Meaning of Exponents

5 min

Teacher Prep
Setup
Students in groups of 2. Display the problem for all to see. 1 minute of quiet think time, followed by partner and whole-group discussions.

Narrative

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. 

Launch

Arrange students in groups of 2. Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss with their partner the things they notice and wonder.

Representation: Internalize Comprehension. Guide information processing and visualization. To support working memory, show the image for a longer period of time. Students may also benefit from being explicitly told not to count the dots, but instead to look for helpful structure within the image.
Supports accessibility for: Memory, Organization

Student Task

What do you notice? What do you wonder?

<p>A figure of a series of dot branches.</p>
A figure of a series of dot branches. In the center is a black dot. Three branches extend from the black dot with one red dot at the end of each branch. There are three branches that extend from each red dot with one green dot at the end of each branch. There are three branches that extend from each green dot with one yellow dot at the end of each branch. There are three branches that extend from each yellow dot with one blue dot at the end of each branch.

 

Sample Response

Students may notice:

  • It looks like a hexagon.
  • There are different colored dots.
  • The black dot is the center.
  • Each dot has three lines off of it.
  • There is a vertical line of symmetry (and two other lines of symmetry).
  • Each dot branches outward to three more dots of a different color.
  • We can multiply the number of dots in one layer by 3 to find the number of dots in the next.

Students may wonder:

  • How many dots are in the outer layer?
  • How many dots would there be in the next layer if we drew it?
  • What would happen if there were only two dots connected to each one?
Activity Synthesis (Teacher Notes)

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?”
Anticipated Misconceptions

Some students may try to count the dots in the two outer layers. To encourage students to use the patterns in the image, ask them if there is an easier way they could use their count from the layer before to determine the next one.

Standards
Building Toward
  • 6.EE.1·Write and evaluate numerical expressions involving whole-number exponents.
  • 6.EE.A.1·Write and evaluate numerical expressions involving whole-number exponents.

20 min

10 min