Unit 4 Grade 9 Applied
Proportional Reasoning: Ratio, Rate, and Proportion
Lesson Outline
BIG PICTUREStudents 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
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning 13
Unit 4: Day 1: Ratio Carousel / Grade 9 Applied75 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 à 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 à Presentation
As a class, summarize the important points for each term students need to know for the carousel 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.
Action! / Groups of 3 à 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 à 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.
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.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning 13
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 Δ ABC / Area Δ DEF / Ratio ABC/DEF24.98 cm² / 6.25 cm² / 4.00:1
2. What do you notice about the ratio of the areas?
3. If Area ΔABC = 64 cm², what is the area of ΔDEF? Explain.
4. If Area ΔDEF = 15 cm², what is the area of ΔABC? Explain.
5. Follow the Midpoint Segments instructions and complete the following chart.
BC / DE / Ratio BC/DE6. 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 BeeDaily 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
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Unit 4: Day 2: Growing Dilemma / Grade 9 Applied75 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
Assessment
Opportunities
Minds On ... / Whole Class à 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 à Anticipation Guide
Students, individually, complete the Before column on the anticipation guide (BLM 4.2.1) and discuss their choices with their partner. / 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.
Action! / Pairs à 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 à 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 à 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.
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?
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning 13
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 / Statement / AfterAgree / 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.
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.
Side Length (S) / Perimeter (P) / Ratio (S:P) / Ratio in Lowest TermsFirst Differences
1
2
3
4
5
4. State the characteristics of this relationship:
a) first differences
b) ratios
c) graph
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.
2. Complete the chart.
3. Graph Area vs. Side Length on the grid provided.
Side Length (S) / Area(A) / Ratio (S:A) / Ratio in Lowest Terms
First Differences
1
2
3
4
5
4. State 3 characteristics of this relationship:
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.
2. Complete the chart.
3. Graph Diagonal Length vs. Side Length on the grid provided.
Side Length (S) / Diagonal (D) / Ratio (S:D) / Ratio in Lowest TermsFirst Differences
1
2
3
4
5
4. State 3 characteristics of this relationship:
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.
2. Complete the chart.
3. Graph Volume vs. Side Length on the grid provided.
Side Length (S) / Volume (V) / Ratio (S:V) / Ratio in Lowest TermsFirst Differences
1
2
3
4
5
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?
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning 13