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
1 Ratio Carousel •
GSP®4 file:
Middle Mania
2
Growing
Dilemma
•
•
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, algebraic).
Investigate and determine what a ratio is using examples and nonexamples of proportional and non-proportional situations (e.g., two
ordered quantities that share a multiplicative relationship).
3
Pondering
Proportions
•
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
•
4
I’d Rather Be
Scaling
•
(e.g., scaling/tables, drawings, constant of proportionality, unit rate, cross
products).
•
•
5
Planning a
Class Trip
•
•
6
That’s
Stretching It
•
•
7
Proportion
Potpourri
8
9
10
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.
Investigate a variety of methods for solving problems using proportions.
Solve problems and make comparisons using unit rates.
Investigate percent as a proportional relationship.
Solve problems involving percents, ratios, 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.
• Solve problems involving percents, fractions, and decimals in a variety
of contexts.
Instructional Jazz
Instructional Jazz
Review and Assessment
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
Expectations
NA1.01, NA1.02,
NA1.03, NA1.04,
NA1.05
CGE 5a, 5e
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
NA1.01, NA1.02,
NA1.03, NA1.04,
NA1.05, MG2.02
CGE 2c, 2d
NA1.01, NA1.02,
NA1.03, NA1.04,
NA1.05
CGE 2a, 2c, 5b
NA1.01, NA1.02,
NA1.03, NA1.04,
NA1.05
CGE 2c, 5e
NA1.01, NA1.02,
NA1.03, NA1.04,
NA1.05, NA1.06
CGE 5a, 7i
NA1.01, NA1.02,
NA1.03, NA1.04,
NA1.05, NA1.06
CGE 5a, 7i
1
Unit 4: Day 1: Ratio Carousel
Grade 9 Applied
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
75 min
Assessment
Opportunities
Minds On ...
Action!
Groups of 3 Æ Graffiti
Use heterogeneous groupings. Prepare chart paper for each of the following
terms: ratio, rate, unit rate, equivalent ratios. Each group uses a differentcoloured marker, cycles through the chart paper stations (2 minutes per chart),
and writes characteristics of the term.
Whole Class Æ Presentation
As a class, summarize the important points for each term students need to know
for the carousel activity.
Groups of 3 Æ 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.
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.
Consolidate
Debrief
Application
Concept Practice
Whole Class Æ 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.
Home Activity or Further Classroom Consolidation
Find examples of ratio, rate, and unit rate in your environment to post on the
bulletin board.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
The student notes
should connect the
prior knowledge
from the graffiti with
the knowledge from
the carousel.
The bulletin board
becomes a class
portfolio showing
that proportions are
pervasive. Collect
further examples in
other lessons in
the unit.
2
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.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
3
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?
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
4
4.1.1: Ratio Carousel (continued)
Station: GSP®4 Middle Mania
1. Follow the Midpoint Triangle instructions and complete the following chart.
Area ∆ ABC
24.98 cm²
Area ∆ DEF
6.25 cm²
Ratio ABC/DEF
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.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
5
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.”
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
6
Middle Mania (GSP®4 File)
Middle Mania.gsp
Midpoint Triangles
Midpoint Segments
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
7
Unit 4: Day 2: Growing Dilemma
Grade 9 Applied
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
75 min
Assessment
Opportunities
Minds On ...
Whole Class Æ 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 Æ 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 Æ 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
Concept Practice
Reflection
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.
Pairs Æ 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 Æ 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.
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
8
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
Agree
Disagree
Statement
After
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.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
9
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)
First
Differences
Ratio
(S:P)
Ratio in
Lowest
Terms
1
2
3
4
5
4. State the characteristics of this relationship:
a) first differences
b) ratios
c) graph
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
10
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)
First
Differences
Ratio
(S:A)
Ratio in
Lowest
Terms
1
2
3
4
5
4. State 3 characteristics of this relationship:
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
11
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)
First
Differences
Ratio
(S:D)
Ratio in
Lowest
Terms
1
2
3
4
5
4. State 3 characteristics of this relationship:
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
12
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)
First
Differences
Ratio
(S:V)
Ratio in
Lowest
Terms
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
Unit 4: Day 3: Pondering Proportions
Grade 9 Applied
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
75 min
Assessment
Opportunities
Minds On ...
Whole Class Æ 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 Æ 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 Æ 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.
Application
Concept Practice
Home Activity or Further Classroom Consolidation
• Complete the questions on worksheet Television Dimensions.
• Complete Pythagorean theorem questions.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
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.
Select appropriate
practice questions.
14
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?
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
15
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?
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
16
Unit 4: Day 4: I’d Rather Be Scaling
Grade 9 Applied
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
75 min
Assessment
Opportunities
Minds On ...
Pairs Æ Peer Coaching
Students compare solutions and help each other with practice questions from
the home activity.
Whole Class Æ Discussion
Show an aerial photograph of a large crowd. Students suggest ways the number
of people could be counted.
Action!
Pairs Æ 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 Æ 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
“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.kuleuv
en.be/pharbio/gpap
er.htm
Consolidate the
aerial photo activity
before beginning
the scaling activity.
Whole Class Æ 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.
Application
Concept Practice
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.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
The teacher
provides
appropriate
problems.
17
4.4.1: I'd Rather Be Scaling
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
18
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
i.
1:400
ii.
12000:1
iii.
iv.
1:250000
Diagram Measurement
6 cm
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
7.2 cm
8 cm
Actual Measurement
0.00375 mm
0.6 mm
19
Unit 4: Day 5: Planning a Class Trip
Grade 9 Applied
Materials
• grocery store
flyers
• BLM 4.5.1
Math Learning Goals
• Investigate a variety of methods for solving problems using proportions.
• Solve problems and make comparisons using unit rates.
75 min
Assessment
Opportunities
Minds On ...
Whole Class Æ 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 Æ Calculating Rates
Students complete BLM 4.5.1.
Curriculum Expectations/Oral Questions/Mental Note: Assess students’
understanding of rates.
Consolidate
Debrief
Whole Class Æ 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.
Application
Concept Practice
Practice
Home Activity or Further Classroom Consolidation
Complete the questions involving rates.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
Ensure that ads for
these items are
available in the
classroom in case
students don’t bring
them in.
Select appropriate
practice questions.
20
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
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
21
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.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
22
Unit 4: Day 6: That’s Stretching It
Grade 9 Applied
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
75 min
Assessment
Opportunities
Minds On ...
Pairs Æ Peer Coaching
Students compare solutions and help each other with practice questions from
the Home Activity.
Action!
Pairs Æ 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 Æ 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
Application
Exploration
Reflection
If you make elastic
meters ahead of
time, edit BLM
4.6.1 to remove
instructions.
Whole Class Æ Discussion
Students share effective strategies for the three types of percent problems.
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.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
Select appropriate
practice questions.
23
4.6.1: Elastic Meter and Percent
elastic
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 %
0%
10%
33%
45%
50%
60%
75%
90%
100%
20%
10%
Measure (cm)
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:
a) 85%
b) 65%
c) 43%
d) 58%
Extrapolate: Estimate the following lengths:
a) 120%
b) 135%
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
24
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?
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
25
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?
•
set up and solve a ratio
•
solve an equation 0.15 ×
= $2.25
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?
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
26
Unit 4: Day 7: Proportion Potpourri
Grade 9 Applied
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
75 min
Assessment
Opportunities
Minds On ...
Action!
Consolidate
Debrief
Application
Reflection
Concept Practice
Pairs Æ 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 Æ 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?
Pairs Æ 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 Æ 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.
Whole Class Æ Discussion
Students share solutions to relay questions that posed problems.
Individual Æ 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.
Home Activity or Further Classroom Consolidation
Prepare for the unit assessment by completing practice questions, creating
reference sheets, and organizing your notes.
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
See page 86, Think
Literacy, CrossCurricular
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.
Reference sheets
could be an
accommodation for
identified
exceptional
students.
27
4.7.1: Double It!
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
28
4.7.2: Review Relay
1. Reduce the ratios to lowest terms:
2. Calculate the following percents:
15:35 =
45% of 220 =
18
=
6
120% × 555 =
1.5% × 1400 =
144:72 =
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?
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
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?
29
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
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?
1:70
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?
TIPS4RM: Grade 9 Applied – Unit 4: Proportional Reasoning
30