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.
Student Lesson in Spanish
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
Activity
None
Student Task Statement
What do you notice? What do you wonder?
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.
Student Response
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Building on Student Thinking
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.
Activity Synthesis
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?”
12.2
Activity
20 mins
Rice on a Chessboard
Standards Alignment
Building On
Addressing
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
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.
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Activity Synthesis
The goal of the discussion is for students to connect the idea of multiplying factors of 2 to get the expression and to define exponent more generally. Display the expression , and explain that this notation is convenient for communicating and computing repeated multiplication. Remind students that the 3 in is called an “exponent.” Explain that it is used to indicate how many factors of 2 to multiply, so means: .
Invite students to share their responses. Ask questions, such as:
“How did you know what represents? How did you find its value?”
“How many times as much as is ? Could you answer this without finding the value of both expressions?”
“How did you organize your work to answer questions 3 and 4?”
“How many grains of rice would the inventor receive at the end of 32 days (half the chessboard)? After 64 days? Does this surprise you?”
12.3
Activity
10 mins
Make 81
Standards Alignment
Building On
Addressing
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
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).
Launch
Remind students that an exponent indicates how many times a number is being multiplied. Display . Ask, “What does this expression mean?” ()
Explain that there are different ways to read expressions with exponents. Point to and explain that we can read this expression as “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.
Speaking, Listening: MLR8 Discussion Supports. To support the transfer of new vocabulary and mathematical language to long-term memory, invite students to chorally repeat ways to read expressions with exponents. Display the expression and the phrase “two to the third power times two.” Read the phrase once, and invite students to repeat as you point to each part of the expression. Ask students to chorally repeat in unison 1–2 times. Advances: Reading, Speaking
Activity
None
Student Task Statement
Here are some expressions. All but one of them equals 16. Find the one that is not equal to 16 and explain how you know.
Write three expressions with exponents so that each expression equals 81.
Student Response
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Building on Student Thinking
When evaluating exponential expressions, students might apply the order of operations incorrectly. For example, they might interpret as instead of 2 multiplied by itself 4 times. Students might also multiply the base and exponent, for instance, writing instead of . If so, ask students to rewrite the expressions without exponents, for example, .
Activity Synthesis
The purpose of the discussion is to highlight that we can use exponents to express numbers in multiple ways.
Review students’ responses to the first question and address any misconceptions. Then invite 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 students only share expressions that include or , challenge them to come up with one more expression that involves an exponent and another operation. For any expression presented, ask other students confirm that it has the value of 81.
Lesson Synthesis
Display the image from the Warm-up and ask students to think about some of the things they may have wondered:
“The first layer has 3 red dots. How can we write an expression with an exponent to represent the number of dots in each layer?” (3 with an exponent that is the number of the layer)
“Based on those expressions, how many dots are in the outer layer?” () “How many would be in the next layer?” () “In the one after that?” ()
“What part of the expression stays the same as we move from layer to layer?” (The 3 stays the same.) “What part changes? In what way does it change?” (The exponent changes. It increases by 1 for each higher layer.)
“What would happen if there were 2 dots or 4 dots connected to each one? What part of the expression would change?” (The number that is being repeatedly multiplied would change to 2 or 4.)
Student Lesson Summary
When we write an expression like , we call the exponent.
If is a whole number, it tells how many factors of 2 we should multiply to find the value of the expression. For example, , and .
There are different ways to say . We can say “two raised to the power of five” or “two to the fifth power” or just “two to the fifth.”
Standards Alignment
Building On
Addressing
Building Toward
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
Ask students to close their books or devices. Then ask students to share what they know about chess. Invite them to describe a chessboard. Explain that many historians believe chess comes from a strategy board game called chaturanga (cheh-tor-AHN-gah) that people in India played more than 1,000 years ago. Over time, the game became very popular and versions of it spread across the world.
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the story, without revealing the questions.
For the first read, read the problem aloud then ask, “What is this situation about?” (A king offered prizes to the inventor of chess but the inventor declined them. He wanted only grains of rice in each square of the chessboard.) Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured. (Two grains of rice in the first square of a chessboard and twice as much in each of the subsequent squares.)
After the third read, reveal the question: “Do you think the inventor was foolish?” and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read. (Calculate the grains of rice for the second square, third square, and so on.)
Consider doing a quick survey of the class to see if they would have chosen one of the prizes if they were the inventor. It is natural for students to agree with the king and be skeptical of the inventor’s proposal.
Tell students they will explore the amount of rice that could be expected in the inventor's proposal. Distribute scientific calculators to students or be ready to display a scientific calculator from a device. Follow with 5 minutes of quiet work time for students to complete the first two questions and then pause for discussion. Encourage students to describe any patterns they see in the table.
If possible, display the first few screens from the applet to help students see how the rice doubles each day, up until the fourth or fifth day. Use the Play and Pause buttons in the lower left corner of the screen.
Then draw students' attention to the third question. Ask, “How would you use the calculator to figure this out?” After a minute of quiet think time, solicit responses. Tell students how to calculate with exponents on the calculator and make the point that exponent notation is much more convenient for calculation and communication than writing out all the repeated factors.
Give students 5 minutes to complete the last two questions, followed by whole-class discussion. Consider playing the entire animation in the applet before discussing the notation.
Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems, and other text-based content. Display an image or video of a chessboard to activate prior knowledge of the context of the doubling problem. Supports accessibility for: Language, Conceptual Processing
Activity
None
Student Task Statement
A wealthy king wanted to reward the inventor of chess for creating a beautiful game. The king offered the inventor a chest of jewels, a palace, or land to rule.
The inventor declined these rewards and said that all he wished for were some grains of rice.
Shocked, the king demanded an explanation.
The inventor explained, “On the first day, give me 2 grains of rice for the first square on the chessboard. On the second day, give me 4 grains of rice for the second square on the board, and 8 on the third day. Keep doubling the grains of rice each day for each of the 64 squares on the board.”
The king and all the members of his court burst out into laughter. “How foolish! You decline the treasures I offer for a few handfuls of rice?”
Do you think the inventor was foolish?
Complete this table to show the number of grains on the first 5 days.
day
expression with repeated multiplication
expression with exponents
number of grains of rice
1
2
2
4
3
8
4
5
What does represent in this situation? Find the value of without a calculator.
Pause for discussion.
How many days would it take for the number of grains of rice to exceed 50,000?
Will the number of grains of rice exceed 1 million before half of the chessboard is reached? Explain or show your reasoning.
Student Response
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Building on Student Thinking
Students might evaluate as . If this happens, instruct students to review the table showing the number of grains accumulated each day. It will be apparent that many more than 12 grains will be accumulated after 6 days.
