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: Problem 1
The grocery store sells beans in bulk. The grocer's sign above the beans says, “5 pounds for $4.”
At this store, you can buy any number of pounds of beans at this same rate, and all prices include tax.
Alberto said, “The ratio of the number of dollars to the number of pounds is 4:5. That's $0.80 per pound.”
Beth said, "The sign says the ratio of the number of pounds to the number of dollars is 5:4. That's 1.25 pounds per dollar."
Are Alberto and Beth both correct? Explain.
Problem 2
Claude and Dora go shopping at the same grocery store as Alberto and Beth.
Claude needs two pounds of beans to make soup. Show Claude how much money he will need.
Dora has $10 and wants to stock up on beans. Show Dora how many pounds of beans she can buy.
Whose rate of change, Alberto’s or Beth’s, do you prefer to use to solve Claude’s problem? Dora’s problem? Explain.
Problem 3
Sanyukta can type 2 pages in 8 minutes.
a.   What are two rates associated with this ratio?
b.   Sanyukta has a deadline in 22 minutes, at which point she needs to be done typing an article. What is the greatest number of pages that Sanyukta can type before her deadline?
c.   Sanyukta has to type a 9-page article. How much time will it take her?
Problem 4
Solve each problem involving distance, speed, and time.
a.   Melanie drives at a constant speed of 35 miles per hour. How far does she travel in 2 hours?
b.   Nathan drives at a constant speed of 40 miles per hour. How long it take him to travel 60 miles?
c.   Oscar drives at a constant speed of 50 miles per hour. How far does he travel in 90 minutes?
d.   Penelope travels 14 miles in 30 minutes. At what constant speed is she driving?


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