Download Unit 4 - Expressions and Equations 2

7th/8th Grade Mathematics Curriculum Guide
2015 – 2016
Unit 4: Expressions and Equations
Time Frame: Quarter 2 – About 27 days
Connections to Previous Learning:
Students in Grade 6 learn the concepts of ratio and unit rate as well as the precise mathematical language used to describe these relationships. They learn to solve problems
using ratio and rate reasoning using a variety of tools such as tables, tape diagrams, double number lines and equations. Students develop an understanding of ratio and
proportion using ratio tables, tape diagrams, and double number lines. In 6th grade, students read, write and evaluate numerical expressions involving variables and whole
number exponents. They apply properties of operations using the appropriate order of operations to generate equivalent expressions. In grade 6, students learned to read, write,
interpret and solve one-variable equations and inequalities in real-life and mathematical situations. This unit focuses on extending the understanding of ratios and proportions
that was explored in Grades 6 and 7. By using coordinate grids and various sets of three similar triangles, students prove that the slopes of the corresponding sides are equal,
thus making the unit rate of change equal. After proving with multiple sets of triangles, students generalize the slope to y = mx for a line through the origin and y = mx+b for a
line through the vertical axis at b. Students learn that proportional relationships are part of a broader group of linear functions, and they are able to identify whether a
relationship is linear. Nonlinear functions are included for comparison.
Focus of this Unit:
Students use their understanding of structure to rewrite general linear expressions in equivalent forms. Expressions include rational coefficients and multiple terms. Students
will apply their understanding of properties when adding, subtracting, factoring, and expanding expressions with and without context.
Students will read, write, interpret, and solve multi-step real-world and mathematical problems using algebraic and numerical expressions, equations, and inequalities. They will
apply their understanding of the solution process to geometry when they solve circumference and area problems. Unit rates were explored in Grade 6 as the comparison of two
different quantities with the second unit a unit of one, (unit rate). In Grade 7 unit rates were expanded to complex fractions and percents through solving multi-step problems
such as: discounts, interest, taxes, tips, and percent of increase or decrease. Proportional relationships were applied in scale drawings, and students should have developed an
informal understanding that the steepness of the graph is the slope or unit rate.
Connections to Subsequent Learning:
Students will apply their understanding of ratios and proportionality to situations involving multi-step ratio and percent problems as well as scale drawings. A more complete
understanding of order of operations and their properties will lay the foundation for the extensive study of functions next year. Understanding that equations can have multiple
solutions will lay a foundation for the study of solving systems of simultaneous linear equations. In high school, students use function notation and are able to identify types of
nonlinear functions. Graphing will be extended to exponential, rational, and quadratic equations and their graphs. Math 1 further develops the concept of solving systems
through standards A.REI.5 and A.REI.6
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 4
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.
Clover Park School District
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7th/8th Grade Mathematics Curriculum Guide
Stage 1 Desired Results
2015 – 2016
Transfer Goals
Students will be able to independently use their learning to…
• Adapt systems to measure anywhere in the universe.
• Apply principals of linear equations to make informed decisions and predictions.
• Recognize when quantities form an algebraic relationship and solve for an unknown quantity.
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Interpret the situations that mathematical relationships represent.
Define, evaluate, and compare equations, tables, words, and graphs in order to model relationships in real-world and mathematical problems.
Solve mathematical problems by using variables to represent the unknown.
Determine the relationships among rational numbers in real life situations.
Meaning Goals
UNDERSTANDINGS Students will understand that…
7.EE
• A variety of strategies are used when solving an equation.
• A written problem situation can be translated into numbers, symbols,
operations, and variables.
• Equations express relationships between quantities.
• Properties of operations are used to generate equivalent expressions.
• Variables can be used to represent numbers in any type of
mathematical problem.
• Understand the difference between an expression and an equation.
• Expressions you simplify and equations you solve for the variable’s
value.
• Write and solve multi-step equations including all rational numbers.
• Some equations may have more than one solution and understand
inequalities.
• Properties of operations allow us to add, subtract, factor, and expand
linear expressions.
8.EE
• Mathematical models are used to represent real-world situations.
• Data can be represented in graphs, tables, words, and equations.
• Linear relationships are characterized by a constant rate of change
Unit 4
ESSENTIAL QUESTIONS
7.EE
• Why is it important to keep an equation balanced?
• How can equations be used to express relationships among quantities?
• How can patterns be modeled through the use of algebraic thinking?
• How can the order of operations be applied to evaluating expressions, and solving from onestep to multi-step equations?
8.EE
• What makes a linear equation?
• How can linear relationships be modeled and used in real-life situations?
• Why is one variable dependent upon the other in relationships?
• What does the solution mean to the problem?
• How do we express a relationship mathematically?
• How do we determine the value of an unknown quantity?
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7th/8th Grade Mathematics Curriculum Guide
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(slope).
There are explicit connections between proportional relationships,
lines and linear equations.
The equation y=mx+b is a straight line and that slope and y-intercept
are critical to solving real problems involving linear relationships.
Changes in varying quantities are often related by patterns which can
be used to predict outcomes and solve problems.
Unit rates can be explained in graphical representation, algebraic
equations, and in geometry through similar triangles.
Mathematical models are used to represent real-world situations.
Inverse operations can be used to isolate/solve for a variable.
Variables are used to represent unknown values
2015 – 2016
Acquisition Goals
Students will know…
7.EE
• The relationship between ordered pairs and coordinates.
• Variables represent an unknown quantity.
• The coefficients of like terms can be combined if exponents and
variables are the same.
• Verifying the solution to any equation requires testing a value for the
given variable.
• Performing the same operation to each side of an equation keeps it
balanced (equal).
8.EE
• The meaning of the solution to an algebraic equation.
• One-variable linear equation
• Linear inequality
• When a relationship is a linear.
• Connections among expressions, graphs, and tables.
Unit 4
Students will be skilled at…
7.EE
• Use Commutative, Associative, Distributive, Identity, and Inverse Properties to add and
subtract linear expressions with rational coefficients. (7.EE.1)
• Use Commutative, Associative, Distributive, Identity, and Inverse Properties to factor and
expand linear expressions with rational coefficients. (7.EE.1)
• Rewrite an expression in a different form. (7.EE.2)
• Choose the form of an expression that works best to solve a problem. (7.EE.2)
• Explain your reasoning for the choice of expression used to solve a problem. (7.EE.2)
• Use commutative, associative, distributive, identity, and inverse properties to calculate with
numbers in any form (whole numbers, fractions and decimals). (7.EE.3)
• Convert between rational number forms (whole numbers, fractions and decimals) to solve
problems as appropriate. (7.EE.3)
• Solve multi-step mathematical problems posed with positive and negative rational numbers in
any form (whole numbers, fractions, and decimals), using tools strategically. (7.EE.3)
• Solve multi-step real-life problems posed with positive and negative rational numbers in any
form (whole numbers, fractions, and decimals), using tools strategically. (7.EE.3)
• Use mental computation and estimation strategies to assess the reasonableness of the answer.
(7.EE.3)
• Translate words or real-life situations into variable equations. (7.EE.4)
• Translate words or real-life situations into variable inequalities. (7.EE.4)
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7th/8th Grade Mathematics Curriculum Guide
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Solve one- or two-step equations with rational numbers fluently. (7.EE.4)
Solve word problems leading to one- or two-step equations with rational numbers. (7.EE.4)
Construct simple equations and inequalities with rational numbers to solve problems. (7.EE.4)
Compare an algebraic solution to an arithmetic solution, identifying the sequence of the
operations used in each approach. (7.EE.4)
Solve word problems leading to one- or two-step inequalities with rational numbers. (7.EE.4)
Graph the solution set of inequalities and interpret it in the context of the problem. (7.EE.4)
Know the formulas for the area and circumference of a circle. (7.G.4)
Use the formulas for area and circumference of a circle to solve problems.
(7.G.4)
Informally, derive the area formula for a circle based on circumference. (7.G.4)
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8.EE
• Identify and contextualize the rate of change and the initial value from tables, graphs,
equations, or verbal descriptions. (8.EE.B.5)
• Sketch a graph when given a verbal description of a situation. (8.EE.B.5)
• Compare graphs, tables, and equations of proportional relationships. (8.EE.5)
• Graph proportional relationships and interpret the unit rate as the slope. (8.EE.5)
• Interpret equations in y=mx+b form as a linear function. (8.EE.B.6)
• Construct a model for a linear equation. (8.EE.B.6)
• Describe the qualities of a equation using a graph (e.g., where the function is increasing or
decreasing). (8.EE.B.6)
• Use similar triangles to explain why the slope m is the same between any two distinct points on
a non-vertical line in the coordinate plane. (8.EE.6)
• Derive the equation y=mx for a line through the origin. (8.EE.6)
• Simplify linear expressions utilizing the distributive property and collecting like terms. (8.EE.7)
• Create a multi-step linear equation to represent a real-life situation. (8.EE.7)
• Solve equations with linear expressions on either or both sides including equations with one
solution, infinitely many solutions, and no solutions. (8.EE.7)
• Give examples of and identify equations as having one solution, infinitely many solutions, or no
solutions. (8.EE.7)
Unit 4
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7th/8th Grade Mathematics Curriculum Guide
Calculators
7.EE.1
7.EE.2
7.EE.3
7.EE.4
7.EE.4a
7.EE.4b
8.EE.5
8.EE.6
8.EE.7
8.EE.7a
8.EE.7b
no
no
no
yes
yes
no
no
yes
yes
yes
Materials Needed for Unit
Holt Course 3
Holt Algebra 1
2015 – 2016
Additional Materials
Holt Course 2
Discovering Algebra
Prerequisite Skills Required for this Unit
Stage 1 Established Goals: Common Core State Standards for Mathematics
7.EE.A Use properties of operations to generate equivalent expressions.
7.EE.A.1 No Calculator Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
7.EE.A.2 No Calculator 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.EE.B Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
7.EE.B.3 No Calculator 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.
7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about
the quantities.
7.EE.B.4a Calculator Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of
these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the
perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?
7.EE.B.4b Calculator Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution
set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your
pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.
8.EE.B Understand the connections between proportional relationships, lines, and linear equations.
8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical 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
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7th/8th Grade Mathematics Curriculum Guide
2015 – 2016
8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.7 Solve linear equations in one variable.
8.EE.C.7a Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by
successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different
numbers).
8.EE.C.7b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property
and collecting like terms.
Vocabulary …
Overview of Assessments & Performance Tasks for Unit
ratio, unit rate, proportion, slope, simple interest, tax, markup, markdown, gratuity, tip, commission, fee,
percent, percent increase, percent decrease, percent error, rate of change
Mid-Unit Assessment
End-of-Unit Assessment
linear equation, linear inequality, distributive property, commutative property, associative property
Formative Assessments for this unit
intercepts, ordered pairs, coordinates, coordinate plane, slope intercept form, constant, increasing,
Suggested Performance Tasks
decreasing, axis, vertical, simplify, like terms, solution, inverse operations
Georgia Department Of Education:
Howard County
• Shop Smart – 7.EE.2
• Drops in a Bucket 8.EE.B.5
• Let’s Paint
Illuminations
Illustrative Mathematics
Inside Mathematics (on P drive)
• Toy Trains 7.EE.3, 7.EE.4a
• Squares & Circles 8.EE.5 & 7
• Rule of 4 for linear equations 8.EE.5
TI Activities
• Solving Equations 1
• Solving Equations 2
• Slope and y-intercept
7.EE.A.1
Unit 4
7.EE.A.1 & 7.EE.A.2
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7th/8th Grade Mathematics Curriculum Guide
Vocab - distributive property, commutative property, associative property, inverse, simplify, like terms
This is a continuation of work from 6th grade using properties of operations (table 3, pg. 90) and combining
like terms. Students apply properties of operations and work with rational numbers (integers and positive /
negative fractions and decimals) to write equivalent expressions.
Example 1:
What is the length and width of the rectangle below?
2015 – 2016
Holt Course 3 Lesson 1-1 Variables and Expressions
Holt Course 3 Lesson 1-2 Algebraic Expressions
Holt Course 3 Lesson 11-1 Simplifying Algebraic Expressions
Additional if necessary
Holt Course 2 Lesson 1-9 Simplifying Algebraic Expressions
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 rectangles. To get the area of 4b,
multiply by 2b, which will be the length of the second rectangle. The final answer will be 2(a + 2b)
Example 2:
Write an equivalent expression for 3(x + 5) – 2.
Solution:
3x + 15 – 2 Distribute the 3
3x + 13 Combine like terms
Example 3:
Suzanne says 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.
Example 4:
Write equivalent expressions for: 3a + 12.
Solution:
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7th/8th Grade Mathematics Curriculum Guide
2015 – 2016
Possible solutions might include factoring as in 3(a + 4), or other expressions such as a + 2a + 7 + 5.
Example 5:
A rectangle is twice as long as its width. One way to write an expression to find the perimeter would be w + w
+ 2w + 2w. Write the expression in two other ways.
Solution:
6w or 2(2w)
Example 6:
An equilateral triangle has a perimeter of 6x + 15. What is the length of each side of the triangle?
Solution:
3(2x + 5), therefore each side is 2x + 5 units long.
7.EE.A.2
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).
7.EE.A.2
Taught above
Example 1:
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
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)
Example 2:
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? What is another way to
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7th/8th Grade Mathematics Curriculum Guide
2015 – 2016
write the expression?
Solution:
Students may create several different expressions depending upon how they group the quantities in the
problem.
Possible student responses are:
Response 1: To find the total wage, first multiply the number of hours Jamie worked by 9. Then, multiply the
number of hours Ted worked by 9. Add these two values with the $27 overtime to find the total wages for the
week. The student would write the expression 9J + 9T + 27.
Response 2: To find the total wages, add the number of hours that Ted and Jamie worked. Then, multiply the
total number of hours worked by 9. Add the overtime to that value to get the total wages for the week. The
student would write the expression 9(J + T) + 27.
Response 3: To find the total wages, find out how much Jamie made and add that to how much Ted made for
the week. To figure out Jamie’s wages, multiply the number of hours she worked by 9. To figure out Ted’s
wages, 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 (9J) + (9T + 27).
Example 3:
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. Which expression is most useful? Explain.
7.EE.B.3
Unit 4
7.EE.3/7.EE.B.4 and 8.EE.C.7
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7th/8th Grade Mathematics Curriculum Guide
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.
Example 1:
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?
• Susan reported that 18 of the 48 people she surveyed get 8 hours sleep a night
• Kenneth reported that 36% of the people he surveyed get 8 hours sleep a night
• Jamal reported that 0.365 of the people he surveyed get 8 hours sleep a night
Solution:
In Susan’s survey, the number is 37.5%, which is the greatest percentage.
Estimation strategies for calculations with fractions and decimals extend from students’ work with whole
number operations. Estimation strategies include, but are not limited to:
• 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),
• 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),
• rounding and adjusting (students round down or round up and then adjust their estimate
depending on how much the rounding affected the original values),
• 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
• using benchmark numbers that are easy to compute (students select close whole numbers for
fractions or decimals to determine an estimate).
7.EE.4
Vocab – inequality
2015 – 2016
Equations:
Holt Course 3 Lesson 1-7 Solving Equations by Addition and
Subtraction
Holt Course 3 Lesson 1-8 Solving Equations by Multiplication
and Division
Holt Course 3 Lesson 2-7 Solving Equations with Rational
Numbers
Holt Course 3 Lesson 2-8 Solving 2-Step Equations
Holt Course 3 Lesson 11-1 Simplifying Algebraic Equations
Holt Course 3 Lesson 11-2 Solving Multi-Step Equations
(Contextual problems only)
Holt Course 3 Lesson 11-3 Solving Equations w/Variables on
Both Sides
Inequalities:
Holt Algebra 1 Lesson 3-1 Graphing and Writing Inequalities
Holt Algebra 1 Lesson 3-2 Solving Inequalities by Adding and
Subtracting
Holt Algebra 1 Lesson 3-3 Solving Inequalities by Multiplying
and Dividing
Holt Algebra 1 Lesson 3-4 Solving Two-Step and Multi-Step
Inequalities
Students write an equation or inequality to model the situation. Students explain how they determined
whether to write an equation or inequality and the properties of the real number system that you used to
find a solution. In contextual problems, students define the variable and use appropriate units.
7.EE.B.4a
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7th/8th Grade Mathematics Curriculum Guide
Students solve multi-step equations derived from word problems. Students use the arithmetic from the
problem to generalize an algebraic solution
Example 1:
The youth group is going on a trip to the state fair. The trip costs $52. Included in that price is $11 for a
concert ticket and the cost of 2 passes, one for the rides and one for the game booths. Each of the passes
cost the same price. Write an equation representing the cost of the trip and determine the price of one pass.
2015 – 2016
Solution:
x =cost of one pass
Example 3:
Amy had $26 dollars to spend on school supplies. After buying 10 pens, she had $14.30 left. How much did
each pen cost including tax?
Solution:
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2015 – 2016
x = number of pens
26 = 14.30 + 10x
Solving for x gives $1.17 for each pen.
Example 4:
The sum of three consecutive even numbers is 48. What is the smallest of these numbers?
Solution:
x = the smallest even number
x + 2 = the second even number
x + 4 = the third even number
x + x + 2 + x + 4 = 48
3x + 6 = 48
3x = 42
x = 14
Example 5:
Solve:
𝑥+3
−2
= -5
Solution:
x=7
7.EE.B.4b
Students solve and graph inequalities and make sense of the inequality in context. Inequalities may have
negative coefficients. Problems can be used to find a maximum or minimum value when in context.
7.EE.B.4b
Included above
Example 1:
Florencia has at most $60 to spend on clothes. She wants to buy a pair of jeans for $22 dollars and spend the
rest on t-shirts. Each t-shirt costs $8. Write an inequality for the number of t-shirts she can purchase.
Solution:
x = cost of one t-shirt
8x + 22 ≤ 60
x = 4.75 ⤏ 4 is the most t-shirts she can purchase
Example 2:
Steven has $25 dollars to spend. He spent $10.81, including tax, to buy a new DVD. He needs to save $10.00
but he wants to buy a snack. If peanuts cost $0.38 per package including tax, what is the maximum number of
packages that Steven can buy?
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7th/8th Grade Mathematics Curriculum Guide
Solution:
x = number of packages of peanuts
25 ≥ 10.81 + 10.00 + 0.38x
x = 11.03 ⤏ Steven can buy 11 packages of peanuts
2015 – 2016
Example 3:
7 – x > 5.4
Solution:
x < 1.6
Example 4:
Solve -0.5x – 5 < -1.5 and graph the solution on a number line.
Solution:
x > -7
8.EE.C.7
Vocab – linear equation, distributive property
Students solve one-variable equations including those with the variables being on both sides of the equals
sign. Students recognize that the solution to the equation is the value(s) of the variable, which make a true
equality when substituted back into the equation. Equations shall include rational numbers, distributive
property and combining like terms.
Example 1:
Equations have one solution when the variables do not cancel out. For example, 10x – 23 = 29 – 3x can be
solved to x = 4. This means that when the value of x is 4, both sides will be equal. If each side of the equation
were treated as a linear equation and graphed, the solution of the equation represents the coordinates of the
point where the two lines would intersect. In this example, the ordered pair would be (4, 17).
10 • 4 – 23 = 29 – 3 • 4
40 – 23 = 29 – 12
17 = 17
Example 2:
Equations having no solution have variables that will cancel out and constants that are not equal. This means
that there is not a value that can be substituted for x that will make the sides equal.
-x + 7 – 6x = 19 – 7x Combine like terms
-7x + 7 = 19 – 7x Add 7x to each side
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7 ≠ 19
This solution means that no matter what value is substituted for x the final result will never be equal to each
other.
If each side of the equation were treated as a linear equation and graphed, the lines would be parallel.
2015 – 2016
Example 3:
An equation with infinitely many solutions occurs when both sides of the equation are the same. Any value of
x will produce a valid equation. For example the following equation, when simplified will give the same values
on both sides.
1
3
(36a – 6) = (4 – 24a)
2
4
If each side of the equation were treated as a linear equation and graphed, the graph would be the same line.
Students write equations from verbal descriptions and solve.
Example 4:
Two more than a certain number is 15 less than twice the number. Find the number.
Solution:
n + 2 = 2n – 15
17 = n
8.EE.B.5
Vocab – slope, intercepts, slope-intercept form
Students build on their work with unit rates from 6th grade and proportional relationships in 7th grade to
compare graphs, tables and equations of proportional relationships. Students identify the unit rate (or slope)
in graphs, tables and equations to compare two proportional relationships represented in different ways.
Example 1:
Compare the scenarios to determine which represents a greater speed. Explain your choice including a
written description of each scenario. Be sure to include the unit rates in your explanation.
8.EE.B.5 & 8.EE.B.6
Direct Variation
Holt Course 3 Lesson 12-1 Graphing Linear Equations
Holt Course 3 Lesson 12-2 Slope of a Line
Holt Course 3 Lesson 12-3 Using Slopes and Intercepts
Holt Algebra 1 Lesson 5-2 Using Intercepts
Holt Algebra 1 Lesson 5-3 Rate of Change and Slope
Holt Algebra 1 Lesson 5-4 The Slope Formula
Holt Algebra 1 Lesson 5-5 Direct Variation
Slope-Intercept Form
Holt Algebra 1 Lesson 5-6 Slope-Intercept Form
Supplemental
Discovering Algebra 2.4 Direct Variation
Discovering Algebra 2.5 Inverse Variation
Discovering Algebra 2.6 Activity Day: Variation with a Bicycle
Discovering Algebra 3.3 Time-Distance Relationships
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Discovering Algebra 3.4 Linear Equations and the Intercept
Form
Discovering Algebra 3.5 Linear Equations and Rate of Change
Discovering Algebra 3.7 Activity Day: Modeling Data
Discovering Algebra 4.2 Writing a Linear Equation to Fit Data
Solution: Scenario 1 has the greater speed since the unit rate is 60 miles per hour. The graph shows this rate
since 60 is the distance traveled in one hour. Scenario 2 has a unit rate of 55 miles per hour shown as the
coefficient in the equation.
Given an equation of a proportional relationship, students draw a graph of the relationship. Students
recognize that the unit rate is the coefficient of x and that this value is also the slope of the line.
8.EE.B.6
Vocab - Vertical
Triangles are similar when there is a constant rate of proportion 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.
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 2/3for the line.
Students write equations in the form y = mx for lines going through the origin, recognizing that m represents
the slope of the line. Students write equations in the form y = mx + b for lines not passing through the origin,
recognizing that m represents the slope and b represents the y-intercept.
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7th/8th Grade Mathematics Curriculum Guide
2015 – 2016
Stage 2 - Evidence
Evaluative Criteria/Assessment Level Descriptors (ALDs):
OTHER ASSESSMENT EVIDENCE:
7.EE.A (SBAC Target C)
Common Assessments (available on P: drive)
Level 4 students should understand that rewriting an expression can shed light on how Spaced Learning Over Time (SLOT - entrance & exit slips)
quantities are related in an unfamiliar problem-solving context with no scaffolding.
Checking for understanding (CFU’s)
Level 3 students should be able to apply properties of operations as strategies to
Lesson Quizzes
factor and expand linear expressions with rational coefficients. They should
Classroom Assessments
understand that rewriting an expression can shed light on how quantities are related
in a familiar problem-solving context with minimal scaffolding.
Level 2 students should be able to apply properties of operations as strategies to
SBA Released
factor and expand linear expressions with integer coefficients. They should also be
able to add and subtract linear expressions with rational coefficients.
Claim 1 Item Specs
Level 1 students should be able to apply properties of operations as strategies to add
and subtract linear expressions with integer coefficients.
7.EE.B (SBAC Target D)
Level 4 students should be able to use variables to represent and reason with
quantities in real-world and mathematical situations with no scaffolding. They should
be able to construct inequalities with more than one variable to solve problems.
Level 3 students should be able to solve and graph solution sets to inequalities with
one variable. They should be able to use variables to represent and reason with
quantities in real-world and mathematical situations with minimal scaffolding. They
should also be able to construct equations with variables to solve problems.
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7th/8th Grade Mathematics Curriculum Guide
Level 2 students should be able to solve multi-step problems with rational numbers
and solve equations in the form of px + q = r or p(x + q) = r, where p, q, and r are
rational numbers. Students should be able to use variables to represent quantities in
familiar real-world and mathematical situations. They should also be able to create
equations with variables to solve familiar problems with a high degree of scaffolding.
Level 1 students should be able to solve multi-step problems with integers or common
fractions with denominators of 2 through 10, 25, 50, or 100 and decimals to the
hundredths place; solve equations in the form of px + q = r, where p, q, and r are
integers; and distinguish between inequalities and equations with integer coefficients
with or without real-world context.
2015 – 2016
8.EE.B (SBAC Target C)
Level 4 students should be able to use similar triangles to explain why the slope is the
same between any two distinct points on a nonvertical line in a coordinate plane.
Level 3 students should understand that slope is a unit rate of change in a proportional
relationship and convert proportional relationships to linear equations in slopeintercept form while also understanding when and why the y-intercept is zero. They
should also be able to use repeated reasoning to observe that they can use any right
triangle to find the slope of a line.
Level 2 students should be able to compare two different proportional relationships
represented in different ways. They should also be able to calculate the slope of a line
and identify the y-intercept of a line.
Level 1 students should be able to graph a proportional relationship on a coordinate
plane.
8.EE.C (SBAC Target D) – Target D is not fully covered in the 7/8 compressed
Level 4 students should be able to analyze and solve problems leading to two linear
equations in two variables in multiple representations.
Level 3 students should be able to classify systems of linear equations as intersecting,
collinear, or parallel; solve linear systems algebraically and estimate solutions using a
variety of approaches; and show that a particular linear equation has one solution, no
solution, or infinitely many solutions by successively transforming the given equation
into simpler forms until an equivalent equation of the form x = a, a = a, or a = b results
(where a and b are different numbers). They should be able to solve and produce
examples of linear equations in one variable, including equations whose solutions
require expanding expressions using the distributive property and collecting like terms.
Level 2 students should be able to analyze and solve systems of linear equations
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7th/8th Grade Mathematics Curriculum Guide
graphically by understanding that the solution of a system of linear equations in two
variables corresponds to the point of intersection on a plane. They should be able to
solve and produce examples of linear equations in one variable with rational
coefficients with one solution, infinitely many solutions, or no solution.
Level 1 students should be able to solve linear equations in one variable with integer
coefficients.
LEARNING ACTIVITIES:
2015 – 2016
Stage 3 – Learning Plan Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
NOTES:
**Days may change depending on any tasks or assessing you choose to do.
7.EE.A.1 & 7.EE.A.2
Day 1:
Holt Course 3 Lesson 1-1 Variables and Expressions
Day 2:
Holt Course 3 Lesson 1-2 Algebraic Expressions
Day 3:
Holt Course 3 Lesson 11-1 Simplifying Algebraic Expressions
7.EE.3/7.EE.B.4 and 8.EE.C.7
Equations:
Day 4:
Holt Course 3 Lesson 1-7 Solving Equations by Addition and Subtraction
Day 5:
Holt Course 3 Lesson 1-8 Solving Equations by Multiplication and Division
Day 6:
Holt Course 3 Lesson 2-7 Solving Equations with Rational Numbers
Day 7:
Holt Course 3 Lesson 2-8 Solving 2-Step Equations
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7th/8th Grade Mathematics Curriculum Guide
Stage 3 – Learning Plan Sample
Day 8:
Holt Course 3 Lesson 11-1 Simplifying Algebraic Equations
2015 – 2016
Day 9:
Holt Course 3 Lesson 11-2 Solving Multi-Step Equations
Day 10:
Holt Course 3 Lesson 11-3 Solving Equations w/Variables on Both Sides
Inequalities:
Day 11:
Holt Algebra 1 Lesson 3-1 Graphing and Writing Inequalities
Day 12:
Holt Algebra 1 Lesson 3-2 Solving Inequalities by Adding and Subtracting
Day 13:
Holt Algebra 1 Lesson 3-3 Solving Inequalities by Multiplying and Dividing
Day 14:
Holt Algebra 1 Lesson 3-4 Solving Two-Step and Multi-Step Inequalities
8.EE.B.5 & 8.EE.B.6
Direct Variation
Day 15:
Holt Course 3 Lesson 12-1 Graphing Linear Equations
Day 16:
Holt Course 3 Lesson 12-2 Slope of a Line
Day 17:
Holt Course 3 Lesson 12-3 Using Slopes and Intercepts
Day 18:
Holt Algebra 1 Lesson 5-2 Using Intercepts
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7th/8th Grade Mathematics Curriculum Guide
Stage 3 – Learning Plan Sample
Day 19:
Holt Algebra 1 Lesson 5-3 Rate of Change and Slope
2015 – 2016
Day 20:
Holt Algebra 1 Lesson 5-4 The Slope Formula
Day 21:
Holt Algebra 1 Lesson 5-5 Direct Variation
Slope-Intercept Form
Day 22:
Holt Algebra 1 Lesson 5-6 Slope-Intercept Form
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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