standard_description: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
problem_activity: Narrative
The purpose of this activity is for students to practice generating two different patterns from two different rules and observe and quantify relationships between corresponding terms. Both sets of rules generate patterns that have the same relationships between corresponding terms. Each of Priya’s terms is 3 times greater than each of Jada’s corresponding terms and each of Jada’s terms is
\(\frac{1}{3}\)
of Priya’s corresponding terms.
When students find and explain patterns related rules and relationships, they look for and express regularity in repeated reasoning (MP8).
Launch
Groups of 2
“You and your partner will each complete some problems about patterns independently. After you’re done, discuss your work with your partner.”
Activity
5 minutes: independent work time
3 minutes: partner discussion
“Look back at your work and make any revisions based on what you learned from discussing with your partner.”
1–2 minutes: independent work time
Monitor for students who:
revise their thinking based on partner discussion
use multiplication expressions to represent the relationships between the patterns
Student Facing
Partner A
Use each rule to create a pattern.
Jada’s rule: Start at 0. Keep adding 2.
[IMG]
Priya’s rule: Start at 0. Keep adding 6.
[IMG]
If the patterns continue:
What number will be in Priya’s pattern when Jada’s pattern has 34? Explain or show your reasoning.
What number will be in Jada’s pattern when Priya’s pattern has 120? Explain or show your reasoning.
What relationships do you notice between the numbers in Priya's pattern and the numbers in Jada's pattern?
Partner B
Use each rule to create a pattern.
Jada’s rule: Start at 0. Keep adding 3.
[IMG]
Priya’s rule: Start at 0. Keep adding 9.
[IMG]
If the patterns continue,
What number will be in Priya’s pattern when Jada’s pattern has 54? Explain or show your reasoning.
What number will be in Jada’s pattern when Priya’s pattern has 198? Explain or show your reasoning.
What relationships do you notice between the numbers in Priya’s pattern and the numbers in Jada’s pattern?
Student Response
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Activity Synthesis
Ask previously identified students to share their thinking.
Display completed patterns from both sets of rules.
“How are the patterns from the pairs of rules the same? How are they different?” (Some of the numbers are in all four patterns. Some numbers aren’t in any pattern like 5 and 11. In both sets of rules, Priya’s numbers are 3 times greater than Jada’s numbers and Jada’s numbers are
\(\frac{1}{3}\)
Priya’s numbers.)
“How would you describe the relationship between Priya's pattern and Jada's pattern?” (Priya's numbers are all 3 times as much as Jada's corresponding numbers.)
“What can I multiply the numbers in Priya’s pattern by to get the corresponding numbers in Jada’s pattern?” (
\(\frac{1}{3}\)
, each number in Jada's pattern is
\(\frac{1}{3}\)
the corresponding number in Priya's pattern.)


aligns: no