standard_description: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^(3) = 5(^(1/3))^(3) to hold, so (5^(1/3))^(3) must equal 5.
problem_activity: Activity
In this activity, students match cards written in scientific notation with their decimal values. The game grants advantage to students who distinguish between numbers written in scientific notation from numbers that superficially resemble scientific notation (e.g.
\(0.43 \times 10^5\)
).
Launch
The blackline master has three sets of cards: set A, set B, and set C. Set A is meant for demonstration purposes, so only a single copy of set A is necessary.
Arrange students in groups of 2. Consider giving students a minute of quiet time to read the directions. Then, use set A to demonstrate a round of the game for the class. Explain to students that a match can be made by pairing any two cards that have the same value, but it is favorable to be able to tell the difference between numbers in scientific notation and numbers that simply look like they are in scientific notation.
When students indicate that they understand how to play, distribute a set of cards (either set B or set C) to each group. Save a few minutes for a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Begin with a whole-class think aloud to demonstrate the steps of the game. Consider providing some groups with cards that contain more accessible values to begin with.
Supports accessibility for: Memory; Conceptual processing
Representing, Conversing: MLR8 Discussion Supports.
Demonstrate the steps of how to play the game. To do this, select a student to play the game with you while the rest of the class observes. This will help clarify the expectations of the task, invite more student participation, and facilitate meta-awareness of the language involving scientific notation.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Your teacher will give you and your partner a set of cards. Some of the cards show numbers in scientific notation, and other cards show numbers that are not in scientific notation.
Shuffle the cards and lay them facedown.
Players take turns trying to match cards with the same value.
On your turn, choose two cards to turn faceup for everyone to see. Then:
If the two cards have the same value
and
one of them is written in scientific notation, whoever says “Science!” first gets to keep the cards, and it becomes that player’s turn. If it’s already your turn when you call “Science!”, that means you get to go again. If you say “Science!” when the cards do not match or one is not in scientific notation, then your opponent gets a point.
If both partners agree the two cards have the same value, then remove them from the board and keep them. You get a point for each card you keep.
If the two cards do not have the same value, then set them facedown in the same position and end your turn.
If it is not your turn:
If the two cards have the same value
and
one of them is written in scientific notation, then whoever says “Science!” first gets to keep the cards, and it becomes that player’s turn. If you call “Science!” when the cards do not match or one is not in scientific notation, then your opponent gets a point.
Make sure both of you agree the cards have the same value.
If you disagree, work to reach an agreement.
Whoever has the most points at the end wins.
Student Response
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Student Facing
Are you ready for more?
What is
\(9 \times 10^{\text-1} + 9 \times 10^{\text-2}\)
? Express your answer as:
A decimal
A fraction
What is
\(9 \times 10^{\text-1} + 9 \times 10^{\text-2} + 9 \times 10^{\text-3} +9 \times 10^{\text-4}\)
? Express your answer as:
A decimal
A fraction
The answers to the two previous questions should have been close to 1. What power of 10 would you have to go up to if you wanted your answer to be so close to 1 that it was only
\(\frac{1}{1,000,000}\)
off?
What power of 10 would you have to go up to if you wanted your answer to be so close to 1 that it was only
\(\frac{1}{1,000,000,000}\)
off? Can you keep adding numbers in this pattern to get as close to 1 as you want? Explain or show your reasoning.
Imagine a number line that goes from your current position (labeled 0) to the door of the room you are in (labeled 1). In order to get to the door, you will have to pass the points 0.9, 0.99, 0.999, etc. The Greek philosopher Zeno argued that you will never be able to go through the door, because you will first have to pass through an infinite number of points. What do you think? How would you reply to Zeno?
Student Response
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Activity Synthesis
The main idea is for students to practice using the definition of scientific notation and flexibly convert numbers to scientific notation. Consider selecting students to explain how they could tell whether two cards had the same value and whether they were written in scientific notation.


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