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TEKSING TOWARD STAAR © 2014

GRADE 7

TEKS/STAAR-BASEDLESSONS

Parent Guide

Six Weeks 3

®MATHEMATICS

TEKSING TOWARD STAAR

TEKSING TOWARD STAAR SCOPE AND SEQUENCEGrade 7 Mathematics

TEKSING TOWARD STAAR 2014 Page 1

SIX WEEKS 3

Lesson TEKS-BASED LESSONSTAAR

CategoryStandard

SpiraledPractice

StudentActivity

ProblemSolving

Skills andConceptsHomework

Lesson 1____ days

7.4D/ solve problems involving….percents of decrease andpercent of decrease and financial literacy problems

Category 2Readiness

SP 41SP 42

SA 1SA 2

PS 1PS 2

Homework 1Homework 2

Lesson 2____ days

7.7A/represent linear relationships using verbal descriptions,tables, ….,that simplify to the form y mx b .

Category 2Readiness

SP 43SP 44

SA 1SA 2

PS 1PS 2


Lesson 3____ days

7.8C/use models to determine the approximate formulas forthe circumference and area of a circle and connect the modelsto the actual formulas

7.5B/describe as the ratio of the circumference of a circle

and its diameter

7.9B/determine the circumference and area of circles

NotTested

Category 3Supporting

Category 3Readiness

SP 45SP 46

SA 1SA 2

PS 1PS 2


Lesson 4____ days

7.6I/determine experimental and theoretical probabilitiesrelated to simple and compound events using data and samplespaces

Category 1Readiness

SP 47SP 48

SA 1SA 2

PS 1PS 2


Lesson 5____ days

7.8B/explain verbally and symbolically the relationshipbetween the volume of a triangular prism and a triangularpyramid both have congruent bases and heights and connectthat relationship to the formulas

7.9A/ solve problems involving the volume of ….triangularprisms and triangular pyramids

NotTested

Category 3Readiness

SP 49SP 50

SA 1SA 2

PS 1PS 2


Lesson 6____ days

7.9C/determine the area of composite figures containingcombinations of rectangles, squares, parallelograms,trapezoids, triangles, semicircles, and quarter circles

Category 3Readiness

SP 51SP 52

SA 1SA 2

PS 1PS 2


Lesson 7____ days

7.9D/solve problems involving the lateral and total surfacearea of a rectangular prism,…rectangular pyramid,..bydetermining the area of the shape’s net


SP 53SP 54

SA 1SA 2

PS 1PS 2


TEKSING TOWARD STAAR SCOPE AND SEQUENCEGrade 7 Mathematics

TEKSING TOWARD STAAR 2014 Page 2

SIX WEEKS 3

Lesson TEKS-BASED LESSONSTAAR

CategoryStandard

SpiraledPractice

StudentActivity

ProblemSolving

Skills andConceptsHomework

Lesson 8____ days

7.11C/write and solve equations using geometry concepts,including the sum of the angles in a triangle and anglerelationships


SP 55SP 56

SA 1SA 2

PS 1PS 2


Lesson 9____ days

7.6G/solve problems using data represented in…dot plots,including part-to-whole and part-to-part comparisons andequivalents

Category 4Readiness

SP 57SP 58

SA 1SA 2

PS 1PS 2


Lesson 10____ days

7.12B/use data from a random sample to make inferencesabout a population

7.6F/use data from a random sample to make inferences abouta population


Not Tested

SP 59SP 60

SA 1SA 2

PS 1PS 2


ReviewAssessment

2 days

Six Weeks 3 Open-Ended Review

Six Weeks 3 Assessment

Teacher Notes:

STAAR Category 2 Grade 7 Mathematics TEKS 7.4D

TEKSING TOWARD STAAR © 2014 Page 1

Parent GuideSix Weeks 3 Lesson 1

For this lesson, students should be able to demonstrate an understanding of how to performoperations and represent algebraic relationships. Students are expected to applymathematical process standards to represent and solve problems involving proportionalrelationships.

Students are expected to solve problems involving percent of decrease and percent ofincrease and financial literacy problems.

The process standards incorporated in this lesson include:

7.1B Use a problem-solving model that incorporates analyzing given information,formulating a plan or strategy, determining a solution, justifying the solution, andevaluating the problem-solving process and the reasonableness of the solution

7.1D Communicate mathematical ideas, reasoning, and their implications using multiplerepresentations, including symbols, diagrams, graphs, and language as appropriate.

Math Background-Solving Problems Involving Percent of Decrease and Increase

We have worked with percent problems previously. Recall, to solve problems involving percent, a

proportion that can be used ispart percent

=whole 100

.

In this lesson, we will focus on percent of increase or percent of decrease. When a quantity increases,the amount of increase can be used to determine the percent of increase. Like all percent problems, a

percent of increase can be found by using the proportion:increase amount percent

=original 100

. Likewise,

when a quantity decreases, the amount of decrease can be used to determine the percent of decrease.

The proportion used would be:decrease amount percent

=original 100

. Remember the original amount before

the increase or decrease is used as the denominator of one of the ratios used in the proportion.

Example: Jim made 80 on the first quiz this six weeks. On the second quiz, he made 90. What wasthe percent of increase on the second quiz score?

The amount of increase: 90 − 80 = 10

increase amount percent

=original 100

10 x=

80 100

1=

8 100

x8x = 100 x = 12.5%

There was a 12.5% increase on his second quiz score.

Example: Beau got 8 wrong on his facts test last week. This week he got 3 wrong on his test. Whatis the percent of decrease in his number of incorrect facts?

Amount of decrease: 8 − 3 = 5

Use the Product ofMeans Property



decrease amount percent

=original 100

5=

8 100

x8x = 500 x = 62.5%

Beau’s errors on his facts test decreased 62.5%.

Example: A dress cost $48 last week. This week the dress costs $36. What is the percent of decreasein the price of the dress?

The amount of decrease: $48 - $36 = $12

decrease amount percent

=original 100

12=

48 100

x 1=

4 100

x4x = 100 x = 25

The original cost is decreased 25% for the reduced cost.

Solving Problems Involving Percent in Financial Literacy

Percent is used often when solving problems involving money. Some of the situations that involvepercent and money are the following:

There is a percent change in a person’s salary. This affects their budget. When creating a monthly or weekly budget, the different categories of expenses are assigned a

percent value. Sales tax is a certain percent of the cost of some items in the state of Texas. Some states do not

have a sales tax. When trying to save money, you compare prices for sales with different percents of discount and

different regular prices.

Example: Mr. Smith buys a new car that is listed for $38,000. The sales tax on the car is 8%. Howmuch will the cost of the car increase when the sales tax is included?

Cost of the car before taxes is $38,000. The sales tax is 8% of the price.

To find 8% of 38,000 use the proportion:8

38000 100

x .

Use the Product of Means Property: 100 x = 304000

x = $3040 The tax on the car will increase the price of the car $3,040.

Use theProduct ofMeans Property

Use theProduct ofMeans Property



Example: Ms. Lang earned $2500 a month. She received a 5% raise. What will be her new earningsa month? Her earnings are $2500 a month. She receives a 5% raise. Her new earnings will be her original earnings plus the raise. In percent, her new earnings will be 100% + 5% or 105% of her original earnings.

Using a proportion to determine 105% of 2500:105

$2500 100

x

Using the Product of Means Property: 100x = $262,500 Solving for x: x = $262,500 100 or $2625

Her new monthly earnings will be $2625.

STAAR Category 2 Grade 7 Mathematics TEKS 7.7A



For this lesson, students should be able to demonstrate an understanding of how to performoperations and represent algebraic relationships. Students are expected to applymathematical process standards to represent linear relationships using multiplerepresentations.

Students are expected to represent linear representations using verbal descriptions,tables,….and equations that simplify to the form of y= mx + b.




7.1E Create and use representations to organize, record, and communicate mathematicalideas

Math Background-Representing Linear Relationships using Verbal Descriptions andTables

In order to represent linear relationships using verbal descriptions in equations of the form y = mx + b,we must first be able to translate phrases into expressions.

Verbal descriptions are going to contain words such as “5 more than”, “6 less than”, “twice”, or “threetimes” a number. For example, if the information given is “ the number of red beads is 6 more thanthe number of blue beads”, choose a variable to represent the number of blue beads.

If x represents the number of blue beads, then y can represent the number of red beads.

The number of red beads is 6 more than the number of blue beads.

y = 6 + x

The equation to represent this linear situation is 6 or 6y x y x . The Commutative Property of

Addition which states that the order of the addends in an expression can be interchanged allows therewriting of the equation to let the right side be x + 6.

Example: The length of a rectangle is 7 units less than twice the width. Write an equation torepresent this situation if y represents the length and x represents the width of the rectangle.

NOTE: 7 less than twice the width should be thought of as twice the width less 7 units. This will helpkeep you from writing the subtraction terms in the wrong order. Subtraction is not commutative sothe order is important. 10 − 2 is not the same as 2 − 10.

The length of a rectangle is twice the width less 7 units.

y = 2x − 7


TEKSI

The equation that represents the relationship between the length and width of the rectangle is2 7y x .

Representing Linear Relationships using Tables

A linear relationship between two quantities can be represented by values in a table. If an equation orverbal description of the relationship is given, a table can be generated with values that satisfy therelationship. If a table of values is given, then an equation or verbal description can be used todescribe the relationship in the table.

Example: The table below shows the relationship between the length and the width of a rectangle.

Write

First,

13

4.

relati

SeconNo, 8

Now,differ

Usingvaluecolumdifferappevalueand scolumother

DoesThe e

Exam

L

Width, x 3 4 5 7

an equation to represent the relationship of the length and width of the rectangle.

check to see if there is a constant equivalency between the ratios of y to x. Is11

3the same as

The answer is NO. There is no need to check the ratios any further. This is NOT a proportional

onship.

d, check to see if there is a constant difference between y and x. Is 11 − 3 the same as 13 − 4? is not the same as 9.

check the ratio of the difference between the y-values of two entries in the columns to theence between the two x-values of those columns.

the first two columns, the difference in the y-values is 13 − 11 or 2. The difference in the x-s is 4 – 3 or 1. The ratio of the y-difference to the x-difference is 2:1. Now use two differentns. Using the second and third columns, the difference in the y-values is 15 − 13 or 2. The

ence in the x-values is 5 − 4 or 1. The ratio of the y-difference to the x-differences is 2:1. Therears to be an equivalency in these ratios. They are both 2:1. Using that ratio value of 2, as theof m in the equation y = mx + b, we have y = 2x + b. Multiply the x value of any column by 2ee what you need to add to that product to get the y-value of that column. Using the firstn, 2(3) + b = 11. 6 + b = 11. b = 5. The equation appears to be y = 2x + 5. Check it in theordered pairs and make sure it works for them also.

Length , y 11 13 15 19

Width, x 3 4 5 7

y = 2x + 5 2(3) + 5 2(4) + 5 2(5) + 5 2(7) + 5

NG T

the expression in the middle column simplify to the y-value below it? Yes, for all four columns.quation that represents the values in the table is y = 2x +5.

ple: Complete the table of values to represent the linear equation y = 3x − 1.

ength , y 11 13 15 19

y =

x 3 4 5 7

3x − 1 3(3) − 1 3(4) − 1 3(5) − 1 3(7) − 1

OWARD STAAR © 2014 Page 2

y 8 11 14 20



Example: Complete the table of values to represent the following verbal description.The number of nickels in a coin collection is 8 more than 5 times the number of pennies.

First, translate the words into an equation to represent the relationship between the number of nickelsand the number of pennies.

The number of nickels in a coin collection is 8 more than 5 times the number of pennies.y = 8 + 5x

The equation that represents the relationship is y = 8 + 5x or y = 5x + 8.

The table to represent the relationship is:

x 2 7 8 10

y = 5x +8 5(2) + 8 5(7) + 8 5(8) +8 5(10) +8

y 18 43 48 58

STAAR Category 3 Grade 7 Mathematics TEKS 7.8C/7.5B/7.9B



For this lesson, students should be able to demonstrate an understanding of how to representand apply geometry and measurement concepts. Students are expected to applymathematical process standards to use geometry to describe or solve problems involvingproportional relationships. Students are expected to apply mathematical process standards tosolve geometric problems.

Students are expected to describe as the ratio of the circumference of a circle to its

diameter. Students are also expected to use models to determine the approximate formulasfor the circumference and area of a circle and connect the models to the actual formulas.Students are also expected to determine the circumference and the area of circles.


7.1A Apply mathematics to problems arising in everyday life, society, and the workplace.



7.1F Analyze mathematical relationships to connect and communicate mathematical ideas

Math Background-Using Models to Determine the Approximate Circumference andArea of a Circle and Connect the Models to the Formulas

A circle is a figure that is a set of points that are all equidistant from a fixed point. The fixed point iscalled the center of the circle. A circle is named by the center point. The distance from the center toany point of the circle is called a radius. The segment that contains two points of the circle and thecenter point is the diameter of the circle. It will be equivalent to two radii. The length around the circleis called the circumference of the circle.

Circle P is shown below. The radius PR and diameter AB are shown.

Determining the Circumference and Area of a Circle

The circumference of a circle, the length around the circle, is the product of the diameter and .

Pi ( ) is a constant irrational number. This means it cannot be expressed as the ratio of two integers.

However, if we need to calculate with it we use a rational approximation of 3.14 or22

7for . They are

close enough to the irrational number that an approximation will be near the actual value.

P

R

AB

STAAR Category 3 Grade 7 Mathematics TEKS 7.8C/7.5B/7.9B


Example: If a circle has a circumference of 3 inches, we can approximate it by replacing with

3.14. The circumference is approximately 9.42 inches.

The STAAR formula chart gives two formulas for circles. Both formulas have as part of the formula.

The circumference is given as c d . The area formula is given as 2A r . Another relationship we

need to recall for a circle is the diameter is twice the radius, or d = 2r.

From Student Activity 1, we learned that the ratio of C to d is approximately 3. The actual value of theratio of C to d for any circle is .

Example: What is the length of the circumference of a circle with a radius of 4 inches?If the radius is 4 inches, then the diameter is 2(4 inches) or 8 inches.The circumference is d or 8. Usually the coefficient is written first, so the circumference is

8 inches. If we approximate this by using 3.14 for , the circumference is 25.12 or approximately 25

inches.

Example: What is the area of a circle with a diameter of 10 centimeters?If the diameter is 10 centimeters, then the radius is (10 centimters) 2 or 5 centimeters.

The area is 2r or (5) 2 or 25 square centimeters. The area of the circle is approximately 25(3.14)

square centimeters or 78.5 square centimeters.

Example: What is the area of a circle whose circumference is 36 inches?

If the circumference is 36 inches, then the diameter is 36 inches. If the diameter is 36 inches, then

the radius is 18 inches. The area will be 18 2 square inches. 324 square inches or approximately

1,017.36 square inches. If someone said the area was approximately 1,020 square inches, would thatbe acceptable?

STAAR Category 1 Grade 7 Mathematics TEKS 7.6I



For this lesson, students should be able to demonstrate an understanding of how to representprobabilities and numbers. Students apply mathematical process standards to probability andstatistics to describe or solve problems involving proportional relationships.

Students are expected to determine experimental and theoretical probabilities related tosimple and compound events using data and sample spaces.


7.1A apply mathematics to problems arising in everyday life, society, and the workplace




Math Background-Determining Experimental and Theoretical Probabilities usingData

Recall from an earlier lesson, the probability of an event is the ratio of the number of favorableoutcomes to the number of possible outcomes.

P(event) =number of favorableoutcomes

number of possibleoutcomes

There are two types of probabilities. One probability is called the theoretical probability. The theoretical

probability ratio is stillnumber of favorableoutcomes


. For example, if you toss a coin, there are 2

possible outcomes, a tail or a head. The theoretical probability of tossing a head is1

2. The theoretical

probability of tossing a tail is also1

2.

The other type of probability is called the experimental probability. The experimental probability ratio

isnumber of favorableoutcomes


. The data used in the ratio is based on an experiment of trials.

For example, if you toss a coin 50 times, you could get 25 heads and 25 tails which would match thetheoretical probability. However, it is more likely you would get data such as 27 tails and 23 heads. Ifthis were the data you recorded for tossing a penny 50 times, the experimental probability of tossing a

head would NOT be1

2. It would

number of heads

number of tosses

or

23

50, which is less than

1

2.


TEKSING TOWARD STAAR ©

The more trials you have, the more likely the experimental probability will be closer to the theoreticalprobability. If you tossed a coin 2000 times, you would expect the number of heads and the number of

tails to be close to 1000 each or1

2of the tosses.

Since a probability is a ratio of two values, it can be expressed as a fraction, decimal, or percent. If the

probability is1

2, then it can be written also as 0.5 or 50%.

Example: A bag contains 5 red tiles, 6 blue tiles, and 4 yellow tiles. Benny randomly selects a tilefrom the bag. What is the theoretical probability he will select a red tile? A yellow tile? A blue tile?

P(r) =number of red tiles

number of tiles

=

5 1

15 3

The theoretical probability of randomly drawing a red tile is1 1

or 33 %3 3

.

P(y) =number of yellow tiles

number of tiles

=

4

15

The theoretical probability of randomly drawing a yellow tile is4 2

or 26 %15 3

.

P(b) =number of blue tiles

number of tiles

=

6 2

15 5

The theoretical probability of randomly drawing a blue tile is2

or 40%5

.

Example: A bag contains 5 red tiles, 6 blue tiles, and 4 yellow tiles. Benny randomly selects a tilefrom the bag. Benny draws a tile from the bag, records it color, and returns the tile to the bag beforedrawing another tile. He does this 40 times. The results of his experiment are recorded in the tablebelow.

Based on Benny’s datafrom the bag will be red

P(r) =number of red dr

number of draw

Based on Benny’s datafrom the bag will be yel

P(y) =number of yellow

number of dr

Color red blue yellow

2014 Page 2

from his experiment, what is the experimental probability the next tile he draws?

aws

s=

12 3

40 10 or 30%

from his experiment, what is the experimental probability the next tile he drawslow?

draws

aws=

10 1

40 4 or 25%

No. of Draws 12 18 10



Based on Benny’s data from his experiment, what is the experimental probability the next tile he drawsfrom the bag will be blue?

P(b) =number of blue draws

number of draws

=

18 9

40 20 or 45%

Example: Using the two examples above, compare the theoretical probability to the experimentalprobability of drawing each color tile.

RED:

The theoretical probability of drawing a red is1 1

or 33 %3 3

. The experimental probability of drawing a

red is3

10or 30%. The theoretical probability is slightly larger than the experimental probability.

YELLOW:

The theoretical probability of drawing a yellow is4 2

or 26 %15 3

. The experimental probability of

drawing a yellow is1

4or 25%. The theoretical probability is slightly larger than the experimental

probability.

BLUE: The theoretical probability of drawing a blue is2

or 40%5

. The experimental probability of

drawing a blue is9

20or 45%. The experimental probability is slightly larger than the theoretical

probability.

The sum of the three theoretical probabilities and the sum of the three experimental probabilities musteach be 1 or 100%. Use that as a check to make sure you have not miscalculated.

Simple events are when there is one event. The event can be tossing a coin, rolling a number cube,drawing a card, spinning a spinner, etc. Compound events are when you have more than one eventoccurring. The events could be tossing a coin and spinning a spinner, tossing a coin and rolling anumber cube, drawing a card and spinning a spinner, spinning 2 different spinners, etc.

Compound events can be events that are independent events. Independent events are events that theresults of one event do NOT affect the results of the other event. An example of independent events istossing a coin and spinning a spinner. If the results of one event do affect the results of the secondevent, then they are dependent events. An example of dependent events is drawing 2 tiles from a bag,one at a time, and NOT replacing the first tile before drawing the second tile. The first draw affects thenumber of tiles in the bag for the second draw.

Where there are two independent events, the probability of certain events occurring is the product ofthe probability of each event occurring. P(A and B)= P(A) P(B)

When there are two dependent events, the probability of certain events occurring is the probability ofthe first event times the probability of the second event occurring given the occurrence of the firstevent. This is written P(A and B)= P(A) P(B/A)



Example: A bag contains 5 red marbles and 10 blue marbles. You are to select a marble, record itscolor, replace the marble in the bag, and then draw a second marble. What is the probability you willdraw 2 marbles that are red?

The P(r) =5 1

15 3 for the first draw. The P(r) =

5 1

15 3 for the second draw.

The P(r and r) =1 1 1

3 3 9 .

These were independent events.

Example: A bag contains 5 red marbles and 10 blue marbles. You are to select a marble, record itscolor, do NOT replace the marble in the bag, and then draw a second marble. What is the probabilityyou will draw 2 marbles that are red?

The P(r) =5 1

15 3 for the first draw. The P(r) =

4 2

14 7 for the second draw. (1 red has been drawn

and NOT replaced so there are only 4 red marbles now and there are only 14 marbles)

The P(r and r) =1 2 2

3 7 21 .

These were dependent events.

The probabilities of drawing 2 red marbles are not the same for the two examples. Replacing themarble back in the bag before drawing the second marble makes the events independent. Whichsituation had the greater probability of occurring?

Determining Theoretical and Experimental Probabilities using Sample Spaces

A sample space of an event is a set of all the possible outcomes of the event. The set can be a list, atree diagram, or a table.

A sample space for tossing a coin is {heads, tails}. A sample space for rolling a 1-6 number cube isthe list: 1, 2, 3, 4, 5, and 6.

To determine the probability of an event using a sample space, use the same ratio

number of favorableoutcomes


. Look at the sample space and count the number of favorable

outcomes for the numerator of the ratio. Count the total number of entries in the set for thedenominator of the ratio.

Example: What is the probability of rolling a 6 on a 1-6 number cube?

A sample space for rolling a number cube is {1, 2, 3, 4, 5, 6}. The number of favorable outcomes ( a

6) is 1. The number of entries in the set is 6. The ratio that represents the probability is1

6.


TEKSING TOWARD STA

Example: If you spin the spinner below, what is the probability you will spin a T?

A sample space for the spinner is {S, T, S, N, T, N, T, M}. The probability or spinning a T is

number of Ts

number of outcomes. P(T)=

3

8

Example: You are rolling a number cube and tossing a coin. What is the probability you will roll a 4and toss a tails?

A sample space for the number cube is {1, 2, 3, 4, 5, 6}

A sample space for tossing a coin is {heads, tails}

P (4)=1

6P(tails) =

1

2P(4 and tails)=

1 1 1

6 2 12

A sample space for both events could be the list: 1/tails; 1/heads; 2/tails; 2/heads; 3/tails; 3/heads;4/tails; 4/heads; 5/tails; 5/heads; 6/tails; 6/heads

There are 12 items in the list and 1 in the list is 4/tails. The probability would be1

12.

Example: A dessert store kept a record of the number of slices of apple pie they sold one day lastweek. They also recorded the choice of topping. The table shows the sample space of the apple pieslices sold that day.

What is the probab

Total the number o

T

TT

N

M

SS

N

Topping Whipped Cream Ice Cream No Topping

AR © 2014 Page 5

ility the next slice of apple pie ordered will have a whipped cream topping?

f slices served. 32 + 25+ 43 = 100. P(WC) =32

100or 32%.

Number Served 32 25 43

STAAR Category 3 Grade 7 Mathematics TEKS 7.8A/7.9A



For this lesson, students should be able to demonstrate an understanding of how to representand apply geometry and measurement concepts. Students are expected to applymathematical process standards to develop geometric relationships with volume. Studentsare expected to apply mathematical process standards to solve geometric problems.

Students are expected to model the relationship between the volume of a triangular prism andtriangular pyramid that have congruent bases and heights and connect that relationship to theformulas. Students are also expected to solve problems involving the volume of triangularprisms and triangular pyramids.






Math Background-Understanding Triangular Prisms and Pyramids with CongruentBases and Heights

A prism is a three-dimensional figure with two parallel, congruent polygon bases. The bases, whichare also two of the faces, can be any polygon. The other faces are rectangles. A prism is namedaccording to the shape of its bases. A triangular prism will have triangular bases and triangular lateralfaces. Do not assume the base is the face the prism is sitting on. The height of a triangular prism isthe perpendicular distance between the two bases. For the prisms we will study it will be one of theedges.

A pyramid is a three-dimensional figure with only one base. The base can be any polygon. The otherfaces are triangles. A pyramid is named according to the shape of its base. For example, the base inthe pyramid below is a triangle, so the figure is a triangular pyramid.

BaseHeight

Lateral Face



The volume of a three-dimensional figure is the capacity of the figure. Volume is measured in cubicunits such as cubic inches, cubic feet, or cubic centimeters.

The volume of a triangular prism will be the total number of cubic units that is the capacity of thefigure. If the base of a triangular prism is the triangle shown below, it will require 15 1-inch cubes and6 half cubes to cover the base in 1 layer. This is a total of 18 full cubes. The area of the base is 18square inches and the volume of the cubes is 18 cubic inches.

If the cubes are stacked 6 layers high, then each layer would be 18 cubic inches. 6 layers would be(18 6) or 108 cubic inches. Just as in a rectangular prism we studied earlier, the volume of a

triangular prism will be (area of the base)(height) . This formula is written V Bh . Just remember

that B represents the area of the base, so use your area formula for that type of polygon if you mustcalculate the area of the base.

A triangular pyramid that has the same height and base as a triangular prism will have a volume that is

1

3the volume of the prism. For the prism whose base is shown above with a volume of 108 cubic

inches, a pyramid that had the same base and height would have a volume of1

(108)3

= 36 cubic

inches. The formula for the volume of a triangular pyramid is1

(area of the base)(height)3

V or

1or

3 3

BhV Bh .

Base

Height



Look at the prism and pyramid below. They have congruent bases and heights.

The volume of the prism is 3 times the volume of the pyramid.

Solving Problems Involving Triangular Prisms and Pyramids

If the volume of a triangular prism and the height of the prism are known, then the area of the basecan be found by dividing the volume of the prism by the height.

The formula for the volume of a prism is V Bh . B can be isolated by dividing both sides by h.

V Bh

h h Remember this is the Division Property of Equality. The formula now is

VB

h .

Likewise, we could have divided both sides by B and have a formula for finding h when we know the

volume and the area of the base.V

hB

Therefore, we now have three formulas we can use for prisms: and andV V

V Bh B hh B

Using the formula for the volume of a pyramid,3

BhV , we can isolate the variables B and h in the

same manner. First multiply both sides by 3. This gives 3V Bh . If you divide both sides by h, you

isolate B. If you divide both sides by B, you isolate h. The three formulas for a pyramid will be:

3 3and and

3

Bh V VV B h

h B

Example: If the volume of a triangular prism is 200 cubic units and the height of the prism is 10 units,give 2 possible sets of dimensions of the base of the prism.

Using the formulaV

Bh

and substituting 200 cubic units for V and 10 units for h, the formula becomes

200 cubic units

10 unitsB .

Base

Height



When cubic units are divided by units, the label becomes square units because

2units units units units units unitsunits units = units

units units

B = 20 square units.

Two triangles that have an area of 20 square units are:

1. Right triangle with leg lengths of 10 and 4 units

2. Triangle with a base length of 8 units and a height of 5 units.

Can you think of any other possible triangles with dimensions that would be correct?

Example: The height of a triangular pyramid is 12 units. It has a right triangle base. One leg of theright triangle is one-half the height and the other leg is three-fourths the height. What is the volume ofthe pyramid?

Analyze: The height of a pyramid is 12 inches. One leg of the right triangle base is1

2h and the other

leg of the right triangle base is3

4h . What is the volume of the pyramid?

Plan: Determine the legs of the right triangle base. Find the area of the base. Use the volume of apyramid using the formula.

Solve: leg 1 =1

2h =

1(12)

2= 6 inches. leg 2 =

3

4h=

3(12)

4= 9 inches. The right triangle base has

legs that are 6 inches and 9 inches. The area of the base is2

2(6 inches)(9 inches) 54 in.= 27 in.

2 2

The volume of the pyramid is2 3

327 in. 12 in. 324 in.108 in.

3 3 3

BhV

Evaluate and Justify: The area of the base is about 30 square inches and the height is about 10inches. 30 times 10 divided by 3 is about 100. 100 is close to 108. Our answer is reasonable.

STAAR Category 3 Grade 7 Mathematics TEKS 7.9C



For this lesson, students should be able to demonstrate an understanding of how to representand apply geometry and measurement concepts. Students are expected to applymathematical process standards to solve geometric problems.

Students are expected to determine the area of composite figures containing combinations ofrectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles.






Math Background-Understanding the Area of Composite Figures

A composite figure is a figure that is created by using two or more simpler figures such as rectangles,parallelograms, trapezoids, triangles, and squares.

To find the area of a composite figure, follow the steps listed below.

Step 1: Divide the composite figure into simpler figures that you can find the area ofStep 2: Find the area of each simpler figure using the formula for the area of each figureStep 3: Find the total of the areas of the simpler figures.

The STAAR Reference Materials chart gives the following area formulas:

Figure Area

Triangle 1

2bh

Rectangle bh

Parallelogram bh

Trapezoid1 2

1( )

2b b h

Circle 2r



Some other formulas for area you might find useful. One is the area of a square, 2A s , where s

represents the side of the square. The area of a rhombus (equilateral parallelogram) can be found by

using the formula, 1 2

1

2A d d , where d

1and d

2represent the lengths of the two diagonals of the

rhombus. The area of a right triangle can be found by using the formula 1 2

1

2A leg leg .

Example: Find the area of the following figure.

Step 1: Divide the figure into simpler figures.

The figure divides into a rectangle and a trapezoid.

Step 2: The dimensions of the rectangle are 10.5 cm and 8 cm. The dimensions of the trapezoid arebases of 9.5 cm and 5.5 cm and a height of 7 cm. (17.5 – 10.5 = 7)

The area of the rectangle is 10.5 8 84 square centimeters.

The area of the trapezoid is1 1

(9.5 5.5)(7) (15)(7) 7.5(7) 52.52 2

square centimeters. (Why is the

label “square centimeters” for both areas?)

Step 3: The total area of the figure is the sum of 84 square centimeters and 52.5 square centimeters.The total area is 136.5 square centimeters.

Some composite figures have options on the simpler figures.

Can you see any other way for Step 1?

8 cm

10.5 cm

17.5 cm 9.5

cm

5.5

cm

10.5 cm

17.5 cm

0.75 cm

5.5

cm

0.75 cm

8 cm



Determining the Area of Composite Figures Involving Circles, Semicircle, and Quarter Circles

A composite figure can also have a circle, semicircle, or quarter circle as part of the simpler figures.

Since the formula for the area of a circle is 2r , the area of a semicircle is2

2

r. The area of a quarter

circle is2

4

r. 3.14 can be used as a rational approximation for , or you can leave your answer in

terms of . Sometimes, the information given will indicate whether you should use the approximation

for .

If the composite figure has a piece of a figure missing, then subtraction may be involved in finding thetotal area of the figure.

Example: Find the shaded area in the figure below.

Step 1: The composite figure is a rectangle with 2 semicircles and 1 circle removed.

Step 2: The rectangle has a base of 24 units and a height of 8 units. The area will be bh or (24)(8).The area of the rectangle is 192 square units.The two semicircles are equivalent to 1 circle. We have 2 full circles removed from the rectangle. The

area of one of the circles is 2(4 ) 16 square units. The diameter of the circle was 8 units so a radius

is 4 units. Two circles would have an area of 2(16 ) or 32 square units.

Step 3: Find the area of the rectangle minus the area of the two circles.(192 32 ) square units. If we substitute 3.14 for , the shaded area is 91.52 square units.

24 units

8units




For this lesson, students should be able to demonstrate an understanding of how to representand apply geometry and measurement concepts. Students are expected to applymathematical process standards to solve geometric problems.

Students are expected to solve problems involving the lateral and total surface area of arectangular prism, rectangular pyramid,… by determining the area of a shape’s net.






Math Background-Understanding Lateral and Total Surface Area of RectangularPrisms and Pyramids

Models can be used to find the surface area of three-dimensional figures. The total surface area of afigure is equal to the sum of the area of all its surfaces. The lateral surface area of a figure is equalto the sum of the areas of all its lateral faces but does not include the area of the base or bases.

One way to find the surface area of a three-dimensional figure is to draw a net for the figure. The areaof each surface can then be calculated and the sum can be found.

Prisms have any type polygons for their bases and rectangles for their lateral faces. The net for aprism will consist of these polygons. Pyramids also have a polygon base but they have triangles fortheir lateral faces. In this lesson we will focus on rectangular prisms and pyramids.

Remember a rectangular prism will have 6 faces that are all rectangles. Any pair of opposite facescan be designated to be the bases of the prism. To determine the lateral area of the prism the basesmust be identified as their areas are not included in the lateral area.

A net is a two dimensional model of a three dimensional shape. Imagine taking a cereal box andopening the box up so that you could “flatten” it out. The figure you would have when all the faces arein the same plane would be a net for the box. A net for a rectangular prism will consist of 3 pairs ofcongruent rectangles.

A cube is a special rectangular prism in which all six faces are congruent squares. The net for a cubewould show these six congruent squares whose side length is an edge of the cube. The total surfacearea (TA) of a cube can be found by calculating the area of one of the squares and multiplying by 6.

TA = 6e 2 . The lateral area (LA) of a cube would be the sum of the areas of the 4 lateral square faces.

LA = 4e 2



Example: Draw a net for a cube with an edge of 8 units. Shade the bases in the net. Find the totalsurface area and the lateral surface area of the cube.

The total surface area is the sum of the 6 squares whose edges are 8 units. Each square has an area of

8 2 or 64 square units. The six squares would have an area of 6(64) or 384 square units.

The lateral surface area is the sum of the 4 non-shaded squares whose edges are 8 units. Each square

has an area of 8 2 or 64 square units. The four squares would have an area of 4(64) or 256 squareunits. Notice if we subtract the area of the two bases (128 square units) from the total surface area(384 square units) we will have the lateral surface area (384 128 256) .

The total surface area of any rectangular prism will be the lateral area plus the area of the two bases.2TA LA B

Example: Draw a net for a rectangular prism with a width of 4 units, a height of 8 units, and a lengthof 6 units. Shade the bases in the net. Find the total surface area and the lateral surface area of theprism.

The total surface area of the prism is the sum of the areas of all six rectangles. The rectangles havedimensions of 6 units by 8 units, 4 units by 6 units, and 4 units by 8 units. There are two of each ofthese rectangles for a total of 6. The areas of the rectangles are 6 8 , 4 6 , and 4 8 square units.

These areas are 48 square units, 24 square units, and 32 square units. Since there are two of each,the areas of the six rectangles have a sum of (96 + 48 + 64) square units or 208 square units.

Cube

8 units

8 units

Net

8 units

4 units

6 units

6 units

8 units

4 units

Prism

Net



The total surface area of the prism is 208 square units.

The lateral surface area is the sum of the four rectangles that are not shaded in the net. They havedimensions of 6 units by 8 units and 4 units by 8 units. There are 2 of each of these rectangles. Thearea of these rectangles is either 48 square units or 32 square units. Two of each would give a sum of(96 + 64) square units or 160 square units.

The lateral surface area of the prism is 160 square units.

Notice the total area decreased by the lateral area would be 48 square units. This is the sum of theareas of the two bases.

Determining the Total and Lateral Surface Areas of a Rectangular Pyramid

A rectangular pyramid has one rectangular base and four isosceles triangular lateral faces. If thevertex where the triangular faces meet is directly above the center of the base, then the pyramid is aright rectangular pyramid. The height of the triangular faces is NOT the height of the pyramid. Ifthe base of the right pyramid is a square, then the four lateral triangular faces will be congruent andthe height of the triangles is called the slant height of the pyramid.

If the base is a non-square rectangular base, then the lateral faces will be congruent in pairs.

The total surface area of a rectangular pyramid is the sum of the areas of the 5 faces (4 lateraltriangular faces and 1 rectangular base). The lateral surface area of a rectangular pyramid is the sumof the areas of the 4 lateral triangular faces.

A net for the pyramid will be a 2-dimensional model of the 3-dimensional pyramid.

Square Pyramid

Slant height

Height ofPyramid

Height ofPyramid

Height of a face



Example: Find the total surface area and the lateral surface area of a square pyramid with a baseedge of 12 units and a slant height of 10 units.

The total surface area is the sum of the area of the square base and the areas of the 4 lateral triangularfaces. The four triangular faces are congruent isosceles triangles with bases of 12 units and heights of

10 units. Each triangle has an area of1

(12)(10)2

or 60 square units. The area of the 4 triangles is 240

square units. The area of the square base is 212 or 144 square units.

The total surface area is 144 + 240 or 884 square units.

The lateral area is the area of the 4 triangles which is 240 square units.

Example: Find the total and lateral surface areas of the rectangular pyramid whose net is shownbelow.

The area of the base is 12(6) square units. The area of the base is 72 square units.

12 u

10 u

Square Pyramid

12 u

10 u12 u

12 u

Net

6 u

12 u 10.4 u

11.7 u



The lateral faces with the 10.4 height will each have an area of1

(10.4)(12)2

. The area of each of these

triangles is 62.4 square units. The two lateral faces with the 10.4 height will have a total area of 124.8

square units. The lateral faces with the 11.7 height will each have an area of1

(11.7)(6)2

. The area of

each of these triangles will be 35.1 square units. These two lateral faces will have a total area of 70.2square units.

The total surface area of the pyramid will be (72 + 62.4 + 70.2) square units which is 204.6 squareunits.

The lateral surface area of the pyramid will be (62.4 + 70.2) square units which is 132.6 square units.

STAAR Category 4 Grade 7 Mathematics TEKS 7.6G

TEKSING TOWARD STAAR © 2014


For this lesson, students should be able to demonstrate an understanding of how to representand analyze data. Students apply mathematical process standards to probability andstatistics to describe or solve problems involving proportional relationships.

Students are expected to solve problems using data represented in…dot plots,….. includingpart-to-whole and part-to part comparisons and equivalents.


7.1A apply mathematics to problems arising in everyday life, society, and the workplace




Math Background-Analyzing Data Represented in Dot Plots

Earlier we learned how to create a dot plot from data. We will review dot plots first. In a dot plot, dotsare used to indicate the frequency of the data points. A dot plot does not work well when the data sethave an extremely large number of points. Stem-and-leaf plots, box plots, or histograms would be abetter way to display the data when there are many data points. In a dot plot, you must have a dot foreach data point.

Remember the median value of the data set is the middle value. On a dot plot, count forward from theleast value and backward from the greatest value until you come to the middle value. If there is nomiddle value, then average the two middle values. The median value does not have to be a value of thedata set. The mode of the dot plot is the value that has the most dots above it. The range of the dotplot is the difference between the greatest value and the least value. The mean value is the average ofall the data points.

Example: Jenny conducted a survey in her class about the number of books the students had read lastsix weeks. The results of her survey are shown in the dot plot below.

1 2

Reading Books

Page 1

3 4 5 6



Analyze the data in the dot plot. Answer the following questions about the data.

How many students did Jenny use in her survey? There are 23 dots. Each dot represents a student.23 students were used in her survey.

What is the median number of books read by the group? The middle dot will be the 12th dot countingfrom 1 forward, or the 12th dot counting backward from 6. The 12th dot is one of the dots above 3.The median number of books read is 3 books.

What is the mode number of books read by the group? The value that has the most dots above it is4. The mode number of books read by the group is 4 books.

What is the range in the number of books read by the group? The difference between 6 and 1 is 5.The range is 5.

What is the mean number of books read by the group? The total number of books read by the 23students is 4(1) + 4(2) + 4(3) + 5(4) + 2(5) + 4(6) = 4 + 8 + 12 + 20 + 10 + 24 = 78.78 23 ≈ 3.4 books.

Solving Problems Using Data in Dot Plots

When solving problems using data in a dot plot, you may be asked to find percentages or ratios whencomparing a data point with other data points or comparing a data point with the entire data set.Recall that a ratio is a comparison of two quantities and can be written in fraction form, decimal form,or percent form. Read carefully to make sure you have the quantities in the correct position of theratio. Comparing 1 to 2 is not the same as comparing 2 to 1.

Example: Look at the dot plot shown below. Answer the questions about the data in the dot plot.

The quiz scores of 15 seventh graders are shown in the plot.

1. What percent of the students made a score higher than 82? The number of students who made a

score higher than 82 is 5. The percent of the students is the percent equivalent to5 1

or15 3

. This

percent is1

33 %3

.

Quiz Scores

78 79 80 81 82 83 84



2. What is the ratio of the number of students who made a score less than 80 to the number ofstudents who made a score 80 or higher? 3 students made less than 80 and 12 students made 80or higher. The ratio would be 3:12 or 1:4.

3. What percent of the students who made 82 or higher made an 84? 8 students made 82 orhigher. 2 students made 84. 2 is 25% of 8.

STAAR Category 4 Grade 7 Mathematics TEKS 7.12B/7.6F



For this lesson, students should be able to demonstrate an understanding of how to representand analyze data and how to describe and apply personal financial concepts. Students applymathematical process standards to use statistical representations to analyze data. Studentsalso apply mathematical process standards to use probability and statistics to describe orsolve problems involving proportional relationships.

Students are expected to use data from a random sample to make inferences about apopulation.


7.1A Apply mathematics to problems arising in everyday life, society, and the workplace



Math Background-Choosing a Random and Varied Sample

When you are studying or gathering information about a group, that group is called the population.The population can be limited to a specific group, such as seventh graders in a school, or it can includethe entire population of this country.

There are times when you want to know something about a large group of people but it is impractical tosurvey every member of the group. Instead, you ask a smaller group, a sample, and apply yourconclusions to the entire population. This process is known as sampling.

Sampling is frequently used in science experiments, social studies, and in surveys. From a sample youcan draw conclusions about a total population based on information you have about a few of itsmembers. When you are taking a sample in order to get an idea about the whole population, you needto get enough data to be sure your conclusions are accurate. Those conclusions are valid only if thesample is representative of the total population. The sample must consider the kind of members in thepopulation and the number of members in the population.

A sample will be representative of a total population if these guidelines are followed:

The group of people who are surveyed should be selected at random. This is called taking a randomsample. The sample should not be selected in a way that might bias the results. A random sampleshould be a select group that closely represents the whole population.

The group surveyed must be varied enough to be representative of the total population.

The size of the group surveyed must be large enough to be representative of the total population.



EXAMPLE: You want to use a toothpaste recommended by dentists. What questions should you askthe advertisers who say two out of three dentists recommend Brand X?

The group of people who are surveyed should be selected at random.

How the dentists who were surveyed were selected?

The group surveyed must be varied enough to be representative of the total population.

How did you make sure that all kinds of dentists were surveyed?

The size of the group surveyed must be large enough to be representative of the totalpopulation.

How many dentists are there? How many did you survey?

Once you have selected your sample, the questions you ask must not sway members of the group torespond in a certain way.

Example: Residents in New City, Texas, were surveyed about the possibility of a new race track beingbuilt in their area. Decide if the following questions may be biased.1. Would you vote for the new track if taxpayer’s money is used to build it?

This is biased. It discourages people from voting for the new track because it will use their taxmoney.

2. Do you favor a new race track?This is not biased. It is just an opinion on whether they want a new race track.

3. Would you like a new race track for our city if it brings in lots of traffic to the area?This is biased. It discourages people from voting for the new track because it will cause trafficcongestion which could affect them personally.

Example: The school board is considering changing the start and end time of the school day for middleschool students. You are to gather data from the students at your school. There are 600 students atyour school. How can you get a varied and large enough random sample so that the results representthe school population?

To be large enough to be representative of the 600 students, you need to survey about 60 students.

To be varied enough, you need to survey students from grade 6, grade 7, and grade 8.

To be random, you can randomly select 20 locker numbers from each grade level hall and then surveythe students assigned to those lockers.

MadMaking Inferences About a Population from Data

After gathering data from a random sample, you can use any statistical representation to makeinferences about the population.

Look at the dot plot below.



The central office at a school district surveyed 20 teachers in the district and asked them about howmany miles they ran a week as exercise.

Miles 0, 0, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 9, 10, 10

Inferences about the Population:

1. Most of the teachers ran at least 3 miles per week for exercise.

2. Most of the teachers ran between 3 and 7 miles per week.

3. 50% of the teachers ran at least 3 miles but less than 6 miles a week.

4. Even though none of the 20 teachers in the survey ran 1 or 2 miles a week, there are probablyteachers in the district that do run 1 or 2 miles a week for exercise.

While some inferences can be made from the dot plot. The survey probably should have included moreteachers. The data doesn’t tell us if the teachers are elementary, middle school, or high schoolteachers. Perhaps, a dot plot for the data based on the type of school the teacher belongs wouldimprove the quality of the data. Questions such as “do elementary teachers run more miles a weekfor exercise than middle school teachers?” could be answered. Another possibility is a dot plot of themale teachers and a dot plot for the female teachers. That would give more inferences about thepopulation.

The data from a random sample and proportional reasoning can be used to make inferences about apopulation.

Example: A quality control officer checks CDs for the quality of their contents. In a shipment of 1000,she randomly pulled 25 CDs and found 1 to be inferior in quality. At this rate, how many CDs in theshipment could be inferior in quality?

1

25represents the ratio of inferior to total sample. The proportion

1

25 1000

x can be solved for the

number of CDs in the order that could be inferior. Using the Property of Means, 25x = 1000, x = 40.

There could be 40 inferior CDs in the 1000 CD shipment. This is probably too high a percentage ofinferior CDs. The company may request another random sample be done, or they might request alarger random sample. The company should be taking steps to improve the superior quality in the CDsand reduce the inferior quality of CDs.

Exercise Miles

0 1 2 3 4 5 6 7 8 9 10Number of Miles per Week

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