standard_description: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^(2) + y^(2))^(2) = (x^(2 )– y^(2))^(2) + (2xy)^(2 )can be used to generate Pythagorean triples.
problem_activity: Narrative
The purpose of this activity is for students to compare two numbers in context, to explain or show their reasoning, and record the results of the comparisons with the symbols >, =, or < (MP2). The numbers may be fractions with the same numerator or the same denominator, or a fraction and a whole number.
Students are likely to generate different comparison statements for the same situation. For example, they may write
\(\frac{5}{8} > \frac{3}{8}\)
or
\(\frac{3}{8}<\frac{5}{8}\)
to represent
\(\frac{5}{8}\)
being the greater fraction. During synthesis, discuss how both statements capture the comparison and are valid.
MLR8 Discussion Supports.
Synthesis: As students share the similarities and differences between the representations and comparison statements, use gestures to emphasize what is being described. For example, show with your fingers the partitions such as fourths or eighths that are the same in the representations that are being compared.
Advances: Listening, Representing
Engagement: Provide Access by Recruiting Interest.
Synthesis: Optimize meaning and value. Invite students to share a connection between activity content and their own lives. Ask “How can I use this in my own life?”
Supports accessibility for: Conceptual Processing
Launch
Groups of 2
“Let’s use what we learned about comparing fractions and recognizing equivalent fractions to solve problems about lengths.”
Activity
“Work independently to solve the problems. For each one, be sure to show your thinking and to write a comparison statement.”
6–8 minutes: independent work time
“Share your responses and reasoning with your partner.”
2–3 minutes: partner discussion
Monitor for:
different representations or reasoning strategies used for the same problem, such as diagrams, fraction strips, number lines, or explanations in words
different statements written for the same problem, such as
\(4=\frac{12}{3}\)
and
\(\frac{12}{3}=4\)
, or
\(\frac{2}{3} > \frac{2}{8}\)
and
\(\frac{2}{8} < \frac{2}{3}\)
Student Facing
For each problem:
Answer the question and explain or show your reasoning.
Represent your answer with a statement that uses the symbols >, <, or =.
A beetle crawled
\(\frac{2}{8}\)
of the length of a log. A caterpillar crawled
\(\frac{2}{3}\)
of the length of the same log. Which insect crawled farther?
A grasshopper is 4 centimeters long. A caterpillar is
\(\frac{12}{3}\)
centimeters long. Which insect is longer?
[IMG]
A ladybug crawled
\(\frac{3}{8}\)
the length of a branch. An ant crawled
\(\frac{5}{8}\)
the length of the same branch. Which insect crawled farther?
A grasshopper jumped
\(\frac{5}{8}\)
the width of the sidewalk. A frog jumped
\(\frac{5}{6}\)
the width of the same sidewalk. Which jumped a longer distance?
Student Response
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Activity Synthesis
Select 2–3 students to share their reasoning strategies or representations for at least one of the situations.
“How did you use what you’ve learned in earlier lessons to compare fractions?” (If the fractions had the same numerator, I thought about the size of the denominators. If they had the same denominator, I compared the numerators.)
Select students who wrote different but equally valid comparison statements (for instance,
\(\frac{2}{8}<\frac{2}{3}\)
and
\(\frac{2}{3} > \frac{2}{8}\)
) to share.
Discuss how to read each statement and ask students whether both accurately represent the comparison.
Emphasize that we can write comparison statements in more than one way, but we need to check that the statements make sense given the numbers we write and the symbols we use.


aligns: no