The purpose of this Warm-up is to give students an opportunity to look for multiplication patterns in an image, which will be useful when students interpret exponential relationships and notation in later activities. While students may notice and wonder many things about the image, including patterns in the dots, lines, and color, how repeated multiplication is shown in the image is the most important discussion point. Students see and make use of structure (MP7) to describe the fact that each dot branches out to three more dots of a different color. These connections mean we have repeatedly growing groups of 3, so we can multiply by 3 to find the number of dots and lines at various stages. 

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the idea of each dot branching out into 3 more dots to form the new layer does not come up during the conversation, ask students to discuss this idea. Consider asking:

• “How many red dots branch out from the black dot in the center?”
• “How many green dots branch out from each red dot?”
• “How could you use these facts to calculate how many green dots there are?”
• “Does this pattern continue? How do you know?”

The purpose of this activity is to introduce a context where exponent notation is naturally useful. The task lends itself to connecting repeated calculations with an expression involving exponents (MP8). The situation described in the task is based on a legend that has a rich mathematical history. If time permits, consider inviting students to research the different versions of the legend and their connections to mathematical communities in Asia over time.

In the digital version of the activity, students use an applet to see how the number of grains of rice increases each day through day 16. The applet allows students to stop at any time and see an expression that represents the amount of rice on any given day. Use the digital version to help students visualize how quickly the amount of rice increases or to simply provide an additional tool for all students to better understand the nature of exponential growth.

If students don't have individual access, displaying the applet for all to see would be helpful after students answer the second question.

In this activity, students practice finding the value of exponential expressions and writing equivalent expressions. To determine equivalence, students need to apply the meaning of exponents and to look for structure (MP7).