LESSON 101 Ratio Problems Involving Name -...

Saxon Math Course 1 L101-401 Adaptations Lesson 101

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Ratio Problems Involving Totals (page 528)

• In some ratio problems a total is needed in order to solve the problem.

1. Fill in the ratio box with things you know.

2. Write a proportion.

Use the row that answers the question asked.Use the row that is already complete.

Example: The ratio of boys to girls in a class was 5 to 4.If there were 27 students in the class, how many girls were there?

4 __

9 =

g ___

27

g = 4 ∙ 27

_______ 9

g = 12

Practice Set (page 530)

a. Sparrows and crows perched on the wire had the ratio of 5 to 3. If the total number of sparrows and crows on the wire was 72, how many were crows?

crowstotal

___ ? ___

72

72 ∙ 3 _______

8 =

Cancel.

b. Raisins and nuts were mixed by weight in a ratio of 2 to 3. If 60 ounces of mix were prepared, how many ounces of raisins were used?

raisinstotal

___ ? ___ ∙ _________ =

Cancel.

c. There are 20 green and blue markers in a ratio of 3 to 2. How many of each color are there?

green

blue

3

1


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1. 2. See the Student Reference Guide.

3. x ÷ 6 = 12 4. radius

5. pounds$

_____

1.65

___ 6.

AC = 12 cm

AB = 1

__ 4 of AC

BC =

___ 4 of AC

7. a. –3 + –4 =

b. +5 + –5 =

c. –6 + +3 =

d. +6 + –3 =

8. a. –3 – –4 = b. +5 – –5 =

c. –6 – +3 = d. –6 – –6 =

e. Change the s of the

subtrahend and a .

9. 10.

Written Practice (page 530)

r p

a.

b.

Use work area. Use work area.

Use work area.


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19. 6n = 21 ∙ 4 20. Nia’s garage is 20 feet long, 20 feet wide, and 8 feet high.

a. bottom layer

b. all layers

11. 12.

13. 1 1 __

2 × 4 14. 6 ÷ 1

1 __

2

15. (0.4) 2 ÷ 2 3 16. x + 2 1 __

2 = 5

5

2 1 __

2

17. 8 __

5 =

40 ___

x 18. 0.06n = $0.15

Written Practice (continued) (page 531)


x =

x =

n =

a.

b.

n =


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21. circumference

A 2 1

__ 2

in. B 5 in. C 7 3 __

4 in. D 9

1 __

4 in.

22. 9 2 – √__

9 × 10 – 2 4 × 2 =

23. What kind of polygon?

24. The sum of the angles of each triangle

is .

25. 15° 0° –8°

26.

27. 1, 2, 3, 4, 5, 6

perfect squarestotal numbers

___

28. (4, 0), (0, –3), (0, 0)

area

29. Divide 18 feet by per yard.

ydft

1

__ 3

___

18

30. 1 gal = qt

qt$

_____

3.80

1 ___


==

Use work area.


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Teacher Notes:• Introduce Hint #60, “Gram/Kilogram

Manipulatives.”

• Review “Equivalence Table for Units” on page 1 in the Student Reference Guide.

• Metric weight manipulatives can be found in the Adaptations Manipulative Kit.

Mass and Weight (page 533)

• Physical objects are composed of matter.

• The amount of matter in an object is its mass.

• Mass does not change with changes in gravity.

• Weight does change with gravity changes.

The weight of an astronaut changes on the moon.

His or her mass does not change on the moon.


a. Half of a kilogram is how many grams?

b. The mass of a liter of water is one kilogram. The mass of two liters of beverage

is about how many grams?

Lg

1 ___

2 __

?

c.

5 lb 10 oz+ 1 lb 9 oz

d.

9 lb 8 oz– 6 lb 10 oz

e. A half-ton pickup truck can haul a half-ton load. Half of a ton is how many pounds?


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1. 84, 90, 92, 92, 92, 96 3. 18 one-point baskets

6 three-point baskets

total points from one- and three-point baskets

points from whole game

points from one- and three-point baskets

points from two-point baskets

number of two-point baskets

4. 4 __

7 =

A 7

__ 4

B 14

___ 17

C 12

___ 21

D 2 __

3

9. 10.


2. average (mean)

8490929292

+ 96

6. least to greatest –1, 1, 0.1, –0.1, 0

7. 10 3 10 2

A 10 9 B 10 6 C 10 5 D 10

8. The area of the square in this figure is 100 mm2.

a. radiusb. diameterc. area of the circle (Use 3.14 for π.)

A = π r 2

5. 4

__ 5 =

___

20

Use work area.Use work area.

a. b. c.

, , , ,

a.

b.


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19. Multiply the probabilities.

90° _____ 360°

=

___ ∙

___ =

20. probability of sector 1 =

___ 4

___ 4

of 100

11. 12.

1 2

__ 3 =

___

3 1 __

2 =

___

+ 4 1 __

6 =

___

13. 5

__ 6 ×

3 ___

10 × 4 = 14. 6

1 __

4 ÷ 100

15.

6.437..

16. ) ________

1. 0 0 0

17. octagonpentagon

18. 4 × 5 2 – 50 ÷ √__

4 + ( 3 2 – 2 3 ) =


Use work area.

.


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21. top layer

all layers

22. x ÷ 4 = 5

23.

10 lb 1 oz 08 lb 4 oz

24.

25. x 1 2 4 5

3x – 5 –2 1 7 ?

26.

27.

29. (3, –1) and (3, 5) 30.

) ___

31


28. gmg

1 ___

___

?

( , )

a.

b.

c.

a. ∠

b. m∠1 = , m∠2 =


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Teacher Note:• Introduce Hint #61, “Perimeter of

Complex Shapes.”

Perimeter of Complex Shapes (page 538)

• Perimeter means to add all the sides.

Some sides will not be labeled.

Subtract to find labels for these sides.

Hint: Sometimes it helps to use two different colors.

Trace over all horizontal lines in one color.

Trace over all vertical lines in another color.

10 – 4 = m 8 – 2 = n

6 in. = m 6 in. = n


Find the perimeter of each complex shape:

a.

12 – 5 = y

8 – 3 = x

perimeter =

b.

20 – = x

15 – = y

perimeter =


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1. ( ) ÷ ( ) = 2. x ÷ 3 = 24

a. x =

b. (22 + 22 + ) ÷ 3 = 24

3. 1 yd = inches

perimeter inches

a. each side

b. area

4. 5 __

3 =

30 ___

5. 6.

7. 100 ÷ 10 2 + 3 × ( 2 3 – √___

16 ) = 8. a. pounds = 1 ton

b. probability of something impossible

9. Fraction Decimal Percent

a. b.

10.


a.

b.


a.

b.

a. b.


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20. a. –3 + –8 =

b. –3 – –8 =

c. –8 + +3 =

d. –8 – +3 =

21. The sum of the angles

in a triangle is 180°.

m∠C = m∠B

11. 12. 10 1

__ 2

÷ 3 1 __

2

13. (6 + 2.4) ÷ 0.04 = 14. 7 1 __

2 + 6

3 __

4 + n = 15

3 __

8

7 1 __

2 =

___

6 3 __

4 =

___

15 3

__ 8 =

___

=

___

15. x – 1 3 __

4 = 7

1 __

2

7 1 __

2 =

___

1 3

__ 4 =

___

16. 10 1

__ 2

(× 2)

___

3 1 __

2 (× 2)

17.

18. Use exponents.

20,500,000

19. prime numbers between 40 and 50


x =

n =

Use work area.

Use work area.

, ,

.

( × ) + ( × )


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22. a. perimeter:

b. area:

c. 20 mm side

___________ longest side

fraction:

decimal:

23.

24. queen of spades52 cards

___

26.

50 – = x

20 – = y 28. (–3, –2) and (5, –2)

29. 16 oz = lb

1 gal = pt

1 __

2 gal = pt

1 __

2 gal = pounds

30. a. 360° ÷ =

b. 180° – =


27.

25.

Use work area.

a.

b.

a.

b.

c. ( , )

a. b.


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Algebraic Addition Activity (page 543)

Activity Sign Game

Level 1

• During the game, positive and negative pairs are neutralized.

• After the game, count the remaining positives and negatives to see what remains. Tell what remains for these games:

Level 2

• Now the positives and negatives are shown in clusters of more than 1.

• The same rules apply as in Level 1. The suggested strategy is to group all the signs first. So +3 combines with +1 to make +4, and –5 combines with –2 to make –7.

• Since there are three more negatives than positives, –3 remain. Tell what remains for these games:

Level 3

• Now the clusters take on a disguise.

• Sometimes a cluster will have no sign, sometimes it will have one sign, sometimes it will even have two signs.

• First, remove the disguise.Positive clusters will have: no sign – – + +Negative clusters will have: + – – +

• Sometimes the cluster has a “shield” (parentheses). Don’t be fooled.

Look through the shield to see the sign–(–3) is really +3–(+3) is really –3

• Tell what remains for these games:

Level 4

• Extend Level 3 to a line of clusters:

–3 + (–4) – (–5) – (+2) + (+6)

• Use the following steps to find the answer:

Step 1: Remove the disguises: –3 – 4 + 5 – 2 + 6

Step 2: Group signs: –9 + 11

Step 3: Find what remains: +2


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1. edges vertices

2.

12 lb 6 oz7 lb 8 oz

3. fishsnails

___ 4. 5. Multiply the probabilities.

1 ___ ∙

1 ___ =

6. 7.

8. 9. 2 __

3 =

A 2

__ 4 B

3 __

4 C

4 __

6 D

3 __

2



a. –2 + –3 – –4 + –5 = b. –3 + (+2) – (+5) – (–6) =

c. +3 + –4 – +6 + +7 – –1 = d. 2 + (–3) – (–9) – (+7) + (+1) =

e. 3 – –5 + –4 – +2 + +8 = f. (–10) – (+20) – (–30) + (–40) =

a.

b.

b. a.


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18. Round to a whole number.

) ______

0.624

19. x ÷ 3 = 20

10. 6 __

8 =

a ___

12 11. 13 – = x

12 – = y

12. 13.

14. 15. 40% off

$6.95

$6.95

16. See the Student Reference Guide.

√____

200

17. ( 1 __ 2

) 3 the probability of three “heads” in three coin tosses


20.

) _____

) _____

) _____

) _____

) _____

450

21. –3 + –5 – –4 – +2 =

a =

and

Use work area.


∙ ∙


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22. 34 + 52 × 4 – √____

100 × 23 = 23.

24. 3 __

4 of 60 played

___ 4

of 60 did not

isof

___

___

25. kmhr

88

___ 1

? ___

26. area

27. average

3.123.23.15

3.1

28. There are 90 two-digit counting numbers.

Since Hector was t of only

o number, the probability of

correctly guessing the n in one

try is .

29. (3, 5), (–1, 5), (–1, –3)

area


30. 2 gal

______ 1

× 4 qt

_____ 1 gal

× 2 pt

____ 1 qt

=

a.

b.

c. .

Use work area.


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Using Proportions to Solve Percent Problems (page 548)

• A ratio box may be used to solve percent problems.

• Remember: A percent may be expressed as a fraction. 30% equals 30 ___ 100

Example: Thirty percent of the students earned an A on the test. If twelve students earned an A, how many students were there in all?

30

____ 100

= 12

___ t t =

12 ∙ 100 _________

30 = 40


a. Forty percent of the cameras in a store are b. Seventy percent of the team members playeddigital cameras. If 24 cameras are not digital, in the game. If 21 team members played, how how many cameras are in the store in all? many team members did not play?

c. Referring to problem b, what proportion d. Joan walked 0.6 mile in 10 minutes. How farwould we use to find the number of members can she walk in 25 minutes at that rate? Write on the team? and solve a proportion to find the answer.

___ =

___ t

0.6 ___

10 =

d ___

e. Ninety percent of the 30 students loved math. How many of the students ?

Answer:

3

4 10

1


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1. 2 1 __

2 = 2.5

mihr

50

___ 1

? ___

2. in.mi

1 ___

___

?

3. Ratio Actual Count 4. Volume = lwh

5. a. +10 + –10 =

b. –10 – –10 =

c. +6 + –5 – –4 =

6. lbkg

___ 1

? ___

7. 8.

9. 10.



a.

b.

c.


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19. The sum of the

angles in a

quadrilateral is 360°.

20.

) _____

) _____

) _____

) _____

) _____

500

11. 12.

4 1 ___

12 =

___

5 1 __

6 =

___

+ 2 1 __

4 =

___

13. 4 __

5 ×

___ ×

3 __

1 = 14.

0.125× 80

15. (1 + 0.5) ÷ (1 – 0.5) = 16. c ___

12 =

3 __

4

17.

$8.75

$8.75 18. one hundred five and five hundredths


Use work area.

c =

∙ m∠B = m∠C =


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21. 1 qt is just less than 1 L

qt = 1 gal

1 gal is just less than L


23. perimeter = 18 cm

24. area

25. ? 0° 12° –5°

== 5

26. Write the answer as a decimal.

27. perimeter

28. x 1 _ 2 1 1 _ 2 2

y 1 1 _ 2 3 4 1 _ 2 6

Rule: Multiply x by to find .

29. ydft

1 ___

? ___

15

15 ft

12 ft ydft

1 ___

? ___

12

30. The probability is because the

past outcome does affect

the future o .


Use work area.

Use work area.b. a. by


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Two-Step Equations (page 553)

• Always do the same thing to both sides of an equation.

• To solve two-step equations:

1. Change the sign.

2. Move to the other side.

3. Then multiply or divide.

4. Check the answer.

Example: 3n – 1 = 20 Changed minus to plus.

3n = 21 Added 1 to both sides of the equation.

n = 7 Divided both sides of the equation by 3.

3(7) – 1 = 20 Replaced n with 7.


a. 3n + 1 = 16 b. 2x – 1 = 9

3n = 2x =

n = x =

c. 3y – 2 = 22 d. 5m + 3 = 33

3y = 5m =

y = m =

e. 4w – 1 = 35 f. 7a + 4 = 25

4w = 7a =

w = a =

– +

+ –

+


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1. (28 + 15 + ) ÷ 3 = 20 2. 2 1 __

2 = 2.5

in.mi

___ 10

2.5

___ ?

3. isof

___

___ 4. nickelquarter

___ = ____

100

5. Numbers from 1–30 that have a 1 in them.

, , , , , ,

, , , , ,

Write the probability as a fraction and a decimal.

6. 8x + 1 = 25

8x =

x =

7. 3w – 5 = 25

3w =

w =

8. a. –15 + +20 =

b. –15 – +20 =

c. (–3) + (–2) – (–1) =

9. tonspounds

___ ? ___ 10. 1 gal =

qt– 1 qt

= pt


x =

w =

a.

b.

c.


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11.

12.

13.

14.

15.

8 1

__ 3 =

___

– 3 1 __

2 =

___

16. 2 1 __

2 ÷ 100

17. 0.014 ÷ 0.5

) _____

18. (6 × 104) + (9 × 102) + (7 × 100)

19. 100 22 ∙ 52

1000 23 ∙ 53

1,000,000 ∙


20.

Use work area.


∙


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21. 6 + 32(5 – √__

4 ) = 22. volume

23. inchesfoot

3 ___ =

____

100 24. area

25. perimeter

26. circumference (Use 3.14 for π.)

A 6 ft

B 6 ft 3 in.

C 6 ft 8 in.

D 7 ft

27. area

28. List the miles in order.

3, , , , , , 10

29. average

30. How many miles did Celina ride

on Thursday then on ?

Answer:


Use work area.

a.

b.

c.


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Teacher Note:• Review Hint #61, “Perimeter of

Complex Shapes.”

Area of Complex Shapes (page 557)

• Perimeter of complex shapes Add all sides.

• Area of complex shapes

1. Divide the shape into two or more parts.

2. Find the area of each part.

3. Add the parts.

• Formulas to remember:

Area of a rectangle A = lw

Area of a triangle A = bh

___ 2

(Be sure to label area in square units.)


a. The same figure has been divided two different ways. Find the length of the unknown side in each figure and find the total area of each figure.

b. The trapezoid is divided into a rectangle and a triangle. Find the area of the trapezoid.

28 cm2

+ 6 cm2

34 cm2


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1. ) _______

2. in.ft

1

__ 2

___

?

3. tulipsroses

___ 4. A pentagon has how many sides?

1 ___ =

____

100

5. kgg

1 ___ 6. a. +15 + –10 =

b. –15 – –10 =

c. (+3) + (–5) – (–2) – (+4) =

7. 103 – (102 – √____

100 ) – 103 ÷ 100 = 8. 6

__ u

= 8 ___

1.2

9.

10.



a. b. c.

u =


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19. GCF of 30 and 45

1 , , , , , ,

, 30

20.

11.

12.

5 3

__ 8

=

___

4 1

__ 4

=

___

+ 3 1

__ 2

=

___

13. 8

__ 3 ∙

5 ___

12 ∙

9 ___

10 = 14.

64.8..

15. The sum of the angles in a triangle is 180°.

16. See the Student Reference Guide.

ptoz

1 __

___

17. 3m + 8 = 44

=

m =

18. Use exponents.

110,000,000


Use work area.

m =

b.

a.

( × ) + ( × )


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21. 1 yd = ft

1 layer

all layers

22. 0.3n = $6.39

23. perimeter

24. area

25.

27. Complete the line plot to display the data. See Investigation 4.

28.

12 lb 3 oz– 8 lb 7 oz

29.

30. 10 gallons

__________ 1

× 31.5 miles

__________ 1 gallon

=


26. Arrange in order.

, , , , , , , ,

a. most frequent:

b. middle age:

c. average rounded:

d. m , m , m

Use work area.

Use work area.

a.

b.

n =


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Teacher Note:• The activity in the Student Edition is

optional.

Transformations (page 561)

• Figures that have the same shape and size are congruent.

• One will fit exactly on top of the other.

• The matching parts are equal in measure.

• To position triangle ABC on triangle XYZ, make three different kinds of moves (transformations).


Name the transformation(s) necessary to position triangle I on triangle II in each exercise.

a. b. c.

d. e.

and and

Transformations

Name Movement

Rotationturning a figure about a certain point

Translationsliding a figure in one direction without turning the figure

Reflectionreflecting a figure as in a mirror or “flipping” a figure over a certain line


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1. sum of the first five positive even numbers 2.

3.

4. area

5. isof

___

___ 6. LCM of 6, 8, 12

7.

Rotate triangle I until its orientation matches t II.

Then translate t I until it is positioned on

t II.

9.

10.


Practice Set (continued) (page 563)

f. Triangle ABC is reflected across the y-axis to become triangle A´B´C´. List the coordinates of the vertices of triangle ABC and its image, triangle A´B´C´.A (–2, 4) A´ (2, 4)B (–1, 1) B´ ( , )C ( , ) C´ ( , )

8. 0.7

___ 20

= n ____

100

Use work area.

Use work area.

Use work area.

n =


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19. volume = lwh

20. 25 – 52 + √___

25 × 2 =

11.

12. 4 3 __

4 + ( 2

1 __

4 –

7 __

8 ) =

2

1 __

4 =

___

– 7

__ 8

=

___

4

3 __

4 =

___

+ =

___

13. 1 1

__ 5

÷ ( 2 ÷ 1 2 __

3 ) = 14. 6.2 + (9 – 2.79) =

9.– 2.79

6.2+ .

15. –3 + +7 + –8 – –1 = 16. Round the answer to the nearest cent.

$2.89$2.89

17. mmm

___ 18. least to greatest

0.3, 0.31, 0.305


Use work area.

, ,


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30. R(–1, 4), S(–3, 1), T(–1, 1) Reflect across the y-axis.

21. 8a – 4 = 60

= 60

= 60

22. A right angle

is .

1 __

3 (right angle) =

23. perimeter

24. area

25. 2 gal = pt

ptlb

___

___

26. total area

27. 1 liter = milliliters

29. 1 1 __

2 = 1.5

kmm

1 ___

1.5 ___

?



first second draw draw

___ 10

∙

___ 9 =

a =

R´(1, 4), S´( , ), T´( , )


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Teacher Note:• Refer students to “Similar and

Congruent Triangles” on page 28 and “Scale Factor” on page 31 in the Student Reference Guide.

Corresponding Parts Similar Figures (page 566)

• If two figures are congruent, their corresponding parts (angles and sides) match exactly.

Example: Triangle ABC and triangle XYZ are congruent.∠A corresponds to ∠X. ___

AB corresponds to ___

XY .

• If two figures are similar, they have the same shape but not necessarily the same size.

• Similar figures have equal matching angles.

• In all similar polygons the ratios of corresponding sides are equal.

• The lengths of corresponding sides are related by a ratio called the scale factor.

scale factor from smaller figure to larger figure = larger side _________ smaller side

scale factor from larger figure to smaller figure = smaller side _________ larger side

Make sure you use corresponding sides.

• To find an unknown corresponding side length, multiply by the scale factor.


a. “All squares are similar.” True or false?

b. “All similar triangles are congruent.” True or false?

c. “If two polygons are similar, then their corresponding angles are equal in measure.”

True or false?

d. These two triangles are congruent. Which side of triangle PQR is the same length as ___

AB ?

Triangles I, II, and III are similar.Triangles I and II are congruent.


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1. prime numbers See the Student Reference Guide.

2. cmkm

2 ___

10 ___

?

3. commercial minutesprogram minutes

8 ___ = 4.

5.

6. Write the probability as a fraction and a decimal.

Pacific statestotal states

___

7. 7w – 3 = 60

=

=

8. 8 __

n =

4 ___

2.5 9.


Practice Set (continued) (page 569)

e. Which two of these triangles appear to be similar? and

f. These two pentagons are similar. The scale factor for corresponding sides is 3. How long is segment AE? How long is segment IJ?

AE = IJ =

b. a.

w = Use work area.n =


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19.

a. scale factor from small to large

b. scale factor from large to small

10.

12. a. numbers less than 4total numbers

___

b. prime numberstotal numbers

___ × 100

14. (6.2 + 9) – 2.79 =

6.2+ 9.

.– 2.79

15. 103 ÷ 102 – 101 =

16. y = 2x

x 2 3 5 10

y

17. ) _______

18.


11.

20. 0.12m = $4.20

13. 200 cm

_______ 1

∙ 1 m _______

100 cm =

b. a.

Use work area.

Use work area.

m =

.

Use work area.

a.

b.


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21. a. +7 + –8 =

b. –7 + +8 =

c. –7 – +8 =

d. –7 – –8 =

23.

24. area of triangle = 1 __

2 bh

25.

26. similar triangles

27.

28.

29. 6 2 __

3 ÷ 100 30. ( 1 ___

10 )

2

0.01


22. The triangles are congruent.

A = 1 __

2 bh

Use work area.

a.

b.

r and t


L E S S O N

110 Name ©

200

7 H

arco

urt

Ach

ieve

Inc.

Symmetry (page 573)

• A line of symmetry divides a figure in half so that the halves are mirror images of each other.

Example : An equilateral triangle has three lines of symmetry.

• Rotational symmetry is when the image of a figure reappears in the same position as it turnsless than one full turn.

Example: A square reappears in its original position as it is turned 90°, 180°, and 270°.


a. Sketch a different line of symmetry for each square.

b. Which of these letters does not have a line of symmetry?

A B C D E F

c. Which two of these letters have rotational symmetry? (Hint: Rotating your paper might help

you find the answer.) ,

L M N O P Q


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1.

2.

3. volume

4. isof

___ _____

360°

5. The equilateral triangle has

lines of symmetry.

It rotational symmetry.

7. area

8.

9. 10.


6. perimeter

)


Use work area.


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19. “All squares are similar.”

True are false?

20. 33 – 32 ÷ 3 – 3 × 3 =

11.

12.

24 1

__ 6

=

___

23 1 __

3 =

___

+ 22 1 __

2 =

___

13. ( 1 1 __

5 ÷ 2 ) ÷ 1

2 __

3 = 14. 9 – (6.2 + 2.79)

6.2+ 2.79

9.– .

15. 0.36m = $63.00 16. Round to the nearest cent.

$24.89 0.065

17. Round to the nearest thousandth.

0.065 ÷ 4

18.

) ______

) ______

) ______

) ______

) ______

) ______

1000


Use work area.

m =

∙


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21. perimeter

22. area

23.

r , r ,

and t

25. circumference to the nearest hundred feet

27. • center on origin • radius 5 units

28. area of the circle in problem 27 (Use 3.14 for π.)


30.

___ ∙

___ ∙

___ ∙

___ =

29. –3 + –4 – –5 – +7 =

24. The triangles are congruent.

perimeter = 24 cm

length of shortest side =

26. Label point M at the midpoint of ___

AB .

Use work area.

( , ), ( , )

LESSON 101 Ratio Problems Involving Name -...

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