Unit 7, Lesson 5
What about Other Bases?
Accelerated 7
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This Math Talk focuses on comparing powers of positive and negative numbers. It encourages students to think about repeated multiplication and to rely on what they know about integers and the meaning of the bases and exponents to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students find equivalent expressions involving exponents.
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Keep all previous problems and work displayed throughout the talk.
Decide mentally whether each statement is true.
To involve more students in the conversation, consider asking:
MLR2: Collect and Display
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The purpose of this activity is to extend the definition of non-positive exponents to other bases. Students notice that patterns that were true for powers of 10 also hold for other bases (MP8).
Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by 2 minutes to share their reasoning with a partner. Conclude with a whole-class discussion.
Complete the table.
A table with two rows. First row, value, 16, 4 blanks, fraction 1 over 2, three blanks. Second row, exponent form, 2 to the fourth power, 8 blanks. Above the top row, arrows point column to column right to left, labeled time 2. Below the bottom row, arrows point column to column, left to right, labeled times question mark.
The goal of this discussion is for students to connect that the exponent rules discovered for powers of 10 work with other bases. Discuss the following questions:
The purpose of the discussion is to check whether students understand that the rules they have developed for doing arithmetic with powers of 10 are valid for any other nonzero rational bases as well. This is also a good opportunity to address possible misconceptions that students have, especially regarding non-positive exponents.
Consider using these discussion questions to emphasize the important concepts in this lesson:
Tell students that the exponent rules with other bases work exactly the same way as they did with a base of 10. They can always check this by expanding the factors of an expression that has exponents. Introduce and explain the visual display prepared earlier. This display should be kept visible to students throughout the remainder of the unit.
We can keep track of repeated factors using exponent rules. These rules also help us make sense of negative exponents and why a number to the power of 0 is defined as 1. These rules can be written symbolically where the base
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