Powers of 10
10 min
Narrative
The goal of this Warm-up is for students to visualize 10n for different exponents before they learn exponential notation in this lesson. Monitor for students who use the symmetry of the diagram to estimate how many line segments there are of each size. For example, the picture can be rotated 10 times around the center and each arm is the same, which means the number of each size segment has one factor of 10. This idea can be applied at a smaller scale to get a second and third factor of 10.
When students analyze the diagram and determine the number of segments of each length, they are observing and making use of the repeated structure of 10 segments joining at the different vertices (MP7, MP8).
Launch
- Groups of 2
- “How many do you see? How do you see them?”
- Display the image.
- 1 minute: quiet think time
Teacher Instructions
- Display the image.
- “Discuss your thinking with your partner.”
- 1 minute: partner discussion
- Record responses.
Student Task
How many do you see? How do you see them?
Sample Response
Sample responses:
- 1: Big snowflake, made up of medium snowflakes, which are made of little snowflakes.
- 10: Medium snowflakes, long spokes, medium spokes on the end of each long spoke.
- 100: Little snowflakes, medium spokes because there are 10 at the end of each long spoke.
- 1,000: Little spokes because there are 10 little spokes that make up each little snowflake, there are 10 little snowflakes in each medium snowflake, and there are 10 medium snowflakes in the 1 big snowflake.
Activity Synthesis (Teacher Notes)
- Invite students to share their estimates for how many of the smallest line segments are in the diagram.
- “How can you find out exactly how many there are?” (I can count the number of long segments and then the number of medium-size segments on 1 long segment and then the number of tiny segments on 1 medium-size segment. Then I multiply those numbers.)
- Invite students to count, and then display the expression: 10×10×10.
- “How does the expression relate to the diagram?” (It’s the total number of tiny segments.)
- “Another way to write 10×10×10 is 103. This is called a power of ten. The number 3 tells us how many factors of 10 there are, or how many times we multiply 10 to get the number.”
Standards
- 5.NBT.2·Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
- 5.NBT.A.2·Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
20 min
15 min
Knowledge Components
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Powers of 10
10 min
Narrative
The goal of this Warm-up is for students to visualize 10n for different exponents before they learn exponential notation in this lesson. Monitor for students who use the symmetry of the diagram to estimate how many line segments there are of each size. For example, the picture can be rotated 10 times around the center and each arm is the same, which means the number of each size segment has one factor of 10. This idea can be applied at a smaller scale to get a second and third factor of 10.
When students analyze the diagram and determine the number of segments of each length, they are observing and making use of the repeated structure of 10 segments joining at the different vertices (MP7, MP8).
Launch
- Groups of 2
- “How many do you see? How do you see them?”
- Display the image.
- 1 minute: quiet think time
Teacher Instructions
- Display the image.
- “Discuss your thinking with your partner.”
- 1 minute: partner discussion
- Record responses.
Student Task
How many do you see? How do you see them?
Sample Response
Sample responses:
- 1: Big snowflake, made up of medium snowflakes, which are made of little snowflakes.
- 10: Medium snowflakes, long spokes, medium spokes on the end of each long spoke.
- 100: Little snowflakes, medium spokes because there are 10 at the end of each long spoke.
- 1,000: Little spokes because there are 10 little spokes that make up each little snowflake, there are 10 little snowflakes in each medium snowflake, and there are 10 medium snowflakes in the 1 big snowflake.
Activity Synthesis (Teacher Notes)
- Invite students to share their estimates for how many of the smallest line segments are in the diagram.
- “How can you find out exactly how many there are?” (I can count the number of long segments and then the number of medium-size segments on 1 long segment and then the number of tiny segments on 1 medium-size segment. Then I multiply those numbers.)
- Invite students to count, and then display the expression: 10×10×10.
- “How does the expression relate to the diagram?” (It’s the total number of tiny segments.)
- “Another way to write 10×10×10 is 103. This is called a power of ten. The number 3 tells us how many factors of 10 there are, or how many times we multiply 10 to get the number.”
Standards
- 5.NBT.2·Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
- 5.NBT.A.2·Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.