6-7th Grade Mathematics Curriculum Guide
2015 – 2016
Unit 2: Ratios, Rates, & Proportions
Time Frame: Quarter 1 – about 33 days
Connections to Previous Learning:
The study of ratios and proportional relationships extends students’ work in measurement and in multiplication and division from the elementary grades. It is expected that
students will have prior knowledge and experience related to concepts and skills such as multiples, factors, and divisibility rules. This background knowledge about relationships
and rules for multiplication and division of whole numbers connects to the understanding of how to complete tables to help support the development of ratio and rate
reasoning.
In Grade 6, students develop an understanding of ratio and proportion using ratio tables, tape diagrams, and double number lines.
Focus of this Unit:
Students learn that a ratio expresses the comparison between two quantities. Special types of ratios are rates, unit rates, measurement conversions, and percentages are
concepts that are applied to a variety of real world and mathematical situations. Students gain a deeper understanding of proportional reasoning through instruction and
practice. They develop and use multiplicative thinking to develop a sense of proportional reasoning as they describe ratio relationships between two quantities.
Students extend their understanding of ratios and develop understanding of proportionality to solve single and multi-step problems involving such real world
contexts as percent of increase or decrease and scale drawing.
Connections to Subsequent Learning:
Ratios and proportional relationships are foundational for further study in mathematics and science and useful in everyday life. Students use ratios in geometry and in algebra
when they study similar figures and slopes of lines, and later when they study sine, cosine, tangent, and other trigonometric ratios in high school.
Students use ratios when they work with situations involving constant rates of change, and later in calculus when they work with average and instantaneous rates of change of
functions. An understanding of ratio is essential in the sciences to make sense of quantities that involve derived attributes such as speed, acceleration, density, surface tension,
electric or magnetic field strength, and to understand percentages and ratios used in describing chemical solutions. Ratios and percentages are also useful in many situations in
daily life, such as in cooking and in calculating tips, miles per gallon, taxes, and discounts. They also are also involved in a variety of descriptive statistics, including demographic,
economic, medical, meteorological, and agricultural statistics (e.g., birth rate, per capita income, body mass index, rain fall, and crop yield) and underlie a variety of measures,
for example, in finance (exchange rate), medicine (dose for a given body weight), and technology (kilobits per second). Students will use this understanding of proportionality to
find scale factors between geometric figures and develop understandings of congruence and similarity. They will use ratio tables to study statistics and probability.
Mathematical Practices
1. Make Sense of Problems and Persevere in Solving Them.
2. Reason Abstractly and Quantitatively.
3. Construct Viable Arguments and Critique the Reasoning of Others.
4. Model with Mathematics.
Unit 5
5. Use Appropriate Tools Strategically.
6. Attend to Precision.
7. Look for and Make Use of Structure.
8. Look for and Express Regularity in Repeated Reasoning.
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6-7th Grade Mathematics Curriculum Guide
Stage 1 Desired Results
2015 – 2016
Transfer Goals
Students will be able to independently use their learning to…
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Make decisions based upon the proportional relationships that they see in their lives.
How can I use proportional relationships to solve ratio and percent problems?
Meaning Goals
UNDERSTANDINGS
Students will understand that…
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A ratio expresses the comparison between two quantities. Special
types of ratios are rates, unit rates, measurement conversions,
and percents.
A ratio or a rate expresses the relationship between two
quantities. Ratio and rate language is used to describe a
relationship between two quantities (including unit rates.)
A rate is a type of ratio that represents a measure, quantity, or
frequency, typically one measured against a different type of
measure, quantity, or frequency.
Ratio and rate reasoning can be applied to many different types
of mathematical and real-life problems (rate and unit rate
problems, scaling, unit pricing, statistical analysis, etc.).
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ESSENTIAL QUESTIONS
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When is it useful to be able to relate one quantity to another?
How are ratios and rates similar and different?
What is the connection between a ratio and a fraction?
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How do rates, ratios, percentages and proportional relationships apply to our world?
When and why do I use proportional comparisons?
How does comparing quantities describe the relationship between them?
How do graphs illustrate proportional relationships?
How can I use proportional relationships to solve ratio and percent problems?
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Rates, ratios, percentages and proportional relationships express
how quantities change in relationship to each other.
Rates, ratios, percentages and proportional relationships can be
represented in multiple ways.
Rates, ratios, percentages and proportional relationships can be
applied to problem solving situations.
Rates, ratios, percentages and proportional relationships can be
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6-7th Grade Mathematics Curriculum Guide
•
applied to problem solving situations such as interest, tax,
discount, etc.
Rates, ratios, percentages and proportional relationships can be
applied to solve multi-step ratio and percent problems.
2015 – 2016
Acquisition Goals
Students will know…
Students will be skilled at…
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A ratio compares two related quantities.
Ratios can be represented in a variety of formats including each,
to, per, for each, %, 1/5, etc.
A percent is a type of ratio that compares a quantity to 100.
A unit rate is the ratio of two measurements in which the second
term is 1.
When it is appropriate to use ratios/rates to solve mathematical
or real life problems.
Mathematical strategies for solving problems involving ratios and
rates, including tables, tape diagrams, double line diagrams,
equations,
equivalent fractions, graphs, etc
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Unit 5
Use ratio language to describe a ratio relationship between two quantities. (6.RP.1)
Represent a ratio relationship between two quantities using manipulatives and/or pictures, symbols
and real-life situations. (a to b, a:b, or a/b) (6.RP.1)
Represent unit rate associated with ratios using visuals, charts, symbols, real-life situations and rate
language. (6.RP.2)
Use ratio and rate reasoning to solve real-world and mathematical problems. (6.RP.3)
Make and interpret tables of equivalent ratios. (6.RP.3)
Plot pairs of values of the quantities being compared on the coordinate plane. (6.RP.3)
Use multiple representations such as tape diagrams, double number line diagrams, or equations to
solve rate and ratio problems. (6.RP.3)
Solve unit rate problems (including unit pricing and constant speed). (6.RP.3)
Solve percent problems, including finding a percent of a quantity as a rate per 100 and finding the
whole, given the part and the percent. (6.RP.3)
th
Compute unit rates involving rational numbers, fractions, and complex fractions. (7.RP.1)
Compute ratios of length in like or different units. (7.RP.1)
Compute ratios of area and other measurements in like or different units. (7.RP.1)
Determine whether two quantities are in a proportional relationship by using a table and or graph.
(7.RP.2)
Identify the constant of proportionality (unit rate) in tables, graphs, diagrams, and verbal descriptions.
(7.RP.2)
Create and solve equations to represent proportional relationships. (7.RP.2)
Use words to describe the location of a point on a graph and its relationship to the origin. (7.RP.2)
Explain what a point on a graph of a proportional relationship means in terms of the situation. (how
does the one quantity relate to the other) (7.RP.2)
Solve multi-step ratio and percent problems. (7.RP.3)
Solve problems involving simple interest and tax. (7.RP.3)
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6-7th Grade Mathematics Curriculum Guide
•
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2015 – 2016
Solve problems involving markups and markdowns, gratuities and commissions, and fees. (7.RP.3)
Solve problems involving percent increase, percent decrease, and percent (margin of) error. (7.RP.3)
Stage 1 Established Goals: Common Core State Standards for Mathematics
6.RP.A Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird
house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For
example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per
hamburger.”
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line
diagrams, or equations.
6.RP.A.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the
coordinate plane. Use tables to compare ratios.
6.RP.A.3b Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many
lawns could be mowed in 35 hours? At what rate were lawns being mowed?
6.RP.A.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part
and the percent.
6.RP.A.3d) Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
7.RP.A Analyze proportional relationships and use them to solve real-world and mathematical problems.
7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units. For
example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction ½ /1/4 miles per hour, equivalently 2 miles per hour.
7.RP.2 Recognize and represent proportional relationships between quantities.
7.RP.2a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing
whether the graph is a straight line through the origin.
7.RP.2b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.2c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the
relationship between the total cost and the number of items can be expressed as t = pn.
7.RP.2d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is
the unit rate.
7. RP.3 Use proportional relationships to solve multi-step ratio and percent problems.
Unit 5
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6-7th Grade Mathematics Curriculum Guide
2015 – 2016
Explanations, Examples, and Comments
Overview of Suggested Assessments & Performance Tasks for Unit
Vocabulary will be listed next to each standard
Mid-Unit Assessment –
End-of-Unit Assessment –
Percent, proportion, rate, unit rate, ratio, quantity, tape diagram, double number line,
numerator, denominator, equivalent, part-to-part, part-to-whole, rational numbers,
constant of proportionality, percent increase, percent decrease, percent error,
Formative assessments for this unit
6.RP.3 – Pre
6.RP.3 – Post
7.RP.2b
7.RP.2a & c
7.RP.3a
Possible Performance Tasks-P drive
th
Georgia (“p” drive - 6 grade performance tasks)
• Ratios and Rates 6.RP.1 6.RP.2, 6.RP.3a & b
• How Many Noses are in Your Arms? 6.RP.1, 6.RP.3d
• Fruit Punch 6.RP.3
• Making Sense of Graphs 7.RP.2b
• Analyzing Tables 7.RP.2a
• Reaching the Goal 6.RP.3c
• Free Throws 6.RP.3c
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7 grade Georgia performance tasks
• Which Deal is Better? 6.RP.3b, 7.RP.1
• Patterns and Percents 7.RP.3
Illuminations:
• Feeding Frenzy 6.RP.3 & 7.RP.2
• In Your Shadow 7.RP.2
• Highway Robbery 6.RP.3 & 7.RP.1 & 2
Illustrative Mathematics
• Cereal 7.RP.1,2,3
• Lawn Mowing 7.RP.1,2,3
• Mixing Paints 7.RP.1,3
• Photographs 7.RP.1,2
Unit 5
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6-7th Grade Mathematics Curriculum Guide
6.RP.A.1
2015 – 2016
Stage 3 MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine which of the materials below
best meet student instructional needs. All materials listed may not be needed.
Vocab – ratio, part-to-part, part-to-whole
A ratio is the comparison of two quantities or measures. The comparison can be part-towhole (ratio of guppies to all fish in an aquarium) or part-to-part (ratio of guppies to
goldfish).
A ratio is a comparison of two quantities which can be written as
A rate is a ratio where two measurements are related to each other. When discussing
measurement of different units, the word rate is used rather than ratio. Understanding rate,
however, is complicated and there is no universally accepted definition. When using the term
rate, contextual understanding is critical. Students need many opportunities to use models to
demonstrate the relationships between quantities before they are expected to work with
rates numerically.
Holt Course 1 Lesson 7-1 Ratios and Rates, 6.RP.A.1 & 2 (1)
Holt Course 2 Lesson 5-2 Rates, 6.RP.A.2 (1)
Additional Resources, if necessary
Holt Course 2 Lesson 5-1 Ratios (1)
Holt Course 1 Lesson 7-2 Using Tables to Explore Equivalent Ratios and Rates (1)
Holt Course 3 Lesson 5-2 Ratios, Rates, and Unit Rates (1)
A comparison of 8 black circles to 4 white circles can be written as the ratio of 8:4 and can be
regrouped into 4 black circles to 2 white circles (4:2) and 2 black circles to 1 white circle (2:1).
Students should be able to identify all these ratios and describe them using “For every….,
there are …”
6.RP.A.2
Vocab – rate, unit rate, numerator, denominator
Unit 5
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2015 – 2016
A unit rate compares a quantity in terms of one unit of another quantity. Students will often
use unit rates to solve missing value problems. Cost per item or distance per time unit are
common unit rates, however, students should be able to flexibly use unit rates to name the
amount of either quantity in terms of the other quantity. Students will begin to notice that
related unit rates are reciprocals as in the first example. It is not intended that this be taught
as an algorithm or rule because at this level, students should primarily use reasoning to find
these unit rates.
In Grade 6, students are not expected to work with unit rates expressed as complex fractions.
Both the numerator and denominator of the original ratio will be whole numbers.
Examples:
o On a bicycle you can travel 20 miles in 4 hours. What are the unit rates in this
situation, (the distance you can travel in 1 hour and the amount of time
required to travel 1 mile)?
Engage NY 7th Module 1, Lessons 11 and 12, 7.RP.1
(2) http://www.engageny.org/resource/grade-7-mathematics-module-1
o
A simple modeling clay recipe calls for 1 cup corn starch, 2 cups salt, and 2 cups
boiling water. How many cups of corn starch are needed to mix with each cup of
salt?
7.RP.1
Vocab – rational numbers
Students continue to work with unit rates from 6th grade; however, the comparison now
includes fractions compared to fractions. The comparison can be with like or different units.
Fractions may be proper or improper.
Unit 5
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6-7th Grade Mathematics Curriculum Guide
2015 – 2016
Example 1:
1
1
If gallon of paint covers of a wall, then how much paint is needed for the entire wall?
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1
1
Solution: gal / wall.
2
6
3 gallons per 1 wall
6
6.RP.A.3a,b, & d:
Vocab – percent, proportion, quantity, tape diagram, double number line, equivalent
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Using the information in the table, find the number of yards in 24 feet.
There are several strategies that students could use to determine the solution to this
problem.
o Add quantities from the table to total 24 feet (9 feet and 15 feet);
therefore the number of yards must be 8 yards (3 yards and 5 yards).
o Use multiplication to find 24 feet: 1) 3 feet x 8 = 24 feet; therefore 1 yard
x 8 = 8 yards, or 2) 6 feet x 4 = 24 feet; therefore 2 yards x 4 = 8 yards.
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Compare the number of black to white circles. If the ratio remains the same, how many
black circles will you have if you have 60 white circles?
Unit 5
6.RP.A.3a,b, & d:
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Engage NY 6 Module 1 Lessons 10-11, 14-15, 6.RP.A.3a (2)
http://www.engageny.org/resource/grade-6-mathematics-module-1
th
Engage NY 6 Module 1 Lessons 18-23, 6.RP.2, 6.RP.3b,& 6.RP.3d (6)
http://www.engageny.org/resource/grade-6-mathematics-module-1
th
Possible Performance Tasks: (“p” drive - 6 grade performance tasks
Georgia
• Ratios and Rates 6.RP.1 6.RP.2, 6.RP.3a & b
• How Many Noses are in Your Arms? 6.RP.1, 6.RP.3d
• Which Deal is Better? 6.RP.3b, 7.RP.1
• Fruit Punch 6.RP.3
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6-7th Grade Mathematics Curriculum Guide
7.RP.A.2
Vocab – Constant of Proportionality
Students’ understanding of the multiplicative reasoning used with proportions continues
from 6th grade. Students determine if two quantities are in a proportional relationship from a
table. Fractions and decimals could be used with this standard.
Note: This standard focuses on the representations of proportions. Solving proportions is
addressed in 7.SP.3.
Example 1:
The table below gives the price for different numbers of books. Do the numbers in the table
represent a proportional relationship?
2015 – 2016
7.RP.A.2
Holt Course 2 Lesson 5-4 Identifying and Writing Proportions, 7.RP.2.a (1)
Engage NY 7th Module 1, Lessons 7-10, 7.RP.2b,c, and d & 7.EE.4.a
(4) http://www.engageny.org/resource/grade-7-mathematics-module-1
Additional Resources, if necessary
Holt Course 3 Lesson 5-1 Ratios and Proportions
Performance Tasks:
th
Georgia “p” drive (6 grade performance tasks)
• Making Sense of Graphs 7.RP.2b
• Analyzing Tables 7.RP.2a
Illustrative Mathematics
• Photographs 7.RP.1,2
Solution:
Students can examine the numbers to determine that the price is the number of books
multiplied by 3, except for 7 books. The row with seven books for $18 is not proportional to
the other amounts in the table; therefore, the table does not represent a proportional
relationship.
Students graph relationships to determine if two quantities are in a proportional relationship
and to interpret the ordered pairs. If the amounts from the table above are graphed (number
of books, price), the pairs (1, 3), (3, 9), and (4, 12) will form a straight line through the origin
(0 books, 0 dollars), indicating that these pairs are in a proportional relationship. The ordered
pair (4, 12) means that 4 books cost $12. However, the ordered pair (7, 18) would not be on
the line, indicating that it is not proportional to the other pairs.
Illuminations:
• Feeding Frenzy 6.RP.3 &
7.RP.2 http://illuminations.nctm.org/Lesson.aspx?id=2854
• In Your Shadow 7.RP.2
http://illuminations.nctm.org/Lesson.aspx?id=1672
• Highway Robbery 6.RP.3 & 7.RP.1 &
2 http://illuminations.nctm.org/Lesson.aspx?id=3128
The ordered pair (1, 3) indicates that 1 book is $3, which is the unit rate. The y-coordinate
when x = 1 will be the unit rate. The constant of proportionality is the unit rate. Students
identify this amount from tables (see example above), graphs, equations and verbal
Unit 5
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6-7th Grade Mathematics Curriculum Guide
descriptions of proportional relationships.
2015 – 2016
Example 2:
The graph below represents the price of the bananas at one store. What is the constant of
proportionality?
Solution:
From the graph, it can be determined that 4 pounds of bananas is $1.00; therefore, 1 pound
of bananas is $0.25, which is the constant of proportionality for the graph. Note: Any point on
the line will yield this constant of proportionality.
Students write equations from context and identify the coefficient as the unit rate which is
also the constant of proportionality.
Example 3:
The price of bananas at another store can be determined by the equation: P = $0.35n, where
P is the price and n is the number of pounds of bananas. What is the constant of
proportionality (unit rate)?
Solution:
The constant of proportionality is the coefficient of x (or the independent variable). The
constant of proportionality is 0.35.
Example 4:
A student is making trail mix. Create a graph to determine if the quantities of nuts and fruit
are proportional for each serving size listed in the table. If the quantities are proportional,
what is the constant of proportionality or unit rate that defines the relationship? Explain how
the constant of proportionality was determined and how it relates to both the table and
graph.
Unit 5
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6-7th Grade Mathematics Curriculum Guide
2015 – 2016
The relationship is proportional. For each of the other serving sizes there are 2 cups of fruit
for every 1 cup of nuts (2:1).
The constant of proportionality is shown in the first column of the table and by the steepness
(rate of change) of the line on the graph.
Example 5:
The graph below represents the cost of gum packs as a unit rate of $2 dollars for every pack
of gum. The unit rate is represented as $2/pack. Represent the relationship using a table and
an equation.
Unit 5
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6-7th Grade Mathematics Curriculum Guide
2015 – 2016
A common error is to reverse the position of the variables when writing equations. Students
may find it useful to use variables specifically related to the quantities rather than using x and
y. Constructing verbal models can also be helpful. A student might describe the situation as
“the number of packs of gum times the cost for each pack is the total cost in dollars”. They
can use this verbal model to construct the equation. Students can check their equation by
substituting values and comparing their results to the table. The checking process helps
student revise and recheck their model as necessary. The number of packs of gum times the
cost for each pack is the total cost. (g x 2 = d)
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6.RP.A.3c:
6.RP.A.3c/7.RP.A.3:
•
Engage NY 6 Module 1 Lessons 24-28, 6.RP.3c (5)
http://www.engageny.org/resource/grade-6-mathematics-module-1
th
If 6 is 30% of a value, what is that value? (Solution: 20)
Additional Resources, if necessary
Holt Course 2 Lesson 6-4 Percent of a Number and Holt Course 3 Lesson 6-2
Estimate with Percents)
Holt Course 2 Lesson 6-4 Lab Explore Percents (1)
Holt Course 1 Lesson 7-7 Percents (1)
Holt Course 1 Lesson 7-8 Percents, Decimals, and Fractions (1)
Holt Course 3 Lesson 6-1 Relating Decimals, Fractions, and Percents (1)
CMP Bits and Pieces I Investigation 6.3 Changing Forms
CMP Bits and Pieces I Investigation 6.4 It’s Raining Cats and Dogs
Holt Course 1 Lesson 7-9 Percent Problems (1)
Holt Course 2 Lesson 6-3 Estimate with Percents (1)
Holt Course 3 Lesson 6-4 Finding a Number When the Percent is Known (1)
7.RP.A.3
Vocab – percent increase, percent decrease, percent error, mark-up, mark-down
In 6th grade, students used ratio tables and unit rates to solve problems. Students expand
their understanding of proportional reasoning to solve problems that are easier to solve with
cross-multiplication.
Students understand the mathematical foundation for cross-multiplication. An explanation of
this foundation can be found in Developing Effective Fractions Instruction for Kindergarten
Through 8th Grade.
Example 1:
3
Sally has a recipe that needs teaspoon of butter for every 2 cups of milk. If Sally increases
4
the amount of milk to 3 cups of milk, how many teaspoons of butter are needed?
Using these numbers to find the unit rate may not be the most efficient method. Students
can set up the following proportion to show the relationship between butter and milk.
Unit 5
7.RP.A.3:
Holt Course 2 Lesson 6-6 Percent of Change (1)
Holt Course 3 Lesson 6-6 Applications of Percents (1)
Holt Course 3 Lesson 6-7 Simple Interest (1)
Engage NY Module 1, Lessons 14, 7.RP.3
(1) http://www.engageny.org/resource/grade-7-mathematics-module-1
Additional Resources, if necessary
CMP Bits and Pieces III Investigation 4 Using Percents
CMP Bits and Pieces III Investigation 5 More about Percents
CMP Bits and Pieces I Investigation 6.3 Changing Forms,
CMP Bits and Pieces I Investigation 6.4 It’s Raining Cats and Dogs
Holt Course 2 Lesson 6-4 Percent of a Number
Holt Course 2 Lesson 6-3 Estimate with Percents
Holt Course 3 Lesson 6-5 Percent Increase and Decrease
Holt Course 3 Lesson 6-7 Simple Interest
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6-7th Grade Mathematics Curriculum Guide
Solution:
1
One possible solution is to recognize that 2 • 1 = 3 so
1
needed would be 1 teaspoons.
8
2
3
4
1
• 1 = x. The amount of butter
2
3
A second way to solve this proportion is to use cross-multiplication • 3 = 2x. Solving for x
1
would give 1 teaspoons of butter.
8
Suggested Performance Tasks:
th
Georgia “p” drive (6 grade performance tasks)
4
Finding the percent error is the process of expressing the size of the error (or deviation)
between two measurements. To calculate the percent error, students determine the
absolute deviation (positive difference) between an actual measurement and the accepted
value and then divide by the accepted value. Multiplying by 100 will give the percent error.
(Note the similarity between percent error and percent of increase or decrease)
Example 2:
Jamal needs to purchase a countertop for his kitchen. Jamal measured the countertop as 5 ft.
The actual measurement is 4.5 ft. What is Jamal’s percent error?
7
th
•
•
2015 – 2016
Reaching the Goal 6.RP.3c
Free Throws 6.RP.3c
• Patterns and Percents 7.RP.3
Illustrative Mathematics
• Cereal 7.RP.1,2,3
• Lawn Mowing 7.RP.1,2,3
• Mixing Paints 7.RP.1,3
• Photographs 7.RP.1,2
Illuminations:
• Big Math and Fries
http://illuminations.nctm.org/Lesson.aspx?id=3170
• It’s Not Heavy, It’s Your
Grade http://illuminations.nctm.org/Lesson.aspx?id=3173
• Summer Daze
http://illuminations.nctm.org/Lesson.aspx?id=2847
The use of proportional relationships is also extended to solve percent problems involving
sales tax, markups and markdowns simple interest (I = prt, where I = interest, p = principal, r =
rate, and t = time (in years)), gratuities and commissions, fees, percent increase and
decrease, and percent error.
Unit 5
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6-7th Grade Mathematics Curriculum Guide
Students should be able to explain or show their work using a representation (numbers,
words, pictures, physical objects, or equations) and verify that their answer is reasonable.
Students use models to identify the parts of the problem and how the values are related. For
percent increase and decrease, students identify the starting value, determine the difference,
and compare the difference in the two values to the starting value.
2015 – 2016
For example, Games Unlimited buys video games for $10. The store increases their purchase
price by 300%?
What is the sales price of the video game?
Using proportional reasoning, if $10 is 100% then what amount would be 300%? Since 300%
is 3 times 100%, $30 would be $10 times 3. Thirty dollars represents the amount of increase
from $10 so the new price of the video game would be $40.
Example 3:
Gas prices are projected to increase by 124% by April 2015. A gallon of gas currently costs
$3.80. What is the projected cost of a gallon of gas for April 2015?
Solution:
Possible response: “The original cost of a gallon of gas is $3.80. An increase of 100% means
that the cost will double. Another 24% will need to be added to figure out the final projected
cost of a gallon of gas. Since 25% of $3.80 is about $0.95, the projected cost of a gallon of gas
should be around $8.15.”
$3.80 + 3.80 + (0.24 + 3.80) = 2.24 x 3.80 = $8.15
Example 4:
A sweater is marked down 33% off the original price. The original price was $37.50. What is
the sale price of the sweater before sales tax?
Solution:
The discount is 33% times 37.50. The sale price of the sweater is the original price minus the
discount or 67% of the original price of the sweater, or Sale Price = 0.67 x Original Price.
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2015 – 2016
Example 5:
A shirt is on sale for 40% off. The sale price is $12. What was the original price? What was the
amount of the discount?
Solution:
The sale price is 60% of the original price. This reasoning can be expressed as 12 = 0.60p.
Dividing both sides by 0.60 gives an original price of $20.
Example 6:
At a certain store, 48 television sets were sold in April. The manager at the store wants to
encourage the sales team to sell more TVs by giving all the sales team members a bonus if
the number of TVs sold increases by 30% in May.
How many TVs must the sales team sell in May to receive the bonus? Justify the solution.
Solution:
The sales team members need to sell the 48 and an additional 30% of 48. 14.4 is exactly 30%
so the team would need to sell 15 more TVs than in April or 63 total (48 + 15)
Example 7:
A salesperson set a goal to earn $2,000 in May. He receives a base salary of $500 as well as a
10% commission for all sales. How much merchandise will he have to sell to meet his goal?
Solution:
$2,000 - $500 = $1,500 or the amount needed to be earned as commission. 10% of what
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6-7th Grade Mathematics Curriculum Guide
amount will equal $1,500.
2015 – 2016
Example 8:
After eating at a restaurant, your bill before tax is $52.60 The sales tax rate is 8%. You decide
to leave a 20% tip for the waiter based on the pre-tax amount. How much is the tip you leave
for the waiter? How much will the total bill be, including tax and tip? Express your solution as
a multiple of the bill.
Solution:
The amount paid =
The total bill would be $67.20.
= 0.28 x $52.50 or $14.70 for the tip and tax.
Stage 2 - Evidence
SAMPLE Assessment Evidence
OTHER ASSESSMENT EVIDENCE:
Formative assessments (available on P: drive)
Level 4 students should be able to solve unfamiliar or multi-step problems by finding
Spaced Learning Over Time (SLOT - entrance & exit slips)
the whole, given a part and the percent; explain ratio relationships between any two
Checking for understanding (CFU’s)
number quantities; and identify relationships between models or representations.
Lesson Quizzes
Level 3 students should be able to use ratio reasoning to solve and understand the
Classroom Assessments
concept of unit rates in unfamiliar or multi-step problems, including instances of unit
pricing and constant speed, and solve percent problems by finding the whole, given a
part and the percent. They should be able to describe a ratio relationship between any
two number quantities (denominators less than or equal to 12).
Level 2 students should be able to understand the concept of unit rate in
straightforward, well-posed problems and solve straightforward, well-posed, one-step
problems requiring ratio reasoning.
Level 1 students should be able to describe a ratio relationship between two whole
Evaluative Criteria/Assessment Level Descriptors (ALDs):
6.RP.A (SBAC Target A)
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number quantities, find missing values in tables that display a proportional
relationship, and plot the pairs of values from a table on the coordinate plane. They
should be able to find a percent as a rate per hundred and convert measurement
units.
2015 – 2016
7.RP.A (SBAC Target A)
Level 4 students should be able to solve real-world problems involving proportional
relationships and measurement conversions in various formats (e.g., verbally,
tabularly, graphically) in a contextual scenario that involves identifying relationships
between elements presented in various formats.
Level 3 students should be able to identify, represent, and analyze proportional
relationships in various formats; find unit rates associated with ratios of fractions; and
use unit rates to solve one-step problems involving rational numbers. They should be
able to analyze a graph of a proportional relationship in order to explain what the
points (x, y) and (1, r) represent, where r is the unit rate, and use this information to
solve problems.
Level 2 students should be able to find whole number proportionality constants in
relationships presented in graphical, tabular, or verbal formats in familiar contexts.
They should also be able to identify proportional relationships presented in equation
formats and find unit rates involving whole numbers.
Level 1 students should be able to identify proportional relationships presented in
graphical, tabular, or verbal formats in familiar contexts.
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6-7th Grade Mathematics Curriculum Guide
LEARNING ACTIVITIES:
Stage 3 – Learning Plan Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
NOTES:
2015 – 2016
**Days may change depending on any tasks or assessing you choose to do.
6.RP.A.1 & 6.RP.A.2
Day 1: Ratios and Rates
• Holt Course 1 Lesson 7-1
Day 2: Ratios, Rates, and Unit Rates
• Holt Course 3 Lesson 5-2
7.RP.1
Day 3: Ratios of Fractions and Their Unit Rates
th
• Engage NY 7 Module 1, Lessons 11
Day 4: Ratios of Fractions and Their Unit Rates
th
• Engage NY 7 Module 1, Lessons 12
6.RP.A.3a
Day 5: The Structure of Ratio Tables: Additive and Multiplicative
th
• Engage NY 6 Module 1 Lessons 10
Day 6: Comparing Ratios Using Ratio Tables
th
• Engage NY 6 Module 1 Lessons 11
Day 7: From Ratio Tables, Equations, & Double Number Line Diagrams to Plots on the Coordinate Plane
th
• Engage NY 6 Module 1 Lessons 14
Day 8: A Synthesis of Representations of Equivalent Ratio Collections
th
• Engage NY 6 Module 1 Lessons 15
6.RP.A.2, 6.RP.A.3b,& 6.RP.A.3d
Day 9: Finding a Rate by Dividing Two Quantities
th
• Engage NY 6 Module 1 Lessons 18
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6-7th Grade Mathematics Curriculum Guide
Stage 3 – Learning Plan Sample
Day 10: Comparison Shopping – Unit Price & Related Measurement Conversions
th
• Engage NY 6 Module 1 Lessons 19
2015 – 2016
Day 11: Comparison Shopping – Unit Price & Related Measurement Conversions
th
• Engage NY 6 Module 1 Lessons 20
Day 12: Getting the Job Done – Speed, Work, & Measurement Units
th
• Engage NY 6 Module 1 Lessons 21
Day 13: Getting the Job Done – Speed, Work, & Measurement Units
th
• Engage NY 6 Module 1 Lessons 22
Day 14: Problem-Solving Using Rates, Unit Rates, & Conversions
th
• Engage NY 6 Module 1 Lessons 23
7.RP.2.a
Day 15: Identifying and Writing Proportions,
• Holt Course 2 Lesson 5-4
7.RP.2b,c, and d & 7.EE.4.a
Day 16: Unit Rate as the Constant of Proportionality
th
• Engage NY 7 Module 1, Lessons 7
Day 17: Representing Proportional Relationships with Equations
th
• Engage NY 7 Module 1, Lessons 8
Day 18: Representing Proportional Relationships with Equations
th
• Engage NY 7 Module 1, Lessons 9
Day 19: Interpreting Graphs of Proportional Relationships
th
• Engage NY 7 Module 1, Lessons 10
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Stage 3 – Learning Plan Sample
6.RP.A.3c
2015 – 2016
Day 20: Percent and Rates per 100
th
• Engage NY 6 Module 1 Lessons 24
Day 21: A Fraction as a Percent
th
• Engage NY 6 Module 1 Lessons 25
Day 22: Percent of a Quantity
th
• Engage NY 6 Module 1 Lessons 26
Day 23: Solving Percent Problems
th
• Engage NY 6 Module 1 Lessons 27
Day 24: Solving Percent Problems
th
• Engage NY 6 Module 1 Lessons 28
7.RP.A.3:
Day 25: Percent of Change
• Holt Course 2 Lesson 6-6
Day 26: Applications of Percents
• Holt Course 3 Lesson 6-6
Day 27: Simple Interest
• Holt Course 3 Lesson 6-7
Day 28: Multi-Step Ratio Problems
th
• Engage NY 7 Module 1, Lessons 14
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6-7th Grade Mathematics Curriculum Guide
Stage 3 – Learning Plan Sample
Daily Lesson Plan
Learning Target:
Warm-up:
Activities:
• Whole Group:
• Small Group/Guided/Collaborative/Independent:
• Whole Group:
Checking for Understanding (before, during and after):
Assessments:
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