-variable, two-step equations and inequalities true 23

(C) write and solve equations using geometry concepts, including the sum of the angles in a triangle, and angle relationships

23

TEKS 7.12 Measurement and data. The student applies mathematical process standards to use statistical representations to analyze data.

(A) compare two groups of numeric data using comparative dot plots or box plots by comparing their shapes, centers, and spreads

18

(B) use data from a random sample to make inferences about a population 18

(C) compare two populations based on data in random samples from these populations, including informal comparative inferences about differences between the two populations

18

TEKS 7.13 Personal fi nancial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one’s life as a knowledgeable consumer and investor.

(A) calculate the sales tax for a given purchase and calculate income tax for earned wages 29

(B) identify the components of a personal budget, including income; planned savings for college, retirement, and emergencies; taxes; and fi xed and variable expenses, and calculate what percentage each category comprises of the total budget

30

(C) create and organize a fi nancial assets and liabilities record and construct a net worth statement 31

(D) use a family budget estimator to determine the minimum household budget and average hourly wage needed for a family to meet its basic needs in the student’s city or another large city nearby

30

(E) calculate and compare simple interest and compound interest earnings 33

(F) analyze and compare monetary incentives, including sales, rebates, and coupons 32

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Two

figur

es t

hat

are

sim

ilar

have

the

sam

e sh

ape,

but

not

nec

essa

rily

the

sam

e si

ze.

Cor

resp

ondi

ng d

imen

sion

s or

ang

les

in s

imila

r fig

ures

are

the

pai

r th

at

in t

he s

ame

rela

tive

posi

tions

in t

he f

igur

e. In

sim

ilar

figur

es,

corr

espo

ndin

g an

gles

hav

e eq

ual m

easu

res

and

the

mea

sure

s of

cor

resp

ondi

ng d

imen

sion

s ar

e pr

opor

tion

al.

If �

ABC is

sim

ilar

to �

DEF

, th

en t

he p

airs

of

corr

espo

ndin

g an

gles

with

equ

al m

easu

res

are:

�

A an

d �

D,

� B

and

� E,

and

� C a

nd �

F. T

he c

orre

spon

ding

sid

es a

re __

_ AB

and

___

DE ,

__

_ B

C a

nd

___

EF ,

and

___

CA

and

___

FD .

Sin

ce t

heir

mea

sure

s ar

e pr

opor

tiona

l, ra

tios

with

cor

resp

ondi

ng s

ides

are

equ

al.

So,

for

�AB

C a

nd �

DEF

: AB

__

_ D

E �

B

C

___

EF a

nd AB

__

_ B

C

�

DE

___

EF .

To s

olve

for

unk

now

n si

de le

ngth

s of

sim

ilar

figur

es,

set

up a

pro

port

ion,

use

a v

aria

ble

for

the

mis

sing

leng

th,

and

then

cro

ss m

ultip

ly t

o so

lve

the

prop

ortio

n.

Guid

ed

Inst

ruct

ion

Two

book

s ar

e si

mila

r in

siz

e. T

he la

rger

boo

k is

8 in

ches

wid

e an

d 1

1 in

ches

tal

l. Th

e sm

alle

r bo

ok is

6 in

ches

wid

e. W

ill t

he s

mal

ler

book

fit

on a

she

lf th

at is

9 in

ches

tal

l?

Use

the

fac

t th

at c

orre

spon

ding

sid

es o

f si

mila

r fi g

ures

are

pro

port

iona

l.

Ste

p 1

D

raw

a d

iagr

am t

o m

odel

the

si

tuat

ion.

Mar

k th

e gi

ven

info

rmat

ion

on t

he

diag

ram

.

Let

h �

the

hei

ght

of t

he s

mal

ler

book

.

Ste

p 2

Id

entif

y co

rres

pond

ing

side

s of

the

sim

ilar

figur

es

to w

rite

a pr

opor

tion.

8 __

6 � 1

1

___ h

Ste

p 3

S

olve

the

pro

port

ion

to f

ind

the

valu

e of

h.

8h

� 6

�

11

8h

�

66

Wor

ds to

Kno

wsim

ilar

corre

spon

ding

prop

ortio

nal

8 in

.

11 in

.

6 in

.

h

Pro

ble

m 1

S 7

.5(A

) Ge

nera

lize

the

criti

cal a

ttrib

utes

of s

imila

rity,

inclu

ding

ratio

s wi

thin

and

betw

een

simila

r sha

pes.

R 7

.5(C

) So

lve m

athem

atica

l and

real-

world

pro

blem

s in

volvi

ng s

imila

r sha

pe a

nd s

cale

draw

ings

.

Sim

ilar

Fig

ure

sLe

sso

n9

Diff

eren

t as

pec

ts o

f 7.

5(C

) ar

e co

vere

d in

diff

eren

t le

sson

s.

8/17

/201

8 6

:27:

04 P

M8/

17/2

018

6:2

7:04

PM


Mat

hem

atic

s • L

evel

G

Cop

ying

is il

lega

l.

Mea

surin

g U

p to

the

Tex

as E

ssen

tial K

now

ledg

e an

d S

kills

54

Less

on

9S

imila

r Fi

gure

s

Not

e th

at b

ecau

se �

L an

d �

R a

re m

arke

d as

con

grue

nt,

sim

ilarit

y is

suc

h th

at �

LMN

is

sim

ilar

to �

RST.

The

refo

re,

the

ratio

s de

fined

in t

his

prob

lem

rel

ate

corr

espo

ndin

g si

des:

LM

and

RS

as

wel

l as

MN

and

ST.

How

doe

s th

e ra

tio o

f LM

to

MN

com

pare

to

the

ratio

of

RS t

o ST?

The

ratio

s ar

e eq

ual.

How

doe

s th

e ra

tio o

f LM

to

RS c

ompa

re t

o th

e ra

tio o

f M

N t

o ST?

The

ratio

s ar

e eq

ual.

Ano

ther

Exa

mpl

e

At t

he p

ark,

the

re a

re t

wo

sim

ilar

foun

tain

s. T

he s

mal

ler

foun

tain

has

a d

iam

eter

of

12

fee

t an

d a

heig

ht o

f 4

fee

t. T

he la

rger

fou

ntai

n ha

s a

heig

ht o

f 8

1

__

4 f

eet.

Est

imat

e th

e di

amet

er o

fth

e la

rger

fou

ntai

n.

Sin

ce t

he f

ount

ains

are

sim

ilar,

we

know

tha

t th

eir

dim

ensi

ons

are

prop

ortio

nal.

4 ft

d

8

ft1 4

12 ft

Sin

ce t

he la

rger

fou

ntai

n’s

heig

ht is

app

roxi

mat

ely

twic

e th

e he

ight

of

the

smal

ler

foun

tain

, its

di

amet

er is

app

roxi

mat

ely

twic

e th

e di

amet

er o

f th

e sm

alle

r fo

unta

in.

12

�

2 �

24

The

larg

er f

ount

ain’

s di

amet

er is

abo

ut

24 f

eet.

So

luti

on

Mas

tery

Edu

catio

n •

Mas

tery

Educ

atio

n.co

m

Cop

ying

is il

lega

l. C

hapt

er 2

• P

ropo

rtio

nal R

elat

ions

hips

53

Sim

ilar

Figu

res

Less

on

9

h �

8.

25

The

smal

ler

book

is

8.25

inch

es t

all.

Will

the

sm

alle

r bo

ok f

it on

a s

helf

that

is 9

inch

es t

all?

ye

s

The

tria

ngle

s sh

own

are

sim

ilar

tria

ngle

s. H

ow d

oes

the

ratio

of

LM t

o M

N c

ompa

re t

o th

e ra

tio o

f R

S t

o ST?

H

ow d

oes

the

ratio

of

LM t

o R

S c

ompa

re t

o th

e ra

tio

of M

N t

o ST?

MN

20

6

8

15

L

TS R

Ste

p 1

Writ

e ea

ch r

atio

in f

ract

iona

l for

m.

LM

___

MN �

15

___

20

RS

___

ST �

6

__

8

LM

___

RS �

15

___ 6

MN

___

ST �

20

___ 8

Ste

p 2

Sim

plify

eac

h ra

tio.

LM

___

MN �

3

__

4

RS

___

ST �

3

__

4

LM

___

RS �

5

__

2

MN

___

ST �

5

__

2

Ste

p 3

Com

pare

the

rat

ios.

LM

___

MN

� R

S

___

ST

LM

___

RS

� M

N

___

ST

So

luti

on

Pro

ble

m 2

9781

6097

9172

8_T

X7_

MU

D_M

ath_

TE

_int

erio

r.in

db

2797

8160

9791

728_

TX

7_M

UD

_Mat

h_T

E_i

nter

ior.

indb

27

8/17

/201

8 6

:27:

28 P

M8/

17/2

018

6:2

7:28

PM

28 Mathematics • Level G Copying is illegal. Measuring Up to the Texas Essential Knowledge and Skills

Mat

hem

atic

s • L

evel

G

Cop

ying

is il

lega

l.

Mea

surin

g U

p to

the

Tex

as E

ssen

tial K

now

ledg

e an

d S

kills

56

Less

on

9S

imila

r Fi

gure

s

★P

ract

ice

DIR

ECTI

ON

S

Rea

d e

ach

qu

esti

on.

Then

cir

cle

the

lett

er f

or t

he

corr

ect

answ

er.

1

Two

sim

ilar

tria

ngul

ar p

astu

res

mee

t at

a

vert

ex.

The

dim

ensi

ons

are

give

n in

yar

ds.

How

muc

h fe

ncin

g w

ould

be

need

ed t

o en

clos

e th

e la

rger

pas

ture

?

11

57 13

y

x

A

28.2

yar

ds

B

42.7

yar

ds

C

47 y

ards

D

59.8

yar

ds

2

The

ratio

of

the

corr

espo

ndin

g di

men

sion

s of

tw

o si

mila

r re

ctan

gles

is 3

: 2

. Th

e ra

tio

of t

he le

ngth

to

wid

th f

or e

ach

is 4

: 3

. If

th

e w

idth

of

the

smal

ler

rect

angl

e is

120

, w

hat

is t

he le

ngth

of

the

larg

er r

ecta

ngle

?

F 13

5

G

160

H

180

J 24

0

3

A t

ree

cast

s a

shad

ow 2

0 fe

et lo

ng.

At

the

sam

e tim

e, a

boy

5.5

fee

t ta

ll ca

sts

a sh

adow

8 f

eet

long

. W

hich

pro

port

ion

CAN

NO

T be

use

d to

fin

d th

e he

ight

of

the

tree

?

A

5 . 5

___ 8 �

20 __ x

B

5 . 5

___ x �

8 __ 20

C

8 __

_ 5

. 5 �

20 __ x

D

20

___

5 . 5

� 8 _ x

4

Elle

n ha

s cr

eate

d a

desi

gn u

sing

tw

o pa

ralle

logr

ams.

Det

erm

ine

whe

ther

the

pa

ralle

logr

ams

in t

he d

esig

n ar

e si

mila

r or

not

.

D

AB

C

3

1577

D

AB

C

5

24

103

F N

o, b

ecau

se t

he a

ngle

s do

not

hav

e eq

ual m

easu

res

G

No,

bec

ause

15

__

24 ≠

3 _ 5

H

No,

bec

ause

5 �

3 ≠

24

� 1

5

J Ye

s, b

ecau

se 3

: 1

5 is

equ

ival

ent

to 5

: 2

4

Mas

tery

Edu

catio

n •

Mas

tery

Educ

atio

n.co

m

Cop

ying

is il

lega

l. C

hapt

er 2

• P

ropo

rtio

nal R

elat

ions

hips

55

Sim

ilar

Figu

res

Less

on

9

Cri

tica

l Th

inki

ngSol

ve e

ach

prob

lem

.

1.

In t

riang

le M

AY,

MA

� 4

fee

t, A

Y �

8 f

eet,

and

MY

� 1

0 f

eet.

It

is s

imila

r to

tria

ngle

RO

W.

Jack

ie s

aid

that

RO

� 8

fee

t, O

W �

4 f

eet,

and

RW

� 2

0 f

eet.

Fin

d th

e er

ror

in h

er r

easo

ning

.

Sam

ple

answ

er: T

wo

side

s in

ROW

are

twic

e as

long

as t

he c

orre

spon

ding

side

s in

MAY

. The

oth

er is

hal

f as

long

. The

sid

es a

re n

ot a

ll pr

opor

tiona

l.

She

shou

ld h

ave

mul

tiplie

d by

2 t

o ge

t O

W �

16

feet

.

2.

Wor

k w

ith a

par

tner

to

crea

te s

imila

r re

ctan

gles

and

ana

lyze

the

are

as.

One

par

tner

cut

sou

t a

smal

l rec

tang

le a

nd r

ecor

d th

e m

easu

rem

ents

. Th

e ot

her

part

ner

cuts

out

a r

ecta

ngle

with

dim

ensi

ons

that

are

dou

bled

. U

se o

bser

vatio

n to

rec

ord

the

num

ber

of s

mal

l rec

tang

les

that

wou

ld f

it in

side

the

are

a of

the

larg

e re

ctan

gle.

Rec

ord

the

area

s of

bot

h re

ctan

gles

.N

ow,

repe

at t

he a

ctiv

ity -

this

tim

e cr

eatin

g a

larg

e re

ctan

gle

with

dim

ensi

ons

that

are

trip

led.

Is t

he r

atio

of

corr

espo

ndin

g di

men

sion

s ra

tios

the

sam

e as

the

rat

io o

f ar

eas?

Rec

ord

your

obse

rvat

ions

in y

our

mat

h jo

urna

l.

3.

You

can

find

the

heig

ht o

f an

obj

ect

with

out

mea

surin

g it

if yo

u ca

n m

easu

re it

s sh

adow

and

anot

her

obje

ct’s

hei

ght

and

shad

ow.

Wor

k w

ith a

par

tner

. Fi

nd a

n ob

ject

tha

t is

too

tal

l to

bem

easu

red

dire

ctly,

but

that

is c

astin

g a

shad

ow w

ith a

leng

th y

ou c

an m

easu

re (

e.g.

, sc

hool

build

ing,

fla

gpol

e, e

tc.).

Then

, m

easu

re t

he h

eigh

t of

one

of

you

and

your

sha

dow.

Use

sim

ilar

tria

ngle

s to

fin

d th

e he

ight

of

the

imm

easu

rabl

e ob

ject

. M

ake

a po

ster

of

your

fin

ding

s an

ddi

scus

s yo

ur r

esul

ts w

ith t

he c

lass

. D

id y

ou a

ll co

me

up w

ith t

he s

ame

ratio

to

use

in y

our

prop

ortio

n?

Che

ck s

tude

nt a

nsw

ers

for

reas

onab

lene

ss.

Stud

ents

will

com

e up

with

diff

eren

t ra

tios

dep

endi

ng o

n tim

e of

day

and

obj

ect

and

per

son

mea

sure

d, a

s w

ell a

s on

the

acc

urac

y of

the

ir m

easu

rem

ents

. A c

erta

in

amou

nt o

f er

ror

may

als

o be

intr

oduc

ed if

the

gro

und

is n

ot le

vel.

4.

Wor

k in

gro

ups.

Ope

n a

spre

adsh

eet

file

and

set

up 4

col

umns

with

the

fol

low

ing

head

ings

: Fu

rnitu

re Ite

m,

D,

W,

H.

Usi

ng t

he s

tand

ard

mea

sure

men

ts f

or f

urni

ture

bel

ow,

set

up a

for

mul

a to

fin

d th

e si

zes

of s

imila

r fig

ures

for

min

iatu

re r

eplic

as.

Use

a 1

/12

(.083)

ratio

. Yo

u w

ill n

eed

two

row

s fo

r ea

ch p

iece

of

furn

iture

: fu

ll si

ze a

nd m

inia

ture

siz

e.M

ake

sure

to

plac

e ea

ch m

easu

rem

ent

in a

sep

arat

e ce

ll so

you

can

app

ly t

he f

orm

ula.

Sta

ndar

d m

easu

rem

ents

: C

ouch

: 36"

D �

84"

W �

36"

H;

Love

seat

: 36"

D �

60"

W �

36"

HAr

mch

air: 3

6"

D �

36"

W �

36"

H;

Cof

fee

tabl

e: 3

0"

W �

48"

L �

18"

HM

ake

sure

stu

dent

s im

ple

men

t th

e fo

rmul

a co

rrec

tly.

Elev

ate

Col

labo

rati

veLe

arnin

g

Dis

cuss

9781

6097

9172

8_T

X7_

MU

D_M

ath_

TE

_int

erio

r.in

db

2897

8160

9791

728_

TX

7_M

UD

_Mat

h_T

E_i

nter

ior.

indb

28

8/17

/201

8 6

:27:

30 P

M8/

17/2

018

6:2

7:30

PM


Mas

tery

Edu

catio

n •

Mas

tery

Educ

atio

n.co

m

Cop

ying

is il

lega

l. C

hapt

er 2

• P

ropo

rtio

nal R

elat

ions

hips

57

Sim

ilar

Figu

res

Less

on

9

1

A s

urve

yor

foun

d a

dist

ance

acr

oss

a po

nd

by m

easu

ring

sim

ilar

tria

ngle

s ACD

and

BCE

on la

nd.

E

D AB

C

�A

CD

is s

imila

r to

�B

CE

If A

D �

10

yard

s, A

C �

45

yard

s, a

nd

BE

� 6

0 ya

rds,

wha

t is

the

dis

tanc

e ac

ross

th

e po

nd?

A

13.3

3 ya

rds

B

95 y

ards

C

270

yard

s

D

360

yard

s

2

A m

an,

who

is t

fee

t ta

ll, le

ans

agai

nst

a bu

ildin

g by

put

ting

his

hand

h in

ches

up

the

side

of

the

build

ing.

In

a ph

oto

of t

he

man

, he

is p

fee

t ta

ll. H

ow m

any

inch

es u

p th

e bu

ildin

g is

the

man

’s h

and

in t

he p

hoto

if

h �

9t?

F 9

G

9p

H

9tp

J p __ 9

3

The

two

righ

t tr

iang

les

show

n ar

e si

mila

r.

6 ft

10 ft

12 ft

x ft

Wha

t is

the

val

ue o

f x?

A

18

B

7.2

C

30

D

8

4

The

two

rect

angl

es s

how

n ar

e si

mila

r.

9 cm

x

24 c

m

16 c

m

Whi

ch r

elat

ions

hip

coul

d be

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