standard_description: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
problem_activity: Narrative
In this activity, students compare pairs of numbers with the same number of digits and the same set of digits (for example, 278 and 872, or 1,356 and 3,156). The goal is to elicit the idea that both the placement and the size of the digits matter in determining the value of a number, and that the digits in certain places matter more than others for making comparisons (MP7).
Required Materials
Materials to Gather
Number cards 0–10
Launch
Groups of 3–4
Give each group a set of cards 0–10. Ask students to remove the cards that show 10.
“What three-digit numbers can we make with 5, 7, and 3?” (357, 375, 537, 573, 735, 753)
Activity
“Work with your group to make pairs of numbers using the digits on the cards. First use 3 cards, then 4, then 5, and lastly 6. Compare each pair of numbers.”
“Think about how you go about comparing each pair of numbers.”
5 minutes: group work time
Monitor for students who:
notice that not all digits need to be compared in order to tell which number in a pair is greater
use place-value language in describing the numbers
make two numbers with the same first digit or the same first and second digits
Student Facing
Your teacher will give you a set of cards, each with a single digit, 0–9.
[IMG]
Use the cards for 2, 7, and 8 to make two different three-digit numbers. Use < or > to compare them.
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Now include the digit 1 to make two different four-digit numbers. Compare the numbers.
\(\boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}}\)
Shuffle the cards. Repeat what you did earlier with new cards.
Four-digit numbers
\(\boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}}\)
Five-digit numbers
\(\boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}}\ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \)
Six-digit numbers
\(\boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \underline{\hspace{0.5in}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} _{\ \huge{,}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \ \boxed{\phantom{\frac{00}{00}}} \)
For each pair you compared, how did you decide which number is greater?
Student Response
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Advancing Student Thinking
When answering the question about how they compared the numbers, students may say that they “just know” which one was greater. Encourage them to think about the features of the numbers that gave them an immediate clue about the size of the numbers, or the features that they didn’t find as useful.
Activity Synthesis
Select students to display their number statements and read them. Ask if the class agrees with their comparison.
“How did you decide which number is greater? Did you compare every digit?”
Select students who wrote numbers with the same first digit (or the same first two digits) to share their number statements. Ask them to explain how they compared the numbers.
If no students mentioned that the digits in some places matter more than those in others, ask them about it.
“Did you pay attention only to some digits but not others?”
“Which ones did you prioritize? Were there any you tended to ignore?”


aligns: no