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.

Problem 3: Grandmother Rachel mixed 20 kilograms of jam containing 15% of sugar with 15 kilograms of jam containing 40% of sugar. How many kilograms of
sugar should be added to the new mixture in order to obtain jam that contains 35%
of sugar?

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Problem 2: One of the workers in the jam factory mixed one barrel of jam containing 14% of sugar with another barrel of jam containing 30% of sugar. The new
mixture weighs 80 kilograms and contains 20% of sugar. How much jam was in the first barrel? How much was in the second barrel?

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The graph of y=x^2 - 2x is at right. Use this graph and any other lines which you may need to add to it to solve the equations given. Briefly explain your method.
x^2 - 2x = 3
x^2 - 2x = x

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Part 2:
The table shows some of the distances and times that Frog traveled. Is he going the same speed the whole time, or is he speeding up or slowing down? How can you tell?
Distance
3.75 cm 7.5 cm 12 cm 15 cm 40 cm
Time
1.5 sec 3 sec 4.8 sec 6 sec 16 sec

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Part 3:
Say Clown walks 15 cm in 12 seconds. Find as many different ways as you can to make Frog walk the same speed as the clown.

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·  Find the number of toothpicks for the fourth, fifth, and tenth tower in each sequence
·   Generalize: How many toothpicks are in the nth tower of each sequence?
·   Find the place of the tower made of 40 toothpicks in each sequence.
(image?)

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What can you say about c if c + d = 10 and c is less than d?

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Mary has the following problem to solve: “Find value(s) for x in the expression: x + x + x = 12” She answered in the following manner.
a. 2, 5, 5
b. 10, 1, 1
c. 4, 4, 4
Which of her answer(s) is (are) correct? (Circle the letter(s) that are correct: a,b,c)

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When is the following true – always, never or sometime?
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Is this equation always, sometimes, or never true?  
a + b = b        Why?

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