Lesson 12

The purpose of this warm-up is to give students an opportunity to look for multiplication patterns in an image. While there are many things students may notice and wonder, the focus of the whole-group discussion should be 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. In the image, there are many other patterns students may see with the dots, lines and dot colors. Record or otherwise validate their observations, but don’t dwell too long here.

Arrange students in groups of 2. Tell students that they will look at a picture, and their job is to think of at least one thing they notice and at least one thing they wonder about the picture. Display the problem for all to see and give 1 minute of quiet think time. Ask students to give a signal when they have noticed or wondered about something. Tell them to share the things they noticed and wondered with a partner.

Some students may try to count the dots in the two outer levels. 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 level before to determine the next one.

After giving students a chance to share what they noticed and wondered with a partner, ask a few students to share with the whole group. Record and display their responses for all to see. After each response, ask the class if they agree or disagree.

The purpose of this task is to show a simple context where exponent notation is naturally useful. The task lends itself to connecting repeated calculations with an expression involving exponents (MP8). This motivates creating a shorthand notation that can be used to answer the questions.

Students might evaluate \(2^5\) as \(2\boldcdot 5\). If this happens, have them make a table showing the number of coins accumulated each day. It will soon be apparent that many more than 10 coins will be accumulated after 5 days.

The goal of the discussion is for students to connect the idea of multiplying \(n\) factors of 2 to get the expression \(2^n\). This notation is convenient for communicating and computing when repeated multiplication is involved. Here are some questions for discussion:

• “How did you know what \(2^5\) represents? How did you evaluate it?”
• “How many times greater is \(2^8\) than \(2^7\)? Could you answer this without evaluating both expressions?”
• “Why does writing expressions with exponents make them easier to work with and understand?”
• “How did you organize your work to answer questions 3 and 4?”
• “Use your calculator to find how much money you would get at the end of 28 days. Does it surprise you?”

In this activity, students apply the meaning of exponents to practice writing and evaluating exponential expressions. Students gain experience experimenting with equivalent numerical expressions and engage in looking for structure (MP7) when they replace a portion of an expression with something equivalent to it.

An exponent is used to indicate multiplying a number by itself. For example, \(2^4\) means \(2 \boldcdot 2 \boldcdot 2 \boldcdot 2\), so \(2^4\) equals 16. 

There are different ways to say \(2^4\). You can say “two raised to the power of four” or “two to the fourth power” or just “two to the fourth.” 

Give students 5 minutes of quiet work time, followed by whole-class discussion.

In expressions with multiplication and exponents, students might parse the expression incorrectly, or apply the order of operations incorrectly. For example, they might interpret \(2^3\boldcdot 2\) as \(4^3\) instead of 2 multiplied by itself 4 times.  Students might also multiply the base and exponent, \(9^2=18\) instead of \(9^2 =81\).

To help students overcome these misconceptions, ask students to rewrite the expressions without exponents.  For example, \(2^3\boldcdot2=2\boldcdot2\boldcdot2\boldcdot2=16\).  Once students have demonstrated conceptual understanding of exponents and why exponents come before multiplication in the order of operations, students will be able to evaluate without rewriting expressions.    

The purpose of the discussion is to help students consider that, just as they did with addition, subtraction, multiplication, and division, they can express numbers in multiple ways using exponents. After reviewing the first question and addressing any misconceptions, ask students to share some of their expressions for the second question. Look for expressions that go beyond the more obvious choices and that involve other operations. If no students found more creative expressions, challenge them to come up with one more expression that involves an exponent and another operation. Ask students to share with the class what they came up with and have other students confirm that they evaluate to 81.

You might also ask students to write expressions that do not equal 81, but look like the error in question 1.