Unit 4 Grade 9 Applied Proportional Reasoning: Ratio, · PDF fileUnit 4 Grade 9 Applied Proportional Reasoning: Ratio, Rate, ... • Solve problems involving ratios, rates, ... If

TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning 1

Unit 4 Grade 9 Applied Proportional Reasoning: Ratio, Rate, and Proportion Lesson Outline

BIG PICTURE Students will: • solve problems involving proportional reasoning.

Day Lesson Title Math Learning Goals Expectations 1 Ratio Carousel

GSP®4 file: Middle Mania

• 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).

NA1.01, NA1.02, NA1.03, NA1.04, NA1.05

CGE 5a, 5e

2 Growing Dilemma

• 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.

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

CGE 3c, 4b, 5a, 5b 3 Pondering

Proportions • 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.

NA1.01, NA1.02, NA1.03, NA1.04, NA1.05, MG2.02

CGE 2c, 2d

4 I’d Rather Be Scaling

• 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.

NA1.01, NA1.02, NA1.03, NA1.04, NA1.05

CGE 2a, 2c, 5b

5 Planning a Class Trip

• Investigate a variety of methods for solving problems using proportions. • Solve problems and make comparisons using unit rates.

NA1.01, NA1.02, NA1.03, NA1.04, NA1.05

CGE 2c, 5e 6 That’s

Stretching It • Investigate percent as a proportional relationship. • Solve problems involving percents, ratios, and decimals in a variety of

contexts.

NA1.01, NA1.02, NA1.03, NA1.04, NA1.05, NA1.06

CGE 5a, 7i 7 Proportion

Potpourri • Investigate that proportionality is a multiplicative process and not an

additive process. • Consolidate concept understanding and procedural fluency for ratio,

proportion, and percents. • Solve problems involving percents, fractions, and decimals in a variety

of contexts.

NA1.01, NA1.02, NA1.03, NA1.04, NA1.05, NA1.06

CGE 5a, 7i

8 Instructional Jazz 9 Instructional Jazz

10 Review and Assessment


Unit 4: Day 1: Ratio Carousel Grade 9 Applied

75 min

Math Learning Goals • 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).

Materials • 2 computers with

GSP®4 • 60 colour tiles

of 2 colours • BLM 4.1.1, 4.1.2

Assessment Opportunities

Minds On ...

Groups of 3 Graffiti Use heterogeneous groupings. Prepare chart paper for each of the following terms: ratio, rate, unit rate, equivalent ratios. Each group uses a different-coloured marker, cycles through the chart paper stations (2 minutes per chart), and writes characteristics of the term. Whole Class Presentation As a class, summarize the important points for each term students need to know for the carousel activity.

Action! Groups of 3 Carousel Prepare sufficient sets of each of the three stations. Students use a pencil and calculator and record their findings on BLM 4.1.1. Using the same groups as in the Minds On section, students rotate through the three stations. Direct the groups to move to the next station after 15 minutes. Learning Skills (Teamwork)/Observation/Checklist: Observe and record students’ collaboration skills.

Consolidate Debrief

Whole Class Summarizing Lead a class discussion using guiding questions (BLM 4.1.1). Using information from the discussion, define ratio, rate, and unit rate, using examples from the activity.

Middle Mania.gsp When assigning groups, sort them by colour. This activity establishes the class’s prior knowledge. Use different colours to distinguish each station set (one of each type). A group of students will complete one colour set of stations. Use two sets of cards (BLM 4.1.2) and the GSP®4 file Midpoint Segments for station setup. The student notes should connect the prior knowledge from the graffiti with the knowledge from the carousel.

Application Concept Practice

Home Activity or Further Classroom Consolidation Find examples of ratio, rate, and unit rate in your environment to post on the bulletin board.

The bulletin board becomes a class portfolio showing that proportions are pervasive. Collect further examples in other lessons in the unit.


4.1.1: Ratio Carousel Station: 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.


4.1.1: Ratio Carousel (continued) Station: What’s in the Bag? At your station 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?


4.1.1: Ratio Carousel (continued) Station: GSP®4 Middle Mania 1. Follow the Midpoint Triangle instructions and complete the following chart.

Area ∆ ABC Area ∆ DEF Ratio ABC/DEF 24.98 cm² 6.25 cm² 4.00:1

2. What do you notice about the ratio of the areas? 3. If Area ∆ABC = 64 cm², what is the area of ∆DEF? Explain. 4. If Area ∆DEF = 15 cm², what is the area of ∆ABC? Explain. 5. Follow the Midpoint Segments instructions and complete the following chart.

BC DE Ratio BC/DE

6. What do you notice about the ratio of the length of the line segments? 7. If the length of BC = 17 cm, what is the length of DE? Explain. 8. If length of DE = 22 cm, what is the length of BC? Explain.


4.1.2: 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.”


Middle Mania (GSP®4 File) Middle Mania.gsp Midpoint Triangles

Midpoint Segments


Unit 4: Day 2: Growing Dilemma Grade 9 Applied

75 min

Math Learning Goals • 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.

Materials • colour tiles (250) • linking cubes

(480) • BLM 4.2.1, 4.2.2


Minds On ...

Whole Class Discussion Lead a review of basic concepts needed for the investigation. Students need to be familiar with the concepts of a square, perimeter, area, length of a diagonal, cube, volume, ratio, and lowest terms. Pairs Anticipation Guide Students, individually, complete the Before column on the anticipation guide (BLM 4.2.1) and discuss their choices with their partner.

Action! Pairs Investigation Distribute 16 colour tiles and 27 linking cubes to each pair. Students work in pairs on the four investigations (BLM 4.2.2). Circulate to prompt, clarify, and focus the students on the task. Learning Skills (Collaboration)/Observation/Anecdotal: Observe the students’ contributions to completing the task.

Consolidate Debrief

Pairs Think/Pair/Share/Discussion Students complete the After column of the anticipation guide and share their choices with their partner, providing reasons for their choices. Adjacent pairs of students compare and discuss the results. Whole Class Note Making Lead a discussion to bring out that the perimeter and diagonal investigation show proportional reasoning and the others do not. Students should be able to explain how first differences, the ratios, and the graphs can all show proportionality.

Discussion around the length of the diagonal must include mention of the Pythagorean theorem. Provide limited resources so that students will infer results for larger models.

Concept Practice Reflection

Home Activity or Further Classroom Consolidation • In your journal, write a personal example of proportional reasoning. • Using the scenarios below, check for proportionality and justify your

response. a) You are paid an hourly wage. If you work 3 times the number of hours,

does your pay triple? b) Student council raffle tickets cost $0.50/each or 3 for $1. If you buy twice

as many tickets, does your cost double?


4.2.1: 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 After Agree Disagree

Statement 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.


4.2.2: Growing Dilemma Investigation 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


4.2.2: Growing Dilemma Investigation (continued) Investigation 2: Area Ratios Use the colour tiles to create squares with the indicated side length.

1. Determine the area for each side length.


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

4. State 3 characteristics of this relationship:

Side Length

(S) Area (A)

Ratio (S:A)


Differences 1

2

3

4

5


4.2.2: Growing Dilemma Investigation (continued) Investigation 3: Diagonal Length Ratios Use the colour tiles to create squares with the indicated side length.

1. Determine the length of the diagonal for each side length.


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

4. State 3 characteristics of this relationship:

Side Length

(S) Diagonal

(D) Ratio (S:D)


Differences 1

2

3

4

5


4.2.2: Growing Dilemma Investigation (continued) 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.


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

4. State 3 characteristics of this relationship: A proportion is a statement of two equal ratios. Conclusion a) Which of the 4 relationships that you have investigated are proportional? b) What else can you conclude about relationships that are proportional?

Side Length

(S) Volume

(V) Ratio (S:V)


Differences 1

2

3

4

5


Unit 4: Day 3: Pondering Proportions Grade 9 Applied

75 min

Math Learning Goals • 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.

Materials • BLM 4.3.1, 4.3.2 • linking cubes • colour tiles • grid paper • relational rods


Minds On ...

Whole Class Discussion Lead a discussion in which students share the informal methods of solving proportions from the Home Activity. Students may need to be reminded about conversions between feet and inches. (Do this in the context of ratios.)

Action! Groups of 4 Exploration Form heterogeneous groups based on students’ preferred learning style, using observations of their previous two days’ work. Students use concrete materials and at least two different informal methods for their exploration (BLM 4.3.1). Tools and Strategies/Observation/Mental Note: Assess the selection of tools and computational strategies.

Consolidate Debrief

Whole Class Discussion/Note Making Members from different groups share their solutions. Ensure that a variety of solution strategies are shared. Discuss which strategies were effective for the various types of problems. Summarize the various methods including equivalent ratios, the constant of proportionality, and algebraic reasoning. The algebraic reasoning may have to be formally taught, using the questions on BLM 4.3.2.

An alternate context may be more appropriate depending on the classroom environment. Direct members from struggling groups to visit and observe groups who are using successful strategies. Pythagorean theorem is required to complete BLM 4.3.2. Teachers may need to review this theorem.


Home Activity or Further Classroom Consolidation • Complete the questions on worksheet Television Dimensions. • Complete Pythagorean theorem questions.

Select appropriate practice questions.


4.3.1: 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?


4.3.2: Television Dimensions 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?


Unit 4: Day 4: I’d Rather Be Scaling Grade 9 Applied

75 min

Math Learning Goals • 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.

Materials • photos of crowds • grid paper • 2 colours of

number cubes • overhead

transparencies • BLM 4.4.1, 4.4.2


Minds On ...

Pairs Peer Coaching Students compare solutions and help each other with practice questions from the home activity. Whole Class Discussion Show an aerial photograph of a large crowd. Students suggest ways the number of people could be counted.

Action! Pairs Estimating Differentiate this activity by providing different photographs with varying density and distribution of people and pairing students who are at similar mathematical development levels. Students draw a grid on the photograph or use an overhead overlay. The grid should have six rows and six columns. Use one coloured number cube to randomly choose a row and the other number cube for the column. Pairs of students choose three grid squares by rolling the number cubes. They count the number of people in one of the grid squares and estimate the total number of people using a proportion. Repeat the process using the total for all three grid squares. Pairs compare their results with other pairs who used the same photograph. Connecting/Oral Questions/Anecdotal: Observe students as they discuss and compare their results with other groups. Pairs Scaling Distribute BLM 4.4.1. Assign each pair of students one of the following enlargements (12 × 10 double both vertically and horizontally, 6 × 10 double only vertically, 12 × 5 double only horizontally, 12 × 15 double horizontally and triple vertically, 18 × 15 triple horizontally and vertically). Pairs of students make an enlarged figure on grid paper and write their scaling instructions below it.

Consolidate Debrief

Whole Class Discussion Discuss the impact of sample size on the accuracy of the estimate. • What limitations does this method have? • Which of your enlarged figures are distorted? Why?

Discuss what constitutes a scale diagram, the constant of proportionality, and work through examples on how to use them. Students record definitions and examples in their notes.

“Will the African Elephant Become Extinct in Your Lifetime?” Impact Math: Data Management and Probability; p. 25, crowd photos on newspaper web sites (e.g., outdoor rock concerts, parades). Grid paper in a variety of scales can be created using Graphpap software. This freeware program can be downloaded from: http://pharm.kuleuven.be/pharbio/gpaper.htm Consolidate the aerial photo activity before beginning the scaling activity.


Home Activity or Further Classroom Consolidation • Complete the worksheet, More Scaling Problems. • Complete the practice problems. • Bring in examples of scale diagrams and aerial photos for the bulletin board. • Bring in grocery flyers with prices of snacks and drinks for tomorrow’s

activity.

The teacher provides appropriate problems.


4.4.1: I'd Rather Be Scaling


4.4.2: 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


Unit 4: Day 5: Planning a Class Trip Grade 9 Applied

75 min

Math Learning Goals • Investigate a variety of methods for solving problems using proportions. • Solve problems and make comparisons using unit rates.

Materials • grocery store

flyers • BLM 4.5.1


Minds On ...

Whole Class Brainstorm and Discussion Pose the trip scenario (BLM 4.5.1) and brainstorm what needs to be considered when planning this trip. Create a mind map to record students’ ideas. Students cut out ads for granola bars and juice boxes from their flyers and post them on the bulletin board. Use these ads to teach a lesson on unit rates and how they can be used to comparison shop. As a class, calculate the unit price to determine the best value for granola bars and juice. Calculate the total cost of purchasing these snacks for the class. Students record these costs on their worksheet (BLM 4.5.1).

Action! Individual Calculating Rates Students complete BLM 4.5.1. Curriculum Expectations/Oral Questions/Mental Note: Assess students’ understanding of rates.

Consolidate Debrief

Whole Class Self-Assessment Discuss student responses on the worksheet. Students self-check work and correct errors. Encourage students to share their methods with the class. Clarify any misconceptions that come up during the discussion.

Ensure that ads for these items are available in the classroom in case students don’t bring them in.

Application Concept Practice Practice

Home Activity or Further Classroom Consolidation Complete the questions involving rates.



4.5.1: Planning a Class Trip 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


4.5.1: Planning a Class Trip (continued) 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.


Unit 4: Day 6: That’s Stretching It Grade 9 Applied

75 min

Math Learning Goals • Investigate percent as a proportional relationship. • Solve problems involving percents, ratios, and decimals in a variety of contexts.

Materials • fabric elastic • BLM 4.6.1, 4.6.2 • measuring tape


Minds On ...

Pairs Peer Coaching Students compare solutions and help each other with practice questions from the Home Activity.

Action! Pairs Investigation Pairs investigate that percent is a proportion (BLM 4.6.1). Debrief the activity to ensure the students understand that percent is proportion and account for discrepancies due to measurement error. Whole Class Guided Lesson Demonstrate a problem of each type, determining the unknown part, the unknown percent, and the unknown whole (BLM 4.6.2). Students solve similar problems on their own and share solutions. Help students make connections between the constant of proportionality, rate of change, proportions, and percents. Problem Solving/Observation/Checkbric: Observe students as they problem solve using a checkbric and record their skill levels for: • reading and following instructions • graphing • accuracy in measuring • ability to interpolate/extrapolate • estimating

Consolidate Debrief

Whole Class Discussion Students share effective strategies for the three types of percent problems.

If you make elastic meters ahead of time, edit BLM 4.6.1 to remove instructions.

Application Exploration Reflection

Home Activity or Further Classroom Consolidation • Create your own percent reference sheet, showing examples of the various

types of percent problems. • Complete questions for additional practice.



4.6.1: Elastic Meter and Percent Part A: Make the elastic meter 1. Take a piece of elastic 32 cm long. Mark a line at 1 cm from one end.

From this point make 10 marks every 3 cm. There will be 1 cm left. (The centimetre at each end of the elastic provides a way to hold and stretch the elastic ruler.)

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) 0%

10% 33% 45% 50% 60% 75% 90%

100%

Use your elastic meter to complete the chart.

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%


4.6.2: 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?


4.6.2: Types of Percent Problems – Guided Lesson (continued) 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


Unit 4: Day 7: Proportion Potpourri Grade 9 Applied

75 min

Math Learning Goals • Investigate that proportionality is a multiplicative process and not an additive

process. • Consolidate concept understanding and procedural fluency for ratio, proportion,

and percents. • Solve problems involving percents, fractions, and decimals in a variety of

contexts.

Materials • overhead

projector • coloured pencils • grid paper • linking cubes • BLM 4.7.1, 4.7.2


Minds On ...

Pairs Peer Editing Students compare solutions and help each other with practice questions from the Home Activity. They share the reference sheets on percents and offer suggestions for improvement, if necessary. Whole Class Discussion Use the bulletin board articles to consolidate the connections between ratios, proportions, and percents and their relevance in everyday life. For example, if an article mentions “3 out of 5...” ask: How else could we say this?

Action! Pairs Investigation Display BLM 4.7.1 on the overhead for students to observe. Students use grid paper and linking cubes to make another version of this shape that is twice as big. Allow students time to explore on their own and make mistakes, as this will help direct the discussion on what constitutes proportionality. Choose models to display that show proportionality and others that don’t. Discuss which shapes look proportionate, and ask students to explain what strategies they used to create the proportionate shapes, as well as the disproportionate ones. Students should understand that proportional shapes are created using multiplicative processes, not additive ones. Representing/Oral Questions/Mental Note: Assess how students explain why or why not their models represent proportional shapes. Groups of 4 Review Relay Form heterogeneous groups. Each group completes the first question (BLM 4.7.2). A group member verifies with the teacher that the answer is correct before receiving the next question; incorrect solutions must be corrected by the group.

Consolidate Debrief

Whole Class Discussion Students share solutions to relay questions that posed problems. Individual Journal Students reflect on how ratio, rates, proportions, and percents are part of their daily life. They give examples of how they might use the skills they have learned in this unit.

See page 86, Think Literacy, Cross-Curricular Approaches 7–12. Make as many copies of BLM 4.7.2 as there are groups. Cut out the questions and create piles of each question number. Students are allowed to use their notes and reference sheets for this activity. Peer tutors can help the teacher to run the relay.

Application Reflection Concept Practice

Home Activity or Further Classroom Consolidation Prepare for the unit assessment by completing practice questions, creating reference sheets, and organizing your notes.

Reference sheets could be an accommodation for identified exceptional students.


4.7.1: Double It!


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

186

=

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?


4.7.2: Review Relay (continued) 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?

Documents

Unit 4 Grade 9 Applied Proportional Reasoning: Ratio, · PDF fileUnit 4 Grade 9 Applied Proportional Reasoning: Ratio, Rate, ... • Solve problems involving ratios, rates, ... If