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: Optional activity
The purpose of this optional activity is to practice using a proportional relationship in another context. The chosen numbers make the unit rate a useful tool to answer the questions.
Launch
Remind students of the value of a quarter and a dime.
Student Facing
4 quarters are equal in value to 10 dimes.
How many dimes equal the value of 6 quarters?
How many dimes equal the value of 14 quarters?
What value belongs next to the 1 in the table? What does it mean in this context?
| number of                         quarters | number of                         dimes |
|--------------------------------------------|-----------------------------------------|
|                     1                      |                                         |
|                     4                      |                   10                    |
|                     6                      |                                         |
|                     14                     |                                         |
Student Response
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Student Facing
Are you ready for more?
Pennies made before 1982 are 95% copper and weigh about 3.11 grams each. (Pennies made after that date are primarily made of zinc). Some people claim that the value of the copper in one of these pennies is greater than the face value of the penny. Find out how much copper is worth right now, and decide if this claim is true.
Student Response
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Activity Synthesis
Ask students for the value that belongs next to the 1 in the table. Invite several students to explain the significance of this number.
Speaking: MLR8 Discussion Supports.
To provide a support students in precision of language when describing proportional relationships from a table, provide sentence frames for students to use, such as: “_____ is equal in value to _____, because . . . .”
Design Principle(s): Cultivate conversation


aligns: no