In this activity, students investigate the paces of two different bugs. Students use quantitative and abstract reasoning as they use the tick-mark diagram at the start of the activity to answer questions about pace, decide on a scale for the axes, and mark and label the time needed to travel 1 centimeter for each bug (MP2).

Monitor for students who use different scales on the axes to share later. For example, some students may count by 1 second on the distance axis while others may count by 0.5 second.

If students confuse pace with speed and interpret a steeper line to mean that the ladybug moves faster, consider:

• Asking, “How do the tick marks on either diagram help show which bug is moving faster?”
• Asking, “With distance on the -axis and time on the -axis, where is the ladybug’s location after 4 seconds on the graph? Which line does that point correspond with?”
• Explaining that moving twice as fast means going at half the pace.

Display the images from the Student Task Statement for all to see. Invite students to share their solutions for how long it takes each bug to travel 12 centimeters. Encourage students to reference one or both images as they explain their thinking.

Then invite previously selected students to share their graphs and explain how they decided on what scale to use. If possible, display these graphs for all to see. There are many correct ways to choose a scale for this situation, though some scales may have made it easier to answer the last question. Highlight these graphs and encourage students to read all problems when making decisions about how to construct a graph.

In this activity, students use the tick-mark diagram and graph representations from the previous activity and add a third bug that is moving twice as fast as the ladybug. Students also write equations for all three bugs. An important aspect of this activity is students making connections between these different representations (MP2).

Monitor for students who use these different strategies to write their equations:

• Reason from the unit rates they can see on their graphs and write equations in the form of , where is the constant of proportionality

• Use similar triangles to write equations in the form of , where is a point on the line

The goal of this discussion is to connect the work of using similar triangles to write equations of a line with the work using unit rate to write equations of a line. Display both images from the previous task. Invite previously selected students to share their equations for each bug and record these for all to see.

Use Compare and Connect to help students compare, contrast, and connect the different approaches and representations. Here are some questions for discussion:

• “Did anyone write the same equations, but would explain it differently?”

• “How does the slope of the line show up in each equation?”

• “How do these different representations show the same information?”

As students share their approaches for writing equations, highlight approaches where students used multiple representations to make sense of their equations. For example, ask students to identify features of the tick-mark diagrams, lines, and equations that show the same information. If time allows, demonstrate how the position of the ladybug in a tick-mark diagram can also be seen in the graph of line , and how using the distance and elapsed time values in the corresponding equation will make it true.