Download Accelerated 7th Grade Math Second Quarter Unit 3: Ratios and

HIGLEY UNIFIED SCHOOL DISTRICT
INSTRUCTIONAL ALIGNMENT
Accelerated 7th Grade Math Second Quarter
Unit 3: Ratios and Proportional Relationships
Topic C: Ratios and Rates Involving Fractions
In Topic C, students extend their reasoning about ratios and proportional relationships to compute unit rates for ratios and rates specified by rational numbers, such as a speed of ½
mile per ¼ hour (7.RP.1). Students apply their experience in the first two topics and their new understanding of unit rates for ratios and rates involving fractions to solve multistep
ratio word problems (7.RP.3, 7.EE.4a).
Big Idea:
•
•
Essential
Questions:
•
•
•
•
•
•
Rates, ratios, and proportional relationships express how quantities change in relationship to each other.
Rates, ratios, and proportional relationships can be represented in multiple ways.
Rates, ratios, and proportional relationships can be applied to problem solving situations.
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 problems?
Proportional to, proportional relationship, constant of proportionality, ratio, rate, unit rate, ratio table, complex fraction, commission, markdown, mark-up, discount
Vocabulary
Grade
Domain
Standard
7
RP
1
AZ College and Career Readiness Standards
A. Analyze proportional relationships and use
them to solve real-­‐world and mathematical
problems.
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 ½ mile in each ¼ hour, compute the unit
rate as the complex fraction ½/¼ miles per hour,
equivalently 2 miles per hour.
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Explanations & Examples
Resources
Explanation:
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.
Eureka Math:
Module 1 Lessons 11-15
Big Ideas:
Sections: 5.1
th
Students interpreted and computed quotients of fractions in 6 grade;
solving word problems involving division of fractions by fractions
(6.NS.A.1). Using visual representations to review this concept may be
necessary.
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7.MP.2. Reason abstractly and quantitatively.
7.MP.6. Attend to precision.
For example, review dividing fractions using a bar model.
•
•
Examples:
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•
If ½ gallon of paint covers 1/6 of a wall, then how much paint
is needed for the entire wall?
Solution:
7
RP
3
A. Analyze proportional relationships and use
them to solve real-world and mathematical
problems.
Use proportional relationships to solve multi-step ratio
and percent problems. Examples: simple interest, tax,
markups and markdowns, gratuities and commissions,
fees, percent increase and decrease, percent error.
7.MP.1. Make sense of problems and persevere in
solving them.
7.MP.2. Reason abstractly and quantitatively.
7.MP.3. Construct viable arguments and critique the
reasoning of others.
7.MP.4. Model with mathematics.
7.MP.5. Use appropriate tools strategically.
7.MP.6. Attend to precision.
7.MP.7. Look for and make use of structure.
7.MP.8. Look for and express regularity in repeated
reasoning.
1
1
(𝑔𝑎𝑙𝑙𝑜𝑛)
𝑥 6 (𝑔𝑎𝑙𝑙𝑜𝑛𝑠) 3𝑔𝑎𝑙𝑙𝑜𝑛𝑠
2
= 2
=
= 3 𝑔𝑎𝑙𝑙𝑜𝑛𝑠 ⁄𝑤𝑎𝑙𝑙
1
1
1 𝑤𝑎𝑙𝑙
(𝑤𝑎𝑙𝑙)
𝑥 6 (𝑤𝑎𝑙𝑙)
6
6
Explanation:
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 more efficient to solve by setting
up a proportion. Students use the properties of equality to solve the
equation.
Eureka Math:
Module 1 Lessons 11-15
Big Ideas:
Sections: 5.1, 5.3
Note: Only include one-step equations in this unit (6.EE.B.7). Students
will learn how to solve multi-step equations in Unit 5.
Examples:
•
Sally has a recipe that needs ¾ teaspoon of butter for every 2
cups of milk. If Sally increases the amount of milk to 3 cups of
milk, how many teaspoons of butter are needed?
Solution:
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.
3
𝑡𝑒𝑎𝑠𝑝𝑜𝑜𝑛𝑠 𝑜𝑓 𝑏𝑢𝑡𝑡𝑒𝑟 𝑥 𝑡𝑒𝑎𝑠𝑝𝑜𝑜𝑛𝑠 𝑜𝑓 𝑏𝑢𝑡𝑡𝑒𝑟
4
=
3 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑚𝑖𝑙𝑘
2 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑚𝑖𝑙𝑘
Multiplying both sides of the equation by 6 cups of milk
results in:
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3
𝑡𝑒𝑎𝑠𝑝𝑜𝑜𝑛𝑠 𝑜𝑓 𝑏𝑢𝑡𝑡𝑒𝑟 𝑥 𝑡𝑒𝑎𝑠𝑝𝑜𝑜𝑛𝑠 𝑜𝑓 𝑏𝑢𝑡𝑡𝑒𝑟
6 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑚𝑖𝑙𝑘 𝑥 4
=
𝑥 6 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑚𝑖𝑙𝑘
2 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑚𝑖𝑙𝑘
3 𝑐𝑢𝑝𝑠 𝑜𝑓 𝑚𝑖𝑙𝑘
9
4
𝑡𝑒𝑎𝑠𝑝𝑜𝑜𝑛𝑠 𝑜𝑓 𝑏𝑢𝑡𝑡𝑒𝑟 = 2𝑥 𝑡𝑒𝑎𝑠𝑝𝑜𝑜𝑛𝑠 𝑜𝑓 𝑏𝑢𝑡𝑡𝑒𝑟
Multiply both sides of the equation by ½.
•
𝑥=
9
1
𝑡𝑒𝑎𝑠𝑝𝑜𝑜𝑛𝑠 = 1 𝑡𝑒𝑎𝑠𝑝𝑜𝑜𝑛𝑠 𝑜𝑓 𝑏𝑢𝑡𝑡𝑒𝑟.
8
8
A retail clothing store advertises the following sale: Shorts are
½ off the original price; pants are 1/3 off the original price,
and shoes at ¼ off the original price (called the discount rate).
If a pair of shoes costs $40, what it the sale price?
Solution:
•
A used car sales person receives a commission of 1/12 of the
sales price of the car for each car he sells. What would the
sales commission be on a car that sold for $21,999?
Solution:
Commission = $21,9999(1/12)= $1833.25
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Accelerated 7th Grade Math Second Quarter
Unit 3: Ratios and Proportional Relationships
Topic D: Proportional Reasoning with Percents
In Topic D, students deepen their understanding of ratios and proportional relationships (7.RP.A.1, 7.RP.A.2, 7.RP.A.3) by solving a variety of percent problems. Topic D builds on
students’ conceptual understanding of percent from Grade 6 (6.RP.3c), and relates 100% to “the whole.” Students represent percents as decimals and fractions and extend their
understanding from Grade 6 to include percents greater than 100%, such as 225%, and percents less than 1%, such as1/2 % or 0.5%.. Students create algebraic representations
and apply their understanding of percent to interpret and solve multi-step word problems related to markups or markdowns, simple interest, sales tax, commissions, fees, and
percent error (7.RP.A.3). They apply their understanding of proportional relationships, creating an equation, graph, or table to model a tax or commission rate that is represented
as a percent (7.RP.A.1, 7.RP.A.2). Students solve problems related to changing percents and use their understanding of percent and proportional relationships to solve these
problems. Students also apply their understanding of absolute value from Unit 1 (7.NS.A.1b) when solving percent error problems.
• Rates, ratios, percentages and proportional relationships can be applied to solve multi-step percent problems.
• Multiple representations can be used to solve percent problems (tape diagrams, double number-line, graphs, equations, etc…)
Big Idea:
• A unified understanding of numbers is developed by recognizing fractions, decimals, and percents as different representations of rational numbers.
• How can you use proportional relationships to solve percent problems?
• How can you express a unit rate as a percent?
• How are the fraction and decimal representation related to the percent?
Essential
• How can you identify the “whole” in a percent problem?
Questions:
• How does the context of a problem determine whether there is a percent increase or decrease?
• What is the difference between the absolute error and the percent error?
Ratio,
proportion, percent increase, percent decrease, percent error, markdown, markup, sales price, discount price
Vocabulary
Grade
Domain
Standard
7
RP
3
AZ College and Career Readiness Standards
A. Analyze proportional relationships and use
them to solve real-world and mathematical
problems.
Use proportional relationships to solve multistep ratio
and percent problems. Examples: simple interest, tax,
markups and markdowns, gratuities and commissions,
fees, percent increase and decrease, percent error.
Explanations & Examples
Explanations:
The use of proportional relationships is 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.
Resources
Eureka Math:
Module 4 Lessons 1-11,
16 and 18
Big Ideas:
Sections:6.3-6.7
Students should be able to explain or show their work using a
representation (numbers, words, pictures, physical objects, or
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7.MP.1. Make sense of problems and persevere in
solving them.
7.MP.2. Reason abstractly and quantitatively.
7.MP.3. Construct viable arguments and critique the
reasoning of others.
7.MP.4. Model with mathematics.
7.MP.5. Use appropriate tools strategically.
7.MP.6. Attend to precision.
7.MP.7. Look for and make use of structure.
7.MP.8. Look for and express regularity in repeated
reasoning.
equations) and verify that their answer is reasonable. Models help
students 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.
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)
Note: Only include one-step equations in this unit (6.EE.B.7). Students
will learn how to solve multi-step equations in Unit 5.
Examples:
•
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.”
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•
A sweater is marked down 33%. Its original price was $37.50.
What is the 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.
Therefore, the sale price is $25.13
•
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 discount = 40% of p and the sale price = 60% of p.
Therefore, .60p=$12. Solving for p gives p=$20. The original
price of the shirt was $20.
•
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After eating at a restaurant, your bill before tax is $52.60. The
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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 = 0.20 x $52.50 + 0.08 x $52.50 = 0.28 x
$52.50.
•
Games Unlimited buys video games for $10. The store
increases their purchase price by 300%? What is the sales
price of the video game?
Solution:
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.
•
Stephanie paid $9.18 for a pair of earrings. This amount
includes a tax of 8%. What was the cost of the item before
tax?
Solution:
One possible solution path follows: $9.18 represents 100% of
the cost of the earrings + 8% of the cost of the earrings. This
representation can be expressed as 1.08c = 9.18, where c
represents the cost of the earrings. Solving for c gives $8.50
for the cost of the earrings.
•
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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?
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•
7
EE
3
B. Solve real-life and mathematical problems
using numerical and algebraic expressions and
equations.
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•
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 and is going to give 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 your solution.
•
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?
Explanation:
Students solve contextual problems and mathematical problems using
rational numbers. Students convert between fractions, decimals, and
percents as needed to solve the problem. Students use estimation to
justify the reasonableness of answers.
Eureka Math:
Module 4 Lessons 1-11,
16 and 18
Big Ideas:
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Solve multi-step real-life and mathematical
problems posed with positive and negative
rational numbers in any form (whole numbers,
fractions, and decimals), using tools strategically.
Apply properties of operations to calculate with
numbers in any form; convert between forms as
appropriate; and assess the reasonableness of
answers using mental computation and estimation
strategies. For example: If a woman making $25
an hour gets a 10% raise, she will make an
additional 1/10 of her salary an hour, or $2.50, for
a new salary of $27.50. If you want to place a
towel bar 9 3/4 inches long in the center of a door
that is 27 1/2 inches wide, you will need to place
the bar about 9 inches from each edge; this
estimate can be used as a check on the exact
computation.
7.MP.1. Make sense of problems and persevere in
solving them.
7.MP.2. Reason abstractly and quantitatively.
7.MP.3. Construct viable arguments and critique
the reasoning of others.
7.MP.4. Model with mathematics.
7.MP.5. Use appropriate tools strategically.
7.MP.6. Attend to precision.
7.MP.7. Look for and make use of structure.
7.MP.8. Look for and express regularity in
repeated reasoning.
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Estimation strategies for calculations with fractions and decimals
extend from students’ work with whole number operations. Estimation
strategies include, but are not limited to:
o front-end estimation with adjusting (using the highest place
value and estimating from the front end making adjustments
to the estimate by taking into account the remaining
amounts),
o clustering around an average (when the values are close
together an average value is selected and multiplied by the
number of values to determine an estimate),
o rounding and adjusting (students round down or round up and
then adjust their estimate depending on how much the
rounding affected the original values),
o using friendly or compatible numbers such as factors
(students seek to fit numbers together - i.e., rounding to
factors and grouping numbers together that have round sums
like 100 or 1000), and
o using benchmark numbers that are easy to compute (students
select close whole numbers for fractions or decimals to
determine an estimate).
Sections: 6.1, 6.2
Note: Only include one-step equations in this unit (6.EE.B.7). Students
will learn how to solve multi-step equations in Unit 5.
Examples:
•
Three students conduct the same survey about the number of
hours people sleep at night. The results of the number of
people who sleep 8 hours a nights are shown below. In which
person’s survey did the most people sleep 8 hours?
o Susan reported that 18 of the 48 people she surveyed
get 8 hours sleep a night
o Kenneth reported that 36% of the people he
surveyed get 8 hours sleep a night
o Jamal reported that 0.365 of the people he surveyed
get 8 hours sleep a night
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Solution:
In Susan’s survey, the number is 37.5%, which is the greatest
percentage.
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Accelerated 7th Grade Math Second Quarter
Unit 4: Scale Drawings – Ratios, Rates and Percents (2 weeks)
Topic A: Ratios of Scale Drawings
In Unit 4, students bring the sum of their experience with proportional relationships to the context of scale drawings (7.RP.2b, 7.G.1). Given a scale drawing, students rely on their
background in working with side lengths and areas of polygons (6.G.1, 6.G.3) as they identify the scale factor as the constant of proportionality, calculate the actual lengths and
areas of objects in the drawing, and create their own scale drawings of a two-dimensional view of a room or building. Students then extend this knowledge to represent the scale
factor as a percent. Students construct scale drawings, finding scale lengths and areas given the actual quantities and the scale factor as a percent (and vice-versa). Students are
encouraged to develop multiple methods for making scale drawings. Students may find the multiplicative relationship between figures; they may also find a multiplicative
relationship among lengths within the same figure. Students use their understanding of scale factor to identify similar triangles which will be explored more fully in Unit 6. They use
similar triangles to explain why the slope is the same between any two distinct points on a non-vertical line in a coordinate plane.
•
Big Idea:
•
Scale drawings can be applied to problem solving situations involving geometric figures.
Geometrical figures can be used to reproduce a drawing at a different scale.
•
•
•
•
How do you use scale drawings to compute actual lengths and area?
How can you use geometric figures to reproduce a drawing at a different scale?
How do you determine the scale factor?
What does the scale factor tell you about the relationship between the actual picture and the scale drawings?
Proportional to, proportional relationship, constant of proportionality, one-to-one correspondence, scale drawing, scale factor, (ratio, rate, unit rate,
equivalent ratio, reduction, enlargement, scalar
Essential
Questions:
Vocabulary
Grade
Domain
Standard
7
G
1
AZ College and Career Readiness Standards
A. Draw, construct, and describe geometrical
figures and describe the relationships between
them.
Solve problems involving scale drawings of geometric
figures, including computing actual lengths and areas
from a scale drawing and reproducing a scale drawing
at a different scale.
7.MP.1. Make sense of problems and persevere in
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Explanations & Examples
Explanation:
This standard focuses on the importance of visualization in the
understanding of Geometry. Being able to visualize and then represent
geometric figures on paper is essential to solving geometric problems.
Scale drawings of geometric figures connect understandings of
proportionality to geometry and lead to future work in similarity and
congruence. As an introduction to scale drawings in geometry,
students should be given the opportunity to explore scale factor as the
Resources
Eureka Math:
Module 1 Lessons 16-22
Module 4 Lessons 12-15
Big Ideas:
Sections: 7.5
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solving them.
7.MP.2. Reason abstractly and quantitatively.
7.MP.3. Construct viable arguments and critique the
reasoning of others.
7.MP.4. Model with mathematics.
7.MP.5. Use appropriate tools strategically.
7.MP.6. Attend to precision.
7.MP.7. Look for and make use of structure.
7.MP.8. Look for and express regularity in repeated
reasoning.
number of times you multiply the measure of one object to obtain the
measure of a similar object. It is important that students first
experience this concept concretely progressing to abstract contextual
situations.
Students determine the dimensions of figures when given a scale and
identify the impact of a scale on actual length (one-dimension) and
area (two-dimensions). Students identify the scale factor given two
figures. Using a given scale drawing, students reproduce the drawing
at a different scale. Students understand that the lengths will change
by a factor equal to the product of the magnitude of the two size
transformations. Initially, measurements should be in whole numbers,
progressing to measurements expressed with rational numbers.
Examples:
•
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Julie shows the scale drawing of her room below. If each 2 cm
on the scale drawing equals 5 ft, what are the actual
dimensions of Julie’s room? Reproduce the drawing at 3 times
its current size.
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•
If the rectangle below is enlarged using a scale factor of 1.5,
what will be the perimeter and area of the new rectangle?
•
Solution:
The perimeter is linear or one-dimensional. Multiply the
perimeter of the given rectangle (18 in.) by the scale factor
(1.5) to give an answer of 27 in. Students could also increase
the length and width by the scale factor of 1.5 to get 10.5 in.
for the length and 3 in. for the width. The perimeter could be
found by adding 10.5 + 10.5 + 3 + 3 to get 27 in.
The area is two-dimensional so the scale factor must be
squared. The area of the new rectangle would be 14 x 1.52 or
31.5 in2.
•
The city of St. Louis is creating a welcome sign on a billboard
for visitors to see as they enter the city. The following picture
needs to be enlarged so that ½ inch represents 7 feet on the
actual billboard. Will it fit on a billboard that measures 14
feet in height?
Solution:
Yes, the drawing measures 1 inch in height, which
corresponds to 14 feet on the actual billboard.
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•
Chris is building a rectangular pen for his dog. The dimensions
are 12 units long and 5 units wide.
Chris is building a second pen that is 60% the length of the
original and 125% the width of the original. Write equations
to determine the length and width of the second pen.
Solution:
•
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What percent of the area of the large disk lies outside the
smaller disk?
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Solution:
7
RP
2b
A. Analyze proportional relationships and use
them to solve real-world and mathematical
problems.
Recognize and represent proportional relationships
between quantities.
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Explanation:
In this unit, students learn the term scale factor and recognize it as the
constant of proportionality. The scale factor is also represented as a
percentage.
Eureka Math:
Module 1 Lessons 16-22
Module 4 Lessons 12-15
Module 1 Lesson 20 could
be used as a project for
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b.
Identify the constant of proportionality (unit
rate) in tables, graphs, equations, diagrams,
and verbal descriptions of proportional
relationships.
Examples:
•
Nicole is running for school president and her best friend
designed her campaign poster which measured 3 feet by
2 feet. Nicole liked the poster so much she reproduced
the artwork on rectangular buttons measuring 2 inches
by 1 1/3 inches. What is the scale factor?
the unit.
Big Ideas:
Sections: 7.5
Solution:
The scale factor is 1/18.
•
Use a ruler to measure and find the scale factor.
Actual:
Scale Drawing:
8
EE
6
B. Understand the connections between
proportional relationships, lines, and linear
equations
Use similar triangles to explain why the slope m is the
same between any two distinct points on a non-vertical
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Solution:
The scale factor is 5/3
Explanation:
Triangles are similar when there is a constant rate of proportionality
between them. Using a graph, students construct triangles between
two points on a line and compare the sides to understand that the
slope (ratio of rise to run) is the same between any two points on a
line. This is illustrated below.
Eureka Math:
Module 4 Lessons 15-19
Big Ideas:
Sections: 4.2, Extension
4.2, 4.3, 4.4, 4.5
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line in the coordinate plane; derive the equation y = mx
for a line through the origin and the equation y = mx + b
for a line intercepting the vertical axis at b.
8.MP.2. Reason abstractly and quantitatively.
8.MP.3. Construct viable arguments and critique the
reasoning of others.
8.MP.4. Model with mathematics.
8.MP.5. Use appropriate tools strategically.
8.MP.7. Look for and make use of structure.
8.MP.8. Look for and express regularity in repeated
reasoning.
The triangle between A and B has a vertical height of 2 and a horizontal
length of 3. The triangle between B and C has a vertical height of 4 and
a horizontal length of 6. The simplified ratio of the vertical height to
the horizontal length of both triangles is 2 to 3, which also represents a
slope of 2/3 for the line, indicating that the triangles are similar.
Given an equation in slope-intercept form, students graph the line
represented.
The following is a link to a video that derives y = mx + b using similar
triangles: https://learnzillion.com/lessons/1473-derive-ymxb-usingsimilar-triangles
Examples:
•
Show, using similar triangles, why the graph of an equation of
the form y = mx is a line with slope m.
Solution:
Solutions will vary. A sample solution is below.
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The line shown has a slope of 2. When we compare the
corresponding side lengths of the similar triangles we have the
2
4
𝑥
𝑦
ratios = = 2 . In general, the ratios would be =
1
2
1
𝑚
equivalently y = mx, which is a line with slope m.
•
2
Graph the equation 𝑦 = 𝑥 + 1. Name the slope and y3
intercept.
Solution: The slope of the line is 2/3 and the y-intercept is
(0.1)
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Accelerated 7th Grade Math Second Quarter
Unit 5: Equivalent Expressions, Equations and Inequalities (7 weeks)
Topic A: Equivalent Expressions
In Grade 6, students interpreted expressions and equations as they reasoned about one-variable equations (6.EE.A.2). This module consolidates and expands upon students’
understanding of equivalent expressions as they apply the properties of operations (associative, commutative, distributive) to write expressions in both standard form (by expanding
products into sums) and in factored form (by expanding sums into products).
To begin this unit, students will generate equivalent expressions using the fact that addition and multiplication can be done in any order with any grouping and will extend this
understanding to subtraction (adding the inverse) and division (multiplying by the multiplicative inverse) (7.EE.A.1). They extend the properties of operations with numbers (learned
in earlier grades) and recognize how the same properties hold true for letters that represent numbers. Knowledge of rational number operations from Unit 1 is demonstrated as
students collect like terms containing both positive and negative integers.
An area model is used as a tool for students to rewrite products as sums and sums as products and can provide a visual representation leading students to recognize the repeated
use of the distributive property in factoring and expanding linear expressions (7.EE.A.1). Students examine situations where more than one form of an expression may be used to
represent the same context, and they see how looking at each form can bring a new perspective (and thus deeper understanding) to the problem. Students recognize and use the
identity properties and the existence of inverses to efficiently write equivalent expressions in standard form (2𝑥 + (−2𝑥) + 3 = 0 + 3 = 3)(7.EE.A.2). By the end of the topic,
students have the opportunity to practice Unit 1 work on operations with rational numbers (7.NS.A.1, 7.NS.A.2) as they collect like terms with rational number coefficients
(7.EE.A.1).
• Any number, measure, numerical expression, or algebraic expression can be represented in an infinite number of ways that have the same value.
• Numbers can represent quantity, position, location, and relationships and symbols may be used to express these relationships.
• Properties of operations allow us to add, subtract, factor, and expand linear expressions.
Big Idea:
• Expressions can be manipulated to suit a particular purpose to solve problems efficiently.
• Mathematical expressions are used to represent and solve real-world and mathematical problems.
• When and how are expressions applied to real world situations?
• How can the order of operations be applied to evaluating expressions?
Essential
Questions:
• How does the ongoing use of fractions and decimals apply to real-life situations?
Vocabulary
• What is the difference between an expression and an equation?
coefficients, like terms, variable, numerical expression, algebraic expression, equivalent expressions, term, coefficient, constant, equation, expanded form,
standard form, distributive property, factor, commutative property, associative property, multiplicative inverse (reciprocal), additive inverse, additive identity,
multiplicative identity
Standard
Domain
Grade
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AZ College and Career Readiness Standards
Explanations & Examples
Resources
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7
EE
1
A. Use properties of operations to generate
equivalent expressions.
Apply properties of operations as strategies to add,
subtract, factor, and expand linear expressions with
rational coefficients.
7.MP.2. Reason abstractly and quantitatively.
7.MP.6. Attend to precision.
7.MP.7. Look for and make use of structure.
Explanation:
Students have had prior experience in generating equivalent
expressions; they should be working toward fluency in manipulating
expressions for efficiency with 7.EE.A.1 in this unit.
In 6th grade, students read, wrote and evaluated numerical
expressions involving variables and whole number exponents. They
applied properties of operations using the appropriate order of
operations to generate equivalent expressions. However, the
th
properties were not formally named in 6 grade, they were introduced
via tape diagrams.
Eureka Math:
Module 3 Lessons 1-6
Module 2 Lessons 18-19
Big Ideas:
Sections: 3.1, 3.2,
Extension 3.2
th
This is a continuation of work from 6 grade using properties of
operations and combining like terms (6.EE.A.2b). Students apply
properties of operations and work with rational numbers (integers
and positive / negative fractions and decimals) to write equivalent
expressions.
Note: An expression in standard form is the equivalent of what is
traditionally referred to as a “simplified” expression. This curriculum
does not utilize the term “simplify” when writing equivalent
expressions, but rather asks students to “put an expression in
standard form” or “expand the expression and combine like terms.”
However, students must know that the term “simplify” will be seen
outside of this curriculum and that the term is directing them to
write an expression in standard form.
Examples:
•
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What is the length and width of the rectangle below?
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Solution:
The Greatest Common Factor (GCF) is 2, which will be the
width because the width is in common to both rectangles. To
get the area 2a multiply by a, which is the length of the first
rectangle. To get the area of 4b, multiply by 2b, the length of
the second rectangle. Therefore, the width of the rectangle is
2 and the length is (a + 2b). The area of the rectangle can be
represented as 2(a + 2b).
( x + 5) − 2 .
•
Write an equivalent expression for 3
•
Suzanne thinks the two expressions
2(3a − 2 ) + 4a and
10a − 2 are equivalent? Is she correct? Explain why or why
not?
Solution:
The expressions are not equivalent. One way to prove this is
to distribute and combine like terms in the first expression to
get 10a – 4, which is not equivalent to the second expression.
A second explanation is to substitute a value for the variable
and perform the calculations. For example, if 2 is substituted
for a then the value of the first expression is 16 while the
value of the second expression is 18.
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•
Write equivalent expressions for:
3a + 12 .
Solution:
Possible solutions might include factoring as in 3( a + 4) , or
other expressions such as
•
A rectangle is twice as long as wide. One way to write an
expression to find the perimeter would be
w + w + 2 w + 2 w . Write the expression in two other ways.
Solution:
•
a + 2a + 7 + 5 .
6 w OR 2( w) + 2(2 w)
An equilateral triangle has a perimeter of 6 x + 15 . What is
the length of each of the sides of the triangle?
Solution:
3(2 x + 5) , therefore each side is 2 x + 5 units long.
•
Find the sum of -3a + 2 and 5a – 3.
Note: Eureka Math uses “any order, any grouping” for
commutative and associative property when done
together.
•
Write the expression in standards form:
4(2𝑎) + 7(−4𝑏) + (3 ∙ 𝑐 ∙ 5)
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Solution:
•
Write the sum and then rewrite the expression in
standard form.
o 2g and the opposite of (1 – 2g)
Solution:
•
Write the product and then rewrite the expression in
standard form.
o
•
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The multiplicative inverse of �
1
1
� 𝑎𝑛𝑑
3𝑥+5
3
Predict how many terms the resulting expression will
have after collecting like terms. Then, write the
expression in standard form by collecting like terms.
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2
1
3
4
𝑔− −𝑔+ 𝑔−
5
6
10
5
Solution:
There will be 2 terms.
•
Model how to write the expression in standard form
using the rules of rational numbers.
𝑥 2𝑥 𝑥 + 1 3𝑥 − 1
+
+
+
2
10
20 5
Solution:
o Method 1:
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o
7
EE
2
A. Use properties of operations to generate
equivalent expressions.
Understand that rewriting an expression in different
forms in a problem context can shed light on the
problem and how the quantities in it are related. For
example, a + 0.05a = 1.05a means that “increase by
5%” is the same as “multiply by 1.05.”
7.MP.2. Reason abstractly and quantitatively.
7.MP.6. Attend to precision.
7.MP.7. Look for and make use of structure.
7.MP.8. Look for and express regularity in repeated
reasoning.
Method 2:
Explanation:
Students understand the reason for rewriting an expression in terms of
a contextual situation. For example, students understand that a 20%
discount is the same as finding 80% of the cost, c (0.80c).
Examples:
• All varieties of a certain brand of cookies are $3.50. A person
buys peanut butter cookies and chocolate chip cookies. Write
an expression that represents the total cost, T, of the cookies
if p represents the number of peanut butter cookies and c
represents the number of chocolate chip cookies.
Eureka Math:
Module 3 Lessons 1-6
Module 2 Lessons 18-19
Big Ideas:
Sections: 3.1, 3.2,
Extension 3.2
Solution:
Students could find the cost of each variety of cookies and
then add to find the total.
T = 3.50p + 3.50c
Or, students could recognize that multiplying 3.50 by the total
number of boxes (regardless of variety) will give the
same total.
T = 3.50(p +c)
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•
Jamie and Ted both get paid an equal hourly wage of $9 per
hour. This week, Ted made an additional $27 dollars in
overtime. Write an expression that represents the weekly
wages of both if J = the number of hours that Jamie worked
this week and T = the number of hours Ted worked this week?
Can you write the expression in another way?
Solution:
Students may create several different expressions depending
upon how they group the quantities in the problem.
o One student might say: To find the total wage, I would
first multiply the number of hours Jamie worked by 9.
Then I would multiply the number of hours Ted worked
by 9. I would add these two values with the $27 overtime
to find the total wages for the week. The student would
write the expression 9 J + 9T + 27 .
o Another student might say: To find the total wages, I
would add the number of hours that Ted and Jamie
worked. I would multiply the total number of hours
worked by 9. I would then add the overtime to that value
to get the total wages for the week. The student would
write the expression 9( J + T ) + 27
o
A third student might say: To find the total wages, I would
need to figure out how much Jamie made and add that to
how much Ted made for the week. To figure out Jamie’s
wages, I would multiply the number of hours she worked
by 9. To figure out Ted’s wages, I would multiply the
number of hours he worked by 9 and then add the $27 he
earned in overtime. My final step would be to add Jamie
and Ted wages for the week to find their combined total
wages. The student would write the expression
(9 J ) + (9T + 27)
•
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Given a square pool as shown in the picture, write four
different expressions to find the total number of tiles in the
border. Explain how each of the expressions relates to the
diagram and demonstrate that the expressions are equivalent.
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Which expression do you think is most useful? Explain your
thinking.
st
Standards 7.EE.B.3 and 7.EE.B.4a will be taught at the end of 1 semester (see Quarter 3 for descriptions of standards and examples); however, they will not be assessed on
Benchmark 2. They will be assessed on Benchmark 3.
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