Subject:
Algebra
Material Type:
Lesson Plan
Level:
Middle School
7
Provider:
Pearson
Tags:
Language:
English
Media Formats:
Text/HTML

# Reasonable Estimations & Exact Solutions ## Overview

Students solve real-world problems by writing and solving equations. Students estimate the solution and determine if the estimate is reasonable before finding the exact solution. They write the solution as a complete sentence.

Students complete a Self Check.

# Key Concepts

Students solve real-world problems by first estimating the solution and assessing the reasonableness of the solution. Next, they write an equation to solve the problem and then use the properties of equality to solve the equation. Students write the solution to the problem as a complete sentence.

# Goals and Learning Objectives

• Write equations to solve multi-step real-life problems involving rational numbers.
• Solve equations using addition, subtraction, multiplication, and division of rational numbers.
• Use estimations strategies to estimate the solution and determine if the estimate is reasonable.
• Write the solution as a complete sentence.

# Lesson Guide

Have students watch the video with a partner. Point out to students that their task is to write an equation to solve the problem posed in the video. Students can watch the video as many times as needed in order to find all information needed and write the equation.

As students are watching and listening to the video, they should take notes about any amounts given and label those amounts. They should make note of what they need to find out and use that information to define the variable in their equation.

Possible equation:

Let x equal the price in dollars of one pound of carrot raisin salad.

$\left(2\frac{1}{2}\cdot 6.10\right)+\left(1\frac{3}{4}\cdot 5\right)+\frac{1}{2}x=28.25$

SWD: Students with disabilities may have difficulty determining the relevant information for this task. Before they watch the video, provide students with a note-taking organizer that includes space for notes on how to write an equation.

# The Deli

Watch the video.

• Write an equation to find the cost per pound for the carrot raisin salad.

VIDEO: Best Deli

# Lesson Guide

Discuss the Math Mission. Students will write equations to represent problem situations, make estimates, and solve the equations.

ELL: After defining the term estimation, explain that in solving math problems, we sometimes look for answers that are precise while other times estimation is sufficient. The context of the problem determines whether the answer should be precise or estimated.

## Opening

Write equations, make estimates, and solve the equations.

# Lesson Guide

Have students work in pairs. Ask a few questions to make sure that students understand the task.

SWD: Make sure all students understand the first task. Have students restate the task back to you in their own words so you can assess their understanding.

# Preparing for Ways of Thinking

As students are working, have them record the ways in which they make their estimates so that they can share these during the Ways of Thinking discussion. Try to identify students who seem to have good estimation skills so that they can provide a good model for the class.

Remind students to always begin by identifying the variable in the equation.

# Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

When students have finished solving the problem, they should go back to the problem and use the context of the problem to check their final answer. If the answer does not check, students need to rethink all of their steps. It may be that the equation they have written does not accurately represent the problem, or it may be they made an error in solving the equation.

# Interventions

Student does not know what the variable should represent.

• What does the question ask you to find?

Student does not know where to begin to make an estimate.

• What are you being asked to find?
• Can you round 6.10 to a whole number?
• Think about 2 • 6.
• Think about $\frac{1}{2}$ of 6.

Student does not understand how to check their answer.

• Use the answer you got and calculate the total cost using the information in the problem.
• Does the total cost you calculated match the total cost in the problem?

• Estimate: Total cost is $28.25. The fruit salad costs about$15; the potato salad costs about $8.00, so the carrot raisin salad should cost around$5.
• Equation solution: Let x equal the cost per pound in dollars of the carrot raisin salad.

$\begin{array}{c}\left(2\frac{1}{2}\cdot 6.10\right)+\left(1\frac{3}{4}\cdot 5\right)+\frac{1}{2}x=\text{}28.25\\ \left(15.25\right)+\left(8.75\right)+\frac{1}{2}x=\text{}28.25\\ 24.00+\frac{1}{2}x=\text{}28.25\\ \frac{1}{2}x=\text{}4.25\\ x=\text{}8.50\end{array}$

# The Deli and Other Situations

Mrs. Ortiz buys 0.8 lb of turkey and 2.5 lb of ham. The total cost is $22.75. The ham costs$6.70 per lb. What is the price per pound of the turkey?

• Write an equation to represent the problem.
• Estimate the solution and decide whether your estimate makes sense.
• Solve the equation and write the answer as a complete sentence. ## Hint:

Use the question in the problem to help you decide what quantity the variable in your equation should represent.

# Lesson Guide

Have students work in pairs. Ask a few questions to make sure that students understand the task.

# Preparing for Ways of Thinking

As students are working, have them record the ways in which they make their estimates so that they can share these during the Ways of Thinking discussion. Try to identify students who seem to have good estimation skills so that they can provide a good model for the class.

Remind students to always begin by identifying the variable in the equation.

# Interventions

Student does not know what the variable should represent.

• What does the question ask you to find?

Student does not understand how to check their answer.

• Use the answer you got and calculate the total cost using the information in the problem.
• Does the total cost you calculated match the total cost in the problem?

• 3(5.95 + 2x) + 3(5.95 + x) = $48.75 • Estimate: The total cost of the salads and extra ingredients is$48.75. The salads alone cost about $36 ($6 × 6). The extra ingredients cost about $13. There were 9 extra ingredients, so each ingredient costs about$1.
• Let x equal the cost of each extra ingredient

$\begin{array}{c}3\left(5.95+2x\right)+3\left(5.95+x\right)=\text{}48.75\\ 17.85+6x+17.85+3x=\text{}48.75\\ 35.70+9x=\text{}48.75\\ 9x=\text{}13.05\\ x=\text{}1.45\end{array}$

Each extra ingredient costs $1.45. # Challenge Problem ## Answes • 3x + 5 = 4x + 1.25 • Estimate: The difference in the tips was about$4. Since one boy babysat for $3 per hour and the other for$4 per hour (a difference of $1 per hour), the difference in the tip should be the same as the number of hours each boy babysat. So, Jack and Marcus each babysat for about 4 hours. • Let x equal the number of hours each boy babysat. $\begin{array}{c}3x+5=4x+1.25\\ 3x-3x+5=4x-3x+1.25\\ 5=x+1.25\\ 3.75=x\end{array}$ Each boy babysat 3.75 hours. ## Work Time # Three Salads A salad costs$5.95, plus an extra charge for each additional ingredient. You order 3 salads that each have 2 additional ingredients, and 3 salads that each have 1 additional ingredient. The total cost is $48.75. What is the cost of 1 additional ingredient? • Write an equation to represent the problem. • Estimate the solution and decide whether your estimate makes sense. • Solve the equation and write the answer as a complete sentence. # Challenge Problem Jack and Marcus both babysat for the same number of hours on Thursday and they both earned the same amount of money. However, Jack was paid$3 per hour and received a $5 tip, and Marcus was paid$4 per hour and received a \$1.25 tip. How many hours did the boys babysit?

• Write an equation to represent the problem.
• Estimate the solution and decide whether your estimate makes sense.
• Solve the equation and write the answer as a complete sentence.

## Hint:

You will use the variable more than once in your equation.

# Mathematics

Facilitate the discussion to help students understand the mathematics of the lesson informally. As you discuss each of the problems, ask questions such as the following:

• How did you arrive at your estimate?
• Does your estimate make sense?
• How did you determine what quantity should be represented by the variable?
• What steps did you use to solve the equation?
• What properties of equality did you use in solving the equation?
• Does your solution make sense?
• How does your estimate compare to the solution?
• Is there another way to solve the problem?

ELL: During class discussions, make sure you provide wait time (5–10 seconds) and acknowledge student responses, both verbally and with gestures.

# Ways of Thinking: Make Connections

## Hint:

• How did you come up with your estimate?
• How close is your estimate to the final solution?
• How did you decide which quantity in the problem should be represented by the variable?
• Can you explain the steps you used to solve the equation?
• Does the solution to the equation make sense in terms of the problem situation?
• Why do you sometimes need to use both the addition property of equality and the multiplication property of equality to solve equations?

# Lesson Guide

Have each student write a summary of the math in this lesson; then write a class summary. When done, if you think the summary is helpful, share it with the class.

# A Possible Summary

In this lesson, we not only solved problems using equations, we first estimated the answers. To do this, we sometimes rounded decimal or fraction numbers, we sometimes had to rewrite numbers in a different form, and we used mental math.

We had several ways to check our answers. We could first compare our answer to our estimate, we could check that the solution to the equation made the equation true, and we could go back to the word problem and see if our answer matches the situation.

To solve equations like 4x = 6, you need to use the multiplication property of equality; to solve equations like x + 6 = 10, you need to use the addition property of equality; but to solve equations like 3x + 6 = 12, you need to use both properties.

# Summary of the Math: Write and Solve Equations

Write a summary about writing and solving equations.

## Hint:

• Do you describe how to make an estimate?
• Do you discuss ways to check answers to problems?
• Do you explain how to determine which quantity in a problem should be represented by the variable?

# Lesson Guide

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery and which students would benefit from review before the assessment. Have students work on the Self Check individually.

# Assessment

Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

# Interventions

Student applies operations in the wrong order—for example, chooses 4x + 7 = 80 as an appropriate equation.

• In this expression, what is the first thing that happens to the number I am thinking of? Then what happens?
• What does x represent? What are you adding 7 to?

Student does not recognize all relevant expressions, for example, the student chooses 4(x + 7) = 80 as the only appropriate equation.

• How else could you write the expression 4(x + 7)?

Student calculates an incorrect value for x.

• If you substitute your value of x into the equation, do you get a true equation?
• How will you check whether your value for x is correct?

• 4(x + 7) = 80 and 4x + 28 = 80 both represent the problem.
• The value of x represents the original number (in the statement, "I am thinking of a number...")

$\begin{array}{c}4\left(x+7\right)=80\\ 4x+28=80\\ 4x+28-28=80-28\\ 4x=52\\ \frac{4x}{4}=\frac{52}{4}\\ x=13\end{array}$

# What Is the Number?

Complete this Self Check by yourself.

I am thinking of a number. When I add 7 and then multiply by 4, the result is 80. What is my number?

• Which of the following equations represent this problem? Select all that apply, and justify your choices.

x + 28 = 80

4(x + 7) = 80

4x + 7 = 80

4x + 28 = 80

• For each equation that you identified, find the value of x and explain what it represents.

# Lesson Guide

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery and which students would benefit from review before the assessment. Have students work on the Self Check individually.

# Assessment

Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

# Interventions

Student assumes that the three numbers are equal. For example, the student selects Total = x + 2x + 3x as an appropriate equation.

• What does consecutive mean?
• What does x represent?
• Can you try some numbers to check that this works?

Student does not multiply all terms in the parentheses. For example, the student selects Total = x + (2x + 1) = (3x + 2) as an appropriate equation.

• What does x represent?
• How do you write “one more than x” using algebra? Now read the question again: What happens next? What happens if you add two of these numbers together?

Student calculates an incorrect value for x.

• If you substitute your value of x into the equation, do you get a true equation?
• How will you check whether your value for x is correct?

Student does not correctly interpret the solution of the equation to solve the problem.

• You have found that x = 27. Read the question again. What are the three consecutive numbers?

• Both x + 2x + 2 + 3x + 6 and x + 2(x + 1) + 3(x + 2) correctly represent the situation.
• The value of x represents the first number, so the three numbers are 27, 28, and 29.

$\begin{array}{c}x+2\left(x+1\right)+3\left(x+2\right)=170\\ x+2x+2+3x+6=170\\ 6x+8-8=170-8\\ 6x=162\\ \frac{6x}{6}=\frac{162}{6}\\ x=27\end{array}$

# Three Consecutive Numbers

The numbers 5, 6, and 7 are examples of consecutive numbers—that is, each number follows the previous one.

Suppose three consecutive numbers are used in the following way to get a total. The first number plus two times the second number plus three times the third number equals the total.

• Which of the following expressions represent this situation? Select all that apply, and justify your choices.

Total = x + 2x + 3x

Total = x + 2x + 2 + 3x + 6

Total = x + 2(x + 1) + 3(x + 2)

Total = x + (2x + 1) + (3x + 2)

• The total is 170. What are the three consecutive numbers? Explain your answer.