The goal is to generalize the common difference and common quotient to understand the relationships between points in linear and exponential functions.

Direct students’ attention to the reference created using Collect and Display. Ask students to share what the pairs had in common and the expected results of subtracting or dividing each expression. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Display a graph of , and ask students where they can see the 3. Draw on the graph a bunch of slope triangles with a horizontal side of length 1 and vertical side of length 3, corresponding to the inputs and outputs in the task.

Then display a graph of , and ask students where they can see the 2. Show that when increases by 1, doubles for a few different inputs and outputs in the task.

Demonstrate, or ask a student to demonstrate, working through writing and with fewer terms.

The goal of this discussion is to bring together multiple ideas of how to use properties of exponents and the distributive property to evaluate expressions. Either display the correct solutions or invite students to check with another student to see if their answers are the same. If there are any differences between solutions, invite students to ask questions or discuss as a whole group. Here are some questions for discussion:

• “When two powers with the same base are being divided, what happens to the exponents?” (The exponents are subtracted.)
• “When two powers with the same base are being divided and you know one of the powers and the quotient, how can you figure out the other power?” (If the unknown power is the denominator, I would subtract the quotient’s power from the numerator’s power. If the unknown power is the numerator, I would add the quotient’s power to the denominator’s power.)
• “When distributing something to multiple parts within parentheses, how do you make sure you are distributing it to all parts?” (I draw arrows from the number being distributed to all of the parts inside of the parentheses to help me keep track.)