1. Proportion and Percents
September 2016
Proportion and Percents

2. Ratios and Rates
Key Skill: SWBAT use ratios to make comparisons.

3. How do we compare Numbers?
A basketball player shoots 8 free throws in a game.
She makes 5 out of 8.
How might we state this fact using ratios?

4. Ways to Write Ratios
5:8
5 to 8
5 out of 8

5. Another Ratio
What is another ratio we could write to describe making 5 out of 8 free throws?

6. Another Ratio
What is another ratio we could write to describe making 5 out of 8 free throws?
5:3 the ratio of made shots to missed shots

7. Making Predictions
Over the course of a season, the 5:8 player will shoot 72 free throws.
Use a ratio to predict how many shots the player will make.

8. Making Predictions
Over the course of a season, the 5:8 player will shoot 72 free throws.
Use a ratio to predict how many shots the player will make.
FT made
FT made
FT taken
FT taken

9. Key Vocabulary Equivalent Ratios are two ratios with the same value.
Example: 2:3 and 4:6 are equivalent ratios.

10. 10.1.2 - Compare Ratios Mixture A has 3 bottles of red and 1 of white:
Mixture B has 2 bottles of red and 1 of white:
Which is a darker red?

11. Solution
We could use either the ratio of red:total paint, OR the ratio of red:white.
3:4 is greater than 2:3, so mixture A is darker, OR
3:1 is greater than 2:1, so mixture A is darker

12. Remember!
If you put the number of red bottles on the left side in the first ratio, you MUST put the number of red bottles on the left side in the second ratio.

13. Using the Ratio
If I want to paint a house using the color made by mixture A, but I will need 16 total bottles of paint.
How many bottles of red do I need?
How many bottles of white do I need?

14. Ratio Table
Red 3 6 9 12
White 1 2 4
Total 8 16

15. Proportional Relationships
Key Skill: SWBAT determine if a proportional relationship applies to given situations.

16. Example In a small bag of 40 M&Ms, we find 9 reds.
In a large bag of 80 M&Ms, we find 18 reds.
What does this tell us?

17. Example In a small bag of 40 M&Ms, we find 9 reds.
In a large bag of 80 M&Ms, we find 18 reds.
What does this tell us?
The ratios between red M&Ms and total M&Ms are equivalent.

18. Key Vocabulary
Proportional relationship is one in which all pairs of corresponding values have equivalent ratios.
Remember, equivalent ratios are ratios with the same value, such as 1:2 and 2:4.

19. Proportions
This is what a proportion looks like:

20. Proportions A Proportion is two ratios set equal to each other.
This is what a proportion looks like:
We can prove they are equal by cross-multiplying: 1 x 9 = 3 x 3
9 = 9

21. Cross-Multiplying
To cross multiply means to multiply the numerator (top) of one ratio by the denominator (bottom) of the other and set the results equal to each other.
We then solve for ‘x’ in the usual way to find the missing value.

22. Example

23. Example
3(x) = 8(11)
3x = 88
x = 29.3

24. Equal Ratios
Key Skill: SWBAT identify proportional shapes and solve for a missing value in a proportion.

25. Key Vocabulary
Similarity: Two geometrical objects are called similar if they have the same exact shape, but not necessarily the same size.
Similar shapes have side lengths that are proportional.

26. Example
Are these two triangles similar?

27. Example
Are these two right triangles similar?
Yes!

28. Example
Are these two right triangles similar?

29. Example
Are these two right triangles similar?
No!

30. Similar Shapes What shapes are always similar?
What shapes might be similar, but might not?

31. Shapes Always similar: circles, squares, regular polygons.
NOT always similar: triangles, rectangles, irregular polygons.

32. Example Use a proportion to find the missing side length: ? 5cm 4cm

33. Example Use a proportion to find the missing side length: ? 5cm 4cm

34. Algebraic Example
If I buy 6 packages of trail mix for $26, how many could I buy for $104?

35. Algebraic Example
If I buy 6 packages of trail mix for $26, how many could I buy for $104?

36. Solving Proportions
Key Skill: SWBAT set up proportions to solve problems.

37. Use a Proportion
If we have a book with a scaled photo of Starry Night that measures 6.9cm high by 8.6 cm long, how long is the real painting?

38. Starry Night
73.7cm
???

39. Use a Proportion
If we have a book with a scaled photo of Starry Night that measures 6.9cm high by 8.6 cm long, how long is the real painting? cm

40. Key Vocabulary
Proportion is an equation that states that two ratios are equal.

41. More Examples

42. Map Scales
Key Skill: SWBAT solve problems involving map scales using proportions.

43. Key Vocabulary
Scale Drawing is created at a specific ratio relative to the actual size of the place or object.

44. Examples
The picture of Starry Night from our last lesson is a scale drawing of the actual Van Gogh painting.
A model of a battle ship might be at a scale of 1/1000, meaning the model is much smaller than the real thing.
A model of a molecule might be 1,000,000,000/1 meaning the model is much larger than the real thing

45. Scale Models

46. Map Scales Maps are scale drawings of the real world.
As you would expect, they are generally much smaller than the real world.

47. Key Vocabulary
Map Scale is the ratio between distance on a map and the actual distance of what the map represents.

48. Example
1 inch = 10 miles

49. Ratio Table
Map Inches 1 2 3 4 5
Actual Miles 10 20 30 40 50

50. Map Scale Problems
I measure the distance between two cities at 4 inches. If the map scale is: inch = 10 miles, what is the actual distance between the two cities?

51. Solution
I measure the distance between two cities at 4 inches. What is the actual distance between the two cities?
inches
inches
miles
miles
x = 40 miles

52. Map Scale Problems
What is the distance between the two cities if the measurement on the map is 4 1/4 inches?

53. Solution
I measure the distance between two cities at 4 1/4 inches. What is the actual distance between the two cities?

54. Remember!
If you put inches in the numerator (on top) on one side, you MUST put inches in the numerator on the other side!

55. Summary Map scales can be viewed as proportions.
We can use proportions to solve for a distance or to solve for a distance on a map.
Question: On one map we measure a distance of 1.5 inches between two cities. On another map we measure a distance of 2 inches between the same two cities. How is this possible?

56. Creating Scale Drawings
Key Skill: SWBAT create scale drawings of rectangles and triangles, both larger and smaller than the originals.

57. Scale Drawings
Remember in prior lessons, we examined representations of objects that were smaller than the originals, but were exactly the same shape.
Starry Night picture versus the painting
Map versus the real world
These are called Similar figures

58. Key Vocabulary
Scale Drawings are created with new dimensions at a specific ratio to the original

59. Key Vocabulary
Scale or “Scale Factor” is the ratio between the drawing and the original.
Drawing is 1/10 the size of the horse

60. Large Scale Factors
Scale drawings can also be LARGER than the original figure.
Scale: about 10,000,000,000:1

61. Creating Scale Drawings
Assume this rectangle measures 6cm by 10cm
Create two scale drawings, with scale factors of 2:1 and 1:2
10cm
6cm

62. Creating Scale Drawings
10cm
6cm
12cm
5cm
3cm
20cm

63. Now with Triangles
Why is creating a scale drawing more complicated with triangles?
Because the angle measures are important!
Angles in rectangles are all 90º
12cm
12cm
4cm

64. Now with Triangles
Create two scale drawings, with scale factors of 2:1 and 1:2
12cm
12cm
4cm

65. Now with Triangles
12cm
12cm
4cm
24cm
6cm
2cm
8cm

66. Classwork Draw the original and scale drawings:
4cm by 8cm rectangle, scale drawings with scale factors of 2:1 and 1:4
3cm by 3cm square, scale drawings with scale factors of 3:1 and 1:3
3cm by 4cm by 5cm right triangle, scale drawings with scale factors of 2:1 and 1:2

67. Similarity
Key Skill: SWBAT use proportions to find the missing measurements of similar triangles.

68. Using Proportions
How did explorers in prior centuries estimate the height of trees and mountains or the width of a canyon?

69. 10.3.2 - Percents & Proportions
Key Skill: WWBAT solve percent problems by using proportions.

70. Key Vocabulary
Percent - “for each 100”
Latin

71. Fractions v Percents
Which is larger?

72. Fractions v Percents Which is larger?
The first equates to 47.4%, the second to 47.7%
Now which is larger?

73. Percent Review
To convert a decimal to a percent, move the decimal point 2 places to the right
0.63 = 63%
To convert a percent to a decimal, move the decimal point 2 places left
63% = 0.63

74. Percent Review
To change a fraction to a percent, we DIVIDE the numerator by the denominator, then move the decimal 2 places to the right
1/4 = 1÷4 = .25 = 25%
To change back to a fraction, place the percentage over 100, then reduce
25% = 25/100 = 1/4

75. Example
What is 80% of 40? Express as a proportion.

76. Example What is 80% of 40? Express as a proportion.
Think about which number is the ‘part’ and which is the ‘whole’

77. Example What is 80% of 40? Express as a proportion. 80 x 40 = 100x

78. Example
What percent of 60 is 12? Express as a proportion.

79. Example What percent of 60 is 12? Express as a proportion.
Again, think about which number is the ‘part’ and which is the ‘whole’.
60x = 1200 and x = 20

80. Example
8 is 40% of what number? Express as a proportion.

81. Example 8 is 40% of what number? Express as a proportion.
Again, think about which number is the ‘part’ and which is the ‘whole’.
40x = 800 and x = 20

82. Tips for Percent Problems
100 will always go on the bottom of one side.
Whatever follows the word “of” will be on the bottom of the other side.

83. Example Lemonade recipe is 25% juice concentrate and 75% water.
How much lemonade could I make with 6 gallons of concentrate?

84. Example Lemonade recipe is 25% juice concentrate and 75% water.
How much lemonade could I make with 6 gallons of concentrate?
25x = 600 and x = 24

85. Unit Rates
Key Skill: SWBAT calculate the unit rate for a variety of items.

86. Which is a better deal? Benny’s Bagels: $2.25 for half a dozen
Bagel ‘R Us: Buy a dozen, get one free for $4.75

87. Benny’s v Bagels ‘R Us
Benny’s
Bagels ‘R Us

88. How Much for 18 Bagels?
What is the proportion?

89. How Much for 18 Bagels?
What is the proportion? or

90. Key Vocabulary
A unit rate is the cost to buy a single unit of something, OR how much you can buy for a single dollar

91. Examples These are unit rates: These are NOT unit rates:
2 lbs of oranges for $1
$10/hour to rent a bicycle
$3.05/gallon of gasoline
These are NOT unit rates:
3 t-shirts for $15
$5 for 500 text messages
$60 for 3 hours of work raking leaves

92. Unit Rates with Fractions
Key Skill: SWBAT calculate the unit rate for a variety of items using fractions.

93. Example
If a person walks 1/2 mile in each 1/4 hour, what is the person’s “unit rate” of walking?

94. Example
If a person walks 1/2 mile in each 1/4 hour, what is the person’s “unit rate” of walking?
The question asks how far will the person walk in one hour:
x = 2 miles/hr
Complex Fraction

95. Example
If 2/5 of a pound of grapes costs $2.20, what is the unit rate?

96. Example
If 2/5 of a pound of grapes costs $2.20, what is the unit rate?
The question asks how much does 1 pound cost?
x = $5.50

97. Example
If a team of workers can build 2.7 homes in 12 months, what is the unit rate?

98. Example
If a team of workers can build 2.7 homes in 12 months, what is the unit rate?
The question asks how long it will take to build one home:
x = 4.4 months

99. Currencies
Key Skill: SWBAT use proportions to determine currency exchange amounts.

100. Currency Conversion
If you are in Japan and see a watch on sale for 7,697 yen, how much does it cost in US$ if the exchange rate is 1yen to $ ?

101. Currency Conversion
If you are in Japan and see a watch on sale for 7,697 yen, how much does it cost in US$ if the exchange rate is 1yen to $ ?

102. Exchange Rates Exchange Rates can be shown 2 ways:
1yen = $
$1 = yen
One view is from the perspective of US residents who use dollars; the other is from Japanese residents who use yen
What is the relationship between and 113.2?

103. Exchange Rates What is the relationship between 0.008835 and 113.2?
They are reciprocals!
In other words,
And