Investigating Student Learning: 6 Grade Chapter 8: …mrbentley.wikispaces.com/file/view/6thISLCh8.pdf7/7/2008 6th 8-1 1 DRAFT Investigating Student Learning: 6th Grade Chapter 8:

7/7/2008 6th 8-1 1

DRAFT

Investigating Student Learning: 6th Grade Chapter 8: Ratio and Proportion

Standard NS 1.2*: Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a

b, a to b, a:b).

Lesson 8.1: Ratios and Equivalent Ratios

Concepts:

Ratio is a comparison of two numbers or quantities. The choice of the two quantities and the order in which they are expressed in the ratio are important. Ratios can compare a part to a whole, a part to a part, or whole to another whole. There are several ways to write ratios.

Use to (3 to 4), use a colon (3:4), or use fraction form ( 34

).

Situation: Ms. Kemper’s Class Mr. Jacob’s Class 12 Girls 14 boys 15 Girls 9 boys

Example 1: Compare a Part to a Whole

Girls to total number of students in Ms. Kemper’s Class: 12 to 26, 12:26, or 1226

.

Example 2: Compare a Part to a Part

Girls to Boys in Ms. Kemper’s Class: 12 to 14, 12:14, or 1214

.

Example 3: Compare a Whole to another Whole (or 2 sets of different types of objects)

Students in Ms. Kemper’s Class to students in Mr. Jacob’s Class: 26 to 24, 26:24, or 2624

or Girls in Ms. Kemper’s Class to Girls in Mr. Jacob’s Class: 12 to 15, 12:15, 1215

.

Although a ratio can be written using ab

notation, it does not necessarily represent a fractional part of a

whole. All fractions are ratios (example 1 above), but not all ratios are fractions (examples 2 and 3).

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DRAFT

Ratios Differ from Fractions Similarities A ratio can be written in fraction form, with the number mentioned first as the numerator and the

second number as the denominator.

A ratio can be simplified the same way fractions are simplified (4 to 8 or 48

is equal to 1 to 2 or 12

).

Differences The denominators of fractions and ratios are chosen differently.

A fraction’s denominator always tells you how a whole is divided. A ratio’s denominator could tell you how a whole is divided (ex. 1), OR a different part of a

whole than the numerator describes (ex. 2), OR the number of parts in another whole (ex. 3). Fractions and ratio differ in the units they use.

A fraction compares things that have the same units (like 3 pizza slices out of 8 slices in a whole pizza.

A ratio may compare things with like OR unlike units (3 pizza slices to 6 sandwiches). We don’t add or subtract ratios as we add or subtract fractions. Essential Question(s): What is a ratio? How do you read and write ratios? How do you find equivalent ratios?

7/7/2008 6th 8-1 3

DRAFTISL Item Bank: 6th Grade

Chapter 8: Ration and Proportion

Standard NS 1.2*: Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a

b, a to b, a:b).

Lesson 8.1: Ratios and Equivalent Ratios What is a ratio? Use ratio to compare the quantities below in different ways. Identify those different ways. Example: Different Ways to Compare: 1) triangles to squares

2) triangles to shapes

3) squares to shapes

Different Ways to Compare: 1)

2)

3) Different Ways to Compare: 1) 2) 3)

Different Ways to Compare: 1) 2) 3)

4 small drinks 6 large drinks Different Ways to Compare: 1) 2) 3)

21 red crayons 16 blue crayons Different Ways to Compare: 1) 2) 3)

7 dogs 5 cats Different Ways to Compare: 1) 2) 3)

19 chairs 32 tables Different Ways to Compare: 1) 2) 3)

7/7/2008 6th 8-1 4

DRAFTUse ratio to compare the quantities below in different ways. Identify those different ways. 52 Fiction Stories 43 Biographies Different Ways to Compare: 1) 2) 3)

26 apples 26 oranges Different Ways to Compare: 1) 2) 3)

Class Boys Girls

Ms. Gold 14 12 Different Ways to Compare: 1) 2) 3)

Team Wins LossesHawks 21 18


Pitcher Strikes Balls Beckett 31 15


Test Answers True False World History

36 24


Restaurant Hot Dogs

Sold Hamburgers

Sold Ed’s Eats 39 57


Store Hip Hop

CDs Sold Rock CDs

Sold Tommy’s

Tunes 42 67


7/7/2008 6th 8-1 5

DRAFTUse ratio to compare the quantities below in different ways. Identify those different ways. Example: Case 1: 40 Crayons Case 2: 50 Crayons

5 red 7 red

Different Ways to Compare: 1) red crayons to total crayons in Case 1

2) red crayons to total crayons in Case 2

3) red crayons in Case 1 to red crayons in Case 2

Box 1: 20 pieces of Fruit Box 2: 30 pieces of Fruit5 apples 7 apples


Shelter 1: 37 Animals Shelter 2: 51 Animals 22 dogs 40 dogs

Different Ways to Compare:

1)

2)

3)

Library 1: 700 Books Library 2: 500 Books 64 Biographies 49 Biographies


1)

2)

3)

Class Boys Ms. Gold 14 Mr. Judd 17


2)

3)

Pitcher Strikes Beckett 31 Francis 22


1)

2)

3)

Team Wins Hawks 21 Kings 24


2)

3)

Store Hip Hop CDs Sold

Tommy’s Tunes

42

Marv’s Music

26


1)

2)

3)

7/7/2008 6th 8-1 6

DRAFTUse ratio to compare the quantities below in different ways. Identify those different ways. Example: Case 1: 40 Crayons Case 2: 50 Crayons

5 red 7 red 8 blue 9 blue

Different Ways to Compare: 1) red crayons to blue crayons in Case 1 2) red crayons to total crayons in Case 1 3) blue crayons to total crayons in Case 1 4) red crayons in Case 1 to red crayons in

Case 2 5) blue crayons in Case 1 to blue crayons in

Case 2

6) red crayons to blue crayons in Case 2 7) red crayons to total crayons in Case 2 8) blue crayons to total crayons in Case 2

Box 1: 20 pieces of Fruit Box 2: 30 pieces of Fruit 5 apples 7 apples

8 oranges 9 oranges Different Ways to Compare: 1) 2) 3) 4) 5)

6) 7) 8)

Class Boys Girls Ms. Gold 14 12 Mr. Judd 17 9

Different Ways to Compare: 1) 2) 3) 4) 5)

6) 7) 8)

Pitcher Strikes Balls Beckett 67 47 Francis 52 37

Different Ways to Compare: 1) 2) 3) 4) 5)

6) 7) 8)

7/7/2008 6th 8-1 7

DRAFTHow do you read and write ratios? Write each ratio in three different ways.

Ratio Use to Use a colon Use fraction form

Example: 8 girls to 5 boys

8 to 5 8:5 8

5

8 girls to 20 total students

7 wins to 9 losses

7 wins to 16 games

23 apples to 18 oranges

18 oranges to 41 total fruit

20 sunny days to 11 rainy days

11 rainy days in the month of Jan.

8 inches to 4 inches

8 inches in a foot

Walked 15.1 miles to ran 11.1 miles

Walked 15.1 miles in a 26.2 marathon

Use the situation to write each ratio using colon form.

Situation

Ratio

Use a Colon

apples to oranges

apples to boxed fruit

Boxed Fruit Basket Fruit

6 apples, 5 oranges 12 berries, 2 melons

boxed fruit to basket fruit

Lemon to chocolate cookies

lemon cookies to cookies in jar

Can of Cookies Jar of Cookies

13 sugar, 12 oatmeal 10 chocolate, 15 lemon cookies in the can to cookie in

the jar

forks to knives

Spoons to Gold Table Settings

Silver Table Settings Gold Table Settings 8 forks, 10 knives 11 spoons, 15 napkins

Silver to Gold Table Settings

boys to girls in Mr. Brown’s

girls to students in Ms. Black’s

Ms. Black’s Class Mr. Brown’s Class 11 boys, 14 girls 13 boys, 13 girls

Ms Black to Mr. Brown’s Class

7/7/2008 6th 8-1 8

DRAFTUse the situation to write the ratio described. (many answers are possible)

Situation

Ratio Description

Use a Colon

part to part

6:5 or 12:2

part to whole group

6:11 or 12:14

Example: Boxed Fruit Basket Fruit

6 apples, 5 oranges 12 berries, 2 melons

Whole group to whole group

11:14 or 14:11

part to whole group


Can of Cookies Jar of Cookies

13 sugar, 12 oatmeal 10 chocolate, 15 lemon

part to part

part to part


Silver Table Settings Gold Table Settings 8 forks, 10 knives 11 spoons, 15 napkins

part to whole group

part to whole group

part to part

Book of Stamps Sheet of Stamps

25 Flag, 15 Liberty Bell 15 Peace, 12 Dove


part to whole group

part to part

Ms. Black’s Class Mr. Brown’s Class 11 boys, 14 girls 13 boys, 13 girls



part to part

March July

14 sunny, 17 rainy days 29 sunny, 2 rainy days

part to whole group

part to part


Larry’s Tree Lot Orlando’s Orchard

37 Pine, 54 Douglas Fir 23 Cedar, 57 Maple

part to whole group

part to whole group

part to part

Kylie’s sock drawer Brooke’s sock drawer

8 white, 12 black 9 brown, 15 beige


Of the 3 Ratio Descriptions, which is most like a regular fraction?

A) part to part B) part to whole C) whole to whole

7/7/2008 6th 8-1 9

DRAFT Which ratio shows a part to whole comparison?

Team Wins Losses Hawks 21 12 Kings 24 13

A) Games Kings played to games Hawks played

B) Hawk wins to Hawk losses

C) King wins to games Kings played

D) Hawk wins to King wins

Which ratio shows a part to part comparison?

Class Boys Girls Ms. Gold 14 12 Mr. Judd 17 9

A) Boys to students in Ms. Gold’s Class

B) Girls to students in Ms. Gold’s Class

C) Students in Ms. Gold’s to Mr. Judd’s Class

D) Boys in Ms. Gold’s Class to boys in Mr. Judd’s Class

Which ratio shows a whole to whole comparison?

Pitcher Strikes Balls Beckett 67 47 Francis 52 37

A) Beckett’s strikes to balls

B) Pitches Francis threw to pitches Beckett threw

C) Beckett’s strikes to Francis’s strikes

D) Beckett’s balls to Francis’s balls

Which ratio shows a part to part comparison?

Restaurant Hot Dogs Sold

Hamburgers Sold

Ed’s Eats 39 57 Dan’s Diner 25 43

A) Dan’s hot dogs sold to total sales

B) Ed’s total sales to Dan’s total sales

C) Ed’s hot dogs sold to hamburgers sold

D) Ed’s hot dogs sold to Dan’s total sold

Which ratio shows a part to whole comparison?


Rock CDs Sold

Tommy’s Tunes

42 65

Marv’s Music

26 12

A) 42:65 B) 12 to 38 C) 38:107 D) 2642

Which ratio shows a whole to whole comparison?

Coin Heads Tails Quarter 19 21 Nickel 22 18

A) 19 21

B) 18 to 40 C) 21:19 D) 40 to 40

7/7/2008 6th 8-1 10

DRAFT Jo Jo’s basketball team won 21 of the 30 games they played. What is the ratio of games won to the total number of games played? Is this ratio? (circle) A) part to part B) part to whole C) whole to whole

At Giant Scoop Ice Cream Parlor, 72 chocolate ice cream cones and 64 vanilla ice cream cones were sold last Saturday. What is the ratio of vanilla to chocolate ice cream cones? Is this ratio? (circle) A) part to part B) part to whole C) whole to whole

Edith’s softball team won 10 of the 12 games they played. What is the ratio of games lost to the number of games won? Is this ratio? (circle) A) part to part B) part to whole C) whole to whole

This weekend, the Ace Airfield recorded 67 jets and 39 prop plane landings. What is the ratio of jet landings to the total number of landings this weekend? Is this ratio? (circle) A) part to part B) part to whole C) whole to whole

Wally the Weatherman predicted 29 sunny days in July and 30 sunny days in August. What is the ratio of rainy days for each month? Is this ratio? (circle) A) part to part B) part to whole C) whole to whole

At Capital Elementary School, there are 350 boys and 368 girls. What is the ratio of boys to the total number of students? Is this ratio? (circle) A) part to part B) part to whole C) whole to whole

On Saturday, Pet Salon groomed 18 dogs and 7 cats. The Posh Paw groomed 16 dogs and 11 cats. What is the ratio of total pets groomed at each Pet care operation? Is this ratio? (circle) A) part to part B) part to whole C) whole to whole

At a baseball game, Jorge walked twice and got 2 hits. What was his ratio hits to the number of at bats? What was Jorge’s ratio of the number of times he got on base to the number of at bats? What was his ratio of his number of times on base to the number of outs he made.

7/7/2008 6th 8-1 11

DRAFTHow do you find equivalent ratios? Complete the ratio table.

Panels x1 Number of triangles

Number of gray squares

4

1 panel

Panels x1 x2 Number of triangles

4


4

2 panels

Panels x1 x2 x3 Number of triangles

4


4

3 panels What if you wanted to find out the number of triangles and squares needed for 10 panels?

Panels x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Number of triangles

2

4

6

?


4

8

12

?

You could continue to draw pictures, but instead, you can use what you know about equivalent fractionssince fractions are a type of ratio. LOOK:

Panels x1 x2 x3 Number of triangles

2

x 2 =

4

OR

x 3 =

6


4

x 2 =

8

x 3 =

12

SO, multiply each number in the ratio by the same nonzero number: 24

x 1010

= 2040

In 10 panels there will be 20 triangles and 40 squares.

7/7/2008 6th 8-1 12

DRAFTFind the missing number that makes the ratio equivalent. Example:

68

= 12x

x = 16

Example:

68

= 4y

x = 3

Example:

6:8 = a:24

a = 18

Example:

6 to 8 = 3 to b

b = 4

52

= 10p

p =

8:10 = r:5

r =

14 to 21 = 2 to s

s =

159

= 3k

k =

16:m = 2:8

m =

50t = 3

5

t =

w to 3 = 28 to 12

w =

72f

= 83

f =

89

= 45c

c =

z to 56 = 5 to 7

z =

7n

= 7755

n =

h:22 = 9:11

h =

6 to 4 = 36 to d

d =

26:j = 13:23

j =

8460

= 7q

q =

1424

= 42v

v =

Are the ratios equivalent? Write = or ≠.

Example: 35

910

So, 35

≠ 910

Think: What do you do to 3 to get 9? Multiply by 3. Is 5 x 3 = 10? NO

Example: 49:28 7:4

So, 49:28 = 7:4

Think: What do you do to 49 to get 7? Divide by 7. Is 28 ÷ 7 = 4? Yes

38

1248

10 to 8 5 to 2

72:45 8:5

4 to 6 16 to 24

35 to 56 5 to 7

610

32

5: 8 40:56

81:36 27:6

2712

93

12:3 4:1

4 to 11 32 to 88

137

3921

7/7/2008 6th 8-1 13

DRAFTSet up equivalent ratios for each situation by first determining the meaning of the unknown number. Use the variable x as the missing number. Fill in the blanks. Do not solve.

Situation Meaning of Variable Proportion Example: Carter’s cupcake recipe calls for 3 tablespoons of sugar for every 5 cupcakes. If he wants to make 10 cupcakes, how many tablespoons of sugar will he need?

Example:

x = the number of Tbsp sugar needed for 10 cupcakes

Example:

3 Tbspsugar5 cupcakes

= Tbspsugar10 cupcakesx

Vicky is mixing paint for her art project. To make the color green, she mixes 6 parts blue to 18 parts yellow. If Vicky only has 6 parts yellow, how many parts blue should she add to end up with the correct green color?

x = the number of blue parts for every ___ parts yellow

6 parts blue

18 parts yellow=

Place 6 parts yellow and the

variable x correctly in the ratio.

Joy runs around the park every morning. She runs 4 laps every 10 minutes. If she keeps this same speed, how many laps should Joy be able to run in 30 minutes?

x = the number of ________ in 30 ______________.

laps

minutes= laps

minutes

A chocolate almond candy bar is made up of 18 whole almonds to every 12 ounces of chocolate. How many ounces of chocolate is needed if 90 whole almonds were used?

x = the number of __________if ________________.

=

Poochi the dog usually gets 2 doggie treats for every 8 tricks she performs. How many tricks must Poochi do before she is able to eat 6 doggie treats?

x = the number of ___________________________________.

=

At the Tully Hill Baseball Play-offs, 200 out of 240 people wore a red shirt to the game. If this ratio remained the same, how many people would be wearing shirts if there were only 60 people at the game?

x =

=

7/7/2008 6th 8-1 14

DRAFT

7/7/2008 6th 8-2 14

DRAFTInvestigating Student Learning: 6th Grade


Standard NS 1.3*: Use proportions to solve problems (e.g., determine the value of N if 4

7 =

21N )….

Lesson 8.2: Proportions

Concepts:

A Proportion is an equation showing that two ratios are equal. There are different ways to write proportion. To write a proportion, some common element must tie the numerators together. Another common

element must tie the denominators together.

Situation: A local Science Camp requires the ratio of students to adult chaperones to be 6 to 1. So, if Hudson Elementary’s 6th grade class has 108 students, they must have at least 18 adult chaperones to be in compliance with camp rules.

Example 1: BOTH numerators could relate one element such as students, while both denominators relate to another element such as adult chaperones:

ScienceCamp ratioScienceCamp chaperone ratioadult

student = .#.#

Hudson Elem ofHudson Elem of chaperones

studentsadult

(Science Camp ratio) (Hudson Elem. ratio)

Example 2: OR both numerators could relate to the science camp requirement, while both denominators relate to Hudson Elementary:

#.student ratioof students

Science CampHudsonElem =

#.adult chaperone ratioof adult chaperones

Science CampHudsonElem

(student ratio) (adult ratio)

Ways to Write

Compare Science Camp ratio to Hudson Elem. ratio

Compare student ratio to adult ratio

Use to

6 to 1 = 108 to 18

6 to 108 = 1 to 18

Use a colon

6:1 = 108:18

6:108 = 1:18

Use fraction form

61

= 10818

6108

= 118

The colon notation is read the same way you read analogies in English: Write: a:b = c:d Say: a is to b as c is to d.

7/7/2008 6th 8-2 15

DRAFTWhen two quantities are proportional, a change in one quantity corresponds to a predictable change in

the other. In a direct proportion, both quantities increase by the same factor, or both quantities decrease by the

same factor. Sometimes you don’t have all the elements of a proportion, so you need to solve an equation to find the

missing piece. You can solve a proportion by finding equivalent fractions. Essential Question(s): What is a proportion? How do you solve proportions using equivalent fractions?

7/7/2008 6th 8-2 16



Standard NS 1.3*: Use proportions to solve problems (e.g., determine the value of N if 4

7 =

21N )….

Lesson 8.2: Proportions What is a proportion?

Show how the relationship is a proportion by circling the common relationship between numerators and boxing the common relationship between denominators.

12 in.

Example:

.1. 1

length of recwidth of rec

= .2. 2

length of recwidth of rec

4 in.

2 in.

6 in.

1

2 12 in.

OR

.1.2

length of reclength of rec

= . 1. 2

width of recwidth of rec

4 in.

2 in.

6 in.

2

1

# .1# .1of squares Grof circles Gr

= # .22# .

of squares Grof circles Gr

OR

# .1# .2

of squares Grof squares Gr

= # .# .

of circles Grof circles Gr

Group 1 Group 2

12

Group 1 Group 2

Mr. Hall’s Class All 6th Grade 16 girls 48 girls 10 boys 30 boys

. '

6Mr Hall s girls

th Grade girls = . '

6Mr Hall s boys

th Grade boys

OR Mr. Hall’s Class All 6th Grade 16 girls 48 girls 10 boys 30 boys

. '. '

Mr Hall s girlsMr Hall s boys

= 66th Grade girlsth Grade boys

7/7/2008 6th 8-2 17

DRAFT

Show how the relationship is a proportion by circling the common relationship between numerators and boxing the common relationship between denominators.

2 cm

.

.length of rec Alength of rec B

= ..

width of rec Awidth of rec B

8 cm

12 cm

3 cm

A B 2 cm

OR

..

length of rec Awidth of rec A

= ..

length of rec Bwidth of rec B

8 cm

12 cm

3 cm

A B

# .# .

of triangles Gr Aof triangles Gr B

= # .# .

of shapes Gr Aof shapes Gr B

Group A Group B OR

# .# .of triangles Gr Aof shapes Gr A

= # .# .of triangles Gr Bof shapes Gr B

Group A Group B

Ms. Bender’s Class All 6th Grade 12 girls 72 girls 14 boys 84 boys

. '. '

Ms Bender s girlsMs Bender s class

= 66

th Grade girlsth Grade class

OR Ms. Bender’s Class All 6th Grade 12 girls 72 girls 14 boys 84 boys

. '

6Ms Bender s girls

th Grade girls = . '

6Ms Bender s class

th Grade class

Dance Rules Actual Participation 3 Judges 45 Judges 12 Dancers 180 Dancers

#

#

Dance Rule Judges

Actual Judges = #

#

Dance Rule Dancers

Actual Dancers

OR Dance Rules Actual Participation 3 Judges 45 Judges 12 Dancers 180 Dancers

#

#

Dance Rule Judges

Dance Rule Dancers = #

#

Actual Judges

Actual Dancers

7/7/2008 6th 8-2 18

DRAFTFor each pair of ratios, explain why a proportion is NOT formed.

12 in.

Example:

42

= 612

Comparing length to width and then width to length. Numerators must compare the

same and denominators the same.

4 in.

2 in.

6 in.

1

2

23

= 64

Group 1 Group 2


16: 26 = 48:30

2 cm

8 to 2 = 3 to 12

8 cm

12 cm

3 cm

A B

1025

= 35

Group A Group B


14:26 = 84:72


3 to 45 = 3 to 12

Contest Rules Actual Contest 4 Prizes 28 Prizes 20 Contestants 140 Contestants

140 to 28 = 20 to 7

7/7/2008 6th 8-2 19

DRAFTFor each proportion, identify the two equivalent ratios being compared.

12 in.

Example:

412

= 26

.1.2

length of reclength of rec

= .1.2

width of recwidth of rec

4 in.

2 in.

6 in.

1

2

23

= 46

Group 1 Group 2


16: 48 = 10:30

2 cm

2 to 8 = 3 to 12

8 cm

12 cm

3 cm

A B

1025

= 25

Group A Group B


14:26 = 84:156


12 to 3 = 180 to 45

Contest Rules Actual Contest 4 Prizes 28 Prizes 20 Contestants 140 Contestants

4 to 28 = 20 to 140

7/7/2008 6th 8-2 20

DRAFT

12 in.

Example: Write the proportion in two ways using a colon. 1) 4:2 = 12:6 2) 4:12 = 2:6 Write the proportion in two ways using fraction form.

1) 42

= 126

2) 4

12=

26

Write the proportion in two ways using to. 1) 4 to 2 = 12 to 6 2) 4 to 12 = 2 to 6

4 in.

2 in.

6 in.

1

2

Write a proportion in two ways using a colon. 1) 2) Write a proportion in two ways using fraction form. 1) 2) Write a proportion in two ways using to. 1) 2)

Group 1 Group 2



2 cm


8 cm

12 cm

3 cm

A B

7/7/2008 6th 8-2 21

DRAFT


Group A Group B



Dance Rules Actual Participation 3 Judges 45 Judges 12 Dancers 180 Dancers Write a proportion in two ways using a colon. 1) 2) Write a proportion in two ways using fraction form. 1) 2) Write a proportion in two ways using to. 1) 2)

Contest Rules Actual Contest 4 Prizes 28 Prizes 20 Contestants 140 Contestants Write a proportion in two ways using a colon. 1) 2) Write a proportion in two ways using fraction form. 1) 2) Write a proportion in two ways using to. 1) 2)

7/7/2008 6th 8-2 22

DRAFTUse equivalent ratios to determine if each pair of ratios can form a proportion. Write yes or no.

Example: 47

, 1221

47

x 33

= 1221

yes

Example: 35

, 1825

35

x 66

= 1830

no

Example: 1832

, 68

1832

÷ 3?

= 68

no

23

, 1015

921

, 37

49

, 1645

128

, 34

97

, 5449

1527

, 59

196

, 3812

3616

, 95

89

, 5672

6035

, 128

137

, 5228

2832

, 78

5639

, 1413

9

11, 99

121

2442

, 47

7

12, 56

84

5496

, 916

4

13, 24

78

7/7/2008 6th 8-2 23

DRAFTFind the missing number in each proportion. Example:

68

= 12x

x = 16

Example:

68

= 4y

y = 3

Example:

8n = 15

24

n = 5

Example:

6p

= 29

p = 27

45

= 20t

t =

69

= 3r

r =

4l = 21

28

l =

25a

= 57

a =

37

= 24m

m =

824

= 6q

q =

3w = 36

9

w =

24k

= 411

k =

97

= 72k

k =

2718

= 6f

f =

3e = 45

27

e =

48g

= 87

g =

125

= 36j

j =

3224

= 3b

b =

5b = 24

60

b =

81z

= 94

z =

811

= 64y

y =

3344

= 4c

c =

2t = 49

14

t =

39f

= 136

f =

1112

= 66s

y =

5640

= 5v

v =

4q = 52

32

q =

24x

= 821

x =

136

= 78w

w =

5472

= 8d

d =

12s = 81

36

s =

75r

= 159

r =

7/7/2008 6th 8-2 24

DRAFTWrite a proportion using x as the missing number. Then solve.

Situation Proportion Solution Example: Carter’s cupcake recipe calls for 3 tablespoons of sugar for every 5 cupcakes. If he wants to make 10 cupcakes, how many tablespoons of sugar will he need?


=

Place 10 cupcakes and the


Example:

35

= 10x

35

x 22

= 610

x = 6 tablespoons of sugar

If Carter wants to make 25 cupcakes, how many tablespoons of sugar will he need?


=

Place 25 cupcakes and the


35

=

x = ___ tablespoons of sugar

If he only has only 12 tablespoons of sugar, how many cupcakes can he make?


=

Place 12 tablespoons of sugar and the variable x correctly in

the ratio.

x = ___ cupcakes

7/7/2008 6th 8-2 25


Situation Proportion Solution Vicky is mixing paint for her art project. To make the color green, she mixes 6 parts blue to 18 parts yellow. If Vicky only has 6 parts yellow, how many parts blue should she add to end up with the correct green color?

6 parts blue18 parts yellow

=

Place 6 parts yellow and the


6

18 =

x = ___ parts blue

If Vicky starts with 24 parts blue, how many parts yellow should she add to end up with the correct green color?


=

Place 24 parts blue and the


x =

If Vicky starts with 2 parts blue, how many parts yellow should she add to end up with the correct green color?


=

x =

7/7/2008 6th 8-2 26


Situation Proportion Solution Joy runs around the park every morning. She runs 4 laps every 10 minutes. If she keeps this same speed, how many laps should Joy be able to run in 30 minutes?

4 laps10 minutes

=

x =

How long should Joy expect 16 laps to take her?

4 laps10 minutes

=

x =

Last week, Joy only ran 2 minutes before she sprained her ankle and could not run anymore. If she was running her usual pace, how many lap had Joy run when she sprained her ankle?

=

x =

7/7/2008 6th 8-2 27


Situation Proportion Solution A chocolate almond candy bar is made up of 18 whole almonds to every 12 ounces of chocolate. How many almonds and ounces of chocolate is needed to make 15 bars?

=

x =

If I only had 144 whole almonds and wanted to make as many candy bars possible, how much chocolate would I need?

=

x =

If I only had 132 ounces of chocolate and wanted to make as many candy bars possible, how many almonds would I need?

=

x =

7/7/2008 6th 8-2 28


Situation Proportion Solution Poochi the dog usually gets 2 doggie treats for every 8 tricks she performs. How many tricks must Poochi do before she is able to eat 6 doggie treats?

=

x =

Last Wednesday, Poochi was wild! She performed a total of 64 tricks. How many doggie treats did Poochi receive on Wednesday?

=

x =

One day Poochi received only 1 treat. How many tricks did she do that day?

=

x =

7/7/2008 6th 8-2 29


Situation Proportion Solution At the Tully Hill Baseball Play-offs, 200 out of 240 people wore a red shirt to the game. If this ratio remained the same, how many people would be wearing shirts if there were only 60 people at the game?

=

x =

If this ratio remained the same, and 1,000 people came wearing red shirts, how many people would be in attendance?

=

x =

Mr. Tully said he would give free admission to the first 25 people wearing red shirts to tomorrow’s game. If the ratio remained the same, how many people would have to walk through the gate before Mr. Tully could give away his 25 free tickets?

=

x =

7/7/2008 6th 8-3 30


Chapter 8: Ratio and Proportion

Standard NS 1.3*: Use proportions to solve problems…Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.

Lesson 8.3: Solving Proportions using Cross Products

Concepts:

A Proportion is an equation stating that two ratios are equivalent. In the proportion a:b = c:d, a and d are the extreme terms and b and c are the mean terms. The product of the extreme terms equals the product of the mean terms, ad = bc.

means

a : b = c : d

extremes

means

3 : 5 = 6 : 10

extremes The product of the extreme terms equals the product of the mean terms, ad = bc, or 3 ⋅10 = 5 ⋅6.

Why it Works! a : b = c : d

Rewrite the proportion ab

= cd

Multiplying both sides of the (bd) = (bd) equation by bd. Simplify the fractions =

ad = bc

ab

cd

abdb

bd cd

3 : 5 = 6 : 10

Rewrite the proportion 35

= 6

10

Multiplying both sides of the (5 ⋅10) = (5 ⋅10) equation by bd. Simplify the fractions =

3 ⋅10 = 5 ⋅6 30 = 30

35

610

3 5 105⋅ ⋅ 5 10 6

10⋅ ⋅

Finding the cross products is the same as multiplying the extremes by the means. a : b = c : d

Rewrite the proportion ab

= cd

ad bc

Cross Multiply ab

= cd

ad = bc

Cross Product

3 : 5 = 6 : 10

Rewrite the proportion 35

= 6

10

3 ⋅10 5 ⋅6

Cross Multiply 35

= 6

10

3 ⋅10 = 5 ⋅6 30 = 30

Use of the multiplication property of equality and simplification shows why the cross product method

works.

7/7/2008 6th 8-3 31

DRAFTIn a proportion, the cross products are equal. If two ratios form a proportion, the cross products are equal. You can solve a proportion by finding equivalent fractions OR by using cross products.

Solving a proportion by finding equivalent fractions.

59

= 63x

59

= 5x79x7

= 3563

x = 35

Solving a proportion by using cross products.

59

= 63x

5 ⋅63 9 ⋅x

59

= 63x

5 ⋅63 = 9 ⋅x 315 = 9x

3159

= 99x

35 = x Essential Question(s): What are cross products? How do you use cross products to see if two ratios are a proportion? How do you solve proportions using cross products?

7/7/2008 6th 8-3 32


Chapter 8: Ratio and Proportion

Standard NS 1.3*: Use proportions to solve problems…Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse.

Lesson 8.3: Solving Proportions using Cross Products What are cross products? Fill in the blanks.

You know that 23

= 69

is a proportion because of your work with equivalent fractions. 23

x 33

= 69

2 i9 3 i6

The cross products can be found by multiplying diagonally, or cross multiplying. 23

= 69

2 i9 = 3 i6 __ = 18 18 is one cross product and ___ is the other cross product.

You know that 74

= 148


x 22

=148

__ i__ __ i__


= 148

7 i__ = __ i14 __ = __ __ is one cross product and ___ is the other cross product.

You know that 1254

= 29


÷ 66

= 29

__ i__ __ i__


= 29

__ i__ = __ i__ __ = __ ___ is one cross product and ___ is the other cross product.

Find the cross products.

37

= 614

__ i __ = __ i __ ___ = ___

836

= 29

__ i __ = __ i __ ___= ___

54

= 1512

__ i __ = __ i __ ___= ___

1521

= 57

__ i __ = __ i __ ___= ___

7/7/2008 6th 8-3 33

DRAFT

First, use equivalent fractions to determine if the two ratios form a proportion. Then, find their cross products. Example: Equivalent Fraction Proportion? Cross Multiply

43

= 2015

43

x 55

= 2015

Yes 4 i15 3 i20

No 43

= 2015

4 i15 = 3 i20 60 = 60

?

Example: Equivalent Fraction Proportion? Cross Multiply

25

= 610

25

x 33

= 615

Yes 2 i10 5 i6

No 25

= 610

2 i10 = 5 i6 20 ≠ 30

?

Equivalent Fraction Proportion? Cross Multiply

45

= 1625

Yes

No 45

= 1625

__ i__ = __ i__ ___ = ___

?


2028

= 57

Yes

No 2028

= 57

__ i__ = __ i__ ___ = ___

?

What do you notice about the cross products in a proportion?

7/7/2008 6th 8-3 34

DRAFT

First, use equivalent fractions to determine if the two ratios form a proportion. Then, find their cross products.


148

= 72

Yes

No 148

= 72

__ i__ = __ i__ ___ = ___

?


74

= 4224

Yes

No 74

= 4224

__ i__ = __ i__ ___ = ___

?


1824

= 34

Yes

No 1824

= 34

__ i__ = __ i__ ___ = ___

?


27

= 1649

Yes

No 27

= 1649

__ i__ = __ i__ ___ = ___

?

What do you notice about the cross products in a proportion?

7/7/2008 6th 8-3 35

DRAFTProve that the two ratios are proportional by using the equivalent fractions method and the cross

product method.

104

= 2510

?

Equivalent Fraction Method

104

x = 3012

Cross Product Method

104

= 3012

59

= 2545

?


59

x = 2545


59

= 2545

WAIT! Look at this!

28

= 312

?


28

x = 312

?


2 i12 = 24 8 i3 = 24

28

= 312

It doesn’t look like this method works! BUT the cross product method does work!

WHY?

28

x 1.51.5

= 312

So, the equivalent fraction method does work, but it’s hard to

recognize because whole numbers are not used. That’s why it is sometimes easier to use the cross product method to check or solve proportions.

7/7/2008 6th 8-3 36

DRAFT

How do you use cross products to see if two ratios are a proportion? Use the cross product method to determine if each pair of ratios can form a proportion. Write yes or no.

Example: 47

, 1221

4 i 21 = 84 7 i 12=84

47

, 1221

yes

Example: 35

, 1825

3 i 25 = 75 5 i 18=90

35

, 1825

No

Example: 96

, 128

9 i 8 = 72 6 i 12=72

96

, 128

Yes

68

, 912

921

, 614

8

18, 3

7

1510

, 64

68

, 1520

1025

, 410

46

, 1016

2012

, 159

186

, 125

4

12, 7

21

129

, 2015

96

, 64

104

, 156

9

15, 6

11

39

, 515

7

12, 56

84

6

15, 8

21

1525

, 915

7/7/2008 6th 8-3 37

DRAFTHow do you solve proportions using cross products? Rewrite the two ratios as an equation with cross products. Example:

123

= 2x

123

= 2x

12 x 2 = 3x

26

= 3n

12p

= 84

12k = 2

8

156

= 4l

6t = 6

9

9y

= 610

9

12= 6

m

10w = 12

15

Use the cross product method to solve the proportion. Example:

123

= 2x

Rewrite as an equation 12 x 2 = 3x

24 = 3x

Solve by isolating the variable 24 = 3x 3 3

8 = x

Example:

123

= 82

12 x 2 = 3 x 8

24 = 24 √

26

= 3n

Rewrite as an equation =

=

Solve by isolating the variable =

n =

26

= 3n

=

= check?

Check

Check

7/7/2008 6th 8-3 38

DRAFT Use the cross product method to solve the proportion.

12p

= 84


=


p =

12p

= 84

=

= check?

12k = 2

8


=


k =

12k = 2

8

=

= check?

156

= 4l


=


l =

156

= 4l

=

= check?

6t = 6

9


=


t =

6t = 6

9

=

= check?

Check

Check

Check

Check

7/7/2008 6th 8-3 39

DRAFTUse the cross product method to solve the proportion.

9y

= 610


=


y =

9y

= 610

=

= check?

912

= 6m


=


m =

912

= 6m

=

= check?

10w = 12

15


=


w =

10w = 12

15

=

= check?

129

= 16x


=


x =

129

= 16x

=

= check?

Check

Check

Check

Check

7/7/2008 6th 8-3 40

DRAFTSolve each proportion by finding the missing number.

46

= 10x

x =

4

12 =

9y

y =

8n = 12

6

n =

16p

= 123

p =

615

= 8t

t =

106

= 12r

r =

8l = 9

6

l =

8a

= 1612

a =

84

= 20m

m =

520

= 12q

q =

12w = 10

8

w =

15k

= 62

k =

155

= 6k

k =

1510

= 8f

f =

8e = 25

10

e =

14g

= 410

g =

1620

= 12j

j =

124

= 5b

b =

6b = 28

14

b =

24z

= 164

z =

2012

= 15y

y =

82

= 7c

c =

10t = 20

25

t =

6f

= 1015

f =

64

= 21s

y =

9

12 =

20v

v =

18q = 4

12

q =

6x

= 1640

x =

217

= 15w

w =

104

= 14d

d =

24s = 8

32

s =

25r

= 106

r =

7/7/2008 6th 8-3 41


Example: 4

18 =

27x

Carter’s cupcake recipe calls for 4 cups of flour for every 18 cupcakes. If he wants to make 27 cupcakes, how many cups of flour will he need? 4 x 27 = 18x 108 = 18x 18 18 6 = x x = 6 cups of flour If Carter wants to make 45 cupcakes, how many how many cups of flour will he need? If he has 14 cups of flour, how many cupcakes can he make?

Vicky is mixing paint for her art project. To make the color orange, she mixes 2 parts red to 6 parts yellow. If Vicky has 15 parts yellow, how many parts red should she add to end up with the correct orange color? If Vicky starts with 21 parts yellow, how many parts red should she add to end up with the correct orange color? If Vicky starts with 9 parts red, how many parts yellow should she add to end up with the correct orange color?

7/7/2008 6th 8-3 42

DRAFT Ned walks around the track every morning. He walks 2 laps every 8 minutes. If he keeps this same speed, how many laps should Ned be able to walk in 20 minutes? How long should Ned expect 11 laps to take him? Last week, Ned only walked 2 minutes before rain disrupted his exercise. If he was running his usual pace, how many laps had Ned walked when it started raining?

A chocolate peanut butter bar is made up of 10 ounces of chocolate to every 6 ounces of peanut butter. How many ounces of chocolate and peanut butter is needed to make 20 bars? If I had 15 ounces of peanut butter and wanted to make as many candy bars possible, how much chocolate would I need? If I only had 35 ounces of chocolate and wanted to make as many candy bars possible, how much Peanut butter would I need?

7/7/2008 6th 8-3 43

DRAFT Sammy the seal usually gets 9 sardines for every 12 tricks he performs. How many tricks must Sammy do before he is able to eat 12 sardines? Last Friday, Sammy put on a show! He performed a total of 32 tricks. How many sardines did Sammy receive on Friday? One day Sammy was not feeling well and receive just 1.5 sardines. How many tricks did he do that day?

On a recent survey at a market, 12 out of 15 people said they owned a cell phone. If this ratio remains the same, how many people would have a cell phone if there were 25 people in the market? If this ratio remained the same, and 40 people had cell phones, how many people would be in the market? If the ratio remained the same and there were 10 people in the market, how many of these customers do NOT own a phone?

7/7/2008 6th 8-5 44


Standard AF 2.2*: Demonstrate an understanding that rate is a measure of one quantity per unit of value of another quantity.

Lesson 8.5: Rates

Concepts:

A rate is a special ratio. A rate is a comparison involving two quantities with different units of measure. Examples:

Situation Rate Ratio In 5 hours, you drive 325 miles 325 miles per 5 hours

(mph) 325 miles5 hours

After running, your heart beats 24 times in 15 seconds

24 heart beats per 15 seconds

24 beats15 seconds

Your car travels 286 miles using 13 gallons of gas

286 miles per 13 gallons (mpg)

286 miles13 gallons

If the second quantity of the rate is one unit, the rate is called a unit rate. When a unit rate is written as a ratio, the number 1 is always in the denominator. Examples:

Unit Rate Ratio miles per hour (mph) number of miles

1 hour

dollars per hour number of dollars1 hour

miles per gallon number of miles1 gallon

dollars per pound number of dollars1 pound

beats per minute number of beats1 minute

All rates are ratios, but not all ratios are rates.

Example: 94 miles2 hours

is a ratio because it compares two quantities and a rate because it compares two

different units.

45 apples65 apples

is a ratio because it compares two quantities but is NOT a rate because it

does not compare two different units of measure.

7/7/2008 6th 8-5 45

Unit rates are easy to use in proportions because they involve less computing than ratios with denominators that are not 1.

Essential Question(s): What is rate? What is unit rate? How do you find unit rate? How do you use unit rate to solve problems?

7/7/2008 6th 8-5 46

ISL Item Bank: 6th Grade Chapter 8: Ratio and Proportion

Standard AF 2.2*: Demonstrate an understanding that rate is a measure of one quantity per unit of value of another quantity.

Lesson 8.5: Rates What is rate?

For each situation, determine what is being compared, then decide if the ratio is a rate or not.

Situation

What is being compared?

Rate?

Example: In five minutes, your heart beats 235 times.

235 beats to 5 minutes

Yes No

Example: Cherri’s heart beat 63 times and Matthew’s heart beat

74 times.

63 beats to 74 beats

Yes No

Yesterday, Guermo caught 3 fish in two hours.

Yes No

Yesterday, Guermo caught 2 fish. Today, he caught 4 fish.

Yes No

Virginia had 82 red blocks and 45 blue blocks.

Yes No

Virginia’s case of blocks had 82 red blocks.

Yes No

In the dining room, there were 16 round tables and 7 square tables.

Yes No

Each of the hotel’s dining rooms had 7 square tables. Yes No

In one 8-hour work day, Chuck’s deli sold 212 pounds of turkey.

Yes No

Chuck’s deli sold 212 pounds of turkey and 186 pounds of chicken.

Yes No

Jose ran 15 miles in 1.5 hours.

Yes No

Jose ran 1 hour on Saturday and 1.5 hours on Sunday.

Yes No

Teddy had 31 pastel color crayons and 43 original color crayons.

Yes No

Teddy had 86 original color crayons in his two boxes of crayons.

Yes No

23 dolphins were seen off the coast of Monterey on Sunday. The next day, 39 dolphins were observed.

Yes No

In one day, 23 dolphins were seen off the coast of Monterey.

Yes No

7/7/2008 6th 8-5 47

What is unit rate? For each situation, determine what is being compared, then write the unit rate measurements as a ratio with 1 in the denominator.

Situation

What is being compared?

Unit Rate Measurement

Example: In five minutes, your heart beats 235 times.

235 beats to 5 minutes

number of beats1 minute

Today, Guermo caught 3 salmon in two hours.

number of _______1 hour

Virginia’s case of blocks had 45 blue blocks.

number of _______1 case

Each of the hotel’s dining rooms had 16 round

tables.

number of _______1 room

In 8 hours, Chuck’s deli sold 188 pounds of chicken.

number of _______1 hour

Jose ran 6 miles in 1.5 hours.

number of _______1 _______

Teddy had 248 pastel color crayons in his eight boxes of crayons.

number of _______1 _______

In 12 days, 132 whales were seen off the coast of Monterey.

number of _______1 _______

In 7 hours, Marcus drove 455 miles

number of _______1 _______

After running, your heart beats 864 times in 9 minutes.

Your car travels 286 miles using 13 gallons of gas

Jeffrey works at the local sandwich shop. Last week he worked 22 hours and made $88.

Marvin’s Market is having a sale on seedless grapes. Murphy spent $2.64 on 3 pounds of grapes.

Katrina loves jumping rope. She can complete 900 turns of the rope or revolutions in 5 minutes.

7/7/2008 6th 8-5 48

DRAFTHow do you find unit rate?

For each situation, write the unit rate measurements as a ratio, write the proportion, then find the unit rate by solving the proportion.

Situation


Proportion

Solution

Unit Rate

Example:

In five minutes, your heart beats 235 times.

number of beats1 minute

235 beats5 minutes

= 1 minute

b

235 beats5 minutes

= 1 minute

b

235 i 1 = 5 i b

235 = 5b

2355

= 55b

47 = b

47 beats1 minute

= 47 beats per minutes

Virginia had 3 cases of blocks. There were a

total of 45 blue blocks in 3 cases.

number of 1 case

45 blocks = 1 case

b

45 blocks = 1 case

b

45 i 1 = ___ i b

___ = ___ b

453

= 33b

___ = b

blocks1 case

= __ blocks per case

Today, Guermo caught 6

salmon in two hours.

number of

1 _______

6 2 _____

= 1 _______

s

6 2 _____

= 1 _______

s

___ i 1 = ___ i s

___ = __ s

2 =

2s

___ = s

salmon

1 hour =

7/7/2008 6th 8-5 49

DRAFTFor each situation, write the unit rate measurements as a ratio, write the proportion, then find the unit rate by solving the proportion.

Situation


Proportion

Solution

Unit Rate

Horton’s Hotel has 4 dining

rooms. The head waiter needs to put the same

number of tables in each room. There are 68 round

tables.

In 8 hours, Chuck’s deli

sold 188 pounds of chicken.

Jose ran 6 miles in 1.5

hours.

7/7/2008 6th 8-5 50


Situation


Proportion

Solution

Unit Rate

Teddy had 248 pastel color

crayons in his eight boxes of crayons.

In 12 days, 132 whales were seen off the coast of

Monterey.

In 7 hours, Marcus drove

455 miles

7/7/2008 6th 8-5 51


Situation


Proportion

Solution

Unit Rate

After running, your heart beats 864 times in

9 minutes.

Your car travels 286 miles

using 13 gallons of gas

Jeffrey works at the local sandwich shop. Last week he worked 22 hours and

made $88.

7/7/2008 6th 8-5 52


Situation


Proportion

Solution

Unit Rate

Marvin’s Market is having a

sale on seedless grapes. Murphy spent $2.64 on 3

pounds of grapes.

Katrina loves jumping rope. She can complete 900 turns of the rope or revolutions in

5 minutes.

6 fat free energy bars have

only 915 calories.

7/7/2008 6th 8-5 53

Find the unit rate.

Situation

Unit Rate

Example:

348 miles in 6 hours

348 miles6 hours

= 1 hour

m

348 i 1 = 6 i b 348 = 6m

3486

= 66m

58 = b

58 miles per hour

91 peanuts in 7 candy bars

108 buttons on 9 shirts

earned $84 in 6 hours

7/7/2008 6th 8-5 54

Find the unit rate.

Situation

Unit Rate

371 feet for 7 seconds

276 miles every 11.5 gallons

At mealtime, 224 fish for 32 seals.

2,750 calories in 11 servings

7/7/2008 6th 8-5 55

DRAFTHow do you use unit rate to solve problems?

For each situation, find and use the unit rate to solve the problem.

Situation

Proportions & Work

Unit Rates

Solution

Example: In four minutes, Jesse’s heart

beats 196 times. In six minutes, Toni’s heart beats

276 times. Who had a faster heart rate?

Jesse Toni

196 beats4 minutes

= 1 minute

j 276 beats6 minutes

= 1 minute

b

196 i 1 = 4 i j 276 i 1 = 6 i t 196 = 4j 276 = 6t

1964

= 44j 276

6 = 6

6t

49 = j 46 = t

Jesse’s rate: 49 beats per minutes Toni’s rate: 46 beats per minutes

Jesse has a faster heart rate than Toni.

In a Fishing Derby, a team from Elk Grove caught 42

fish in 3 hours. Another team from Sacramento caught 65 fish in 5 hours. Which team caught more fish per hour?

Virginia has 4 cases of blocks

with a total of 92 green blocks. Cornell has 7 cases of

blocks with a total of 168 green blocks. Who has more

green blocks per case?

7/7/2008 6th 8-5 56

DRAFTFor each situation, find and use the unit rate to solve the problem.

Situation

Proportions & Work

Unit Rates

Solution

Luxury Motel has 78 beds in its 26 rooms. Royalty Motel has 68 beds in its 17 rooms. Which motel has more beds

per number of rooms?

In 8 hours, Chuck’s deli sold 188 pounds of chicken. In 5 hours, Danny’s deli sold 114 pounds of chicken. Which deli sold more pounds of

chicken per hour?

Kaye ran 9 miles in 1.5 hours. Zoe ran 19.5 miles in 3 hours.

Who ran faster?

7/7/2008 6th 8-5 57


Situation

Proportions & Work

Unit Rates

Solution

Teddy had 248 pastel color crayons in his eight boxes of

crayons. Roman has 87 pastel color crayons in his three

boxes of crayons. Who has more pastel color crayons

per box?

In 12 days, 132 whales were seen off the coast of

Monterey. In 14 days, 182 whales were seen off the coast of Maui. Where were a larger

number of whales seen per day?

In 7 hours, Marcus drove 455 miles. In 4 hours, Izzy drove 258 miles. Who drove more

miles per hour? Who drove at a slower rate?

7/7/2008 6th 8-5 58


Situation

Proportions & Work

Unit Rates

Solution

After running, Russell’s heart

beat 864 times in 9 minutes. Fiona’s heart beat

1,176 times in 12 minutes. Who has the

faster heart rate?

A truck travels 247 miles

using 13 gallons of gas. A SUV travels 285 miles using

19 gallons of gas. Which vehicle gets better gas mileage per gallon?

Jeffrey works at the local sandwich shop. Last week he worked 22 hours and made $99. Tamika works at the

local burger joint. Last week she worked 34 hours and

made $144.50. Who made more dollars per hour?

7/7/2008 6th 8-5 59


Situation

Proportions & Work

Unit Rates

Solution

Marvin’s Market is selling 3 pounds of seedless grapes for

$2.64. Gina’s Grocery is selling 5 pounds of seedless

grapes for $4.10. Which store has the best buy on grapes

per pound?

Katrina and Juan love

jumping rope. Katrina can complete 900 turns of the rope or revolutions in 5

minutes. Juan can complete 525 turns of the rope or

revolutions in 3 minutes. Who can complete more jump rope revolutions per minute?

6 Skinny Energy Bars have

only 915 calories. 4 Eat Right Energy Bars have 624

calories. Which energy bar has a fewer number of

calories per bar?

7/7/2008 6th 8-6 60


Standard AF 2.3*: Solve problems involving rates…. Lesson 8.6: Unit Price

Concepts:

The unit price is the price for one unit. Unit price is price rate or cost per unit.

To find the unit price of any item, set up a proportion: total pricetotal units

= price 1 unit

When similar items are priced differently, comparing the unit price is often used to find the better buy. Example:

The same type of oatmeal cookies are being sold at two different bakeries: Bob’s Bake Shop Carey’s Cookies $9.00 for 12 cookies $10.95 for 15 cookies

Which bakery has the better buy?

Solution: Bob’s Bake Shop Carey’s Cookies

$9.0012 cookies

= 1 cookie

c $10.9515 cookies

= 1 cookie

c

9.00 i 1 = 12 i c 10.95 i 1 = 15 i c 9.00 = 12b 10.95 = 15b

9.0012

= 1212

c 10.9515

= 1515

c 0.75 = c 0.73 = c The unit price is $0.75 per cookie. The unit price is $0.73 per cookie.

Carey’s Cookies has the better buy.

Essential Question(s): What is unit price? How do you find unit price? How do you use unit price to find the better buy?

7/7/2008 6th 8-6 61


Standard AF 2.3*: Solve problems involving rates…. Lesson 8.6: Unit Price What is unit price?

For each of the items, identify the unit price measurement.

Item

Unit Price Measurement

Item


Example:

Price: $2.85 for 3 packs of

gum

price per 1 pack

Price: $3.50 for 2 candy apples

price per 1 _______

Price: $28.20 for 12 lbs

price per 1 ______

Price: $5.34 for 6 bananas

price per 1 _______

Price: $3.50 for 7 bags of

marbles

price per 1 _______

Price: $39.75 for 5 boxes

of chocolate

price per 1 _______

Price: $18.30 for 2.5 lbs

of meat

price per 1 _______

Price: $10.35 for 3

sausages

price per 1 _______

Price: $6.45 for 3 loaves of bread

price per 1 _______

Price: $11.70 for 6 slices

of cake

price per 1 _______

7/7/2008 6th 8-6 62

For each of the items, identify the unit price measurement.

Item


Item


Price: $15.48 for 4 packages of hot dogs

price per 1 ______

Price: $14.80 for 8 cups of

coffee

price per 1 _______

Price: $10.68 for 3 bunches of grapes

price per 1 ______


of crayons

price per 1 ______

Price: $6.75 for 3 six-packs

of soda

price per 1 ______

Price: $6.86 for a 14 pounds watermelon

price per 1 _______

$12.20 for 4 half gallons of

milk

price per 1 ______

Price: $3.57 for 3 cartons of eggs

price per 1 _______

Price: $63.25 for 5 square

yards of top soil

price per 1 ______

Price: $52.56 for 24 boards of wood

price per 1 _______

Price: $46.35 for 15 gallons of gas

price per 1 ______

Price: $4.65 for 3 scoops

of ice cream

price per 1 _______

7/7/2008 6th 8-6 63

How do you find unit rate? Find the unit price for each item.

Item

Unit Price

Item

Unit Price

Example:

Price: $2.85 for 3

packs of gum

$2.85 3 packs

= 1 pack

g

2.85 i 1 = 3 i g 2.85 = 3g

2.853

= 33g

$0.95 = g

$0.95 per pack of gum


Price: $28.20 for 12

lbs


of chocolate

Price: $3.50 for 7 bags of marbles


Price: $18.30 for 2.5 lbs of meat

Price: $10.35 for 3

sausages


Price: $11.70 for 6 slices

of cake

7/7/2008 6th 8-6 64

Find the unit price for each item.

Item

Unit Price

Item

Unit Price

Price: $15.48 for 4

packages of hot dogs

Price: $14.80 for 8 cups of

coffee



of crayons

Price: $6.75 for 3 six-

packs of soda


Price: $12.20 for 4 half gallons of milk


Price: $63.25 for 5 sq.

yards of top soil

Price: $52.56 for 24 boards of wood

Price: $46.35 for 15

gallons of gas

Price: $4.65 for 3 scoops

of ice cream

7/7/2008 6th 8-6 65

How do you use unit price to find the better buy?

Find each unit price. Then use this information to determine the better buy.

Item

Unit Price

Better Buy

Example:

Price: $2.85 for 3 packs of gum

$2.853 packs

= 1 pack

g

2.85 i 1 = 3 i b

2.85 = gb

2.853

= 33g

$0.95 = g


Price: $2.85 for 5 packs of gum

$4.605 packs

= 1 pack

g

4.60 i 1 = 5 i b

4.60 = 5g

4.605

= 55g

$0.92 = g


$0.92 per pack is the

better buy

Item

Unit Price

Better Buy



7/7/2008 6th 8-6 66


Item

Unit Price

Better Buy



Item

Unit Price

Better Buy

Price: $39.75 for 5 boxes of chocolate

Price: $16.10 for 2 boxes of chocolate

7/7/2008 6th 8-6 67


Item

Unit Price

Better Buy



Item

Unit Price

Better Buy



7/7/2008 6th 8-6 68


Item

Unit Price

Better Buy



Item

Unit Price

Better Buy

Price: $10.35 for 3 sausages

Price: $19.80 for 6 sausages

7/7/2008 6th 8-6 69


Item

Unit Price

Better Buy



Item

Unit Price

Better Buy

Price: $11.70 for 6 slices of cake

Price: $8.20 for 4 slices of cake

7/7/2008 6th 8-6 70

Compare the price of the items. Determine the better or best buy.


Better Buy:


_____________________________

Price: $14.80 for 8 cups of coffee

Better Buy:

Price: $6.15 for 3 cups of coffee

_____________________________


Better Buy:


_____________________________

Price: $7.60 for 8 boxes of crayons

Better Buy:

Price: $5.58 for 6 boxes of crayons

_____________________________

7/7/2008 6th 8-6 71


Price: $6.75 for 3 six-packs of soda

Better Buy:

Price: $4.70 for 2 six-packs of soda

_____________________________


Better Buy:


_____________________________

$12.20 for 4 half gallons of

milk

$17.94 for 6 half gallons of milk

Best Buy: __________________

$29.20 for 10 half gallons of milk



Best Buy: _____________________


7/7/2008 6th 8-6 72


The price for 5 sq. yards of top soil is $63.25. The price for 8 sq. yards of top soil is $99.20. Which is the better buy?

Better Buy: _____________________________

At the Lumber Mill, 24 boards of wood costs $52.56. The Wood Builder sells 20 boards of wood for $42.20. Which store has the better buy?

Better Buy: _____________________________

Gill’s Gasoline sells 15 gallons of gas for $46.35. At Pat’s Pump,

12 gallons of gas costs $38.88. Down the road, Oil Up sells 10 gallons of gas for $31.50. Which gasoline station has the best buy?

Best Buy: _____________________________

At Giant Scoop, 3 scoops of ice cream costs $4.65. The Ice Cream

Factory sells 4 scoops for $5.80. Homemade Holly’s sells 2 scoops of ice cream for $3.16. Which ice cream shop has the best buy?

Best Buy:_____________________________

7/7/2008 6th 8-8 73


Standard NS 1.3*: Use proportions to solve problems (e.g.,…find the length of a side of a polygon similar to a known polygon)….

Lesson 8.8: Similar Figures

Concepts:

Similar figures have the same shape but not necessarily the same size.

These polygons are similar because they all have the same shape. Congruent figures have the same size and shape. These polygons are congruent because These polygons are similar they all have the same size and shape because they all have the same shape. even if they are oriented differently. Congruent figures are also similar, but similar figures are not congruent. ≅ means “is congruent to” ∼means “is similar to” Two similar polygons have corresponding (or matching) angles that have the same measure (congruent) and

corresponding sides that are proportional. Example:

All corresponding angles are matching or congruent: ; ;A D B E C F∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠

All corresponding sides are proportional: ABDE

= BCEF

= CAFD

*

63

= 147

= 126

= 21

ABC DEF∼ Say: triangle ABC is similar to triangle DEF

* Note: ABDE

cannot be ABED

.

A

B

CD

F

E

55° 55°

25°25° 100°

100° 6

12

14

3

6

7

7/7/2008 6th 8-8 74

We can use proportions to solve problems involving similar figures. Example: GHI JKL∼ Find m K∠ . Because these are similar triangles, m K∠ corresponds and is equal to m H∠ . Knowing that the sum of interior angles of a triangle is 180°, you can find m H∠ : 40° + 90° + m H∠ = 180° 130° + m K∠ = 180° – 130° – 130° 0 + m H∠ = 50° m H∠ = 50°, m H∠ = m K∠ , so m K∠ = 50° Find x. Because these are similar triangles, corresponding sides are proportional:

GHJK

= GIJL

or GHGI

= JKJL

14x

= 1218

1412

= 18x

14 i18 = 12 i x 14 i18 = 12 i x 252 = 12x 252 = 12x

25212

= 1212

x 25212

= 1212

x

21 = x 21 = x

G

H

I J L

K

40°

40°

x14

12 m

18 m

Essential Question(s): What are similar figures? How do you use corresponding congruent angles in similar figures to find missing angles? How do you use proportion to solve problems involving similar figures?

7/7/2008 6th 8-8 75


Standard NS 1.3*: Use proportions to solve problems (e.g.,…find the length of a side of a polygon similar to a known polygon)….

Lesson 8.8: Similar Figures

What are similar figures? ABC and DEF are similar figures. Fill in the blanks by identifying corresponding congruent angles and corresponding proportional sides.

Corresponding Congruent Angles Corresponding Proportional Sides

A∠ ≅ ∠ ___ AB is proportional to DE . B∠ ≅ ∠ ___ AC is proportional to ____. C∠ ≅ ∠ ___ BC is proportional to ____.

DA

B C E

F

XYZ and PQR are similar figures. Fill in the blanks by identifying corresponding congruent angles and corresponding proportional sides.


X∠ ≅ ∠ ___ XY is proportional to ____. Y∠ ≅ ∠ ___ YZ is proportional to ____. Z∠ ≅ ∠ ___ XZ is proportional to ____.

X

Y

Z

R P

Q

GHI and JKL are similar figures. Fill in the blanks by identifying corresponding congruent angles and corresponding proportional sides.


G∠ ≅ ∠ ___ GH is proportional to ____. H∠ ≅ ∠ ___ HI is proportional to ____. I∠ ≅ ∠ ___ IG is proportional to ____.

H

G

I

K L

J

7/7/2008 6th 8-8 76

MNO and PQR are similar figures. Fill in the blanks by identifying corresponding congruent angles and corresponding proportional sides.

Corresponding Congruent Angles Corresponding Proportional Sides M∠ ≅ ∠ ___ MN is proportional to ____. N∠ ≅ ∠ ___ NO is proportional to ____. O∠ ≅ ∠ ___ OM is proportional to ____.

N

PM

R QO

The two figures below are similar figures. Fill in the blanks by identifying corresponding congruent angles and corresponding proportional sides.

Corresponding Congruent Angles Corresponding Proportional Sides A∠ ≅ ∠ ___ AB is proportional to ____. B∠ ≅ ∠ ___ BC is proportional to ____. C∠ ≅ ∠ ___ CD is proportional to ____. D∠ ≅ ∠ ___ DA is proportional to ____.

B

H

G CA

D

F

E

The two figures below are similar figures. Fill in the blanks by identifying corresponding congruent angles and corresponding proportional sides.


I∠ ≅ ∠ ___ , J∠ ≅ ∠ ___ IJ is proportional to ____. JK is proportional to ____.K∠ ≅ ∠ ___ , L∠ ≅ ∠ ___ KL is proportional to ____. LM is proportional to____.M∠ ≅ ∠ ___ , N∠ ≅ ∠ ___ MN is proportional to ____. NI is proportional to ____.

O M

N

J K

I

L

P

Q

R

S

T

7/7/2008 6th 8-8 77

Are the two figures similar?

Figures

Corresponding

Angles Congruent?

Corresponding Sides

Proportional?

Are the two figures

similar? Example:

F

B

A C

D

E

80°

60° 35°

30°

85°

70°

7 m

6 m

8 m

5 m

5 m

3 m

Example:

A∠ ≅ D∠ ? No B∠ ≅ E∠ ? No C∠ ≅ F∠ ? No

Example: ABDE

= ACDF

? 53

= 85

? 25 = 24 ? Yes No (circle)

Example:

No

J G

H I

K

4 in

L M

N

2 in 2 in

4 in

6 in

3 in 3 in

6 in

G∠ ≅ K∠ ? ____ H∠ ≅ L∠ ? ____ I∠ ≅ M∠ ? ____J∠ ≅ N∠ ? ____

GHKL

= HILM

? = ?

___ = ___ ? Yes No (circle)

X W

U

6 ft

S

V

T

7 ft 10 ft

9 ft

3 ft

9 ft

8 ft 2 ft

M N

R

O

Q

P

7 ft

3 ft

5 ft

3 ft

M∠ ≅ K∠ ? ____ N∠ ≅ L∠ ? ____ O∠ ≅ M∠ ? ____P∠ ≅ N∠ ? ____Q∠ ≅ N∠ ? ____R∠ ≅ N∠ ? ____

MNST

= NOTU

? = ?

?

Yes No (circle)

___ = ___

PQVW

= QRWX

? = ?


7/7/2008 6th 8-8 78

Are the two figures similar?

Figures

Corresponding

Angles Congruent?

Corresponding Sides

Proportional?

Are the two figures

similar?

V U

W

Y X

Z

13 cm

8 cm

12 cm

5 cm

6 cm 8 cm 40°

55°

35°

50°

U∠ ≅ X∠ ? ____ V∠ ≅ Y∠ ? ____ W∠ ≅ Z∠ ? ____

UV

Y=

XVWYZ

? = ?


VWYZ

= WUZX

? = ? ___ = ___ ? Yes No (circle)

J G

H I

K

6 yd

L M

N

4 yd

6 yd

10 yd

6 yd 6 yd

10 yd

4 yd

K∠ ≅ G∠ ? ____ L∠ ≅ H∠ ? ____ M∠ ≅ I∠ ? ____ N∠ ≅ J∠ ? ____

KL = GH

LMHI

? = ? ___ = ___ ? Yes No (circle)

9.6 m

C B

F

A

E

D

I H

L

G K

J

16 m 12 m

4.8 m

11.2 m

12

3 m 6 m

14

20 m 15 m

2.4 m

A∠ ≅ G∠ ? ____ B∠ ≅ H∠ ? ____ C∠ ≅ I∠ ? ____ D∠ ≅ J∠ ? ____E∠ ≅ K∠ ? ____F∠ ≅ L∠ ? ____

EFKL

= FALG

? = ?


EDKJ

= DCJI

? = ?


7/7/2008 6th 8-8 79

How do you use corresponding congruent angles in similar figures to find missing angles?

Example: ABC and DEF are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m B∠ .

B∠ ≅ E∠ . m E∠ = 50°. So m B∠ = 50°

A

B

C

DF

E

40°

40° 50°

ABC and DEF are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m D∠ .

D∠ ≅ ∠ ___ . m∠ ___ = ___°. So m D∠ = ___°

A

B

C

D F

E

35°

35°

55°

GHI and JKL are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m G∠ .

G∠ ≅ ∠ ____. m∠ ____ = ____°. So m G∠ = ____°

H

G

I

K L

J

30°

75° 75°

7/7/2008 6th 8-8 80

MNO and PQR are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m Q∠ .

Q∠ ≅ ∠ ____. m∠ ____ = ____°. So m Q∠ = ____°

N

P

M

R QO

80° 45°

55°

The two figures below are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m Y∠ .

Y∠ ≅ ∠ ____. m∠ ____ = ____°. So m Y∠ = ____°

S

X W

T

V U Y Z

The two figures below are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m F∠ .

F∠ ≅ ∠ ____. m∠ ____ = ____°. So m F∠ = ____°

B

H

G

CA

D

F

E

110°

110°

70°

70°

The two figures below are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m M∠ .

M∠ ≅ ∠ ____. m∠ ____ = ____°. So m M∠ = ____°

O

M

N

J K

I L

P

Q

R

72°72°

72°

72°

72°

72°

7/7/2008 6th 8-8 81

Example: ABC and DEF are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m D∠ . D∠ ≅ A∠ . m A∠ + m B∠ + m C∠ = 180° m A∠ + 40° + 90° = 180° m A∠ + 130° = 180° – 130° = – 130° m A∠ + 0° = 50° m A∠ = 50° , so D∠ = 50°

A

B C

D

F E

40°

STU and VWX are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m W∠ .

W∠ ≅ ___∠ . m ___∠ + m S∠ + m U∠ = 180°

S

T

U W

V

X

30°

105°

DEF and GHI are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m G∠ .

G∠ ≅ ___∠ .

FE

D

I

HG

65°

7/7/2008 6th 8-8 82

The two figures are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m Y∠ . Y∠ ≅ ___∠ . m ___∠ + m M∠ + m L∠ + m O∠ = ____°

O

M

N

Y

X

Z

L

W

60°

55° 135°

The two figures are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m J∠ .

J∠ ≅ ___∠ .

C

A B

I H

J

D

K

120°

The two figures are similar figures. All corresponding sides are proportional. Use corresponding congruent angles to find m R∠ . R∠ ≅ ___∠ .

L

J

K

S

Q

R

M T

72°

7/7/2008 6th 8-8 83

How do you use proportion to solve problems involving similar figures? Example: ABC and DEF are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find EF .

ACDF

= BCEF

4 38 x=

4x = 24

4 244 4

x=

x = 6 So, EF = 6 inches

A

B

C

DF

E

40°

40° 50°3 in.

4 in.

10 in.

8 in.

5 in.

GHI and JKL are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find KL .

LJIG

= KLHI

=

___ = ___

= ___ = ___ So, KL = ____ centimeters

H

GI

K L

J

18 cm

30°75°

75° 15 cm

6 cm

18 cm

6 cm

x

7/7/2008 6th 8-8 84

PQR and MNO are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find MO .

= =

___ = ___

= ___ = ___ So, MO = _______________

7.5 yd 1.5 yd

2 yd 10 yd3 yd

N

P

M

R Q O

80° 45°

55°

The two rectangles are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find TU .

= =

___ = ___

= ___ = ___ So, TU = _______________

12 dm

9 dm3 yd

S

X WT

V

U

Y Z

10 dm

7/7/2008 6th 8-8 85

The two parallelograms are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find IF .

= =

___ = ___

= ___ = ___ So, IF = _______________

8 m

12 m

I

LM

H

F

G

K

J 6 m

8 m

6 m

The two trapezoids are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find SR .

= =

___ = ___

= ___ = ___ So, SR = _______________

20 ft

10 ft 20 ft

Q

T

SN

P

O

U

R5.5 ft

21 ft

7/7/2008 6th 8-8 86

The two figures are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find QT .

= =

___ = ___

= ___ = ___ So, QT = _______________

7.4 mm

35 mm

21 mm

L

J

KS

Q

R

M T

30 mm

11 mm

The two figures are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find AF .

= =

___ = ___

= ___ = ___ So, AF = _______________

31.5 m

C B

H

F

E

A

35 m

41.4 m

D

J

I

G

K

L

27 m

63 m

45 m

27 m 35 m

7/7/2008 6th 8-8 87

ABC and DEF are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find EF . A∠ = 35° AB = 15 cm D∠ = 35° DE = 5 cm B∠ = 55° BC = 9 cm E∠ = 55° EF = ? cm C∠ = 90° CA = 12 cm F∠ = 90° FD = 4 cm

ABDE

= BCEF

=

___ = ___

= ___ = ___ So, EF = _______________

JKL and MNO are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find OM .

J∠ = 65° JK = 15 in. M∠ = 65° MN = 5 in. K∠ = 55° KL = 15 in. N∠ = 55° NO = 5 in. L∠ = 65° LJ = 18 in. O∠ = 65° OM = ? in. = =

___ = ___

= ___ = ___ So, OM = _______________ QRS and TUV are similar figures. All corresponding angles are congruent. Use corresponding proportional sides to find TU .

Q∠ = 30° QR = 5 cm T∠ = 30° TU = ? cm R∠ = 45° RS = 2.5 cm U∠ = 45° UV = 3.5 cm S∠ = 105° SQ = 3.5 cm V∠ = 105° VT = 5 cm = =

___ = ___

= ___ = ___ So, TU = _______________

7/7/2008 6th 8-8 88

The two figures are similar figures. Find m C∠ . Find EF .

A

B

CD

F

E

55°

100°

12 cm

6 cm

14 cm

6 cm

3 cm

The two figures are similar figures. Find m K∠ . Find HI .

G

H

I

J

L

K

40°

18 in

16 in

12 in

20 in

7/7/2008 6th 8-8 89

The two figures are similar figures. Find m F∠ . Find CD .

12 yd

10 yd 8 yd

B

H

G

C

A

D

E

110°

F

8 yd 6 yd

The two figures are similar figures. Find m Y∠ . Find ST .

T

U

Z

W

Y

X

65° 4 dm

V

S 12.5 dm

14 dm

11.25 dm5 dm

7/7/2008 6th 8-8 90

7/7/2008 6th 7-1 (Part 2) 91


Chapter 7 Measurement and Geometry

Standard AF 2.1: Convert one unit of measure to another (e.g. from feet to miles,…).

Lesson 7.1: Customary Units of Measurement Concepts: The customary system of measurement is most commonly used in the United States. Customary units of measure developed out of practical consideration, over long periods of time. Thus,

the relationships among the units are arbitrary. ie. 12 inches = 1 foot; 3 feet = 1 yard Customary measures were often related to body parts and were not standardized but were conveniently

accessible. ie. foot: originally equal to the length of a foot from heel to the tip of the toes.

cubit: originally equal to the length of the forearm from the tip of the middle finger to the elbow. span: originally equal to the length of the fully extended and from the tip of the thumb to the tip of the little finger. digit: The unit of length of the width or breadth of a finger. palm: The unit of length of the hand from the wrist to the base of the fingers.

length – the distance from one point to another point. Common customary units of length are measured in feet, inches, yards, and miles.

Customary Units of Length 1 foot (ft) = 12 inches (in.) 1 yard (yd) = 3 feet (ft)

1 mile (mi) = 5,280 feet (ft) 1 mile (mi) = 1.760 yards (yd)

weight – the heaviness of an object or the pull of gravity on an object. Common customary units of weight are ounces, pounds, and tons and are associated with “dry” measurement.

Customary Units of Weight 1 pound (lb) = 16 ounces (oz) 1 ton (T) = 2,000 pounds (lbs) capacity – the amount that can be contained or held. Common customary units of capacity are fluid ounces, cups, pints, quarts, and gallons, and are associated with “wet” measurement.

Customary Units of Capacity 1 cup (c) = 8 fluid ounces (fl oz) 1 pint (pt) = 2 cups (c) 1 quart (qt) = 2 pints (pt) 1 gallon (gal) = 4 quarts (qt) 1 pint (pt) = 16 ounces (fl oz)

7/7/2008 6th 7-1 (Part 2) 92

DRAFTAll measurements are approximates. Using smaller units to measure an object results in a more precise measurement. Customary measures may be converted from one measure to another. e.g. 1 foot is the same as 12 inches Conversion means to change from larger units to smaller units or smaller units to larger units. When you convert from larger units to smaller units you multiply. When you convert from smaller units to larger units you divide. You can use proportion to change one unit of measure to another. The parts of the proportion must

correspond.

12 inches 24 inches1 foot 2 feet

=

Using proportion to solve conversions is easy to use because the method is the same if you convert from a smaller or larger unit.

Essential Question(s): How do you use proportions to convert from one customary unit of measurement to another customary unit of measurement?

7/7/2008 6th 7-1 (Part 2) 93


Chapter 7 Measurement and Geometry Standard AF 2.1: Convert one unit of measure to another (e.g. from feet to miles,…). Lesson 7.1: Customary Units of Measurement Match each customary unit with its abbreviation.

quart

mi c

pound

yd

foot

oz

gallon

qt

ft

fluid ounce

gal

ton

lb

inch

m

pint

p

ounce

in.

T

cup

ou

mile

i

fl oz

7/7/2008 6th 7-1 (Part 2) 94

DRAFTHow do you convert from one customary measurement to another customary unit of measurement

using proportions? Customary Units of Length Conversion Table 1 foot (ft) = 12 inches (in.) 1 yard (yd) = 3 feet (ft)

1 mile (mi) = 5,280 feet (ft) 1 mile (mi) = 1.760 yards (yd)

Use the conversion table above to find the related customary unit of length and fill in the table

Problem Related Customary Unit of Length

3 feet = ___ inches

1 foot = 12 inches

12 feet = ___ yards

3 feet = 1 yard

or 1 yard = 3 feet

2 miles = ___ feet

1 mile = 5,280 feet

or 5,280 feet = 1 mile

18 yards = ___ feet

48 inches = ____ feet

10,560 feet = ___ miles

72 inches = ___ yards

36 inches = 1 yard

or 1 yard = 36 inches

12 yards = ___ inches

12 feet = ___ inches

5,280 yards = ___ miles

7/7/2008 6th 7-1 (Part 2) 95

DRAFT

Problem Related Customary Unit Set up as a Proportion

36 ft = ____yd

3 ft = 1 yd or 3 ft1 yd

3 ft 36 ft1 yd ydx

= or 3 36 1 x

=

24 in. = ___ ft

12 in. = 1 ft or 12 in.1 ft

15,840 ft = ___ mi

5,280 ft = 1 mi. or 5, 280 ft1 mi

6 yd = ____ ft

1 yd = 3 ft or 1 yd3 ft

9 ft = ____ in.

1 ft = 12 in. or 1 ft12 in.

4 mi = ______ ft

1 mi = 5,280 ft or 1 mi5,280 ft

60 in. = ___ ft

12 yd = ____ ft

144 ft = ____yd

9 mi = ______ ft

2,520 yd = ____ in.

7/7/2008 6th 7-1 (Part 2) 96

DRAFT


Solution

24 ft = ____yd

3 ft = 1 yd or 3 ft1 yd

3 ft 24 ft1 yd ydx

=

3 24 1 x=

3x = 24

3 243 3x=

x = 8

x = 8 yd

36 in. = ___ ft

12 in. = 1 ft or 12 in.1 ft

4 mi = ____yd 1 mi = 1,760 yd or 1 mi1,760 yd

84 in. = ___ ft 12 in. = 1 ft or 12 i n.1 ft

31,680 ft = ___ mi 1 ft = 12 in or 1 ft12 in.

7 mi = ___ yd 1 mi = 1,760 yd or 1 mi1,760 yd

7/7/2008 6th 7-1 (Part 2) 97

DRAFT

Problem Related Customary Unit Set up as a Proportion Solution

54 ft = ____yd

96 in. = ___ ft

78 ft = ____yd

17 ft = ___ in.

216 in. = ___ ft

9 mi = ___ ft

7/7/2008 6th 7-1 (Part 2) 98

DRAFT


18 ft = ____yd

72 in. = ___ ft

45 ft = ____yd

96 in. = ___ ft

14 ft = ___ in.

12 mi = ___ ft

7/7/2008 6th 7-1 (Part 2) 99

DRAFT

Customary Units of Weight Conversion Table 1 pound (lb) = 16 ounces (oz) 1 ton (T) = 2,000 pounds (lbs)

Use the conversion table above to find the related customary unit of weight and fill in the table

Problem Related Customary Unit of Weight

48 oz = ____ lb

16 oz = 1 lb

8,000 lb = ___T

2,000 lb = 1 T

32 oz = ____ lb

12,000 lb = ____ T

3 T = ____ lb

10 lb = ____ oz

6 T = _____ lb

9 lb = _____ oz

18 T = ____ lb

28,000 lb = ____ T

7/7/2008 6th 7-1 (Part 2) 100

DRAFTProblem Related Customary Unit Set up as a Proportion

5 lb = ____ oz

1 lb = 16 oz or 1 lb16 oz

1 lb 5 lb

16 oz oz=

x or 1 5

16

x=

8,000 lb = ___ T

2,000 lb = 1 T or 2,000 lb1 T

17 T = ___ lb

1 T = 2,000 lb or 1 T2,000 lb

128 oz = ____ lb

16 oz = 1 lb or 16 oz1 lb

16,000 lb = ____ T

2,000 lb = 1 T or 2,000 lb1 T

29 lb = ______ oz

1 lb = 16 oz or 1 lb16 oz

6 T = ___ lb

64 oz = ____ lb

21 lb = ____oz

9 T = ______ lb

12,000 lb = ____ T

7/7/2008 6th 7-1 (Part 2) 101

DRAFTProblem Related Customary Unit Set up as a

Proportion Solution

48 oz = ____lb


16 oz 48 oz1 lb lb

=x

16 48 1 x

=

16x = 48

16 4816 16

=x

x = 3

x = 3 lb

14,000 lb = ___ T 2,000 lb = 1 T or 2,000 lb1 T

2,000 lb 14,000 lb

1 T T=

x

7 lb = ____oz

17 T = ___ lb

224 oz = ___ lb

9 lb = ___ oz

7/7/2008 6th 7-1 (Part 2) 102


Proportion Solution

80 oz = ____lb


28,000 lb = ___ T 2,000 lb = 1 T or 2,000 lb1 T

9 lb = ____oz

21 T = ___ lb

112 oz = ___ lb

7 lb = ___ oz

7/7/2008 6th 7-1 (Part 2) 103

DRAFT

Customary Units of Capacity Conversion Table 1 cup (c) = 8 fluid ounces (fl oz) 1 pint (pt) = 2 cups (c) 1 quart (qt) = 2 pints (pt) 1 gallon (gal) = 4 quarts (qt) 1 pint (pt) = 16 ounces (fl oz)

Use the conversion table above to find the related customary unit of capacity and fill in the table

Problem Related Customary Unit of Capacity

72 fl oz = ____ c

8 fl oz = 1 c

14 c = ___ pt 2 c = 1 pt

6 pt = ___ qt

16 qt = ___ gal

8 c = ___ fl oz

8 pt = ___ c

18 qt = ___ pt

7 gal = ___ qt

6 c = ___ fl oz

22 pt = ___ qt

____ fl oz = 3 c

___ gal = 16 qt

7/7/2008 6th 7-1 (Part 2) 104

DRAFT


8 c = ____pt

2 c = 1 pt or 2 c1 pt

2 c 8 c1 pt pt

=x

or 2 8 1 x

=

24 qt = ___ gal

4 qt = 1 gal or 4 qt

1 gal

18 pt = ___ qt

2 pt = 1 qt or 2 pt1 qt

24 fl oz = ____ c

8 fl oz = 1 c or 8 fl oz1 c

6 gal = ____ qt

4 c = ______ fl oz

15 pt = ___ c

96 qt = ____ gal

19 qt = ____pt

56 fl oz = ____ c

14 gal = ____ qt

7/7/2008 6th 7-1 (Part 2) 105

DRAFT


32 fl oz = ___ c

8 321 x=

8x = 32 by using cross

products

8 38 8x=

or 8 fl oz = 1 c

8 fl oz1 c

8 fl oz 32 fl oz1 c c

=x 2

x = 4

12 c = ___ pt

2 c1 pt

or 2 c = 1 pt

7 gal = ___ qt

34 pt = ___ qt

52 qt = ___ gal

14 qt = ___ pt

7/7/2008 6th 7-1 (Part 2) 106

DRAFT


72 fl oz = ___ c

8 721=

x

8x = 72 by using cross

products

8 78 8=

x 2

x = 9

28 c = ___ pt

11 gal = ___ qt

18 pt = ___ qt

64 qt = ___ gal

19 qt = ___ pt

7/7/2008 6th 7-2 (Part 2) 107

DRAFTInvestigating Student Learning: 6th Grade Chapter 7: Measurement and Geometry

Standard AF 2.1: Convert one unit of measure to another (e.g. from feet to miles,…). Lesson 7.2: Metric Units of Measurement Concepts: The metric system is the most commonly used system of measurement in the world. Because of this, the

metric system is officially called the International System of Units. The metric system uses the meter as the basic unit of length.

Unit Relation to Basic Unit Kilometer (km) 1,000 meters

Hectometer (hm) 100 meters Dekameter (dam) [variant Decameter (dkm)] 10 meters

Meter (m) Basic Unit Decimeter (dm) 0.1 meter Centimeter (cm) 0.01 meter Millimeter (mm) 0.001 meter

The metric system uses the gram as the basic unit of mass.

Unit Relation to Basic Unit Kilogram (kg) 1,000 grams

Hectogram (hg) 100 grams Dekagram (dag) variant Decagram (dkg) 10 grams

Gram (g) Basic Unit Decigram (dg) 0.1 grams Centigram (cg) 0.01 grams Milligram (mg) 0.001 grams

The metric system uses the liter as the basic unit of capacity.

Unit Relation to Basic Unit Kiloliter (kL) 1,000 liters

Hectoliter (hL) 100 liters Dekaliter (daL) variant Decaliter (dkL) 10 liters

Liter (L) Basic Unit Deciliter (dL) 0.1 liter Centiliter (cL) 0.01 liter Milliliter (mL) 0.001 liter

tric system of measuring, the units are related to each other in terms of powers of 10. In the me

ie. 1 m = 10 dm = 100 cm = 1,000 mm or 1 km = 10 hm = 100 dam = 1,000 m

7/7/2008 6th 7-2 (Part 2) 108

DRAFT

Roots in the metric system have consistent meanings:

Roots Meaning Example milli- one thousandth 1 millimeter is 0.001 meter centi- one hundredth 1 centimeter is 0.01 meter deci- one tenth 1 decimeter is 0.1 meter deka- one ten 1 dekameter is 10 meters hecto- one hundred 1 hectometer is 100 meters kilo- one thousand 1 kilometer is 1,000 meters

All measurements are approximates. Using smaller units to measure an object results in a more precise measurement. Conversion means to change from larger units to smaller units or smaller units to larger units. Converting within the metric system is simple because it is easy to multiply and divide by powers of ten.

kilo 0.001

milli

centi

deci

meter gram liter

deka

hecto

Base 1

1,000

100

10

0.1

0.01

x

Larger

Smaller

÷

When you convert from larger units to smaller units you multiply.

When you convert from smaller units to larger units you divide. You can use proportions to change one unit of measure to another. The parts of the proportion must

correspond. 10 mm 20 mm1 cm 2 cm

=


Essential Question(s): How do you convert from one metric unit of measure to another metric unit of measure using proportions?

7/7/2008 6th 7-2 (Part 2) 109

DRAFT

ISL Item Bank: 6th Grade Chapter 7: Measurement and Geometry

Standard AF 2.1: Convert one unit of measure to another (e.g. from feet to miles,…). Lesson 7.2: Metric Units of Measurement Match each customary unit with its abbreviation.

kilometer

mL

dec

hectogram

dm

millimeter

mm

deciliter

mg

hg

gram

kL

liter

m

cL

milligram

km

centiliter

gr

decimeter

L g

kiloliter

dL

meter

dm

7/7/2008 6th 7-2 (Part 2) 110

DRAFTMetric units of length

1 kilometer = 10 hectometer 1 hectometer = 10 dekameter 1 dekameter = 10 meters 1 meter = 10 decimeters 1 decimeter = 10 centimeters

1 meter = 1,000 millimeters

1 meter = 100 centimeters

1 kilometer = 1,000 meters

1 centimeter = 10 millimeters

Use the table to find the related metric unit of length and fill in the table

Problem Related Metric Unit of Length

8 kilometers = _____ meters

1 kilometer = 1,000 meters

600 centimeters = _____meters

100 centimeters = 1 meter

5,000 millimeters = _____meters

8 meter = _____ centimeters

35 centimeters = _______ millimeters

125 kilometers = ____ meters

3 meters = ____ millimeters

49 dekameters = ____ meters

50 centimeters = _____ decimeters

14,000 meters = _____ kilometers

4 kilometers = ________ meters

12 meters = __________ millimeters

7/7/2008 6th 7-2 (Part 2) 111

DRAFT


Set up as a Proportion

400 cm = ____m

100 cm = 1 m or 100 cm1 m

100 cm 400 cm

1 m m=

x or 100 400

1 x=

5,000 m = ___ km

1,000 m = 1 km or

1,000 m1 km

12 m = ___ cm

1 m = 100 cm. or 1 m100 cm

14 km = ____ m

1 km = 1,000 m or

1 km1,000 m

9 m = ____ mm

1 m = 1,000 mm or

1 m1,000 mm

450 mm = ______ cm

10 mm = 1 cm or 10 mm1 cm

60 cm = ___ mm

40 dam = ____ m

1,200 cm = ____ m

50 dm = ______ m

7/7/2008 6th 7-2 (Part 2) 112


Proportion Solution

500 cm = ____ m

100 cm = 1 m or 100 cm1 m

100 cm 500 cm1 m m

=x

100 500

1 x=

100x = 500

100 500100 100

=x

x = 5

x = 5 yd

4,000 m = ___ km

1,000 m = 1 km or 1,000 m1 km

1,000 m 4,000 m

1 km km=

x

72 dam = ____m 1 dam = 10 m or 1 dam10 m

1 dam 72 dam10 m m

=x

54 m = ___ cm 1 m = 100 cm or 1 m100 cm

1 m 54 m100 cm cm

=x

9 m = ___ mm 1

7 km = ___ m

7/7/2008 6th 7-2 (Part 2) 113

DRAFTProblem Related Customary Unit Set up as a Proportion Solution

18 km = ____m

300 cm = ___ m

45 cm = ____mm

6 m = ___ mm

14,000 m = ___ km

12 m = ___ cm

7/7/2008 6th 7-2 (Part 2) 114

DRAFTMetric units of weight

1 kilogram = 1,000 grams

1 gram = 100 centigrams

1 gram = 1,000 milligrams

1 centigram = 10 milligrams

1 metric ton (t) = 1,000 kilograms Use the table to find the related metric unit of weight and fill in the table.


8 kilograms = _____ grams

1 kilogram = 1,000 grams

600 centigrams = _____ grams

100 centigrams = 1 gram

5,000 milligrams = _____ grams

8 grams= _____ centigrams

16,000 kilograms = _____ metric tons

700 centigrams = _____ grams

12,000 milligrams = _____ grams

670 centigrams = _____ milligrams

19 metric tons = ______ kilograms

13,000 grams = ______ kilograms

57 grams = _____ milligrams

45 grams = _____ centigrams

7/7/2008 6th 7-2 (Part 2) 115

DRAFT

Problem Related Metric Unit of Weight Set up as a Proportion

5 kg = ____ g

1 kg = 1,000 g or 1 kg1,000 g

1 kg 5 kg

1,000 g g=

x or 1 5

1,000

x=

8,000 g = ___ kg

1,000 g = 1 kg or 1,000 g1 kg

17 g = ___ mg

1 g = 1,000 mg or 1 g1,000 kg

128,000 mg = ____ g

1,000 mg = 1 g or 1,000 mg1 g

16,000 kg = ____ t

290 g = ______ cg

6 t = ___ kg

73 g = ____ cg

32 cg = ____ mg

9 g = ______ mg

12,000 mg = ____ g

7/7/2008 6th 7-2 (Part 2) 116


Proportion Solution

35 kg = ____ g

1 kg = 1,000 g or 1 kg1,000 g

1 kg 35 kg1,000 g g

=x

1 35

1,000 x=

x = 35,000

x = 35,000 g

17,000 g = ___ kg 1,000 g = 1 kg or 1,000 g 1,000 g 17,000 g

1 kg kg=

x1 kg

1,000 17,000

1 x=

1,000 x = 17,000

1,000 17,0001,000 1,000

=x

x = 17

x = 17 kg

9 cg = ____ mg 1 cg = 10 mg or 1 cg10 mg

23 T = ___ kg

7,000 mg = ___ g

13 g = ___ mg

7/7/2008 6th 7-2 (Part 2) 117

DRAFT


Solution

8,000 g = ____kg

1,000 g = 1 kg or 1,000 g1 kg

25 g = ___ mg 1 g = 1,000 mg or 1 g1,000 mg

1 g 25 g1,000 mg mg

=x

400 cg = ____ g

21 t = ___ kg

14,000 mg = ___ g

47 g = ___ mg

7/7/2008 6th 7-2 (Part 2) 118

DRAFT Metric units of capacity

1 kiloliters = 1,000 liters

1 liter = 100 centiliters

1 liter = 1,000 milliliters

1 centiliter = 10 milliliter

Use the table to find the related metric unit of length and fill in the table


8 kiloliter = _____ liters

1 kiloliter = 1,000 liters

600 centiliter = _____liters

100 centiliter = 1 liter

5,000 milliliters = _____ liters

8 liters= _____ centiliters

52,000 liters = _____ kiloliters

400 milliliters = _____ centiliters

16 liters = _____ centiliters

27 liters = _____ milliliters

34 centiliters = _____ milliliters

200 centiliters = _____ liters

25 kiloliters = _____ liters

12,000 milliliters = _____ liters

7/7/2008 6th 7-2 (Part 2) 119

DRAFT


36 kL = ____ L

1 kL = 1,000 kL or 1 kL1,000 L

1 kL 36 kL

1,000 L L=

x or 1 36

1,000

x=

24 L = ___ mL

1 L = 1,000 mL or 1 L1,000 mL

500 cL = ___ L

100 cL = 1 L or 100 cL1 L

6 L = ____ mL

1 L = 1,000 mL or 1 L1,000 mL

9,000 L = ____ kL

1,000 L = 1 kL or 1,000 mg1 L

12 L = ______ cL

1 L = 100 cL or 1 L100 cL

6,000 mL = ___ L

12,000 L = ____ kL

50 L = ____ cL

9,000 mL = ______ L

5 L = ____ mL

7/7/2008 6th 7-2 (Part 2) 120


24 kL = ____ L

1 kL = 1,000 kL or 1 kL1,000 L

1 kL 24 kL1,000 L L

=x

1 24

1,000 x=

x = 24,000

x = 24,000 L

3,600 cL = ___ L

100 cL = 1 L or 100 cL1 L

50 mL = ____ cL 10 mL = 1 cL or 10 mL1 cL

23,000 mL = ___ L 1,000 mL = 1 L or 1,000 mL1 L

9 L = ___ mL 1 L = 1,000 mL or 1 L1,000 mL

7 kL = ___ L 1 kL = 1,000 L or 1 kL1,000 L

7/7/2008 6th 7-2 (Part 2) 121


54 kL = ____ L

1 kL = 1,000 L or 1 kL1,000 L

1 kL 54 kL1,000 L L

=x

1 kL 54 kL

1,000 L L=

x

x = 54,000

x = 54,000 L

24,000 mL = ___ L

32 L = ____ mL

98 L = ___ cL

210 cL = ___ mL

9,000 mL = ___ L

7/7/2008 6th 7-3 (Part 2) 122

DRAFTInvestigating Student Learning: 6th Grade Chapter 7: Measurement and Geometry

Standard AF 2.1: Convert one unit of measure to another (e.g. from feet to miles,…). Lesson 7.3: Converting Between Measurement Systems Concepts: You can convert between measurement systems using approximations. ≈ means approximately. Customary and Metric Unit Equivalents

Length 1 in. ≈ 2.5 cm 1 m ≈ 40 in. 1 m ≈ 3.3 ft 1 m ≈ 1.1 yd 1 mi ≈ 1.6 km 1 km ≈ 0.6 mi

Weight and Mass 1 oz ≈ 30 g 1 kg ≈ 2.2 lb 1 metric ton (t) ≈1.1 tons (T)

Capacity 1 L ≈ 1.1 qt 1 gal ≈ 4 L

From the table above you can see the following relationships An inch is about 2.5 as long as a centimeter A meter is a little longer than a yard A kilogram is little more that twice as heavy as a pound A liter is a little more than a quart

A mile is about 112

times as long as a kilometer.

To change from a larger unit to a smaller unit, multiply. To change from a smaller unit to a larger unit, divide.

You can use proportions to change one unit of measure to another. The parts of the proportion must correspond.

1 in. 3 in.2.5 cm 7.5 cm

=


Essential Question(s): How do you convert between measurement systems using proportions?

7/7/2008 6th 7-3 (Part 2) 123

DRAFT

ISL Item Bank: 6th Grade Chapter 7: Measurement and Geometry

Standard AF 2.1: Convert one unit of measure to another (e.g. from feet to miles,…). Lesson 7.3: Converting Between Measurement Systems Find the measurements that are close to each other. 1 cm 1 inch 2.5 cm 5 cm

5 cm 1 foot 10 cm 30 cm

1.5 km 1 mile 2.5 km 3.5 km

1 L 1 qt 100 mL 2.5 L

1 lb 2 kg 4 lb 6 lb

11 in. 1 m 3 ft 3 yd

4 L 1 gal 6 L 8 L

50 g 2 oz 60 g 70 g

3 qt 3 L 5 qt 7 qt

Which is more?

1 cm or 1 in.

1 L or 1 qt

1 kg or 1 lb

1 g or 1 oz

4 in or 20 cm

5 kg or 20 lb

3 gal or 2 L

7 m or 4 ft

2 m or 2 yd

7/7/2008 6th 7-3 (Part 2) 124

DRAFT Customary and Metric Unit Equivalents

Length

1 in. ≈ 2.5 cm 1 m ≈ 40 in. 1 m ≈ 3.3 ft 1 m ≈ 1.1 yd

1 mi ≈ 1.6 km 1 km ≈ 0.6 mi

Find the related customary unit of length and fill in the table

Problem Related Customary Unit of Length

15 inches ≈ ___ centimeters

1 in. ≈ 2.5 cm

25 feet ≈ ___ meters 1 m ≈ 3.3 ft

1.8 miles ≈ ___ kilometers

23 meters ≈ ___ feet

5 cm ≈ ___ in.

80 inches ≈ ___ meters

4 m ≈ ___ yd

3.2 kilometers ≈ ___ miles

3 meters ≈ ___ inches

5 mi ≈ ___ km

250 cm ≈ ___ in.

7/7/2008 6th 7-3 (Part 2) 125



Complete the table.


36 ft ≈ ____ m

3.3 ft ≈ 1 m or 3.3 ft1 m

3.3 ft 36 ft1 m m

≈x

or 3.3 36 1 x

≈

11 m ≈ ____ yd

1 m ≈ 1.1 yd or 1 m1.1 yd

36 km ≈ ____ mi

1 km ≈ 0.6 mi or 1 km0.6 mi

5 m ≈ ____ in. 1 m ≈ 40 in. or 1 m40 in.

15 in. ≈ ____ cm

9.9 ft ≈ ____ m

50 cm ≈ ____ in.

7.7 yd ≈ ____ m

18 m = ≈ ft

7/7/2008 6th 7-3 (Part 2) 126



Complete the table.


80 in. ≈ ____m

40 in. ≈ 1 m or 40 in.1 m

40 in. 80 in.1 m m

≈x

40 80 1 x

≈

40x ≈80

40 8040 40

≈x

x ≈ 2

x ≈ 2 m

32 mi. ≈ ____ km

1 mi ≈ 1.6 km or 1 mi1.6 km

1 mi 32 mi

1.6 km km≈

x

1 32

1.6 x≈

x ≈51.2

x ≈ 51.2 m

50 cm ≈ ____ in.

2.5 cm ≈ 1 in. or 2.5 cm1 in.

2.5 cm 50 cm1 in. in.

≈x

20 m ≈ ____ ft

1 m ≈ 3.3 ft or 1 m 3.3 ft

1 m 20 m3.3 ft in.

≈x

14 m ≈ ____ yd

1 m ≈ 1.1 yd or 1 m 1.1 yd

1 m 14 m1.1 yd yd

≈x

7/7/2008 6th 7-3 (Part 2) 127



Complete the table.


10 km ≈ ____mi

1 km ≈ 0.6 mi or 1 km0.6 mi

1 km 10 km0.6 mi mi

≈x

1 10

0.6 x≈

x ≈6

x ≈ 6 mi

35 m ≈ ____ in.

1 km ≈ 40 in. or 1 m40 in.

1 m 35 m40 in. in.

≈x

5 in. ≈ ____ cm

1 in. ≈ 2.5 cm or 1 in.2.5 cm

1 in. 5 in.2.5 cm cm

≈x

240 in. ≈ ____ m

40 in. ≈ 1 m or 40 in. 1 m

40 in. 240 in.1 m m

≈x

99 ft ≈ ____ m

3.3 ft ≈ 1 m or 3.3 ft 1 m

55 yd ≈ ____ m

1.1 yd ≈ 1 m or 1.1 yd 1 m

7/7/2008 6th 7-3 (Part 2) 128



Complete the table.


25 km ≈ ____mi

1 km ≈ 0.6 mi or 1 km0.6 mi

1 km 25 km0.6 mi mi

≈x

1 25

0.6 x≈

x ≈15

x ≈ 15 mi

121 yd ≈ ____ m

1.1 yd ≈ 1 m or 1.1 yd 1 m

80 km ≈ ____mi

39.6 ft ≈ ____ m

48 mi ≈ ____ km

125 cm ≈ ____ in.

7/7/2008 6th 7-3 (Part 2) 129


Weight and Mass 1 oz ≈ 30 g 1 kg ≈ 2.2 lb 1 metric ton (t) ≈ 1.1 tons (T)

Find the related customary unit of weight and mass and fill in the table.

Problem Related Customary Unit of Weight and Mass

15 oz ≈ ___ g 1 oz ≈ 30 g

36 grams ≈ ___ ounces

7 metric tons ≈ ___ Tons

125 kilograms ≈ ___ ounces

120 g ≈ ___ oz

34 kg ≈ ___ oz

7 Tons ≈ ___ metric tons

4.8 lb ≈ ___ kg

22 oz ≈ ___ g

22 t ≈ ___ T

7/7/2008 6th 7-3 (Part 2) 130



Complete the table.


7 oz ≈ ____ g 1 oz ≈ 30 g or 1 oz30 g

1 oz 7 oz30 g g

=x

or 1 730

x

=

25 kg ≈ ____ lb 1 kg ≈ 2.2 lb or 1 kg2.2 lb

15 t ≈ ____ T 1 t ≈ 1.102 T or 1 t1.1 T

90 g ≈ ____ oz 30 g ≈ 1 oz or 30 g1 oz

44 lb ≈ ____ kg

3 T ≈ ____ t

6.6 kg ≈ ____ lb

83 oz ≈ ____ g

423 T ≈ ____ t

900 g ≈ ____ oz

155 oz ≈ ____ g

7/7/2008 6th 7-3 (Part 2) 131



Complete the table.


Solution

150 g ≈ ____oz

30 g ≈ 1 oz or 30 g1 oz

30 g 150 g1 oz oz

≈x

30 150 1 x

≈

30x = 150

30 15030 30

=x

x = 5

x = 5 oz

15 kg ≈ ____ lb 1 kg ≈ 2.2 lb or 1 kg2.2 lb

23 oz ≈ ____ g

1 oz ≈ 30 g or 1 oz30 g

95 t ≈ ____ T 1 t ≈ 1.102 T or 1 t1.1 T

660 lb ≈ ____ kg

7/7/2008 6th 7-3 (Part 2) 132



Find the related customary unit of capacity and fill in the table.

Problem Related Customary Unit of Capacity

15 L ≈ ___ qt 1 L ≈ 1.1 qt

55 qt ≈ ___ L

15 gal ≈ ___ L

16 L ≈ ___ gal

28 L ≈ ___ qt

44 qt ≈ ___ L

32 gal ≈ ___ L

32 L ≈ ___ gal

7/7/2008 6th 7-3 (Part 2) 133



Complete the table.

Problem Related Metric Unit of Length Set up as a Proportion

50 L ≈ ____ qt

1 L = 1.1 qt or 1 L1.1 qt

1 L 50 L

1.1 qt qt=

x or 1 50

1.1

x=

44 qt ≈ ____ L 1.1 qt = 1 L or 1.1 qt1 L

25 gal ≈ ____ L 1 gal = 4 L or 1 gal4 L

96 L ≈ ____ gal

12 L ≈ ____ qt

121 qt ≈ ____ L

14 L ≈ ____ qt

88 qt ≈ ____ L

9 gal ≈ ____ L

48 L ≈ ____ gal

7/7/2008 6th 7-3 (Part 2) 134



Complete the table.


Solution

33 qt ≈ ____ L

1.1 qt ≈1 L or 1.1 qt1 L

1.1 qt 33 qt1 L L

≈x

1.1 33 1 x

≈

1.1x ≈ 33

1.1 33 1.1 1.1

≈x

x ≈ 30

x ≈ 30 L

15 L ≈ ____ qt

21 gal ≈ ____ L

52 L ≈ ____ gal

7/7/2008 6th 8-9 135


Standard NS 1.3*: Use proportions to solve problems…. Lesson 8.9: Scale Drawings

Concepts:

A scale drawing is a drawing that is the same shape but not the same size as the object it shows. A scale drawing shows the object larger or smaller than its actual size. Scale drawings often show objects that are too large or too small to be shown in their actual sizes. Blueprints and maps are examples of scale drawings. The scale for a scale drawing is a ratio between two sets of measurements—the measure of the drawing and the actual measure. To solve a problem involving a scale drawing, set up a proportion using scale as one of the ratios:

scale drawingscale actual

= drawing length actual length

A scale drawing of a bedroom is shown below. The length of the bedroom on the drawing is 8 centimeters. What is the actual length of the bedroom?

Scale: 1 centimeter = 3 meters

scale (drawing)scale (actual)

1 centimeter3 meters

= 8 centimetersmetersx

drawing length actual length

1 3

= 8 x

1 i x = 3 i 8 x = 24 The actual length of the bedroom is 24 meters.

doorbed

window

dresser

A scale drawing should be similar to the actual figure. Scale drawings are an application of similar figures, with corresponding angles congruent and corresponding sides proportional.

Essential Question(s): What is a scale drawing? How do you use proportion to solve problems involving scale drawings?

7/7/2008 6th 8-9 136


Standard NS 1.3*: Use proportions to solve problems (e.g.,…find the length of a side of a polygon similar to a known polygon). Use cross multiplication as a method for solving such problems….

Lesson 8.9: Scale Drawings What is a scale drawing?

Look at the scale drawing below. Then answer each question.

Office Drawing

2 in.

3 in. Scale: 1 inch = 5 feet

What does the drawing represent? _____________________________________________________

What is the length of the office drawing? _______________________

What is the width of the office drawing? ________________________

What is the scale of the office drawing? _________________________

door window

window desk


Bathroom Drawing

2 cm


What does the drawing represent? ______________________________________________________

What is the length of the bathroom drawing? _______________________

What is the width of the bathroom drawing? ________________________

According to the scale, what does 1 centimeter represent? _________________________________

According to the scale, what would 2 centimeters represent? _________________________________

door

window

bathtub

toilet

4 cm

7/7/2008 6th 8-9 137


Basketball Court Drawing

5 cm 5 5 in.

9.4 in. Scale: 2 inches = 20 feet

Is an actual basketball court larger or smaller than the above drawing? _____________________

What is the length of the basketball court drawing? _______________________

What is the width of the basketball court drawing? ________________________

What is the scale of the basketball court drawing? _________________________

According to the scale, what would 2 inches represent? ______________________________________


Bedroom Drawing

7.5 cm Scale: 1 centimeters = 2.5 meters

What is the length of the bedroom drawing? _______________________

What is the width of the bedroom drawing? ________________________


According to the scale, what would 2 centimeters represent? ________________________________

4.5 cm

door bed

window

dresser

7/7/2008 6th 8-9 138


Soccer Field Drawing

Scale: 1 inch = 25 yards

What is the length of the soccer field drawing? _______________________

What is the width of the soccer field drawing? ________________________

What is the length of half of the soccer filed drawing? _______________________

What is the length of the penalty area drawing? _______________________

What is the width of the penalty area drawing? ________________________

According to the scale, what does 1 inch represent? _________________________________

According to the scale, what would 2 inches represent? _________________________________

Goal Area

1.5 in.

4 in.

Penalty Area

0.75 in.

1 in.


3rd Floor Drawing

Scale: 2 centimeters = 15 meters

What is the length of the 3rd Floor drawing? _______________________

What is the width of the 3rd Floor drawing? ________________________

What is the length of the Office 1 drawing? _______________________

What is the width of the Office 1 drawing? ________________________

What is the length of the Office 3 drawing? _______________________

What is the width of the Office 3 drawing? ________________________

What is the length of the Bathroom drawing? _______________________

What is the width of the Bathroom drawing? ________________________



Elevator 3.5 cm

Bathroom

Office 1 Office 2

Office 3

6 cm 6 cm

2 cm

1 cm

6.5 cm

3 cm

7/7/2008 6th 8-9 139


Apartment Drawing

8 cm Scale: 1 centimeters = 2 meters

What is the length of the apartment drawing? _______________________

What is the width of the apartment drawing? ________________________

What is the length of the kitchen drawing? _______________________

What is the length of the bathroom drawing? _______________________

What is the width of the bathroom drawing? ________________________



door

Bedroom

Bathroom Kitchen

Living Room

2 cm

4 cm

3 cm

2 cm 2 cm

3 cm


House Drawing

Scale: 1 inch = 24 feet

What is the length of the house drawing? _______________________

What is the width of the house drawing? ________________________

What is the width of the Bedroom 1 drawing? _______________________

What is the width of the Bathroom 2 drawing? ________________________

What is the length of the Garage drawing? _______________________

What is the width of the Bedroom 3 drawing? ________________________

According to the scale, what would 2 inches represent? _________________________________

Bedroom 1

1.25 in.

Garage

Bedroom 2

Bathroom 1

Bathroom 2

KitchenDining Room

Great RoomBedroom 3

Bat

hroo

m 3

2.5 in.

1.25 in.

1 in.

1.25 in. 1 in. 2.25 in.

1.5

in.

7/7/2008 6th 8-9 140

How do you use proportion to solve problems involving scale drawings?

A scale drawing of an office is shown below. Use x to write the proportion for each requested measure:

2 in.


Example: What is the actual length of the office?

Scale Length

Proportion

drawing actual

1 inch5 feet

= 3 inchesfeetx

drawing actual

1 5

= 3 x

What is the actual width of the office?

Scale Width

Proportion

drawing actual

1 inch5 feet

= inchesfeetx

drawing actual

1 5

= x

door window

window desk

A scale drawing of a bathroom is shown below. Use x to write the proportion for each requested measure:

2 cm


What is the actual length of the bathroom?

Scale Length

Proportion

drawing actual

centimeter meters

= centimetersmetersx

drawing actual

= x

What is the actual width of the bathroom?

Scale Width

Proportion

drawing actual

centimeter meters

= centimeters meters

drawing actual

=

door

window

bathtub

toilet

4 cm

7/7/2008 6th 8-9 141

A scale drawing of a basketball court is shown below. Use x to write the proportion for each requested measure:

5 cm 5 5 in.


What is the actual length of the basketball court?

Scale Length

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

What is the actual width of the basketball court?

Scale Width

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

A scale drawing of a bedroom is shown below. Use x to write the proportion for each requested measure:


What is the actual length of the bedroom?

Scale Length

Proportion

drawing actual

centimeter meters

= centimetersmeters

drawing actual

=

What is the actual width of the bedroom?

Scale Length

Proportion

drawing actual

centimeter meters


drawing actual

=

bed

4.5 cm

door

dresser

window

7/7/2008 6th 8-9 142

A scale drawing of a soccer field is shown below. Use x to write the proportion for each requested measure:

Scale: 1 inch = 25 yards What is the actual length of the soccer field?

Scale Length

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

What is the actual width of the soccer field?

Scale Width

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

What is the actual length of the penalty area?

Scale Length

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

What is the actual width of the penalty area?

Scale Width

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

Goal Area

1.5 in.

4 in.

Penalty Area

0.75 in.

1 in.

7/7/2008 6th 8-9 143

A scale drawing of the 3rd Floor is shown below. Use x to write the proportion for each requested measure:

Scale: 2 centimeters = 15 meters What is the actual length of the 3rd Floor?

Scale Length

Proportion

drawing actual

centimeter meters

= centimetersmeters

drawing actual

=

What is the actual width of the 3rd Floor?

Scale Length

Proportion

drawing actual

centimeter meters


drawing actual

=

What is the actual length of Office 1?

Scale Length

Proportion

drawing actual

centimeter meters

= centimetersmeters

drawing actual

=

What is the actual width of the Bathroom?

Scale Length

Proportion

drawing actual

centimeter meters


drawing actual

=


Scale Length

Proportion

drawing actual

centimeter meters

= centimetersmeters

drawing actual

=

What is the actual width of Office 3?

Scale Length

Proportion

drawing actual

centimeter meters


drawing actual

=

Elevator 3.5 cm

Bathroom

Office 1 Office 2

Office 3

6 cm 6 cm

2 cm

1 cm

6.5 cm

3 cm

7/7/2008 6th 8-9 144

A scale drawing of an Apartment is shown below. Use x to write the proportion for each requested measure:


What is the actual length of the Apartment?

Scale Length

Proportion

drawing actual

centimeter meters

= centimetersmeters

drawing actual

=

What is the actual width of the Apartment?

Scale Length

Proportion

drawing actual

centimeter meters


drawing actual

=

What is the actual length of the Kitchen?

Scale Length

Proportion

drawing actual

centimeter meters

= centimetersmeters

drawing actual

=

What is the actual width of the Kitchen?

Scale Length

Proportion

drawing actual

centimeter meters


drawing actual

=

What is the actual length of Bathroom?

Scale Length

Proportion

drawing actual

centimeter meters

= centimetersmeters

drawing actual

=

What is the actual width of Bathroom?

Scale Length

Proportion

drawing actual

centimeter meters


drawing actual

=

door

Bedroom

Bathroom Kitchen

Living Room

2 cm

4 cm

3 cm

2 cm 2 cm

3 cm

7/7/2008 6th 8-9 145

A scale drawing of a house is shown below. Use x to write the proportion for each requested measure:


What is the actual length of the house?

Scale Length

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

What is the actual width of the house?

Scale Width

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

What is the actual length of the Garage?

Scale Length

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

What is the actual width of the Garage?

Scale Width

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

What is the actual length of the Bedroom 3?

Scale Length

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

What is the actual width of Bathroom 2?

Scale Width

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

Bedroom 1

1.5 in.

Garage

Bedroom 2

Bathroom 1

Bathroom 2

KitchenDining Room

Great RoomBedroom 3

Bat

hroo

m 3

2.5 in.

1.25 in.

1 in.

1.5 in. 1 in. 2.25 in.

1.5

in.

1.25 in.2.5 in.0.5 in.

7/7/2008 6th 8-9 146

(Making a scale drawing)

Mr. Rose is making a scale drawing of his backyard. He is using the scale 12

inch = 4 feet. The actual

width of his backyard is 42 feet. What width should the backyard be on the scale drawing?

Scale Width

Proportion

drawing actual

0.5 inches 4 feet

= inches42 feet

x drawing actual

=

Maile is making a scale drawing of her art studio. She is using the scale 1 inch = 8 feet. The actual length of her art studio is 32 feet. What length should the art studio be on the scale drawing?

Scale Length

Proportion

drawing actual

inches feet

= inchesfeet

drawing actual

=

An architect is making a scale drawing of his house. He is using the scale 2 centimeter = 14 meters. The actual length of his living room is 35 meters. What length should the living room be on the scale drawing?

Scale Length

Proportion

drawing actual

centimeter meters

= centimetersmeters

drawing actual

=

Coach Moore is making a scale drawing of his team’s junior football field. He is using the scale 12

inches = 4 yards. The actual length of the junior field is 60 yards. What length should the junior

field be on the scale drawing?

Scale Length

Proportion

drawing actual

inches yards

= inchesyards

drawing actual

=

Mrs. Florence is making a scale drawing of a playground. She is using the scale 1 centimeter = 22 meters. The actual width of the playground is 110 meters. What width should the playground be on the scale drawing?

Scale Width

Proportion

drawing actual

centimeter meters

= centimetersmeters

drawing actual

=

7/7/2008 6th 8-9 147

(Finding actual length) A drawing of a bedroom uses a scale of 1 inch equals 5 feet. The length of the bedroom on the drawing is 3 inches. What is the actual length of the bedroom?

Scale Length

Proportion

drawing actual

1 inches 5 feet

= 3 inches feetx

drawing actual

=

Juanita has a blueprint drawing of her property. The drawing uses a scale of 12

inch equals 3 yards.

The length of Juanita’s property on the drawing is 4 inches. What is the actual length of the property?

Scale Length

Proportion

drawing actual

inches yards

= inchesyards

drawing actual

=

A drawing of a map uses a scale of 2 centimeters equals 11 kilometers. How many kilometers are represented by 8 centimeters on the map?

Scale Length

Proportion

drawing actual

centimeter kilometers

= centimeterskilometers

drawing actual

=

The Park Ranger has a scale drawing of Hyde Park. The drawing uses a scale of 1 centimeters equals 25 meters. The width of Hyde Park on the drawing is 13 centimeters. What is the actual width of Hyde Park?

Scale Width

Proportion

drawing actual

centimeter meters

= centimetersmeters

drawing actual

=

A drawing of a map uses a scale of 12

inch equals 5 miles. How many miles are represented by

7 inches on the map?

Scale Length

Proportion

drawing actual

inches miles

= inchesmiles

drawing actual

=

7/7/2008 6th 8-9 148

A scale drawing of an office is shown below. What is the actual length of the office?

2 in.


Example:

Scale Length

Proportion

Solution

1 inch5 feet

= 3 inches feetx

1 5

= 3 x

1 i x = 5 i 3 x = 15 The actual length of the bedroom is 15 feet.

What is the actual width of the office?

Scale Width

Proportion

Solution

inch feet

= inches feetx

= x

1 i x = x = The actual width of the bedroom is _______.

door window

window desk

A scale drawing of a bathroom is shown below. What is the actual length of the bathroom?

2 cm


Scale Length

Proportion

Solution

cm m

= cm mx

= x

What is the actual width of the bathroom?

Scale Width

Proportion

Solution

=

=

door

window

bathtub

toilet

4 cm

7/7/2008 6th 8-9 149

A scale drawing of a basketball court is shown below. What is the actual length of the basketball court?

5 cm 5 5 in.


Scale Length

Proportion

Solution

in. ft

= in. ft

=

What is the actual width of the basketball court?

Scale Width

Proportion

Solution

=

=

A scale drawing of a bedroom is shown below. What is the actual length of the bedroom?


Scale Length

Proportion

Solution

=

=

What is the actual width of the bedroom?

Scale Width

Proportion

Solution

=

=

bed

4.5 cm

door

dresser

window

7/7/2008 6th 8-9 150

A scale drawing of a soccer field is shown below. What is the actual length of the soccer field?

Scale: 1 inch = 25 yards

Scale Length

Proportion

Solution

=

=

What is the actual width of the soccer field?

Scale Width

Proportion

Solution

=

=

What is the actual length of the penalty area?

Scale Length

Proportion

Solution

=

=

What is the actual width of the penalty area?

Scale Width

Proportion

Solution

=

=

Goal Area

1.5 in.

4 in.

Penalty Area

0.75 in.

1 in.

7/7/2008 6th 8-9 151

A scale drawing of the 3rd Floor is shown below. What is the actual length of the 3rd Floor?

Scale: 2 centimeters = 15 meters

Scale Length

Proportion

Solution

=

=

What is the actual width of the 3rd Floor?

Scale Width

Proportion

Solution

=

=


Scale Length

Proportion

Solution

=

=

What is the actual width of the Bathroom?

Scale Width

Proportion

Solution

=

=


Scale Length

Proportion

Solution

=

=

What is the actual width of Office 3?

Scale Width

Proportion

Solution

=

=

Elevator 3.5 cm

Bathroom

Office 1 Office 2

Office 3

6 cm 6 cm

2 cm

1 cm

6.5 cm

3 cm

7/7/2008 6th 8-9 152

A scale drawing of an Apartment is shown below. What is the actual length of the Apartment?


Scale Length

Proportion

Solution

=

=

What is the actual width of the Apartment?

Scale Width

Proportion

Solution

=

=

What is the actual length of the Kitchen?

Scale Length

Proportion

Solution

=

=

What is the actual width of the Kitchen?

Scale Width

Proportion

Solution

=

=

What is the actual length of Bathroom?

Scale Length

Proportion

Solution

=

=

What is the actual width of Bathroom?

Scale Width

Proportion

Solution

=

=

door

Bedroom

Bathroom Kitchen

Living Room

2 cm

4 cm

3 cm

2 cm 2 cm

3 cm

7/7/2008 6th 8-9 153

A scale drawing of a house is shown below. What is the actual length of the house?


Scale Length

Proportion

Solution

=

=

What is the actual width of the house?

Scale Width

Proportion

Solution

=

=

What is the actual length of the Garage?

Scale Length

Proportion

Solution

=

=

What is the actual width of the Garage?

Scale Width

Proportion

Solution

=

=

What is the actual length of the Bedroom 3?

Scale Length

Proportion

Solution

=

=

What is the actual width of Bathroom 2?

Scale Width

Proportion

Solution

=

=

Bedroom 1

1.5 in.

Garage

Bedroom 2

Bathroom 1

Bathroom 2

KitchenDining Room

Great RoomBedroom 3

Bat

hroo

m 3

2.5 in.

1.25 in.

1 in.

1.5 in. 1 in. 2.25 in.

1.5

in.

1.25 in.2.5 in.0.5 in.

7/7/2008 6th 8-9 154

(Making a scale drawing)

Mr. Rose is making a scale drawing of his backyard. He is using the scale 12

inch = 4 feet. The actual

width of his backyard is 42 feet. What width should the backyard be on the scale drawing?

(Scale) (Length) (Proportion) (Solution)

0.5 inch4 feet

= inches42 feetx =

Maile is making a scale drawing of her art studio. She is using the scale 1 inch = 8 feet. The actual length of her art studio is 32 feet. What length should the art studio be on the scale drawing?

An architect is making a scale drawing of his house. He is using the scale 2 centimeter = 7 meters. The actual length of his living room is 14 meters. What length should the living room be on the scale drawing?

Coach Moore is making a scale drawing of his team’s junior football field. He is using the scale 12

inches = 4 yards. The actual length of the junior field is 60 yards. What length should the junior

field be on the scale drawing?

Mrs. Florence is making a scale drawing of a playground. She is using the scale 1 centimeter = 22 meters. The actual width of the playground is 110 meters. What width should the playground be on the scale drawing?

7/7/2008 6th 8-9 155

(Finding actual length) A drawing of a bedroom uses a scale of 1 inch equals 5 feet. The length of the bedroom on the drawing is 3 inches. What is the actual length of the bedroom?

Juanita has a blueprint drawing of her property. The drawing uses a scale of 12

inch equals 3 yards.

The length of Juanita’s property on the drawing is 4 inches. What is the actual length of the property?

A drawing of a map uses a scale of 2 centimeters equals 11 kilometers. How many kilometers are represented by 8 centimeters on the map?

The Park Ranger has a scale drawing of Hyde Park. The drawing uses a scale of 1 centimeters equals 25 meters. The width of Hyde Park on the drawing is 13 centimeters. What is the actual width of Hyde Park?

A drawing of a map uses a scale of 12

inch equals 5 miles. How many miles are represented by

7 inches on the map?

7/7/2008 6th 8-9 156

Use the paper ruler to measure each drawing. Then use each scale to find the actual measure.

1 2 3 4 5 60 in.

1 inch = 1 foot

14

inch = 10 feet

12

inch = 4 feet

14

inch = 2 foot

12

inch = 14

foot

7/7/2008 6th 8-9 157

Use the paper ruler to measure each drawing to the nearest centimeter. Then use each scale to find the actual measure.

1 centimeter = 1 meter

3 centimeters = 1

2 meter

1 centimeter = 12 meters

1 centimeter = 0.25 meters

7/7/2008 6th 8-10 158


Standard NS 1.3*: Use proportions to solve problems…. Lesson 8.10: Problem-Solving Application: Using Maps

Concepts:

A map is a special type of scale drawing. A map usually represents a region of the earth’s surface. A map shows a flat, abstract view of a place as if looking down from an airplane. Components of a map include a compass rose, map key, and distance scale. The map scale shows the ratio of the map distance to the actual distance. To solve a problem involving a map scale, set up a proportion using the map scale as one of the ratios:

map scale actual scale

= map length actual length

Essential Question(s): How do you use proportion to solve problems involving map scale?

7/7/2008 6th 8-10 159


Standard NS 1.3*: Use proportions to solve problems…. Lesson 8.10: Problem-Solving Application: Using Maps How do you use proportion to solve problems involving map scale?

Henderson County

What does the map represent? _____________________________________________________

What is the map distance between Jackson to Newton? _______________________

What is the map scale? _________________________

Lakeport

Turner

Needles

Jackson

Carrie

Key

City

Road

Capital

Bridge

Tunnel N

S

E W Newton

Scale 0 1 2 3 4 5 miles

1 inch = 5 miles

Using scale as one of the ratios, write a proportion for the actual distance (d) between Jackson to Newton, and then find the actual distance. (Scale) (Distance) (Proportion) (Solution)

1 inch5 miles

= 2 inches milesd

=

The actual distance from Jackson to Newton is ________________________________.

Using scale as one of the ratios, write a proportion for the actual distance (d) between Lakeport and Needles, and then find the actual distance. (Scale) (Distance) (Proportion) (Solution)

1 inch5 miles

= inchesmiles

=

The actual distance from Lakeport to Needles is ________________________________.

7/7/2008 6th 8-10 160

Bountyville

What does the map represent? _____________________________________________________

What is the shortest map distance between the library and the school? _______________________

What is the map scale? _________________________

Key

Building

Road

Bridge

Tunnel

N

S

E W

Scale 0 1 2 miles

1 inch = 2 miles

1.5 in. 1.5 in. 1.5 in.

0.5

in.

0.5

in.

0.5

in.

Library School

Police

City Hall

Using scale as one of the ratios, write a proportion for the actual distance (d) between the library and the school, and then find the actual distance. (Scale) (Distance) (Proportion) (Solution)

1 inch2 miles

= inches miles

=

The actual distance from library to the school is ________________________________.

Using scale as one of the ratios, write a proportion for the shortest actual distance (d) between the school and the police station by way of the tunnel. Then find the actual distance. (Scale) (Distance) (Proportion) (Solution)

=

=

The actual distance from the school to the police station by way of the tunnel is ___________________.

Using scale as one of the ratios, write a proportion for the shortest actual distance (d) between the library and the school if you avoided the bridge. Then find the actual distance. (Scale) (Distance) (Proportion) (Solution)

= The actual distance from the library to the school if you avoided the bridge is ___________________.

7/7/2008 6th 8-10 161

Lucca County

What does the map represent? _____________________________________________________ What is the map scale? _________________________

McFadden

Justice

Tebow

Daniels

Ryan

Key

City

Capital Mountains

River

N

S

E W

Brennan

Scale 0 25 50 75 miles

1 inch = 75 miles

The map distance from Tebow to Justice is 11

4inches. Using scale as one of the ratios, write a

proportion for the actual distance (d) between Tebow and Justice, and then find the actual distance.

(Scale) (Distance) (Proportion) (Solution)

=

The actual distance from Tebow to Justice is ________________________________.

The map distance from Daniels to Tebow is 1 inch. Using scale as one of the ratios, write a proportion for the actual distance (d) from Daniels to Justice by way of Tebow. Then find the actual distance.


=

The actual distance from Daniels to Justice by way of Tebow is ________________________________.

The map distance from Justice to Ryan is 314

inch. The map distance from Ryan to McFadden is 1.5

inches. Using scale as one of the ratios, write a proportion for the actual distance (d) from Justice to McFadden by way of Ryan. Then find the actual distance.


=

The actual distance from Justice to McFadden by way of Ryan is _______________________________.

7/7/2008 6th 8-10 162

The actual distance from the city of Tule to the city of Nighthawk is 45 miles. A map of the cities uses a scale of 1 inch equals 5 miles. If the same scale is used, what distance should the two cities be on the map?


=

=

The map distance from Tule to the city of Nighthawk is __________inches.

A map uses a scale of 1 inch equals 50 miles. If the same scale is used, how many inches will be needed to represent 300 miles?


=

The map distance of 300 miles is _______________________________.

The actual distance from the city of Green Acres to Petticoat Junction is 90 miles. A map of the towns uses a scale of 1 inch equals 12 miles. If the same scale is used, what distance should the two cities be on the map?


The map distance from Green Acres to Petticoat Junction is _______________________________.

A map uses a scale of 12

inch equals 35 miles. If the same scale is used, how many inches will be

needed to represent 175 miles?



inch equals 20 miles. If the same scale is used, how many inches will be

needed to represent 500 miles?


7/7/2008 6th 8-10 163

Using the paper ruler, find the actual distance. Measure to the nearest 14

inch.

1 2 3 4 5 60 in.

Peterson County

Fallout

Champion

BordersSun City

Gusto

Key

City

Road

Capital

Bridge

Tunnel N

S

E W Camelot


1 inch = 5 miles

Using scale as one of the ratios, write a proportion for the actual distance (d) between Sun City and Camelot, and then find the actual distance.


inch miles

= inch miles

=

The actual distance from Sun City and Camelot is ________________________________.

Using scale as one of the ratios, write a proportion for the actual distance (d) between Sun City and Borders by way of Camelot. Then find the actual distance.


inch miles

= inch miles

=

The actual distance from Sun City and Borders by way of Camelot is ___________________________.

Using scale as one of the ratios, write a proportion for the actual distance (d) between Borders and Fallout, if you avoid the tunnel. Then find the actual distance.


=

=

The actual distance from Borders and Fallout, if you avoid the tunnel is _________________________.

7/7/2008 6th 8-10 164

Using the paper ruler, find the actual distance. Measure to the nearest 14

inch.

1 2 3 4 5 60 in.

Joyful County

Satisfaction

Peace

Fortune

Happiness

Contentment

Key

City

Capital Mountains

River

Highway Bridge

Tunnel

N

S

E W Pleasure

Scale 0 50 100 150 miles

1 inch = 150 miles

Using scale as one of the ratios, write a proportion for the actual distance (d) between Happiness and Fortune, and then find the actual distance.


=

=

The actual distance from Happiness to Fortune is ________________________________.

Using scale as one of the ratios, write a proportion for the shortest actual distance (d) between Happinessand Satisfaction if you traveled through a tunnel. Then find the actual distance.


= =

The actual distance from Happiness to Satisfaction if you traveled through a tunnel is ______________.

Using scale as one of the ratios, write a proportion for the shortest actual distance (d) between Happinessand Satisfaction if you traveled over a bridge. Then find the actual distance.

The actual distance from Happiness to Satisfaction if you traveled over a bridge is ______________.

7/7/2008 6th 8-10 165

Collins Town

Key

Building

Road

Bridge

Tunnel

N

S

E W

Scale 0 0.25 0.5 miles

1 inch = 0.5 miles

1.5 in. 1.5 in. 1.5 in.

0.5

in.

0.5

in.

0.5

in. Courthouse

School

Aquarium

Concert Hall Sports Arena

Park

Toby started at the Courthouse and walked 2 blocks north and one block west. At what building did he arrive? How many miles did Toby actually walk?

Estelle was at the park. She started her riding her bike north. Then she rode east over a bridge and soon came to the Concert Hall. How many miles did Estelle ride her bike? One day, Duncan’s class took a field trip to the Aquarium. The bus started off heading west from school, then turned north passing the Concert Hall. The bus then turned west, going through a tunnel, before turning south and ending at the Aquarium. How many miles did the bus travel?

7/7/2008 6th 8-10 166

A map uses a scale of 1 inch equals 63 miles. How many miles are represented by 7 inches on this map? A) 9 miles B) 421 miles C) 441 miles D) 0.11 miles

The actual distance from the town of Marbury to Jonesville is 120 miles. A map of the towns uses a scale of 1 inch equals 25 miles. If the same scale is used, what distance should the two cities be on the map? A) 4.8 inches B) 0.21 inches C) 5 inches D) 48 inches

On a map, the neighborhood park and the public library are 2 12

inches apart. The map uses a scale of 1

inch equals 3 miles. How many actual miles is it from the park to the library?

A) 162

miles B) 1.2 miles C) 8.3 miles D) 7.5 miles


inch equals 50 miles. If the same scale is used, how many inches will be needed

to represent 225 miles? A) 22,500 inches B) 2.25 inches C) 9 inches D) 4.5 inches

The actual distance from the Town Square to City Hall is 114

miles. A map of the town uses a scale of

1 inch equals 0.5 miles. If the same scale is used, what distance should the two cities be on the map? A) 0.4 inches B) 0.625 inches C) 2.5 inches D) 2.25 inches

A map uses a scale of 2 inches equals 15 miles. How many miles are represented by 8 inches on this map? A) 120 miles B) 1.1 miles C) 16 miles D) 60 miles


inch equals 22 miles. If the same scale is used, how many inches will be needed

to represent 88 miles? A) 1 inch B) 4 inches C) 6.25 inches D) 8.5 inches

DRAFT

Revised 7/7/2008 6th Ch. 6 Multi Test Created collaboratively with grade 6 ISL teachers 1

6th Grade Name: _____________________________ Chapter 8 & Chapter 7 (part 2) Free Response Math Test Date: ____________ 1) Kit’s soccer team won 11 of the 16 games

(8-1) they played. Write the ratio of games won (NS 1.2*) to games lost three different ways.

6) (8-2) (NS 1.3*) Write a proportion for comparing the lengths and widths of Rectangles 1 and 2.

12

4

2

12

6

2) What number makes the ratio equivalent? (8-1) (NS 1.2*) 3 : 8 = x : 72

7) Find the missing number in the proportion. (8-3) (NS 1.3*) 4

x = 6

15

3) Write a ratio which shows a part to whole comparison? (8-1) (NS 1.2*)

Store Alternative Singles Sold

Rock Singles Sold

Super CDs 23 21 Mick’s Music 32 38

8) Ned walks around the park every afternoon. (8-3) He walks 3 laps every 8 minutes.

(NS 1.3*) If he keeps this same speed, how many Laps will he walk in 36 minutes??

4) What is the missing number in the proportion?(8-2) (NS 1.3*) 7

x= 42

24

9) Use cross-products to determine if the pair (8-3) of ratios form a proportion. Write yes if it (NS 1.3*) is a proportion or no if it is not.

46

, 1016

5) Gracie is making stew for dinner. The (8-2) recipe says to use 4 ounces of meat for (NS 1.3*) every 3 cups of soup. If Gracie uses 15 cups of soup, how Many ounces of meat should she buy in order to follow the recipe?

10) What is the unit rate measurement for (8-5) Chuck selling 12 insurance policies (AF 2.2*) in 8 hours?

DRAFT


6th Grade Chapter 6 Free Response Math Test

11) Find the unit rate: (8-5) (AF 2.2*) Lena caught 18 fish in 6 hours.

QRS is similar to TUV . (8-8) (NS 1.3*)

R

S

Q

T

V

U 12 cm

14 cm

6 cm6 cm 3 cm

55°100°

12) On his cross-country trip, Manny drove (8-5) 441 miles over 7 hours. In 5 hours, Tina (AF 2.2*) drove 320 miles. What were Manny’s and Tina’s speed? Who drove at a faster rate?

16) What is the measure of R∠ ? 17) In the picture above what is the length of UV ?

13) Melanie bought 7 pounds of apples at the (8-6) market for $4.69. What is the unit price? (AF 2.3)

18) If the two figures below are similar, then (NS 1.3*) what is the corresponding proportional side to AB ?

B H

G

C A

D

F

E110°

70°

14) The table shows prices for local Ice Cream Shops.

Store

Number of Scoops

Price

Fred’s Freezer 4 $5.08 Iggy’s Ice Cream 3 $3.90

Cal’s Creamer 2 $2.58

(8-6) Which ice cream shop has the best buy? (AF 2.3)

19) How many inches in 334

feet? (7-1) (AF 2.1)

15) (8-6) (AF 2.3)

What is the measurement that should be used to find the unit price for a Baker’s Dozen?

Brittany’s Bakery

aker’s Dozen: 13 Donuts for $ 7.15 Special B

20) Write a proportion that can be used to find the related customary unit of length. Include the units in your answer.

(7-1) (AF 2.1) 9 yd = ____ ft

=

DRAFT


6th Grade Chapter 8 & 7 (part 2) Free Response Math Test 21) (7-1) (AF 2-1) Find the capacity of a 12 pint container in quarts.

Customary Units of Capacity 1 cup (c) = 8 fluid ounces (fl oz) 1 pint (pt) = 2 cups 1 quart (qt) = 2 pints 1 gallon (gal) = 4 quarts

Customary and Metric Unit Equivalents Length

1 in. ≈ 2.5 cm 1 m ≈ 3.3 ft 1 m ≈ 1.1 yd

1 mi ≈ 1.6 km 25) Use the table above to write a proportion to (7-3) find the missing length. Include the units in your answer. (AF 2.1) 5 miles ≈ ____ kilometers ≈

22) Use the table above to find the related metric(7-2) unit of length. (AF 2.1) 15 hectometers = ____ dekameters


26) Travis is making a scale drawing of his (8-9) garden. He is using a scale of (NS 1.3*) 2 centimeters = 6 meters. The actual width of Travis’s garden is 15 meters. What width should the garden be on the scale drawing?

23) Write a proportion that can be used to find the related metric unit of length.

(7-2) (AF 2.1) 36 cm = ____mm = mmmm

cm cm

27) A scale drawing of a bedroom is shown. (8-9) (NS 1.3*) Find the actual length of the room?

dresser bed

7 in.


Customary and Metric Unit Equivalents


24) Use the table above to convert 66 pounds to (7-3) kilograms. (AF 2.1)

28) A scale drawing of a garden is shown. (8-9) (NS 1.3*) Find the actual width of the room?

11 in.

6 in. Scale: 1

21 in. = 4 ft

DRAFT


29) A map shows the library 13 centimeters (8-10) from the school. The map uses a scale of 1 (NS 1.3*) centimeters equals 22 meters. What is the actual distance from the library to the school?

30) A map uses a scale of 1 inch equals 40 miles.(8-10) If the same scale is used, how many inches (NS 1.3*) will be needed to represent 260 miles.

Jackson County

31) What is the actual distance from Leisure to Vessel if you went through the capital, Markton? (8-10) (NS 1.3*)

Creighton

Blackhole

Vessel

Torrent

Leisure

Key

City

Road

Capital

Bridge

Tunnel N

S

E W Markton


1 inch = 5 miles

DRAFT


6th Grade Chapter 8 & 7 (part2) Free Response Math Test Answer Key 1) Kit’s soccer team won 11 of the 16 games

(8-1) they played. Write the ratio of games won (NS 1.2*) to games lost three different ways.

1) 11:5 2) 11 to 5 3) 115

6) (8-2) (NS 1.3*) Write a proportion for comparing the lengths and widths of Rectangles 1 and 2.

Possible Answers:26

= 412

, 62

= 124

, 24

= 612

or 42

= 126

12

4

2

12

6

2) What number makes the ratio equivalent? (8-1) (NS 1.2*) 3 : 8 = x : 72

x = 27


x = 6

15

x = 10

3) Write a ratio which shows a part to whole comparison? (8-1) (NS 1.2*)

Possible Answers:23:44,21:44, 32:70, or 38:70

Store Alternative Singles Sold

Rock Singles Sold


8) Ned walks around the park every afternoon. (8-3) He walks 3 laps every 8 minutes.

(NS 1.3*) If he keeps this same speed, how many Laps will he walk in 36 minutes??

13.5 laps

4) What is the missing number in the proportion?(8-2) (NS 1.3*) 7

x= 42

24

x = 4

9) Use cross-products to determine if the pair (8-3) of ratios form a proportion. Write yes if it (NS 1.3*) is a proportion or no if it is not.

46

, 1016

No, 4 i16 ≠ 6 i 10

5) Gracie is making stew for dinner. The (8-2) recipe says to use 4 ounces of meat for (NS 1.3*) every 3 cups of soup. If Gracie uses 15 cups of soup, how Many ounces of meat should she buy in order to follow the recipe?

20 ounces of meat

10) What is the unit rate measurement for (8-5) Chuck selling 12 insurance policies (AF 2.2*) in 8 hours?

1.5 policies per hour

(Or 0.66 hours per policy)

DRAFT


6th Grade Chapter 6 Free Response Math Test

11) Find the unit rate: (8-5) (AF 2.2*) Lena caught 18 fish in 6 hours.

3 fish per hour

QRS is similar to TUV . (8-8) (NS 1.3*)

R

S

Q

T

V

U 12 cm

14 cm

6 cm6 cm 3 cm

55°100°

12) On his cross-country trip, Manny drove (8-5) 441 miles over 7 hours. In 5 hours, Tina (AF 2.2*) drove 320 miles. What were Manny’s and Tina’s speed? Who drove at a faster rate?

Manny – 63 mph; Tina – 64 mph; Tina drove at a faster rate

16) What is the measure of R∠ ?

25° 17) In the picture above what is the length of UV ?

7 centimeters

13) Melanie bought 7 pounds of apples at the (8-6) market for $4.69. What is the unit price? (AF 2.3)

$0.67/lbs.

18) If the two figures below are similar, then (NS 1.3*) what is the corresponding proportional side to AB ?

GE

B H

G

C A

D

F

E110°

70°

14) The table shows prices for local Ice Cream Shops.

Store

Number of Scoops

Price

Fred’s Freezer 4 $5.08 Iggy’s Ice Cream 3 $3.90

Cal’s Creamer 2 $2.58


Fred’s Freezer @ $1.27 / scoop


feet? (7-1) (AF 2.1)

334 feet = 45 inches

15) (8-6) (AF 2.3)

What is the measurement that should be used to find the unit price for a Baker’s Dozen?

price per 1 donut

Brittany’s Bakery

aker’s Dozen: 13 Donuts for $ 7.15 Special B

20) Write a proportion that can be used to find the related customary unit of length. Include the units in your answer.

(7-1) (AF 2.1) 9 yd = ____ ft

1 yd 9 yd 3 ft x ft =

DRAFT


6th Grade Chapter 8 & 7 (part 2) Free Response Math Test 21) (7-1) (AF 2-1) Find the capacity of a 12 pint container in quarts.

12 pints = 6 quarts



1 in. ≈ 2.5 cm 1 m ≈ 3.3 ft 1 m ≈ 1.1 yd

1 mi ≈ 1.6 km 25) Use the table above to write a proportion to (7-3) find the missing length. Include the units in your answer. (AF 2.1) 5 miles ≈ _8___ kilometers 1 mi 5 mi

≈

22) Use the table above to find the related metric(7-2) unit of length. (AF 2.1) 15 hectometers = ____ dekameters

150 dekameters


26) Travis is making a scale drawing of his (8-9) garden. He is using a scale of (NS 1.3*) 2 centimeters = 6 meters. The actual width of Travis’s garden is 15 meters. What width should the garden be on the scale drawing?

5 centimeters

23) Write a proportion that can be used to find the related metric unit of length.

(7-2) (AF 2.1) 36 cm = ____mm 1 36

10 x = mmmm cm cm

27) A scale drawing of a bedroom is shown. (8-9) (NS 1.3*) Find the actual length of the room?

21 feet

dresser bed

7 in.


Customary and Metric Unit Equivalents



30 kilograms

28) A scale drawing of a garden is shown. (8-9) (NS 1.3*) Find the actual width of the room?

16 feet

11 in.

6 in. Scale: 1

21 in. = 4 ft

DRAFT


6th Grade Chapter 8 & 7 (part 2) Free Response Math Test

29) A map shows the library 13 centimeters (NS 1.3*) from the school. The map uses a scale of 1 centimeters equals 22 meters. What is the actual distance from the library to the school?

286 meters

33) A map uses a scale of 1 inch equals 40 miles.(8-10) If the same scale is used, how many inches (NS 1.3*) will be needed to represent 260 miles.

6 12 inches or 6.5 inches

Jackson County

31) What is the actual distance from Leisure to Vessel if you went through the capital, Markton? (8-10) (NS 1.3*)

21.25 miles

Creighton

Blackhole

Vessel

Torrent

Leisure

Key

City

Road

Capital

Bridge

Tunnel N

S

E W Markton


1 inch = 5 miles

DRAFT


6th Grade Name: _____________________________ Chapter 8 & Chapter 7 (part 2) Multiple Choice Math Test Date: ____________ 1) Don’s baseball team won 9 of the 12 games (8-1) they played. What is the ratio of games lost(NS 1.2*) to the number of games won? A) part to part C) whole to whole

B) part to whole

6) (8-2) (NS 1.3*) Identify the correct proportion for comparing the Rectangles 1 and 2.

A) 42

= 612

C) 26

= 412

B) 412

= 62

D) 212

= 46

12

4

2

12

6

2) Which number makes the ratio equivalent? (8-1) (NS 1.2*) 5 : 7 = 40 : x A) x = 35 C) x = 56

B) x = 20 D) x = 54


x = 6

10

A) x = 13 C) x = 20

B) x = 15 D) x = 90 3) Which ratio shows a part to whole comparison? (8-1) (NS 1.2*)

A) 55:48 C) 103 to 65

B) 26 to 65 D) 39:55


R & B CDs Sold


8) Tricia walks around a track every day after (8-3) work. She walks 3 laps every 12 minutes. (NS 1.3*) If she keeps this same speed, how many minutes will it take her to walk 11 laps? A) 20 minutes C) 40.3 minutes

B) 2.75 minutes D) 44 minutes

4) What is the missing number in the proportion?(8-2) (NS 1.3*)

3x = 45

27

A) 5 C) 4

B) 9 D) 7

9) Which equation represents the two ratios (8-3) cross-products? (NS 1.3*) 12

p = 8

4

A) 12 i 8 = 4p C) 12p = 8 i 4

B) 8p = 8 i 12 D) 12 i 4 = 8p

5) Theo’s donut recipe calls for 3 tablespoons (8-2) of powdered sugar for every 8 donuts. If (NS 1.3*) he wants to make 24 donuts, how many tablespoons of powdered sugar will he need? A) 64 C) 10

B) 8 D) 9

10) Which situation describes a rate? (8-5) (AF 2.2*)

A) Harold ran 2 hours on Monday and 1.5 hours on Tuesday.

B) Olga ate 12 orange colored pasta and 8 yellow colored pasta.

C) Yolanda created 14 gingerbread houses in 3 hours.

D) Victor read 3 biographies and 4 mysteries from the library.

DRAFT


6th Grade Chapter 6 Multiple Choice Math Test

11) Find the unit rate: (8-5) (AF 2.2*) In 4 hours, Carlos lemonade stand sold sold 5 gallons of lemonade.

A) 45

gallons per hour C) 20 gallons per hour

B) 1.25 gallons per hour D) 0.8 gallons per hour

JKL is similar to MNO . (8-8) (NS 1.3*)

M

N

O

J

L

K40°

6 in. 12 in.

16 in.

20 in.

12) On her cross-country trip, Melissa drove (8-5) 603 miles over 9 hours. In 5 hours, Zach (AF 2.2*) drove 330 miles. What were Melissa’s and Zach’s speed? Who drove at a faster rate? A) Melissa – 67 mph; Zach – 66 mph; Melissa drove at a faster rate

B) Melissa – 54.27 mph; Zach – 16.5 mph; Melissa drove at a faster rate

C) Melissa – 67 mph; Zach – 66 mph; Zach drove at a faster rate

D) Melissa – 14.9 mph; Zach – 15.5 mph; Zach drove at a faster rate

16) In the picture above what is the length of LJ ?

A) 4 inches C) 4.5 inches

B) 8 inches D) 8.5 inches

17) What is the measure of O∠ ?

A) 40° C) 45°

B) 50° D) 60°

13) Myron bought a 6 pound melon at the

(8-6) market for $2.34. What is the unit price? (AF 2.3) A) $0.39 C) $0.04

B) $2.56 D) $3.09

18) If ABC and DEF are similar figures, (8-8) then….. (NS 1.3*)

A) A∠ and D∠ are proportional

B) B∠ and F∠ are congruent

C) AB and DE are congruent

D) BC and EF are proportional 14) The table shows prices for local Ice Cream Shops.

Store Number of Scoops Price Frozen Feasts 2 $3.02 Creamy Cones 3 $4.38

Icy Igloo 4 $5.88


A) Frozen Feasts B) Creamy Cones C) Icy Igloo


feet? (7-1) (AF 2.1) A) 13 inches C) 37 inches B) 36 inches D) 39 inches

15) (8-6) (AF 2.3)

What is the measurement that should be used to find the unit price? A) price per dollar C) price per 8 cups

B) price per cents D) price per 1 cup

Connie’s Coffee

offee Rate: $14.80 for 8 cups of coffee Special C

20) Which proportion can be used to find the (7-1) related customary unit of length? (AF 2.1) 9 yd = ____ ft

A) 1 yd 9 yd3 ft ftx

= C) 12 yd 9 yd1 ft ftx

=

B) 3 yd 9 yd1 ft ftx

= D) 1 yd 9 yd12 ft ftx

=

DRAFT


6th Grade Chapter 8 & 7 (part 2) Multiple Choice Math Test 21) (7-1) (AF 2-1) Find the capacity of a 16 gallon container in quarts. A) 32 quarts C) 64 quarts

B) 4 quarts D) 2 quarts



1 in. ≈ 2.5 cm 1 m ≈ 3.3 ft 1 m ≈ 1.1 yd

1 mi ≈ 1.6 km 25) Use the table above to write a proportion (7-3) to find the missing length. (AF 2.1) 5 miles ≈ ____ kilometers

A) 1.6 mi 5 mi1 km kmx

≈ C) 1 mi 5 mi

1.6 km kmx≈

B) 1.6 mi 1 mi5 km kmx

≈ D) 5 mi 1 mi

1.6 km kmx≈

22) Use the table above to find the related metric(7-2) unit of length. (AF 2.1) 12 dekameters = ____ meters

A) 1.2 meters C) 120 meters

B) 12 meters D) 1,200 meters


26) A scale drawing of a bedroom is shown. (8-9) (NS 1.3*) Which describes the actual length of the room? A) 5 inches C) 10 feet

B) 4 feet D) 20 feet

dresser

bed

5 in.


23) Which proportion can be used to find the related metric unit of length? (7-2) (AF 2.1)

25 cm = ____mm

A) 1 cm 25 cm

100 mm mmx= C)

100 cm 25 cm1 mm mmx

=

B) 10 cm 25 cm1 mm mmx

= D) 1 cm 25 cm

10 mm mmx=

27) Tonya is making a scale drawing of her (8-9) backyard. She is using a scale of (NS 1.3*) 1 centimeter = 8 meters. The actual width of Tonya’s backyard is 20 meters. What width should the backyard be on the scale drawing?

A) 2.5 centimeters C) 16 centimeters

B) 4 centimeters D) 0.4 centimeters

Customary and Metric Unit Equivalents Weight and Mass

1 oz ≈ 30 g 1 kg ≈ 2.2 lb 1 metric ton (t) ≈1.1 tons (T)


A) 88.0 kg C) 37.8 kg B) 42.2 kg D) 18.2 kg

28) A scale drawing of a showroom is shown. (8-9) (NS 1.3*) Which describes the actual width of the room? A) 160 feet C) 20 feet

B) 64 feet D) 8 inches

14 in.

8 in. Scale: 1

22 in. = 20 ft

DRAFT


29) A map uses a scale of 12

inch equals 30 miles.

(8-10) If the same scale is used, how many inches (NS 1.3*) will be needed to represent 180 miles.

A) 4 inches C) 3 inches

B) 6 inches D) 12 inches

30) On a map, the candy store and movie

(8-10) theater are 6 centimeters apart. The map (NS 1.3*) uses a scale of 2 centimeters equals 55 meters. What is the actual distance from the candy store to the movie theater? A) 330 cm C) 165 m

B) 21.8 m D) 218 m

Henderson County

31) What is the actual distance from Newton to Lakeport? (8-10) (NS 1.3*) A) 3.75 miles C) 18.75 miles

B) 11.25 miles D) 12 miles

Lakeport

Turner

Needles

Jackson

Carrie

Key

City

Road

Capital

Bridge

Tunnel N

S

E W Newton


1 inch = 5 miles


DRAFT 6th Grade Chapter 8 & 7 (part2) Multiple Choice Math Test Answer Key

1. A 2. C 3. B 4. A 5. D 6. C 7. B 8. D 9. D 10. C 11. B 12. A 13. A 14. B 15. D 16. C 17. B 18. D 19. D 20. A 21. C

22. C 23. D 24. D 25. C 26. D 27. A 28. B 29. C 30. C 31. C

Item Analysis for: 6th Grade Chapter 8 & 7 Part 2 Tests Teacher: ______________________ Date Given: ________ 8-1 8-2 8-3 8-5 8-6 8-8 7-1 7-2 7-3 8-9 8-10

Test Item # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Total 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Total Items Correct

Documents

Investigating Student Learning: 6 Grade Chapter 8: …mrbentley.wikispaces.com/file/view/6thISLCh8.pdf7/7/2008 6th 8-1 1 DRAFT Investigating Student Learning: 6th Grade Chapter 8: