Unit 8: Proportional Reasoning. Rates & Scaled Diagrams

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1 Unit 8: Proportional Reasoning Rates & Scaled Diagrams

2 Rates In Grade 8, you explored the difference between a rate and a unit rate In this unit, students will represent a rate in different ways, determine a unit rate, and will then use unit rates to solve problems and make decisions.

3 What is a rate? A rate is a comparison between two things with different units Examples: Price text person month 3000 text 120 km month 6L 50 miles hour 100 km gallon Other examples: Fuel consumption, speed, food prices, monthly fees, etc...

4 A rate is a ratio that is used to compare different kinds of quantities. A unit rate describes how many units of the first type of quantity corresponds to one unit of the second type of quantity. Some common unit rates are: miles (or kilometers) per hour, cost per item, earnings per week, month or year In each case the first quantity is related to 1 unit of the second quantity.

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6 Converting rate to unit rate Rate can be converted to unit rate simply by dividing the first term by second term. Consider an example: Q1. If a car travels 45 km in 30 min, what is the rate at which the car is travelling? Solution: If we express the rate in terms of kilometres and hours, the rate would simply be: 45km 0.5 hr If we want the unit rate (that is, kms per 1 hr ) we simply divide the top number by the bottom number to get: 90 km hr If we want to express in minutes, then the rate would be: 45 km = 1.5 km 30 min hr

7 Q2: If Madison bought 2.5 kg of rice for $7.50, then what is the unit price of rice? Solution: The denominator should be 2.5 kg and numerator should be $7.50. The unit price would be: $7.50 = $3 $3 = $ kg 2.2lb kg 1 kg = 2.2 lb lb

8 Practice Write the following ratio as a unit rate (km/hr): 474km in 6 hours: 474 km = 79 km 6 hrs hr Write the unit rate (cost/muffin): $2 for 5 muffins $2.00 = $ muffins muffin A 16-oz jar of peanut butter costs $4.39. To the nearest cent, find the unit price in cents per ounce. 439 cents = 27 cents 16 oz oz

9 Rates and unit rates are used to solve many real world problems. For example consider the problem: It takes 30 minutes for a tap to fill one bucket. How much time would it take to fill 6 buckets? 30 min x 6 buckets = 180 min bucket

10 Comparing unit rates can help you to make decisions In the month of May, Xander sent 4216 texts. During a one-week period, Xavier sent 1008 texts. Who sent more per day? Find unit rate per day Xander days Xavier = 136 day days = 144 day Xavier sends more texts per day b/c he has the higher unit rate.

11 You can use the unit price to compare prices and see which is the better buy. At Walmart, paper towels are sold in a 2 roll package for $2.49 and $12.99 for a 12 pack Which is the better buy (ie has the lower unit price) 2 roll $ rolls 12 roll = $1.25 roll $12.99 = $ rolls roll The 12 pack is the better buy b/c is has the lower unit price

12 How much would you save by buying one 12 pack as opposed to six 2 pack rolls? 12 pack 6(2 pack) $ ($2.49)= $ $1.95 One would save $1.95 (WOOP!) What else should you consider besides the unit price when buying stuff?

13 Examples 1. Randell buys a package of 15 pencils for $4.50 at Walmart. Angela buys a box of 50 pencils at Staples for $ Which is the better buy? Walmart Staples $4.50 = 30 cents $14.00 = 28 cents 15 pencil pencil 50 pencil pencil Staples is better by 2 cents/pencil

14 2. Each of your finger nails grows at about 0.05 cm/week. Each of your toenails grows at about 0.65 cm/year. Do your toenails or finger nails grow faster? Fingernails 0.05 cm x 52 weeks = 2.60 cm week year year Fingernails grow much quicker

15 3. Matt can buy a 12 kg turkey from a butcher for $ A local supermarket has a sale on turkeys for $1.49/lb. There are 2.2 lb in 1 kg. Which store has the lower price? $42.89 $1.49 x 2.2 lb 12 kg lb kg = $3.57/kg = $3.28/kg The supermarket is cheaper

16 Practice for you to try Pg : 1a, 2b, 4, 7ac

17 Problems involving rates During a Terry Fox Run, student volunteers distribute 250 ml cups of water to participants as they cross the finish line. Each volunteer has a cooler that can hold 64 L of water. How many cups of water can each volunteer dispense? 250 ml = 0.25 L 64 L 0.25 L/cup = 256 cups Each volunteer can give out 256 cups of water.

18 Loose-leaf paper costs $1.49 for 200 sheets or $3.49 for 500 sheets. What is the least you can pay for 100 sheets? 1600 sheets?

19 Questions from a sample exam Pepsi is sold at a local store in cases of 24 for $5.99. What is the unit price per can? $5.99 = $25 cents 24 cans can

20 Spray foam cans come in cases of 8 for $ A carpenter needs to order 132 cans of spray foam. If she can only order by the case, how much money will she spend? 132 cans = 16.5 cans 8 cases case She has to buy 17 cases 17 $62.00/case = $1054

21 Seth needs 45 pencils for school this year. He goes to the store and can buy the pencils in packages of 5 for $0.75 or packs of 15 for $2.10. Which option is the better buy and how much does he save? A) 5 pack saves $0.45 B) 5 pack saves $1.35 C) 15 pack saves $0.45 D) 15 pack saves $ x 9 = $6.75 $2.10 x 3 = $6.30 C is the better option and the correct answer

22 Graphs to represent rates You will be expected to draw and interpret graphs which illustrate various rates, such as gas rates, speed, etc. Consider the following example:

23 Using the graph, discuss the population change in the province from

24 What does a positive slope represent? What does a negative slope represent? Decrease in population What does the slope of a horizontal line mean? Increase in population No change in population What does the steepness of the slope mean? How much the population changes per year

25 The average monthly temperature in Corner Brook, NL, in C, is recorded in the chart below. The sampling period for this data covers 30 years.

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27 Between which two months is the rate of change in temperature the greatest? The least? Between October & November Between July & August

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37 Examples 1. A circle has been transformed so that its image radius is 14 cm. If the scale factor is 0.4, what is the radius of the original circle? K = model = 0.4 = 14 = 0.4r = > r = 35cm original r

38 2. If the Titanic was 883 feet long, the model below is a 1 to 570 scale model, how long is the model? K = model --> 1 = x --> x = 883 = 1.55ft Original

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41 For you to do!! PG. 465: #5 PG. 471: #1a, 2, 13a

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48 What is the relationship between the scale factor (K) and the area? The area will increase/decrease by a factor of K2

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50 2. Travis and Matt painted a mural on the wall, measuring 12 ft by 8 ft using an overhead projector. If the original sketch had an area of 216 in2, what is the scale factor? A= 216 in2 12 ft = 144 in 8 ft = 96 in A' = 144 x 96 = 13, 824 in2 K2 = A' = 13, 824 = 64 --> K = 8 A 216

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54 Da Last (used to be) Topic!! Investigate the relationship between the scale factor and the surface area and the scale factor and volume of a 3D shape

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56 What is the relationship between the surface area and the scale factor? Increases/Decreases by a factor of K2 K2 = A' A What is the relationship between the volume and the scale factor? Increases/Decreases by a factor of K3 K3 = V' V

57 Examples 1. The surface area of the original sphere is 42 cm 2 What's the surface area of the image if a scale factor of 3.5 is applied to the radius of the original sphere? K2 = A' --> 3.52 = A' --> A' = 42(3.52) = 514.5cm2 A 42

58 Examples 2. The volume of the original sphere is 150 cm3 What's the volume of the image if a scale factor of 5 is applied to the radius of the original sphere? K3 = V' --> 53 = V' --> V' = 150(53) 18,750cm3 V 150

59 3. The volume of a can (cylinder) is 355 ml. What is the volume of its image if a scale factor of 3:4 is applied to the radius and height of the original can? K3 = V' --> (0.75)3 = V' --> V' = 355(0.75)3 = ml V 355

60 4. What is the scale factor for a sphere when the volume of the original is 250 mm3 and its image is mm3? K3 = = > K =

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62 Other questions 1. The surface area of a cone is 36 ft2. What is the surface area of its image if a scale factor of 1:4 is applied? 2. Find the volume of a cylinder if its image has a volume of 450 cm3 and a scale factor of 2:3. Round your answer to the nearest cubic centimetre. 3. What is the scale factor of the following pairs of similar spheres? (a) Volume of the original is 450 mm3 and its image is mm3. (b) Surface area of the original is 248 in2 and its image is in2.