Formative Assessments

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Studies on algebraic thinking typically include assessment problems that are used by researchers to elicit a range of students' understanding and misconceptions. These problems are collected here and can be useful tools for teachers' formative assessment of students' comprehension of algebraic concepts.

Border Problem (linear relationship, geometric context)
Here is a 10 by 10 grid. How many squares are in the border?

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Hours and Wage Problem (linear relationship)
Mary’s basic wage is $20 per week. She is also paid another $2 for each hour of overtime she works.
a) What would her total wage be if she worked 4 hours of overtime in 1 week? 10 hours of overtime?

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Refund Problem (direct variation/proportionality)
In some states, a deposit is charged on aluminum pop cans and is refunded when the cans are returned. In New York, the deposit is 5 cents a can.
a) What would be the refund for returning 6 (10 or 12) cans?

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Challenge your students to think about the meaning of equal sign through the following questions. For each of the following tasks, decide if “=” is used appropriately. If not, decide on an appropriate sign.
a.     6 = 6
b.     12-8 = 4
c.     2 = 10 ÷ 5

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Harold has some money. Sally has four times as much money as Harold. Harold earns $18.00 more dollars. Now he has the same amount as Sally. Can you figure out how much money Harold has altogether? What about Sally?

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Two students have the same amount of candies. Briana has one box, two tubes, and 7 loose candies. Susan has one box, one tube, and 20 loose candies. If each box has the same amount and each tube has the same amount, can you figure out how much each tube holds? Each box?

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Open discussion problem: Which phone plan is better? Plan #1: You pay $0.10 per minute for all calls. Plan #2: You pay $0.60 per month plus $0.05 per minute for calls.

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Open discussion problem: Mike and Robin each have some money. Mike has $8 in his hand and the rest of his money is in his wallet. Robin has altogether exactly three times as much money as Mike has in his wallet.

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Sayoo-jan has some money in her wallet. Her father gave her three times more than she already has. She now has 1000 yen. How much money did she have before her father gave her any?

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Group P: For lower secondary school (age 14-15 years)
1. Pick out those statements that are equations from the following list and
write down why you think the statement is an equation.
a) k = 5
b) 7w-w
c) 5t - t = 4t
d) 5r -1 = 11

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