Lesson 7

The purpose of this warm-up is to remind students that the translation of a line is parallel to the original line, and to plant the seed that a line can be taken to a parallel line by translating it. They inspect several lines to decide which could be translations of a given line. Then they describe the translations by specifying the number of units and the direction the original line should be translated.

Arrange students in groups of 2. Give students 2 minutes of quiet think time and access to geometry toolkits. Ask them to share their responses with a partner afterwards.

Students may think that lines \(i\) and \(h\) can’t be images of line \(f\) because the part of \(i\) and \(h\) we can see is shorter than the part of \(f\) we can see. Tell these students that all of the lines go on indefinitely in both directions.

Invite students to share how they know that lines \(h\) and \(i\) are translations of \(f\). The main reason to bring out is that they are parallel to \(f\). It might be worth demonstrating with a transparency that we can translate \(f\) to match up with \(h\) and \(i\), but to match up with the other lines, we would need to rotate (or reflect) the transparency. If using a transparency to demonstrate the number of units to translate \(f\) up or down, it is helpful to draw a dot on a specific point on both the underlying graph and on the transparency.

The goal of this activity is to get students to think about translations of lines in a context. Students examine two scenarios. In the first, there is a proportional relationship between the number of hours Diego works and his total earnings. In the second, Diego starts with $30 saved and then saves all of his earnings. Graphically, the two lines showing these relationships are parallel (the second is a vertical translation of the first for the extra $30 Diego has already saved). The lines have the same slope (Diego’s hourly rate of pay is the same) but different \(y\)-intercepts (one has a \(y\)-intercept of 0 while the other has a \(y\)-intercept of 30). Students will observe this structure in the equations that they write for the two lines. 

Even though babysitters are most often paid for increments of 1 hour or \(\frac12\) hour and in increments of $1, this uses a continuous line to represent this relationship. If a student brings up the idea that it would be better to represent the relationship using discrete points than a line, acknowledge the observation but suggest that a continuous line is an acceptable representation. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation (MP4).

Monitor for students who use these approaches as they graph the two earnings scenarios:

• Making a table (showing earnings for different numbers of hours worked) and then graphing
• Plotting points directly
• Writing an equation that represents the situation and then graphing the solutions to the equation

Select students using these approaches and invite them to present during the discussion.

Invite selected students to share their methods for graphing Diego’s earnings, sequenced in this order: 

• Making a table (showing earnings for different numbers of hours worked) and then graphing
• Plotting points directly
• Writing an equation that represents the situation and then graphing the solutions to the equation

Notice that making a table has the advantage of revealing the arithmetic relationship between Diego’s earnings in the two situations. No matter how many hours, Diego works, he has $30 more in the second situation. Plotting the points directly shows geometrically that the vertical distance is always the same between pairs of points for the same number of hours worked (this vertical distance represents the $30). The equations are the most abstract and contain all of the arithmetic and geometric information once we can interpret them. If \(h\) is the number of hours Diego works and \(m\) is how much money he has saved, then the two equations can be written as \(m = 10h\) and \(m = 30 + 10h\). The graphs of these lines have slope 10 (and they are parallel!), but the second equation has a \(y\)-intercept of 30 while the first has a \(y\)-intercept of 0. The graph of \(m = 30+10h\) is the same as the graph of \(m = 10h\) except that every point is moved up by the same amount (this explains why they are parallel).

Have students connect the equations and situations they have graphed in the activity by asking students what the graph of \(m = 20 + 10h\) would look like (it would lie between the graphs of \(m = 30 + 10h\) and \(m = 10h\)). What situation does this equation represent? (Diego started with $20 saved and then earned $10 for each hour of baby sitting.)

The previous activity examined parallel lines and equations that define them, focusing on their common attributes (slope) and their different attributes (\(y\)-intercept). This activity further examines parallel lines, including situations where the \(y\)-intercept is negative. In addition, students match lines represented in many different ways including

• graphically
• verbal description
• table of values
• equations

You will need the Translating a Line blackline master for this activity.

Once all groups have completed the matching, discuss the following:

• “Were any of the matches more difficult than others? What made them difficult?”
• “Did any groups have to adjust an initial guess that turned out to be wrong? What adjustments were made and why?”
• “What clues did you look for to see which equation went with a graph?”

Solicit all versions of the missing equation. Ensure that students understand that \(y=\frac12x-4\) and \(y=\text -4+\frac12x\) are equivalent.