Percent, Ratio, and Rate
Suggested Time: 5 Weeks
PERCENT, RATIO, AND RATE
Unit Overview
Focus and Context
In this unit, students will work with percents, ratios, and rates, and
solve problems using proportional reasoning. Percents, ratios and
rates, just like fractions and decimals, are comparisons of quantities.
Significant work was done with fractions, decimals, and percents in
Grade 7. However, students worked only with percents between 1%
and 100%. This will now be extended to include percents between 0%
and 1% and greater than 100%. The use of percents is ever-present in
the retail and business worlds, as students will see through problems
involving sales tax and discount. They will also solve problems using
combined percents and percent increase and decrease.
Fractions, decimals and percents are different representations of the
same underlying value. Students will now extend this representation to
include ratios and rates.
A ratio is a comparison of two or more quantities with the same unit,
whereas a rate is a comparison of two quantities measured in different
units. A unit rate, which provides a useful strategy for comparing rates,
is a quantity associated with a single unit of another quantity. Students
will use proportional reasoning to solve problems involving ratios and
rates. It also has a place in other areas of mathematics. For example,
work with similar triangles, dilations, and solving algebraic equations all
involve proportional reasoning.
Math Connects
We live in a world of percentages, ratios, rates and proportional
reasoning. Computations with percents are frequently encountered in
real-life situations, from sales tax and discounts to data analysis. On a
daily basis, students encounter percents in the context of test scores,
sports statistics, weather reports, public opinion surveys, nutrition facts
on food packages, and deciding how big a tip to leave in a restaurant.
They should be aware of the role of percents in currency conversions,
interest charges, commission, and wage increases.
Ratios and rates have as many applications. The ratios of a recipe’s
various ingredients are important to ensure the intended outcome.
Rates exist in everyday situations involving speed, fuel consumption,
Internet downloading, and measuring heart rate. The ability to reason
with proportions has a host of applications in everyday life. When
consumers are making price comparisons to determine the better buy,
or when a worker who earns $300 for 8 hours uses this to determine
the number of hours required to earn $1000, proportional reasoning
occurs.
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GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
Process Standards
Key
Curriculum
Outcomes
[C] Communication
[CN] Connections
[ME] Mental Mathematics
and Estimation
STRAND
[PS]
[R]
[T]
[V]
Problem Solving
Reasoning
Technology
Visualization
OUTCOME
PROCESS
STANDARDS
Number
Demonstrate an understanding of
percents greater than or equal to 0%.
[8N3]
Number
Demonstrate an understanding of ratio
and rate. [8N4]
C, CN, V
Number
Solve problems that involve rates, ratios
and proportional reasoning. [8N5]
C, CN, PS, R
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
C, PS, R, V
143
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N3 Demonstrate an
understanding of percents greater
than or equal to 0%.
[CN, PS, R, V]
In Grade 7, students worked with percents from 1% to 100%. They
converted between percent form, fractional form and decimal form,
and problem solving involved finding a percent of a number. Previous
work will now be extended to include percents between 0% and 1% and
percents greater than 100%. Problem solving situations will be more
varied. Students will apply knowledge of percents to find a number
when a percent of it is known, and solve problems involving percent
increase and decrease, combined percents, and finding the percent of a
percent.
Achievement Indicator:
8N3.1 Provide a context where
a percent may be more than
100% or between 0% and 1%.
Discuss with students the relevance of percents in real world
applications.
Compile with students a list of situations where percents are used. This
list may include, but is not limited to:
• test marks (78% on a science test)
• sales tax (13% tax on all sales)
• discount (25% off all purchases)
• probability (10% chance of rain)
• athletic statistics (scored 25% of shots on goal)
Discuss with students situations that may result in percents
• greater than 100%
i) What percent would a student receive on a test containing bonus
questions if they got all questions correct?
ii) What percent has the cost of soda pop increased when today’s cost
is compared to the cost 50 years ago?
• between 0% and 1%.
i) Ask a hockey fan – What is the percent chance of your favourite
team winning the Stanley Cup playoffs?
ii) What is the percent chance that it will snow in August?
144
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Group Discussion
• Topic 1: When your coach tells you to “give 110%”, what does he
mean?
(8N3.1)
• Topic 2: What is the chance that the principal will give you the day
off school because of your smile?
(8N3.1)
• Topic 3: A newspaper article includes 200% in its headline. Give a
situation to which the article may be referring.
(8N3.1)
Journal
• Paul bragged that he received a 105% on his math test. Is this mark
possible? Give an example to support your answer.
(8N3.1)
Math Makes Sense 8
Lesson 5.1: Relating Fractions,
Decimals, and Percents
Lesson 5.2: Calculating Percents
• Jill predicted that the chance of Maple Academy winning the
championship game against Evergreen Collegiate is 0.50%. Which
school do you think Jill attends? Explain your choice.
(8N3.1)
ProGuide: pp.4-11, 12-17
Student Book (SB): pp.234-241,
242-247
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
145
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N3 Continued
Achievement Indicators:
8N3.2 Represent a given
fractional percent using grid
paper.
8N3.3 Represent a given
percent greater than 100 using
grid paper..
8N3.4 Determine the percent
represented by a given shaded
region on a grid and record it
in decimal, fractional and
percent form.
Students have been introduced to whole percents from 1% to 100% in
previous grades. It is assumed they represented whole number percents
using grid paper. In Grade 8, this is expanded to percents between 0%
and 1%, percents greater than 100%, as well as other fractional percents.
Begin with using a hundreds grid chart to represent percents. Each small
square represents 1%. For fractional percents that are easily recognizable,
i.e. 0.5%, the hundreds grid will be sufficient (shade ½ of one small
square). To represent 29.5% using grid paper the hundreds grid block is
sufficient, since 0.5% would represent half a block. (See diagram below)
In this diagram , out of the 100
blocks, 29 full blocks and half of
another block are shaded. T his
w ould represent 29.5% .
However, other percents may need to utilize the hundredths grid chart.
At this point, the focus for achievement indicator 8N3.4 will be on
representing the percent using grid paper. Decimal and fractional form
of percents will be explored later in the unit.
Consider 30.15%; a hundreds chart is needed to indicate the 30%, but
0.15 of a block is harder to recognize. To accomplish this we introduce
a smaller hundreds chart to the side of the original chart. This smaller
chart will represent the hundredths partitions of the 31st square. By
shading 15 of the blocks in this smaller chart we indicate the 0.15%.
In this diagram , out of the
100 block s, 30 full block s
and part of another block
are shaded. S ince it is
harder to recognise 0.15
of the block being
shaded, the hundredths
grid chart is used and 15
of those 100 block s are
shaded.
Continued
146
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Refer to the NL government website for Representing Percents.
www.ed.gov.nl.ca/edu/k12/curriculum/guides/mathematics/
(8N3.2, 8N3.3, 8N3.4)
Math Makes Sense 8
Lesson 5.1: Relating Fractions,
Decimals, and Percents
Lesson 5.2: Calculating Percents
ProGuide: pp.4-11, 12-17
CD-ROM: Master 5.21, 5.22
SB: pp.234-241, 242-247
Practice and HW Book: pp.102104, 105-106
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
147
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N3 Continued
Achievement Indicators:
8N3.2 Continued
8N3.3 Continued
Fractional percents less than 1% can also be represented using the
hundredths chart, as no full blocks would be shaded in the hundreds
chart. The diagram below represents 0.28%.
In th is d ia g ra m , p a rt
o f a b lo ck is sh a d e d .
T h e h u n d re d th s
ch a rt is u se d a n d 2 8
b lo c ks o u t o f 1 0 0
a re sh a d e d .
8N3.4 Continued
Percents greater than 100% are represented using more than one
hundreds grid chart. The diagram below represents 240%.
In th is d ia g ra m , tw o fu ll h u n d re d s ch a rts
a n d 4 0 b lo cks o f a n o th e r h u n d re d ch a rt a re
sh a d e d .
148
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Math Makes Sense 8
Lesson 5.1: Relating Fractions,
Decimals, and Percents
Lesson 5.2: Calculating Percents
ProGuide: pp.4-11, 12-17
CD-ROM: Master 5.21, 5.22
SB: pp.234-241, 242-247
Practice and HW Book: pp.102104, 105-106
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
149
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N3 Continued
Achievement Indicators:
8N3.5 Express a given percent
in decimal or fractional form.
8N3.6 Express a given decimal
in percent or fractional form.
8N3.7 Express a given fraction
in decimal or percent form.
8N3.4 Continued
In previous grades, students have shifted between percent, fraction and
decimal equivalents for whole number percents between 1% and 100%.
They will apply these skills to fractional percents between 0% and 1%,
percents greater than 100%, as well as other fractional percents.
Fractional percents between 0% and 1% must be developed at a sensible
pace. There is sometimes a tendency among students to see the percent
0.1% as the decimal 0.1. It is important to distinguish the difference
in these two forms. Similarly, students may confuse 34 % with 75%.
The hundreds and hundredths grid charts will help distinguish these
differences. Given a shaded region on a grid, students will be expected
to express the shaded region in fraction, decimal or percent form.
Another strategy that can be used when dealing with percents greater
than 100% and between 0% and 1% is patterning.
For example:
150
Percent
0.3%
Decimal
0.003
Fraction
3%
0.03
3
100
30%
0.3
3
10
300%
3
3
1
Percent
70%
Decimal
0.7
Fraction
7%
0.07
7
100
0.7%
0.007
7
1000
0.07%
0.0007
7
10000
3
1000
7
10
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Copy and complete the following table.
(8N3.5, 8N3.6, 8N3.7)
Math Makes Sense 8
Percent
148%
7
20
Decimal
Fraction
%
Lesson 5.1: Relating Fractions,
Decimals, and Percents
Lesson 5.2: Calculating Percents
26.4%
2.65
0.003
0.254
8
5
1
250
ProGuide: pp.4-11, 12-17, Master
5.6a, 5.6b
CD-ROM: Master 5.21, 5.22
3
8
SB: pp.234-241, 242-247
• As a decimal 140% = 1.40. Use patterning to write the following
percents in decimal form.
Practice and HW Book: pp.102104, 105-106
(i) 14%
(ii) 1.4%
(iii) 0.14%
(8N3.5, 8N3.6, 8N3.7)
9
, use patterning to write the following
• As a fraction 0.09% = 10000
percents in fraction form.
(i)
(ii)
(iii)
(iv)
0.9%
9%
90%
900%
(8N3.5, 8N3.6, 8N3.7)
Journal
• Your friend was absent from school when your teacher explained
fractional percents. When he was studying for his test, he said that
1
2
% was 0.5 as a decimal. How would you help him to understand
the mistake he made?
(8N3.5, 8N3.6, 8N3.7)
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
151
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N3 Continued
Achievement Indicators:
8N3.8 Solve a given problem
involving percents.
In grade 7, students explored various strategies to calculate the percent of
a number. These strategies are found in Unit 3 of the grade 7 curriculum
guide and teachers should review these for themselves before proceeding.
In grade 8, the focus will be on problems involving calculating the whole
when the percent is given, and percent increase and decrease. Problems
may also include finding percents when given the part and the whole,
which is the same as changing from fraction form to percent form.
First Method
A visual model can be used to develop this notion using benchmark
percentages like 10%, 25% or 50%.
25% of a number is 80. What is the number?
0
80
?
0%
25%
100%
0
80
160
240
320
0%
25%
50%
75%
100%
Place 80 above the 25% mark
on a number line that runs from
0% to 100%.
Write appropriate multiples of
80 above the appropriate
multiples of 25% until you
reach 100%.
The matching multiples, 320
and 100%, are equivalent.
Second Method
5% of a number is 20. What is the number?
Since 5% of a number is 20, then 1% must be 4. (20 ÷ 5 = 4 )
Multiply 4 by 100; the answer 400 must be 100%.
Therefore, the number is 400.
Later, students may use properties of proportions to find wholes given
parts of wholes.
Continued
152
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Trina received an 80% on a recent math test. If she answered 48
questions correctly, how many questions were on the test? (8N3.8)
Math Makes Sense 8
• Adam increased his song list by 40%. If he had 300 songs originally,
how many songs does he now have?
(8N3.8)
Lesson 5.2: Calculating Percents
Lesson 5.3: Solving Percent
Problems
• Shawn earned $85 and spent $15. What percent of his money did he
spend?
(8N3.8)
• Last week the canteen sold 60 sandwiches. This week they sold 48
sandwiches. Calculate the percent change. How can you check that
the percent change is correct?
(8N3.8)
Journal
• Catherine said that her amount of homework increased 400% when
it went from one half hour of work to two hours of work. Do you
agree? Explain.
(8N3.8)
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Lesson 5.4: Sales Tax and
Discount
ProGuide: pp.12-17, 18-25, 26-33
CD-ROM: Master 5.22, 5.23,
5.24
SB: pp.242-247, 248-255, 256262
Practice and HW Book: pp.105106, 107-109, 110-111
153
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N3 Continued
Achievement Indicators:
8N3.8 Continued
Problems involving percent increase and decrease are present in many
applications. Consider the following example.
The enrolment in junior high last year was 120 students. This year
enrolment increased by 15%. What is the enrolment this year?
15% of 120 = (0.15 )(120 ) = 18
Add 18 to 120 : 18 + 120 = 138 The enrolment this year is 138 students.
Another application of percent increase and decrease is to find the
amount of change as a percentage rather than the final/initial amounts.
The formula for this calculation is:
 Final Amount − Original Amount 
% of change = 
 × 100
Original Amount


Example 1: A tree which was 3.7 m high last year is measured and found to
now be 4.8 m tall. What is the percent change in the height of the tree?
 Final Amount −Original Amount 
% of change = 
 ×100
Original Amount


−
4.8
3.7


% of change = 
 ×100
3.7


 1.1 
% of change = 
 ×100
 3.7 
The height of the tree increased by 29.7% in one
% of change = 29.7%
year.
Example 2: A large bag of potato chips used to cost $2.99. The store offered
the chips at the new price of $2.65 during the Christmas season. What is the
percent change in the price of the chips over the Christmas season?
 Final Amount −Original Amount 
% of change = 
 ×100
Original Amount


 2.65 − 2.99 
% of change = 
 ×100
2.99


 −0.34 
% of change = 
 ×100
 2.99 
% of change = − 11.4%
Notice that the percent change is a negative number. This means that the
change is a decrease.
154
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Performance
• “Beat that Percent” Game
(8N3.8)
Math Makes Sense 8
Lesson 5.2: Calculating Percents
Goal
The goal of the game is to obtain 10 points before your
opponent(s).
How to Play
1. Shuffle the cards. Deal four cards to each player.
2. The aces count as 1, the face cards count as 0 and numbered
cards count as their face values.
3. Each player chooses two of the cards to form a two digit
number that represents a percent. The remaining two cards
form a two digit number.
4. Calculate the percent of the number.
5. Compare results with your opponent(s). The one with the
greatest value gets a point.
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
Lesson 5.3: Solving Percent
Problems
Lesson 5.4: Sales Tax and
Discount
ProGuide: pp.12-17, 18-25, 26-33
CD-ROM: Master 5.22, 5.23,
5.24
SB: pp.242-247, 248-255, 256262
Practice and HW Book: pp.105106, 107-109, 110-111
155
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N3 Continued
Achievement Indicator:
8N3.9 Solve a given problem
involving combined percents.
A common example of combined percents is addition of percents, such
as GST + PST. Students encounter combined percentages everyday
when they buy items at stores. Tax is charged in Newfoundland and
Labrador by both the federal and provincial government. Currently, the
Federal government charges 6% (GST – Goods & Services Tax) and the
Provincial government charges 7% (PST – Provincial Sales Tax). This
means a total of 13% tax is charged on purchases in Newfoundland and
Labrador. This is called HST, or Harmonized Sales Tax.
Jason purchases a hockey stick that has a sticker price of $74.99. Find the
total price Jason must pay for the stick, and also find the amount of that
total price that is GST and what amount is PST.
Total Price = Sticker Price + Sales Tax
Total Price = 74.99 + (0.13 )(74.99 )
Total Price = 74.99 + 9.75
Total Price = 84.74
The total price is $84.74.
GST Calculation
PST Calculation
(74.99 )× (0.06 ) = 4.50
(74.99 )× (0.07 ) = 5.25
GST = $4.50
PST = $5.25
Notice that GST + PST = $9.75,
which was the calculated HST.
156
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Sheri regularly travels across Canada. She plans on purchasing a
new laptop. The laptop sells for $2150 in both provinces. In which
province should Sheri purchase the laptop? Explain.
(8N3.9)
Note: A table of provincial tax rates is found in the student book on page
256.
Math Makes Sense 8
Lesson 5.4: Sales Tax and
Discount
Journal
ProGuide: pp. 26-32
• Your friend lives in Ontario. You plan a trip together to Quebec City
and want to wear matching jackets during the trip. The jacket costs
$59.90 in each province. Write an email to your friend to convince
her in which of the three provinces, Ontario, Newfoundland and
Labrador, or Quebec, the jackets should be purchased, and why.
(8N3.9)
CD-ROM: Master 5.24
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
SB: pp.256-262
Practice and HW Book: pp.110111
157
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N3 Continued
Achievement Indicator:
8N3.10 Solve a given problem
that involves finding the
percent of a percent.
There are situations in everyday life that involve applying percentage
calculations more than once before an answer is found. An example
of this is when stores hold a “NO TAX” sale. It can be discussed with
students why the price they pay at such a sale is always a little bit less
than the sticker price. They might have believed that the price they pay
would simply be the price on the sticker if there was no tax. However,
by law, stores have to charge tax. They will first discount the price by
the tax rate. Then, they add the tax back on to that discounted amount.
Since they are calculating tax on a smaller amount the final price will be
a little less than the original sticker price.
Brenda finds the perfect coat for winter at the local mall on a day the mall is
having a NO TAX sale. The coat is marked $125. How much will Brenda
pay for the coat?
First the store must discount the coat by 13%.
125 − (0.13 )(125 )
125 − 16.25
108.75
Now the store must add 13% tax to this discounted price.
108.75 + (108.75 )(0.13 )
108.75 + 14.14
122.89
Notice $122.89 is a little less than the original price of $125.
Combined percents are not limited to consumer purchases problems.
Students should be exposed to other types of problems as well.
158
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Two stores offer different discount rates as follows:
Store A: 50% off one day only.
Math Makes Sense 8
Store B: 25% off one day followed by 25% off the reduced price the
second day.
Lesson 5.4: Sales Tax and
Discount
Which store has the better sale?
(8N3.10)
ProGuide: pp.26-32
CD-ROM: Master 5.24
Journal
• A jacket cost $100. The discount on the jacket is 15%. However you
must also pay 15% sales tax. Would the jacket cost you $100, less
than $100 or more than $100? Explain your reasoning. (8N3.10)
SB: pp.256-262
Practice and HW Book: pp.110111
• Charlie works part-time at a local fast food restaurant. On his next
pay check, he will receive a 5% increase in pay on top of a 10%
performance bonus. Charlie tells his friends he is receiving a15%
raise in pay. Is he correct? Explain.
(8N3.10)
Problem Solving
• Cyril collects hockey cards. He had 150 cards in his collection.
His birthday was in June and his friends gave him hockey cards as
presents which increased his collection by 20%. At Christmas his
hockey card collection increased by another 15%. How many cards
are in his collection after this 15% increase?
(8N3.10)
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
159
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N4 Demonstrate an
understanding of ratio and rate.
[C, CN, V]
Students have had previous exposure to ratios. In grade 6, they defined,
represented and interpreted ratios presented to them concretely. In grade
7, they related ratio to fractions and percent, and solved proportions
within problem solving situations involving percent. In this unit,
students will build on and extend their knowledge of ratio. They will
also be introduced to rate. They will describe and record rates using reallife examples. Exposure to problem solving situations using unit rates
and unit prices should lead them to make connections between math
and everyday life.
Achievement Indicators:
8N4.1 Express a two-term
ratio from a given context in
the forms 3:5 or 3 to 5.
It may be necessary to remind students that a ratio is a comparison of
two numbers. Consider the following examples.
13 people rode the rollercoaster; 2 of them were girls. The ratio of
people to girls was 13:2, and this gives an example of a whole-to-part
ratio.
In an orchard with 80 trees, 43 of the trees were apple trees and the rest
were pear trees. The ratio of apple trees to pear trees was 43:37, which
represents a part-to-part ratio.
8N4.2 Express a three-term
ratio from a given context in
the forms 4:7:3 or 4 to 7 to 3.
These examples can be developed into three-term ratios.
If 13 people rode the rollercoaster and 2 of them were girls, the ratio can
also be expressed as Boys : Girls : Total People = 11:2:13, or 11 to 2 to 13.
Similarly, in an orchard with 80 trees, if 43 of the tress were apple trees
and the rest were pear trees, a resulting three-term ratio compares Apple
to Pear to Total Trees, or 43 to 37 to 80 = 43:37:80.
160
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Discussion
• In your classroom state these ratios:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Boys to girls
Girls to boys
Boys to total students
Boys to girls to total students
Window to doors
Desks to chairs
(8N4.1,8N4.2)
Paper and Pencil
• Write a part:part:whole ratio for each situation.
(i) A bag contains 3 jujubes and 5 lollipops.
(ii) A fishing basket holds 6 trout and 5 smelt.
(iii) In the harbour there are two types of boats: dories and
longliners. There are 40 boats in total and seven of them are
longliners.
(8N4.2)
Math Makes Sense 8
Lesson 5.5: Exploring Ratios
ProGuide: pp.34-38
CD-ROM: Master 5.25
SB: pp.264-268
Practice and HW Book: pp.112114
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
161
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N4 Continued
8N5 Solve problems that involve
rates, ratios and proportional
reasoning.
[C, CN, PS, R]
Achievement Indicators:
8N4.3 Express a part-to-part
ratio as a part-to-whole
fraction.
Once students have an understanding that a part-to-part ratio compares
one part of a set to another part of a set, while a part-to-whole ratio
compares one part of a set to the whole set, they should be able to
convert a part-to-part ratio to a part-to-whole ratio. For example, 1 can
concentrate of frozen juice to 4 cans of water can be represented as
which is the ratio of concentrate to solution, or
water to solution.
8N5.1 Explain the meaning of
a
b within a given context.
8N5.2 Provide a context in
which ba represents a:
•
•
•
•
•
fraction
rate
ratio
quotient
probability.
8N4.4 Identify and describe
ratios from real-life examples,
and record them symbolically.
162
4
5
1
5
,
, which is the ratio of
It should be emphasized that only part-to-whole ratios can be expressed
as fractions because the denominator is always referencing the whole.
Students could be asked to explain a ratio such as in the context of a
real-life example. It could be described as a part-to-whole ratio, where
the numerator represents a part of the whole, and the denominator
represents the whole. For example, Daniel gets a hit 2 out of every 9
times he goes to bat. The ratio of hits to bats is 2:9.
Teachers should note that probability will be studied in a later unit:
Data Analysis and Probability. Ratios can then be re-examined to
determine probabilities of events.
Ratios are encountered frequently when describing real world situations.
Students should be encouraged to write ratios in words first. This
may assist them in writing the terms of a ratio in the correct order of
comparison when expressing them in number form.
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Write each ratio as a fraction in simplest form.
(i) 14 to 6
(ii) 4:22
(iii) 18:12
(iv) 25 to 20
(v) 18:21
(vi) 18:3
(vii) 7:21
(viii) 20 to 9
(ix) 4:10
(x) 84 to 16
Math Makes Sense 8
Lesson 5.5: Exploring Ratios
(8N4.3)
ProGuide: pp.34-38
CD-ROM: Master 5.25
SB: pp.264-268
Practice and HW Book: pp.112114
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
163
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N4 and 8N5 Continued
Achievement Indicators:
8N4.5 Express a given ratio as
a percent.
Students were exposed to converting from a fraction to a percent
in previous grades. They revisited this earlier in the unit when they
expressed a fraction in percent form.
To effectively solve ratio problems, comparing ratios is necessary. The
following are effective strategies that can be used for ratio comparison.
8N5.3 Solve a given problem
involving ratio.
• Use equivalent ratios
• Use unit ratios
• Use percents
Relate equivalent ratios to earlier work with equivalent fractions.
Remind students that finding an equivalent fraction involves
multiplying the numerator and denominator by the same non-zero
number. A brief review of reducing fractions to simplest form may be
necessary.
A unit ratio has a term of 1. To compare 20:5 with 140:20, note that
20:5 = 4:1 and 140:20 = 7:1. It should then be clear that 140:20 is the
greater ratio.
Students will now apply work with ratios to problem solving situations.
There are many types of problems involving ratios, and students
should be exposed to a variety. In many problem solving situations,
one term of a proportion is missing and must be determined. Students
must be made aware that a proportion is a relationship in which two
ratios are equal. Proportional reasoning problems can be solved using
several different methods. A possible solution method is given with the
following example.
The ratio of indoor basketballs to outdoor basketballs at the recreation centre
is 6:3. If the recreation centre has 45 basketballs, how many of them are
indoor basketballs?
Let x represent the number of indoor basketballs. It is necessary to compare
the number of indoor balls to the total number of balls, resulting in the partto-whole ratio 6:9. Use a proportion and solve with equivalent fractions.
6
9
=
x
45
Since 9 × 5 = 45 , multiply the numerator by 5 as well. This results in
x = 6 × 5 . Therefore, there are 30 indoor basketballs.
164
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Math Makes Sense 8
Lesson 5.5: Exploring Ratios
Lesson 5.6: Equivalent Ratios
Lesson 5.7: Comparing Ratios
Lesson 5.8: Solving Ratio
Problems
ProGuide: pp.34-38, 39-45, 4956, 57-63, Master 5.6a, 5.6b
CD-ROM: Master 5.25, 5.26,
5.27, 5.28
SB: pp.264-268, 269-275, 279286, 287-293
Practice and HW Book: pp.112114, 115-117, 118-121, 122-123
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
165
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N4 and 8N5 Continued
Students have worked with ratios, which compare quantities with the
same unit. Now the focus shifts to rates, which involve quantities with
different units. However, the mathematics used to talk about rates is
the same as the mathematics used to talk about ratios. Both represent
comparisons. Problems involving rates can be solved using the same
techniques as those involving ratios.
Achievement Indicators:
8N4.6 Express a given rate
using words or symbols,
e.g., 20 L per 100 km or
20L/100 km.
8N4.7 Identify and describe
rates from real-life examples,
and record them symbolically.
8N5.1 Explain the meaning of
a
b within a given context.
8N5.2 Provide a context in
which ba represents a:
•
•
•
•
•
fraction
rate
ratio
quotient
probability.
8N4.8 Explain why a rate
cannot be represented as a
percent.
166
Students must be reminded to include the units when writing rates.
Because a rate compares quantities measured in different units, without
the units a rate has no meaning.
Students should be familiar with numerous and various examples of
rates already, even if they couldn’t previously identify them as rates.
Have the class brainstorm to identify as many real-life examples of rates
as possible. Some examples that students should relate to include speed
(km/h), text messaging rates ($/month), and school schedules (periods/day
or days/cycle).
It is important to continue emphasizing that a rate compares two
different things. In any context students may provide in which ba
represents a rate, two quantities in different units must be compared.
The distinction between ratios and rates is subtle. As students work with
rates, they should be encouraged to continue to examine the similarities
and differences and make connections between ratios and rates.
A fundamental difference between the two is the ability to represent a
ratio, but not a rate, as a percent. Students should recall from previous
work with ratios in this unit that part-to-whole ratios can be expressed
as a percent, whereas part-to-part ratios cannot. For example, if Daniel
gets a hit 2 out of every 9 times at bat, this is a ratio that can be written as a percent. If the hits are thought of as successful at bats, parts of
a whole are being compared to the whole. A percent compares part of a
whole to the whole. Because the units in a rate are different, there isn’t a
whole to make a comparison to. Therefore, a rate cannot be represented
as a percent.
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Identify the rates in the following situations, and express them using
words and symbols.
(i)
When Denise bought gasoline, she paid $27.44 for 11.2 litres.
Find the price of gasoline per litre.
(ii) Jacob filled his 60-gallon bathtub in 5 minutes. How fast was
the water flowing?
(iii) On her vacation, Charmaine’s flight lasted 4.5 hours. She
traveled 954 miles. Find the average speed of the plane.
(8N4.6)
Math Makes Sense 8
Lesson 5.9: Exploring Rates
Lesson 5.10: Comparing Rates
Group Discussion
• Discuss the best way to measure each of the following.
(i) The speed you travel on the highway
(ii) How many eggs a family uses in:
a day
a week
a month
(iii) Hockey players are rated depending on achievements per
minutes played. Name some appropriate achievements.
(8N4.6, 8N4.7)
ProGuide: pp.64-69, 70-76
SB: pp.294-299, 300-306
Practice and HW Book: pp.124126
Journal
• Use examples to explain how ratios and rates are the same, and then
use examples to explain how ratios and rates are different.
(8N4.8, 8N5.1, 8N5.2)
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
167
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N5 Continued
Achievement Indicator:
8N5.4 Solve a given problem
involving rate.
Proportional reasoning can be developed through activities that
compare and determine the equivalence of ratios and rates, and
solving proportions in a wide variety of problem-based contexts. It is
important that students see the usefulness of proportions. The topic
is rich in problem-solving opportunities and lends itself to real-world
applications. For example, ratios, rates and proportions are commonly
used for scale models, altering a recipe, and comparison shopping.
Problem solving with rates often involves comparing rates. When
writing equivalent rates, students should check to be sure the
positioning of the units in the terms within each rate is the same. For
example, a rate equivalent to 100 km/h should be able to be written as
a fraction with the measurement of distance in the numerator and the
measurement of time in the denominator.
To solve problems involving distance, time and average speed, or to
determine the better buy in consumer situations, it is often beneficial to
use unit rates. A unit rate illustrates two measurements that are directly
proportional, where one term is 1.
It is important for students to be aware that when they are comparing
unit rates, the numbers must be in the same units. For example, if
comparing one quantity measured in grams with another measured
in kilograms, the options are to change both measurements to grams
or kilograms. The unit of measurement used for such a unit rate is
often the student’s choice. A review of conversion from one unit of
measurement to another may be necessary here.
Sample problems with two possible solution methods follow on the next
two-page spread.
Continued
168
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Paper and Pencil
• Jane found a good deal on soft drinks. She could buy 12 packs for
$2.99. She needs 72 cans for her party. Explain how she can calculate
the cost.
(8N5.4)
• Which is the better buy: 1.2 L of orange juice for $2.50, or 0.75 L of
orange juice for $1.40? Explain why it is the better buy. (8N5.4)
Math Makes Sense 8
Lesson 5.9: Exploring Rates
Lesson 5.10: Comparing Rates
Interview
• When making lemonade Sue uses 5 scoops of powder for 6 cups of
water, and Sarah uses 4 scoops of powder for 5 cups of water.
i) Are the situations proportional to each other? Explain why or why
not.
ii) In which situation is it likely the lemonade will be more
flavourful? What assumptions did you make?
(8N5.4)
• Explain why 1: 20,000,000 is another way to describe the ratio of 1
cm representing 200 km on a map.
(8N5.4)
ProGuide: pp.64-69, 70-76,
Master 5.7a, 5.7c
CD-ROM: Master 5.29
SB: pp.294-299, 300-306
Practice and HW Book: pp.124126, 127-128
Journal
• Discuss whether or not the following could be solved using a
proportion:
David is 6 years old and Ellen is 2 years old. How old will Ellen be
when David is 12 years old?
(8N5.4)
Portfolio
• A statue of John Cabot was made from a model. The height of the
model was 25 cm. Find the height in metres of the statue if it was
made using a scale of 1:15 (scale represents ratio of model to actual
height).
(8N5.4)
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
169
PERCENT, RATIO, AND RATE
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
8N5 Continued
Achievement Indicator:
8N5.4 Continued
The local drugstore is advertising cases of macaroni and cheese at a sale
price of $8.99, with 12 boxes in a case. The grocery store across the
street is selling the same macaroni and cheese at a price of $5 for 6
boxes. Which is the better deal?
Solution #1: Unit Rates
$0.75
Drug Store: 12$8.99
boxes = 1box
Grocery Store:
$5.00
6 boxes
=
$0.83
1box
Solution #2: Equivalent Rates
$8.99
Drugstore: 12boxes
Grocery Store:
$5.00
6 boxes
= 12$10.00
boxes
The sale the drugstore offers is The drugstore offers the better
the better buy.
deal.
Fred received a gift card for his birthday. He used it to download some
new music for his MP3 player. Fred downloaded 12 songs in 15 minutes.
At this rate, how many songs could he download in 1 hour?
Solution #1: Unit Rates
12 songs
15min
=
0.8 songs
1min
Solution #2: Equivalent Rates
12 songs
15min
=
x songs
60min
Since there are 60 minutes in 1 To create an equivalent rate,
hour, multiply 0.8 by 60. He can multiply by 4 and x = 48 .
download 48 songs in an hour.
170
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
PERCENT, RATIO, AND RATE
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Project
• Research a local or national long-distance running event, and
compare the performance of winners from different years. Compare
the distance travelled to the time it takes to complete the race.
(8N5.4)
• Determine the fuel economy for your family vehicle. You can prepare
and use a log such as the one that follows to track fuel purchases,
kilometres driven, and fuel economy over several weeks.
(8N5.4)
Amount of
Gas
Purchased
(L)
Beginning
Odometer
Reading
(km)
Ending
Odometer
Reading
(km)
Total
Distance
Travelled
Fuel
Efficiency
Math Makes Sense 8
Lesson 5.9: Exploring Rates
Lesson 5.10: Comparing Rates
ProGuide: pp.64-69, 70-76,
Master 5.7a, 5.7c
CD-ROM: Master 5.29
SB: pp.294-299, 300-306
Practice and HW Book: pp.124126, 127-128
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE
171
PERCENT, RATIO, AND RATE
172
GRADE 8 MATHEMATICS DRAFT CURRICULUM GUIDE