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.

For the equation 2(x + 1) + 4 = 12, identify all possible steps that could be done next.

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3( x + 2) = 12
x + 2 = 4
a) What step did the student use to get from the first line to the second line?
b)  Do you think that this way of starting this problem is (a) a very good way; (b) OK to do, but not a very good way; (c) not OK to do?

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Solve this equation in two different ways:
4(x + 2) = 12

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If m is a positive number, which of these is equivalent to (the same as) m + m + m + m?
a) 4m     b) m4     c) 4(m + 1)     d) m + 4

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Here are two equations:
213x + 476 = 984
213x + 476 + 4 = 984 + 4
•Without solving either equation, what can you say about the answers to these equations?
a) Both answers are the same   b) Both answers are different     c) I can’t tell without doing the math

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Question 1a: Write a story problem that can be formulated by the equation: x+5=8.
Question 1b: Reword your story problem so that it can be formulated by the equation 2x+5=8.
Question 1c: Reword your story problem so that it can be formulated by the equation 2x-5=8.

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A school with 345 students had a sports day. The students chose between football, swimming, and bike riding. Twice as many students chose basketball as bicycling, and there are 30 fewer students who chose swimming than football. 120 students went swimming. How many students chose football?

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Problem 1: Grandmother Rachel mixed 20 kilograms of jam containing 15% sugar with 15 kilograms of jam containing 40% sugar. How many kilograms of
sugar should be added to the new mixture in order to obtain jam that contains 35% sugar?

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One of the workers in the jam factory mixed one barrel of jam containing 14% sugar with another barrel of jam containing 30% sugar. The new
mixture weighs 80 kilograms and contains 20% sugar. How much jam was in the first barrel? How much was in the second barrel?

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The Pet Problem: There are 280 animals signed up to be in the pet show. There are four times as many dogs as cats and there are twice as many birds as cats. How many of each animal signed up for the pet show?

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