standard_description: Know and apply the Binomial Theorem for the expansion of (x + y)^(n )in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)
problem_activity: Narrative
The purpose of this activity is for students to measure lengths using a ruler that is marked with half inches and quarter inches. Building on their understanding of equivalent fractions, students recognize that lengths that line up with a half-inch mark can be read as one-half of an inch or two-fourths of an inch. They also recognize that lengths that are whole numbers of inches can be expressed as fractions.
MLR8 Discussion Supports.
Synthesis: Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking
Required Materials
Materials to Gather
Materials from a previous activity
Rulers (inches)
Required Preparation
Each student needs a ruler marked with half inches and quarter inches from the warm-up.
Launch
Groups of 2
Display the problem.
“Think about how Kiran and Jada record the length of the worm. Use the ruler to explain how both of their measurements make sense.”
1 minute: quiet think time
2–3 minutes: partner discussion
Monitor for students that say they both make sense because
\(4\frac{1}{2}\)
is equivalent to
\(4\frac{2}{4}\)
and select them to share their responses.
Make sure each student has a ruler marked with half inches and quarter inches from the previous activity.
Activity
“Now, work with your partner to measure the length of some worms. Some lengths can be recorded in more than one way. Be ready to explain how you recorded each length.”
3–5 minutes: partner work time
Student Facing
Kiran and Jada are discussing the length of a worm.
[IMG]
[IMG]
Kiran says that the worm is
\(4\frac{2}{4}\)
inches long.
Jada says that the worm is
\(4\frac{1}{2}\)
inches long.
Use the ruler to explain how both of their measurements are correct.
Measure the length of the following worms.
A
[IMG]
B
[IMG]
C
[IMG]
D
[IMG]
Student Response
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Advancing Student Thinking
If students start measuring from the edge of their ruler instead of from 0, consider asking:
“In the image for the first problem, how is the ruler lined up to the worm?”
“Why does it make sense to not line up the left end of the ruler with the left end of the worm?”
Activity Synthesis
Invite students to share how they recorded their measurements and why they chose to record them in the way they did.
“Let’s think more about the first worm you measured. What are all the ways that we could record its length?” (4 inches.
\(\frac{8}{2}\)
inches.
\(\frac{16}{4}\)
inches.)
“What are all the ways that we could record the length of the last worm?” (
\(3\frac{1}{2}\)
inches.
\(3\frac{2}{4}\)
inches.
\(\frac{7}{2}\)
inches.
\(\frac{14}{4}\)
inches.)
“We can use different numbers to describe the same length because the numbers are equivalent.”


aligns: no