In this Warm-up, students compare values in order to write true equations or inequalities. Students are supported by considering individual values in this activity, which reminds them of the mathematical concepts before extending those ideas to comparing functions in the associated Algebra 1 lesson. Students make sense of problems and persevere in solving them when comparing values (MP1).

The purpose of the discussion is to recognize methods students used to compare the values. Select students to share their solutions and reasoning. After each solution is shared, ask if there are other methods students used to determine the solution. In particular, listen for students who:

• Manipulate fractions to get common denominators.
• Rewrite fractions in decimal form.
• Solve for the values of a function or expression and compare.
• Recognize that the function is decreasing, so greater inputs will result in lesser outputs.

If no students share the method using the decreasing function, point it out.

In this activity, students invent situations that could be represented by given graphs. It is not essential that students describe situations that would be perfectly linear. Particular attention should be given to discussion about how the graphs increase and decrease. In the associated Algebra 1 lesson, students look at more complex functions and the average rate of change over certain intervals is essential to the comparison. Students are supported in this activity by gaining a better understanding of real situations when the functions increase or decrease. These situations can be referred to when discussing the concept with students in future lessons and activities.

When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

In this activity, students draw graphs that represent given situations (MP2). Particular focus should be given to describing the functions using the terms “increasing” and “decreasing” as well as recognizing regions when one function is greater than the other or when they are equal.