Students use tables and graphs to estimate solutions to linear equations and inequalities and use symbolic reasoning to find exact solutions.

 Content ACE Materials Resources Problem 2.4 (1 day) 20-25, 43, 64-68

### Focus Question

What strategies do you find useful to find solutions for linear equations?

### Prior Knowledge

• Linear Functions, Equations, and Inequalities
• Mathematical Modeling
• The overarching goals of this Problem are to review and extend student understanding of and skill in solving linear equations and inequalities by using approximation and exact reasoning strategies.

Students developed methods for solving linear equations in Moving Straight Ahead. Many students will need to review those methods. At this point, we expect students to use informal methods to solve inequalities. Students will learn formal methods in the Grade 8 Unit It’s in the System, on systems of equations and inequalities.

• Video Launch

### Launch Questions

• Engage students in the context of the Problem, charging for canoe rental, by asking whether students have ever been at a water attraction that rented boats. If so, do they recall how the rental cost worked? Did cost depend on time?

Focus on the story line of Sandy’s Boat House and its charging function  c=2.25q +2.50 . Explain to students that c is the charge, in dollars, for renting a canoe for q quarters of an hour from Sandy’s Boat House. Set the scene of students applying for jobs at the boat house. The owner needs to check to see whether they can be trusted to calculate correct rental charges using the cost function given.

### Explore

• Question A gives students an unstructured challenge to answer the standard questions about linear functions and equations.

Questions B, C, and D call on students to demonstrate their understanding and skill in using specific strategies. In those parts, the goal is not to answer the questions first stated in Question A but to explain how graphs, tables, and/or exact reasoning could be used to answer the questions.

Use Question D to revisit the fact-family work from Grades 6 and 7. The linear equation  2.25 q +2.5=9.25  can be solved using a fact-family strategy. Because the fact family for  a+b=c  includes  a=cb , you can write  2.25 q =9.252.50 , or  2.25 q =6.75 . Because the fact family for  a b=c  includes  b=c a , you can write  q =6.75 2.25  or  q =3 .

Although it is fairly easy to solve equations of the form  a x+b=c  by using fact families, it is usually easier to use the rules of equality for equations of the form  a x+b=c x+d . Use Question E to practice these ideas.

As students work, make sure they have viable strategies both for estimating solutions and for finding exact solutions. Ask questions as necessary to help students make sense of estimating and finding solutions.

Plan your Summarize as you observe what students can do. The important goal for students is to solve equations. If your students seem to understand equation-solving methods and are using them correctly and efficiently, you can focus on inequalities in the Summarize. If students have difficulty with solving equations, focus on these and delay work on inequalities.

• What evidence will you use in the summary to clarify and deepen understanding of the Focus Question?
• What will you do if you do not have evidence?

### Summarizing

• If you have a function in the form  y =mx +b  and you know the value of one variable, either x or y, how can you use a graph to estimate the value of the other variable?
Find the value you know on the axis for that variable and then use the line to find the corresponding value on the axis of the other variable.
• How can you use a table to estimate the value of the other variable?
Use the table to find how much one variable changes for a unit change in the other variable. Then use the rate of change to find the unknown variable.
• How can you use reasoning to find the exact value of the other variable?
Substituting the value of the known variable into the equation  y =m x+b  will give you the exact value of the other variable.