Unit 4 Grade 9 Applied

Proportional Reasoning: Ratio, Rate, and Proportion

Lesson Outline

BIG PICTURE Students will: solve problems involving proportional reasoning.

Math Learning Goals Expectations

Investigate ratio as a tool for comparing quantities, both qualitative and quantitative.

Estimate answers, and devise and explain informal solutions (e.g., constant of proportionality, unit rate, equivalent ratios) in a variety of contexts (e.g., numerical, geometric, measurement, probability, algebraic).

Investigate and determine what a ratio is using examples and non-examples of proportional and non-proportional situations (e.g., two ordered quantities that share a multiplicative relationship).

Determine the characteristics of the graph of a proportional relationship.

Explore and develop an understanding of proportions, estimate answers, and devise and explain informal solutions (e.g., constant of proportionality, unit rate) in a variety of contexts (e.g., numerical, geometric, measurement, probability, algebraic).

Solve problems using the Pythagorean relationship to connect proportional reasoning to contexts.

Investigate a variety of methods for solving problems using proportions (e.g., scaling/tables, drawings, constant of proportionality, unit rate, cross products).

Solve problems involving ratios, rates, and directly proportional relationships in a variety of contexts.

Use estimation and proportional reasoning to determine the population size based on a random sample.

Solve problems and make comparisons using unit rates. Investigate percent as a proportional relationship. Solve problems involving percents, ratios, fractions, and

decimals in a variety of contexts. Investigate that proportionality is a multiplicative process and

not an additive process. Consolidate concept understanding and procedural fluency

for ratio, proportion, and percents.

LR1.03 LR2.02 LR2.03 MG2.02 NA1.01 NA1.02 NA1.03 NA1.04 NA1.05 NA2.02

4.1: Who Eats the Most? (Refer to the cards on 4.2: Who Eats the Most) The cards at this station give information on the average weight and the average daily food intake for a variety of animals. 1. Put the cards in order of how much they eat from highest to lowest. Record each animal with

the corresponding data beneath each animal.

2. Now put the cards in order of how much they eat relative to their weight (from highest to

lowest). Record each animal with the corresponding rate beneath each animal.

3. Explain what strategies you used to complete question 2. 4. Which order do you think best represents who eats the most? Explain.

4.2: Who Eats the Most? Cards

Vampire Bat Queen Bee

Daily Food Intake: 28 g Weight: 28 g

Daily Food Intake: 9 g Weight: 0.113 g

Tiger Hamster

Daily Food Intake: 6.4 kg Weight: 227 kg

Daily Food Intake: 11 g Weight: 100 g

Elephant Hummingbird

Daily Food Intake: 180 kg Weight: 4100 kg

Daily Food Intake: 2 g Weight: 3.1 g

Blue Whale Giant Panda

Daily Food Intake: 4.5 tons Weight: 118 tons

Daily Food Intake: 15 kg Weight: 125 kg

Adapted from: NCTM “World’s Largest Math Event 2000.”

4.3: What’s in the Bag? You have a bag with two different-coloured tiles. 1. Without looking, pull a tile out of the bag. Make a tally mark in the appropriate column in the

table below. 2. Put the tile back into the bag and shake it up. 3. Repeat steps 1 and 2 a total of 20 times.

Colour 1: Colour 2:

Tally

Total

4. What appears to be the ratio of colour 1 to colour 2 in your bag? 5. Answer the following questions using the information you have collected.

Justify your answers. a) If you had 30 of colour 1 in your bag, how many of colour 2 would you expect to have? b) If you had 20 of colour 2 in your bag, how many of colour 1 would you expect to have? c) If you had a total of 80 tiles in your bag, how many of each colour would you expect to

have? d) If you had 40 of colour 1 in your bag, how many tiles in total would you expect to have?

4.4: Ratios and Rates Definitions

Ratio – a comparison of quantities with the same units. Example – in a deck of cards the ratio of aces to face cards is 4 to 12.

**Note: a ratio is in simplest form when there are no common factors.

Therefore our ratio above in simplest form is 1 to 3. Ratio Notations – There are 3 basic forms to represent ratio.

1. 5 to 8 2. 5 : 8

3. 8

5

Rate – a comparison of two quantities with different units. Example – 400 kilometres per 4 hours is a rate of speed. - also written in the form 400 km/4 h. Unit Rate – is a rate in which the second term is 1.

** Note: the unit rate for 400 km /4 h would be 100 km/h. This can be found by dividing.

Equivalent Ratios – two ratios which have the same value. This can be

done by multiplying or dividing both terms by the same value.

Examples: Ratio Equivalent Ratio

2

4

1

2 or

8

16

1 : 3 2 : 6 or 5 : 15

4.5: Ratios and Rate Practice #1

Show your work for each question. Be sure to put all answers in simplest form.

Using Ratios 1. a) For planting shrubs, a landscaper mixes topsoil and peat moss in a ratio of

4:1. He has two 20 L bags of topsoil. How much peat moss does he need?

b) What will be the total volume of the mixture? 2. The ratio of males to females in Brian’s homeroom is 14 : 16. The ratio of males

to females in the school is 350 : 400. Brian argues that these ratios are equivalent. Is he correct? Explain.

3. To color 4 L of base paint, add 20 mL of tint. What is the ratio of tint to base

paint in the mixture?

Using Rates 4. After a 6 h snowstorm, there were 24 cm of new snow on the roads.

At what average rate did the snow fall during the storm?

5. A grocery store charges $1.19 for a 500 g package of sugar or $2.49 for a 2 kg

package. a) The labels on the grocery store shelf often show a unit price for items. What

is the unit price in dollars per 100 g for each package of sugar?

b) Which is the better buy? Explain. 6. a) During a trip from Chatham to Ottawa, Nelson traveled 700 km and used 80 L

of fuel. What was this car’s fuel consumption rate in Litres per 100 km for highway driving?

b) Driving back and forth to work within Chatham, Nelson traveled 490 km and used 65 L of fuel. What was the fuel consumption rate for driving in the city?

4.6: A Global Village

In a global village of 100 people there are:

61 people from Asia

13 people from Africa

12 people from Europe

08 people from South America including Central America, Mexico and the Caribbean

05 people from the United States and Canada

01 person from Oceania (Australia, New Zealand, and Pacific islands)

On the grid represent the global village of 100 by colouring in the squares. Create a legend by marking the colour beside each statistic

Record the following ratios:

a. People from Europe : People from Asia ________

b. People from America : People from Oceania ________

c. People from Africa : People from United States and Canada ________

There are approximately 1200 students in a school. If the student body resembled the global village described, how many people would be from each of the 6 regions. Show your work and describe the strategy that you used.

2006 Winter Olympics: During the Winter Games, the media tell the world about Olympic events and topical issues. In Torino, almost 10 000 men and women provided images, words, and photos of the Olympic Games. If the media resembled the global village described, how many would be from each of the 6 regions? Fill in the table with your solution. Show your work. Do you think that these numbers represent the actual numbers of the media from each region? Explain your thinking. Connect and Reflect: The World Cup of Soccer played in Germany during the summer of 2006. The distribution of the 32 teams in the tournament representing the same regions of the world is listed below:

Region # of Teams

Asia 4 Teams

Europe 14 Teams

United States and Canada 1 Team

Africa 5 Teams

South America 7 Teams

Oceania 1 Team

How well does the distribution of the teams represent the populations for the 6 regions? Explain your thinking.

Region Media People

Asia

Europe

United States and Canada

Africa

South America

Oceania

4.7: Best Buy

You can buy trail mix at the health food store in four different-sized packages. The table shows the package size and cost:

Size Cost

100g $1.00

300g $2.00

500g $3.00

1kg $4.50

Task: Use a geoboard to find the unit rates per 100g and determine the best buy. On the geoboard, create unit rate triangles for each of the four trail mix package sizes. The vertical scale is the cost and the horizontal scale is the weight.

Use the following horizontal scale: 1 space = 100 g

Use the following vertical scale: 1 space = $0.50

Use geobands to make rate triangles for each of the four packages on your geoboard. Record your triangles below:

Determine and record the unit rate (per 100g) for each package of trail mix:

Rank the four packages from best value to least value:

1. 2. 3. 4.

Which package would you recommend? Give reasons for your choice. Explain how the geoboard helped you with this problem.

Size Cost/100g

100g

300g

500g

1kg

4.8: Fish Tales Have you ever wondered how scientists estimate how many fish there are in a lake? Try this “capture-recapture” activity. The shoebox is your “lake.” It has an unknown quantity of fish (colour tiles). You cannot see how many fish are in the lake. Instructions:

• Take a handful of the fish.

• “Tag” the fish by placing a dot sticker on each of the colour tiles.

• Count the number of tagged fish and place them back into the lake.

• Mix the tiles up and redistribute the tagged fish in your lake.

• Each group member: − Takes one handful of fish − Counts the total number of captured fish − Counts the number of tagged fish

• Gather everyone’s data and use the information to estimate the number of fish in the lake.

Questions

1. What is the ratio of tagged fish to untagged fish?

2. What is the ratio of the total number tagged and the total population in the lake?

3. Compare your answer to the actual value. How close was your estimate?

4. Where else could this method for approximation of total population be used?

4.9: Ratios and Rate Practice #2

1. What is the better deal? HINT: Find the Unit Rate.

a) Flour: $1.15/kg OR $5.79 for a 5-kg bag

b) Linguini: $0.004/g OR $1.39 for a 500 g box

c) Dog Food: $2.09/kg OR $14.95 for an 8 kg bag

d) Pens: 10 pens for $1.99 OR 15 pens for $3.15

e) DVDs: $2.49 for 10 OR $5.99 for 25

2. Given the following ingredients: ½ cup white flour ¼ cup rolled oats ½ cup brown sugar 100 g butter

a) Write the ratio of flour : sugar : oats in simplest form.

b) Why can butter not be included in this ratio?

c) If you need to make four times the amount of this recipe, how much of each ingredient should you use?

3. Write each rate as a UNIT RATE. a) 30 mm of rain in 3 h _____________________

b) $4.99 for 24 cans of pop _____________________

c) $50 for 4 h of work _____________________

d) 14 laps in 4 min _____________________

e) 30 pages read in 20 min _____________________

4. Convert the following currencies given the exchange rate:

$1 US = $1.13 CDN

a) $200 US = _________ CDN b) $100 CDN = _________ US

c) $532 US = _________ CDN d) $42 CDN = _________ US

5. Two different websites allow users to download songs. Which Website gives the better deal? Show your calculations

Music at Work Nautical Disaster 200 songs costs $32.00 220 songs for $34.00

4.10: Blast from the Past

These review questions will help prepare you for the investigations which follow. 1. Find the perimeter of the square.

2. Find the area of the square.

3. Find the diagonal length of the square.

4. Find the volume of the cube.

5 cm 5 cm

5 cm

5cm

4.11: Anticipation Guide Instructions Check Agree or Disagree, in ink, in the Before category beside each statement before you

start the Growing Dilemma task.

Compare your choice with your partner.

Revisit your choices at the end of the investigation.

Before Statement

After

Agree Disagree Agree Disagree

1. If you double the length of a square, then the perimeter also doubles.

2. If you double the length of a square, then the area also doubles.

3. If you double the length of a square, then the length of the diagonal also doubles.

4. If you double the sides of a cube, then the volume also doubles.

Conclusion – Answer these questions once you finish your investigations a) Which of the 4 relationships that you have investigated are proportional?

b) What else can you conclude about relationships that are proportional?

A ___________________ is a statement of two equal ratios.

Investigation 1: Perimeter Ratios

Use the colour tiles to create squares with the indicated side length.

1. Determine the perimeter for each side length.

2. Complete the chart.

3. Graph Perimeter vs. Side Length on the grid provided.

4. State the characteristics of this relationship: a) first differences b) ratios c) graph

Side Length

(S)

Perimeter (P)

Ratio (S:P)

Ratio in Lowest Terms First

Differences

1

2

3

4

5

Investigation 2: Area Ratios

Use the colour tiles to create squares with the indicated side length.

1. Determine the area for each side length.

2. Complete the chart.

3. Graph Area vs. Side Length on the grid provided.

4. State 3 characteristics of this relationship: a) first differences b) ratios

c) graph

Side Length

(S)

Area (A)

Ratio (S:A)

Ratio in Lowest Terms First

Differences

1

2

3

4

5

Investigation 3: Diagonal Length Ratios 1. Determine the length of the diagonal for each side length.

NOTE: Stop the decimals at 3 spots. Do not Round

2. Complete the chart.

3. Graph Diagonal Length vs. Side Length on the grid provided.

4. State 3 characteristics of this relationship: a) first differences b) ratios

c) graph

Side Length

(S)

Diagonal (D)

Ratio (S:D)

Ratio in Lowest Terms First

Differences

1

2

3

4

5

Investigation 4: Volume Ratios

Use the linking cubes or tiles to create cubes with the indicated side length.

1. Determine the volume of the cube for each side length.

2. Complete the chart.

3. Graph Volume vs. Side Length on the grid provided.

4. State 3 characteristics of this relationship: a) first differences b) ratios

c) graph

Side Length

(S)

Volume (V)

Ratio (S:V)

Ratio in Lowest Terms First

Differences

1

2

3

4

5

4.12: Television Viewing Use a different method to complete each part of the question.

You should be prepared to explain your methods to the class.

Did you know that there is an optimal distance for a person to be from a television for ideal viewing? The ratio of the size of the television screen to the distance a person should sit from it is 1:6. a) How far away should a person sit from a 20-inch television?

b) If the room is 17 feet long, can a person sit at an optimal distance from a 27-inch television? Explain your reasoning.

c) What is the largest television that can be used in the 17-foot room for a person to sit an

optimal distance from it?

Basic Television Information Traditional televisions have a ratio of width to height of 4:3.

High definition televisions (HDTV) have a ratio of width to height of 16:9.

Television sizes are given as the length of the diagonal of the screen, i.e., a 27-inch television is 27 inches from one corner to the diagonally opposite corner.

Problem 1 Darren wants to buy a new television. He finds a traditional television at the store and measures the width of it to make sure it fits in his home. He measures the width to be 24 inches but he forgets to measure the height and the diagonal.

a) What is the height of the television? b) What is the size of the television? (the length of the diagonal)

Problem 2 Sasha is buying a new HDTV. She finds one and measures the width to be about 35 inches.

a) What is the height of the television? b) What is the size of the television? c) What is the optimal viewing distance for Sasha’s new HDTV?

4.13: More Scaling Problems 1. Dandelion Inquiry

In testing a new product for its effectiveness in killing dandelions,

it is necessary to find an area containing many dandelions,

count them, apply the product, and count the dandelions

again at a later time. How might this be accomplished

without counting every single dandelion? Design a

technique different from the one used in class.

2. Interpreting Scale Diagrams

Recall that scale = diagram measurement : actual measurement a) Finding the scale

The actual length of this cell is 0.32 mm across. What scale was used to draw this diagram?

b) Using the scale

This diagram was drawn using a scale of 1:7. What is the actual height of this penguin?

c) Complete the table.

Scale Diagram Measurement Actual Measurement

i. 1:400 6 cm

ii. 12000:1 0.00375 mm

iii. 7.2 cm 0.6 mm

iv. 1:250000 8 cm

4.14: Planning a Class Trip (optional) Your class is planning a trip to a local attraction. Use the information below to make some decisions about timelines and budget.

Admission to the attraction is $15.00/person.

The attraction opens at 10:00 a.m. and closes at 4:00 p.m.

You want to serve juice boxes and granola bars on the bus on the way to the attraction.

It is 85 km from your school to the attraction.

The rented bus travels at an average speed of 70 km/h.

Diesel fuel costs 75.9 cents/L.

The bus uses 35 L of fuel every 100 km.

Snacks Record the cost for snacks that you calculated as a class.

Juice boxes

Granola bars

Timelines You want to arrive at the attraction when it opens and need to return to the school by 3:15 p.m. Determine the departure times from school in the morning and the attraction in the afternoon. You may find it helpful to highlight key information from the list above.

Departure time from school

Departure time from attraction

Fuel Determine the total amount of fuel required for the trip. Determine the total cost of fuel for the trip.

Total cost of fuel

Total Cost of Trip Assume that every student in your class is participating in this trip. In addition to the costs mentioned above, the bus company also charges a flat fee of $85 to pay the driver.

a) What is the total cost for your class to go on this trip? b) What is the cost per person? c) If three people decide not to go, how would the cost per person change? Explain your

answer.

4.15: Elastic Meter and Percent

Part A: Make the elastic meter 1. Take a piece of elastic 18 cm long. Mark a line at 1 cm from one end.

From this point make 10 marks every 1.5 cm. There will be a little left on the end.

2. On the first line write 10%; 2nd line, 20%; 3rd line, 30% (…up to 100%).

Part B: Use the elastic meter 3. Estimate from the bottom to the top where 60% of the right edge of your desk would be.

Put a very small pencil mark here. (Please erase it after the experiment.) 4. Stretch out the elastic meter from the bottom to the top.

Use the 60% mark on the elastic meter to correct your estimate. 5. Use a measuring tape to measure this length. Record it in the appropriate place in the

following chart.

Percent % Measure (cm)

Use your elastic meter to complete the chart.

0%

10%

33%

45%

50%

60%

75%

90%

100%

6. Graph your data on the grid below. Be sure to label your axes. Choose an appropriate scale. 7. On your graph draw a line of best fit.

Interpolate: Use line of best fit to estimate the lengths of the following percents:

Extrapolate: Estimate the following lengths:

a) 120% b) 135%

elastic

20%

10%

a) 85% b) 65%

c) 43% d) 58%

4.16: Types of Percent Problems – Guided Lesson 1. Determining an unknown part:

a) Do together: HDTVs are on sale for 25% off. What is the discount on a television that normally costs $885?

set up and solve a ratio

solve an equation 0.25 885 =

b) Do on your own: If you purchase a CD for $18.99, how much tax would you pay?

(15% for both GST and PST)

2. Determining an unknown percent:

a) Do together: Shuva purchased a new MP3 player on sale. It was $219.50 originally, but she paid $142.68, not including tax. What was the percent discount on the MP3 player?

set up and solve a ratio

solve an equation 219.50 = (219.50 – 142.68)

b) Do on your own: David was shopping for a new pair of shoes. He found a pair that was

$89.99 on sale for $22.50 off. What was the percent discount on the shoes?

3. Determining the unknown whole:

a) Do together:

Cayla wanted to return a defective calculator, but her dog Buster had chewed up the receipt. She could still see that the 15% tax came to $2.25. What was the cost of Cayla’s calculator?

b) Do on your own:

Himay was very happy because his new cell phone was on sale for 40% off and was only $65.00. What was the original price of Himay’s phone?

set up and solve a ratio

solve an equation 0.15 = $2.25

4.17: Review Relay

1. Reduce the ratios to lowest terms: 15:35 =

18

6 =

144:72 =

2. Calculate the following percents: 45% of 220 =

120% 555 =

1.5% 1400 =

3. The driving distance from Thunder Bay to Vancouver is approximately 2500 km.

How long would it take you to drive from Thunder Bay to Vancouver at 90 km/hour without making any stops?

4. If the ratio of the Canadian dollar to the US dollar is $1.40:$1.00, how much Canadian money is equivalent to US$250?

5. Measure the length indicated in centimetres. What is the actual length of the shark, in metres?

Scale Diagram

1:70

6. You want to purchase a new shirt that costs $22.50.

a) How much tax will you have to pay including GST and PST?

b) What is the total cost of your shirt?

7. You are shopping for DVDs at the video store with a $30.00 gift certificate that you received from a friend. You find a great DVD that was $34.50 on sale for 25% off.

Do you have enough money to buy the DVD including GST and PST?

8. You are working at Tecky Television Sales. Recall that the HDTV’s width:height ratio is 16:9. A customer wants to know:

a) If he has an entertainment centre that has an opening that is 48 inches wide, how high will the cabinet opening have to be?

b) If the cabinet opening is 48 inches by

32 inches, will a 50-inch HDTV fit inside?