## Printing Books

• Lesson
6-8
1

Students explore the relationships among lines, slopes, and y-intercepts in the context of printing their algebra textbooks. Students use a spreadsheet to facilitate their exploration. This activity is based on an idea from Navigating Through Algebra in Grades 6-8 (NCTM, 2001).

Distribute the Printing Books activity sheet to each student. As a class, read the scenario and three options. The three options are:

1. Local printing company: The algebra textbook can be printed by a local printer for a cost of $9.50 per book with an initial cost of$5,000 for typesetting.
2. Local copy center: The algebra book can be duplicated at a local copying center for $0.05 per page plus$2.00 per book for binding.
3. The school district: The school district’s own copying center can reproduce the textbook at a cost of $0.035 per page plus an initial cost of$3,000 for typesetting.

Before students begin to work through each of the three options, ask them to circle the one they predict will be the best solution to the problem. Put students in pairs to discuss their predictions. On the chalkboard, record the names of the students under the option number they have selected (1, 2, or 3). Leave this information on the chalkboard to discuss toward the end of the lesson.

This problem involves a number of different tasks. First, tables and graphs for each situation need to be made. Using a spreadsheet makes this problem much easier. For example, you can use the Printing Books Spreadsheet. In order to create a graph, you will need to right click on the hyperlink, and choose "save target as" to save this file to your computer. Then you will be able to use the Chart Wizard to create graphs.

The figure below shows the costs of the three plans for up to 2500 books. The copy center's plan will be the most expensive, and the printing company's the least expensive if all 2,250 books are produced at one location.

Below is a graphical version of this data. Students will note the slopes of each of the printing options. Students may also note the point of intersection of all three lines.

If each high school can use a different printing company, which choice should each make? The figure below displays additional data and a graph on which the zoom feature allows us to see more clearly where the costs for different plans intersect.

At this point students should revisit their predictions from the beginning of the lesson. Using the information recorded on the chalkboard, ask the students who predicted the correct option to explain their predictions.

Pose the following change to the scenario to the class:

Suppose that these additional conditions apply:

• Western High School will need 400 textbooks next year.
• Eastern High School will need 550 textbooks next year.
• Northern High School will need 1400 textbooks next year.

Students should return to their partners from the beginning of the lesson and discuss this new "twist" to the scenario. Each pair of students should submit a written proposal to the Board of Education defending their choice.

On the basis of these data, students may write something similar to the following:

Western High School will need 400 textbooks for next year, so the cheapest way of having these books made would be to use the local copy center. It would cost $7,300 dollars for these books. Eastern High School needs 550 books. The cheapest place to go would be the school district's in-house copy center. It would cost$9,256.25. Northern High School needs 1400 books. The cheapest way to get these books would be to go with the printing company. This would cost $18,300. All together, these three orders would cost$34,856.25. If all 2350 books were produced by one company, the cheapest choice would be the printing company. This would cost \$27,325. It actually would cost less to produce all the books together, rather than letting the individual schools order their texts.

### Reference

Adapted from Friel, Susan, et al, Navigating Through Algebra in Grades 6 - 8, from the Navigations Series, NCTM (2001).

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### Learning Objectives

Students will:

• Connect the concept of linearity with real-world contexts.
• Identify ways that the table, the graph, and the equation provide information to solve the problem.
• Explore relationships among lines, slopes, and y-intercepts.

### NCTM Standards and Expectations

• Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.
• Relate and compare different forms of representation for a relationship.
• Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.
• Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
• Use graphs to analyze the nature of changes in quantities in linear relationships.

### Common Core State Standards – Mathematics

Grade 7, Expression/Equation

• CCSS.Math.Content.7.EE.A.2
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.''

Grade 8, Expression/Equation

• CCSS.Math.Content.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.

Grade 8, Expression/Equation

• CCSS.Math.Content.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.

Grade 8, Functions

• CCSS.Math.Content.8.F.B.4
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.