SEVENTH GRADE > 7.EE.4 TEACHER GUIDE

TEACHER GUIDE TO CLARIFICATION

7.EE.4

 

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

 

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.

a.  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. 

b.  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. 

Use Variables to Represent Quantities Leading to Equations or Inequalities

Each of the Expressions and Equations standards for 7th grade deal with teaching students how to set up and solve equations.  7.EE.1 is about solving using properties. 7.EE.2 focuses on  writing equations and expressions in multiple forms to increase understanding and 7.EE.3 puts 1 and 2 together for the purposes of solving real-life problems.  7.EE.4 begins to include inequalities as well.

Also 7.EE.4 expects students to represent the unknown with a variable because as problems get more complex, algebraic methods become more valuable. 

For example:

Two cyclists are riding toward each other along a road (each at a constant speed). At 8 am, they are 63 miles apart. They meet at 11 am. If one cyclist rides at 12.5 miles per hour, what is the speed of the other cyclist? 

First solution: The first cyclist travels 3 x 12.5 = 37.5 miles. The second travels 63 - 37.5 = 25.5 miles, so goes

         = 8.5 miles per hour.

 

Another solution uses a key hidden quantity, the speed at which the cyclists are approaching each other, to simplify the calculations: since       = 21, the cyclists are approaching each other at 21 miles per hour, so the other cyclist is traveling at 21 - 12.5 = 8.5 miles per hour.

The steps in solving the equation mirror the steps in the numerical solution. As problems get more complex, algebraic methods become more valuable. For example, in the cyclist problem above the numerical solution requires some insight in order to keep the cognitive load of the calculations in check. By contrast, choosing the letter s to stand for the unknown speed, students build an equation by adding the distances travelled in three hours (3s and 3 x 12.5) and setting them equal to 63 to get 3s + 3(12.5) = 63.

It is worthwhile exploring two different possible next steps in the solution of this equation:

3x + 37.5 = 63 and 3(x + 12.5) = 63

The first is suggested by a standard approach to solving linear equations; the second is suggested by a comparison with the numerical solution described earlier.

Students also set up and solve inequalities, recognizing the ways in which the process of solving them is similar to the process of solving linear equations:

 

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 solution.

 

Students also recognize one important new consideration in solving inequalities: multiplying or dividing both sides of an inequality by a negative number reverses the order of the comparison it represents. It is useful to present contexts that allow students to make sense of this.

For example,

If the price of a ticket to a school concert is p dollars then the attendance is 1000 - 50p. What range of prices ensures that at least 600 people attend?

Students recognize that the requirement of at least 600 people leads to the inequality 1000 - 50p      600. Before solving the inequality, they use common sense to anticipate that the answer will be of the form p < ?, since higher prices result in lower attendance (Note that inequalities using     and     are included in this standard, in addition to < and >.)

Common Core Standards Writing Team. (2011, April 22). 

Progressions for the Common Core State Standards in Mathematics (draft).  6-8, Expressions and Equations. Tucson, AZ: Institute for Mathematics and Educations, University of Arizona.

 

Examples

Students 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. Examples

 · Amie had $26 dollars to spend on school supplies. After buying 10 pens, she had $14.30 left.  How much did each pen cost?

 · The sum of three consecutive even numbers is 48. What is the smallest of these numbers?

 · Solve:       + 5 = 20.

· Florencia has at most $60 to spend on clothes. She wants to buy a pair of jeans for $22 and spend the rest on t-shirts. Each t-shirt costs $8. Write an inequality for the number of t-shirts she can purchase.

 · Steven has 25 dollars. He spent $10.81, including tax, to buy a new DVD. He needs to set aside $10.00 to pay for his lunch next week. If peanuts cost $0.38 per package including tax, what is the maximum number of packages that Steven can buy? 

 

Write an equation or inequality to model the situation. Explain how you determined whether to write an equation or inequality and the properties of the real number system that you used to find a solution.

 

Solve     x + 3 and graph your solution on a number line.

Kansas Association of Teachers of Mathematics (KATM) Flipbooks.  Questions or to send feedback: melisa@ksu.edu.

Retrieved from Math Flipbooks.

 

In contrast with the simple linear expressions they see in Grade 6, the more complex expressions students seen in Grade 7 afford shifts of perspective, particularly because of their experience with negative numbers: for example, students might see 7 – 2 (3 - 8x) as 7 – (2 (3 – 8x)) or as 7 + (-2) (3 + (-8) x)  (MP7). 

 

As students gain experience with multiple ways of writing an expression, they also learn that different ways of writing expressions can serve different purposes and provide different ways of seeing a problem.  For example,

a + 0.05a =1.05a means that “increase by 5%” is the same as “multiply by 1.05.”7.EE.2  In the example below , the connection between the expressions and the figure emphasize that they all represent the same number, and the connection between the structure of each expression and a method of calculation emphasize the fact that expressions are built up from operations on numbers.

Instructional Strategies:

To assist students’ assessment of the reasonableness of answers, especially problem situations involving fractional or decimal numbers, use whole-number approximations for the computation and then compare to the actual computation. Connections between performing the inverse operation and undoing the operations are appropriate here. It is appropriate to expect students to show the steps in their work. Students should be able to explain their thinking using the correct terminology for the properties and operations. Continue to build on students’ understanding and application of writing and solving one-step equations from a problem situation to multi-step problem situations. This is also the context for students to practice using rational numbers including: integers, and positive and negative fractions and decimals. As students analyze a situation, they need to identify what operation should be completed first, then the values for that computation. Each set of the needed operation and values is determined in order. Finally an equation matching the order of operations is written. For example, Bonnie goes out to eat and buys a meal that costs $12.50 that includes a tax of $.75. She only wants to leave a tip based on the cost of the food. In this situation, students need to realize that the tax must be subtracted from the total cost before being multiplied by the percent of tip and then added back to obtain the final cost.

C = (12.50 - .75)(1 + T) + .75 = 11.75(1 +T) + .75 where C = cost and T = tip. Provide multiple opportunities for students to work with multi-step problem situations that have multiple solutions and therefore can be represented by an inequality. Students need to be aware that values can satisfy an inequality but not be appropriate for the situation, therefore limiting the solutions for that particular problem.

 

Kansas Association of Teachers of Mathematics (KATM) Flipbooks.  Questions or to send feedback: melisa@ksu.edu.

Retrieved from Math Flipbooks.

For additional information go to Achieve the Core

Coherence and Connections: Need to Know

Grade Below

Grade Level

Grade Above

6.EE.6

6.EE.7

6.EE.8

7.RP.2

7.EE.47.NS.3

8.EE.7b

8.EE.8

This cluster is connected to the Grade 7 Critical Area of Focus #2, Developing understanding of operations with rational numbers and working with expressions and linear equations, and to Critical Area of Focus #3, Solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume.

 

 

Examples of Major Within-Grade Dependencies
Because rational number arithmetic (7.NS.1–3) underlies the problem solving detailed in 7.EE.3 as well as the solution of linear expressions and equations (7.EE.1–2, 4), this work should likely begin at or near the start of the year. 

 

The work leading to meeting standards 7.EE.1–4 could be divided into two phases, one centered on addition and subtraction (e.g., solving x + q = r) in relation to rational number addition and subtraction (7.NS.1) and another centered on multiplication and division (e.g., solving px + q = r and p(x + q) = r) in relation to rational number multiplication and division (7.NS.2). 

 

PARCC Model Content Frameworks: Mathematics Grades 3-11 (version 5). (2017, November).

Retrieved from: https://files.eric.ed.gov/fulltext/ED582070.pdf

Illinois Assessment of Readiness Mathematics Evidence Tables. Retrieved from:

https://www.isbe.net/Documents/IAR-Grade-7-Math-Evidence-State.pdf

 

Evidence

Statement Key

Evidence Statement Text

Clarifications

MP

7.EE.4a-1

 

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. a. 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. 

1,2,6,7

Calculator

yes

7.C.5

 

Given an equation, present the solution steps as a logical argument that concludes with the set of solutions (if any). Content Scope: Knowledge and skills articulated in 7.EE.4a

1,2,3,

6,7

yes

7.EE.4a-2

 

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. a. Fluently solve equations of the form px + q = r and p(x+q) = r, where p, q, and r are specific rational numbers.

i) Each task requires students to solve two equations (one of each of the given two forms). Only the answer is required. 

6,7

no

7.EE.4b

 

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. b. 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. 

i) Tasks may involve <, > , ≤ or ≥ . 

1,2,5,

6,7

no

Classroom Resources
 

7.EE.4 Daily Discourse

 

Mathematics Assessment Project
http://map.mathshell.org/materials/download.php?fileid=1261

HOT Questions

1.  Mandy’s monthly earnings consist of a fixed salary of $2800 and an 18% commission on all her monthly sales. To cover    

     her planned expenses, Mandy needs to earn an income of at least $6400 this month.

 

     Part A: Write an inequality that, when solved, will give the amount of sales Mandy needs to cover her planned expenses.

     2800 + .18x ≥ 6400

 

     Part B: Graph the solution of the inequality on the number line.

2.  Johnny was given the following problem:

 

     Amanda has $40 to spend on flowers. She wants to buy a pair of red rose flowers for $18 and spend the rest on lily    

     flowers. Each lily flower costs $11. Write an inequality for the number of lily flowers she can purchase.

 

     His inequality is as follows:

             x = lilies

             11x + 18     40                                             Do you agree with him? Why or why not?

            This is his work:

             11x +18    40

             -18     -18      

 

 

   

             X     2

 

             The inequality is incorrect.  He should has written 11x + 18 ≤ 40

3.  Rebecca bought one gold fish ($32) and one star fish ($12). She spends the rest of her money on guppy fish. She starts with    

     $80. Each guppy costs $6. Write an inequality for the number of guppies she can purchase.

     32 + 12 + 6x ≤ 80

 
 
 
 
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