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 activity prompts students to use number lines to illustrate the decomposition of a fraction into sums of other fractions, reinforcing their work from an earlier lesson. Along the way, students recognize that one way to decompose a fraction greater than 1 is to write it as a sum of a whole number and a fraction less than 1. This insight prepares students to interpret and write mixed numbers in later activities.
Action and Expression: Develop Expression and Communication
. Provide access to colored pencils. Invite students to trace or draw each “jump” with a different color. Then invite students to circle or write out each addend in the corresponding color.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Organization
Launch
Display or draw this number line:
[IMG]
“What number does the point describe?” (8)
“What do you think the ‘jumps’ represent?” (The numbers that are put together to go from 0 to 8)
“What equations can we write to represent the combination of jumps?” (
\(6 + 2 =8\)
or
\(2 + 6 = 8\)
)
“Let’s look at the jumps on some other number lines and see what they may represent.”
Activity
Groups of 2
“Work independently on the activity for a few minutes. Afterwards, share your responses with your partner.”
5–7 minutes: independent work time
2 minutes: partner discussion
Student Facing
On each number line, draw two “jumps” to show how to use sixths to make a sum of
\(\frac{8}{6}\)
. Then, write an equation to represent each combination of jumps.
[IMG]
[IMG]
Noah draws the following diagram and writes:
\(\frac{8}{6} = \frac{6}{6} + \frac{2}{6}\)
and
\(\frac{8}{6} = 1 + \frac{2}{6}\)
. Which equation is correct? Explain your reasoning.
[IMG]
On each number line, draw “jumps” to show how to use thirds to make a sum of
\(\frac{7}{3}\)
. Then, write an equation to represent each combination of jumps.
[IMG]
[IMG]
Write
\(\frac{7}{3}\)
as a sum of a whole number and a fraction.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
Students may not make a connection between the fractions in the problems and the number lines because the latter show no fractional labels. Consider asking students to label the number line to show fractions or asking, “Where do you think
\(\frac{1}{6}\)
would be labeled on this number line?”
Activity Synthesis
Invite students to share their equations. Record them for all to see.
Focus the discussion on part b: writing
\(\frac{7}{3}\)
as a sum of a whole number and a fraction. Students are likely to write
\(1 + \frac{4}{3}\)
and
\(2 + \frac{1}{3}\)
.
Explain that
\(2 + \frac{1}{3}\)
can be written as
\(2\frac{1}{3}\)
, which we call a
mixed number
. This number is equivalent to
\(\frac{7}{3}\)
.
“Why might it be called a mixed number?” (It is a mix of a whole number and a fraction.)


aligns: no