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1 Number Sense and Numeration PRESENTED BY TROTTER Mathematics, Grade 7 Introduction Welcome to today s topic Parts of Presentation, questions, Q&A Housekeeping NOT the Chat Room Your questions Satisfaction Meter 1

2 What you will learn At the end of this presentation, you will be able to explain what ratios and equivalent ratios are use ratios to solve problems Agenda s in everyday life Definition and terms Comparing ratios Applications of ratios 2

3 Agenda s in everyday life Definition and terms Comparing ratios Applications of ratios s in everyday life Seetha and her little brother Valen are throwing a ball into a basketball hoop. Valen misses the basket a lot more than Seetha does. In fact, for every time Valen gets the ball into the basket, Seetha gets it in three times. There are lots of ways to say this: Valen gets 1/3 as many baskets as Seetha. Valen gets as many baskets as Seetha. The ratio of Valen s baskets to Seetha s baskets is 1:3. 3

4 s in everyday life A ratio describes how two numbers relate to each other. It can be expressed as a statement, a fraction, or percentage. We use ratios to compare quantities and to help us compare prices when we are shopping. We use ratios to compare how much snow and rain has fallen in different years. s in everyday life s are used in the following subjects: Mathematics, simple and complex Economics Physics Chemistry Biology Computer science 4

5 Agenda s in everyday life Definition and terms Comparing ratios Applications of ratios Agenda Definition and terms : Part to whole : Part to part Equivalent ratio Proportion 5

6 Definition : a comparison of two or more quantities with the same unit. For example, 2 red circles as part of 3 circles in total. The 2 and the 3 are the terms of the ratio. The unit is the circle. Part to whole ratio: ratio that compares a part of the whole to the whole, e.g. 2:3 This ratio compares circles. Circles are the unit. Part to part ratio: ratio that compares a part of the whole to another part, e.g. 2 squares:3 circles This ratio compares shapes. Shapes are the unit. Definition A comparison of two or more quantities with the same unit. For example: 2 red circles as part of 3 circles in total. The 2 and the 3 are the terms of the ratio. The unit is the circle. 6

7 Definition Part to whole ratio A ratio that compares a part of the whole to the whole, e.g., 2:3 This ratio compares circles. Circles are the unit. Definition Part to part ratio A ratio that compares a part of the whole to another part, e.g., 2 squares:3 circles This ratio compares shapes. Shapes are the unit. 7

8 Definition Equivalent s s having the same value. Example: 1:2 and 2:4 TIP 1(x2):2(x2)= 2:4 Scale factor: You make equivalent ratios by multiplying or dividing all of the terms of a ratio by a number. The number is known as the scale factor. Definition Proportion An equation stating that two ratios are equal. Example: 1:2 = 2:4 You do the same operation when you are finding equivalent fractions. Example: 8

9 Examples What is the part to part ratio of circles to squares? a) 4:3 b) 4:7 Examples What is the part to part ratio of circles to squares? a) 4:3 b) 4:7 9

10 Examples What is the part to whole ratio of circles to shapes? a) 4:3 b) 4:7 Examples What is the part to whole ratio of circles to shapes? a) 4:3 b) 4:7 10

11 Examples Which is the equivalent ratio to 4:5? a) 20:25 b) 7:8 Examples Which is the equivalent ratio to 4:5? a) 20:25 b) 7:8 11

12 Agenda s in everyday life Definition and terms Comparing ratios Applications of ratios Comparing s s can be compared to one another. To compare different ratios the terms of the ratios must have the same units. For example, you could compare the ratios of boys to girls in two different classes because you are comparing boys to girls in both classes. This would not work if you tried to compare boys and girls to students and teachers. The units are different. 12

13 Comparing s A 2:3 B 5:7 Class A has a ratio of 2 boys to 3 girls and Class B has a ratio of 5 boys to 7 girls. If the classes are the same size, which class has more boys? Comparing s To compare ratios, one of the term numbers must be the same! Let s look at the ratios: Class A 2:3 (boys to girls) Class B 5:7 (boys to girls) 13

14 Comparing s To compare the boys in the class we will need to use equivalent ratios to get the number of girls in each class to be the same. Comparing s The ratios of boys to girls are: Class A 2:3 and Class B 5:7 Remember learning about multiples? Multiples of a number are found by multiplying the number by 1, 2, 3, 4, 5, and so on. When we find multiples for two numbers, the ones that are the same are common multiples. What is the lowest common multiple of 3 and 7? 14

15 Comparing s Class A 2:3 and Class B 5:7 21 is common to both 3 and 7 Class A 2(x 7) : 3(x 7) = 14:21 Class B 5(x 3) : 7(x 3) = 15:21 For two given classes of the same size, Class B would have more boys. Agenda s in everyday life Definition and terms Comparing ratios Applications of ratios 15

16 Applications Why do we need to compare ratios? We compare ratios to answer questions and make decisions. Let s look at the previous problem. Applications Class A 2:3 and Class B 5:7 21 is common to both 3 and 7 Class A 2(x 7) : 3(x 7) = 14:21 Class B 5(x 3) : 7(x 3) = 15:21 We can draw conclusions from our comparisons, such as Class B has more boys than Class A. 16

17 Applications Let s revisit Class A for another example. Class A has a ratio of 2 boys:3 girls. How many male and female students would there be in a class of 25? Applications First, let s look at the ratio as a class. 2 boys:3 girls = 5 students This ratio represents 5 students. If the class has 25 students then the number of boys and girls must equal divided by 5 gives us a scale factor of 5 because each number in the original ratio was multiplied by 5 to get the equivalent ratio. 17

18 Applications If the scale factor is 5, then 2(x 5) : 3(x 5) = 5(x 5) 10: 15 = 25 Therefore, there are 10 males and 15 females in the class. Agenda s in everyday life Definition and terms Comparing ratios Applications of ratios 18

19 Equivalent ratios have the same value. They are multiplied or divided by the scale factor. 5:6 is equivalent to a) 25:30 b) 10:11 c) 15:20 Equivalent ratios have the same value. They are multiplied or divided by the scale factor. 5:6 is equivalent to a) 25:30 b) 10:11 c) 15:20 5:6 = 5(x 5):6(x 5) = 25:30 Another example of an equivalent ratio is: 5:6 = 5(x 10):6(x 10) = 50:60 19

20 The part to part ratio of circle to squares is a) 7:12 b) 7:5 c) 5:7 The part to part ratio of circle to squares is 7 circles to 5 squares = 7:5 a) 7:12 b) 7:5 c) 5:7 20

21 The part to whole ratio of circles to shapes is a) 5:4 b) 4:5 c) 5:9 The part to whole ratio of circles to shapes is 5 circles to 9 shapes = 5:9 a) 5:4 b) 4:5 c) 5:9 21

22 The ratio 3:9 is equivalent to 1:3 and 6:18. The ratio 3:9 is equivalent to 1:3 and 6:18. 3:9 = 3( 3) : 9( 3) = 1:3 and 3:9 = 3(x 2) : 9(x 2) = 6:18 22

23 You are offered 1:8 or 2:10 of a pizza. Which would give you more pizza? a) 1:8 b) 2:10 c) They are equal You are offered 1:8 or 2:10 of a pizza. Which would give you more pizza? a) 1:8 b) 2:10 c) They are equal 23

24 The 1 piece of the 8 slices takes up less area than the 2 pieces of the 10 slices. 1:8 = 10:80 2:10 = 16:80 (x10) (x8) 88 cm Maria makes a model of a building that is at a scale of 2 cm to 1 m. 2 cm:1 m The model is 88 cm high. How high is the building? 24

25 88 cm Maria makes a model of a building that is at a scale of 2 cm to 1 m. 2 cm:1 m The model is 88 cm high. How high is the building? 2 cm:1 m = 2 cm (x 44) :1 m (x 44) = 88 cm:44 m. The building is 44m high. Resources Math League Oswego City School District Regents Exam Prep Center Math/ratio/Prac.htm 25

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