standard_description: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
problem_activity: Narrative
This optional activity gives students an additional opportunity to practice using number lines to decompose fractions into sums of other fractions and to record the decompositions as equations.
The fractions on the cards (shown here) contain no whole numbers or mixed numbers, but some students may use them to help them find the second addend (and to avoid counting tick marks on the number line). Some may also choose to label each number line with whole numbers beyond 1 to facilitate their reasoning and equation writing.
\(\frac{1}{3} \qquad \frac{2}{3} \qquad \frac{3}{3} \qquad \frac{4}{3} \qquad \frac{5}{3} \qquad \frac{6}{3} \qquad \frac{7}{3} \qquad \frac{8}{3}\)
Required Materials
Materials to Copy
Make Two Jumps
Required Preparation
Create a set of cards from the blackline master for each group of 2.
Launch
Groups of 2
Give each group a set of fraction cards from the blackline master.
Display the four number lines.
“What do you notice? What do you wonder?”
30 seconds: quiet think time
30 seconds: partner discussion
Activity
“Label each point on the number line with a fraction it represents.”
1–2 minutes: independent work time
“You will make two jumps on the number line to go from 0 to the point and write an equation to represent your moves.”
Explain how to use the cards and how to complete the task. Consider demonstrating with an example and allowing students to ask clarifying questions before they begin.
Monitor for students who:
use whole numbers and mixed numbers in their equations and those who don’t
label their number lines with whole numbers beyond 1
Student Facing
Here are four number lines, each with a point on it.
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For each number line, label the point. This is your target. Make two forward jumps to get from 0 to the target.
Pick a card from the set given to you. Use the fraction on it for your first jump. Draw the jump and label it with the fraction.
From there, draw the second jump to reach the target. What fraction do you need to add? Label the jump with the fraction.
Write an equation to represent the sum of your two fractions.
Student Response
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Activity Synthesis
Select previously identified students to share their responses to the first couple of diagrams (or more if time permits). Start with students who used no whole numbers or mixed numbers. Ask them to explain why they chose to write the numbers the way they did.
Consider discussing the merits and challenges (if any) of expressing the fractions as whole or mixed numbers.


aligns: no