The purpose of this Warm-up is to remind students that the translation of a line is parallel to the original line. They begin by inspecting several lines to decide which are translations of a given line. Students then describe the translations by specifying the number of units and the direction, in preparation to see the equation as a translation of .

Invite students to share how they determined that lines and are translations of . Emphasize that lines and are parallel to , and line matching up with the other lines would require a rotation or reflection. If possible, demonstrate the transformations using a clear transparency or tracing paper. If using a transparency or tracing paper to demonstrate the translations, it is helpful to draw a dot for a specific point on both the underlying graph and on the transparency as a reference point.

The goal of this activity is for students to think about translations of lines in a context by examining two scenarios. Graphically, the two lines representing the relationships are parallel. The lines have the same slope but different vertical intercepts. Students will observe this structure in the equations that they write for the two lines (MP7). 

Even though babysitters are often paid for increments of 1 hour or ½ hour and in increments of $1, a continuous line is used to represent this relationship. If a student brings up the idea that it would be better to represent the relationship using discrete points rather 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, starting with an approach that looks at the arithmetic relationship, followed by an approach that looks at the geometric relationship, and concluding with a more abstract approach that combines the first two:

• 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

In the digital version of the activity, students use an applet to graph two linear relationships. The applet allows students to quickly and accurately create a graph of two scenarios. The digital version may reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.

The purpose of this discussion is for students to connect the graphical translation of a line to a context. Invite previously selected students to share their approaches to graphing each line. Sequence the discussion in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “How much more money does Diego have after 1 hour of babysitting in the second situation compared to the first? After 2 hours? After 5 hours? After hours? (Diego always has $30 more.)

• “Where can this extra $30 be seen in each approach?” (Table: The number of dollars saved after a given number of hours is always 30 more in the second situation; Graph: The first line is translated up by 30 units, and the vertical intercept is at ; Equation: The second equation is the same as the first but with a “.”

• “Where can the $10 that Diego earns each hour be seen in each approach?” (Table: The number of dollars saved goes up by 10 each time the number of hours goes up by 1; Graph: The slope of both lines is 10; Equation: Both equations have the term “.”)

This activity continues to examine parallel lines, including situations where the vertical intercept is negative. Students sort cards with lines represented graphically, algebraically, with a table of values, or with a verbal description. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Once all groups have completed the Card Sort, discuss the following:

• “Which matches were tricky? Explain why.”

• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

• “What clues did you look for to see which equation went with each graph?”

Invite students to share their version of the missing equation. Record responses for all to see. Ensure students understand that and are equivalent.