Prepared by the Junior High Assessment Committee

Department of Education

Testing and Evaluation

Prepared by the Junior High Assessment Committee

March, 2002

Introduction

This bank of questions was prepared for the grade 8 mathematics teachers of Nova Scotia to provide them with examples of assessment questions that probe students’ comprehension in mathematics. All these example question relate to the following specific curriculum outcomes in the Atlantic Canada mathematics curriculum for grade 8:

Teachers should note that each question is in its own box; solutions and/or comments about the solutions are provided directly below each question. Teachers should feel free to use these questions on their assessments or for other purposes. While all these questions relate to common fraction outcomes, they can also serve as models for the development of questions related to other specific outcomes.

These are only examples of possible strategies:

(A)  Compare both fractions to one-half (a benchmark) to get greater than because it is greater than while is less than .

(B)  Since both numerators are 3, one just has to consider the denominators. The 5 means that the whole is cut up into more pieces than for 4, so each fifth piece is smaller that each fourth piece, making greater than .

(C)  Convert both to decimal equivalents to get 0.538 and 0.45, so is greater than .

Other examples of strategies include: drawing them both as parts of the same shape, plotting them as points on a number line, converting them both to the same denominator, and converting them both to the same numerator.

There are many possible answers for both (A) and (B).

Examples for (A): and

(A)  (B) 17 (C) cup (amount for batch

(A) (B)

(A)  Let the black rectangle represent 1. Show 3 blue pieces to represent . These can be cut in half if they are covered with 6 yellow pieces (equivalent). Then 3 yellow pieces represent half of . 3 yellow pieces is .

(B)  Let the black rectangle represent 1. Show 6 brown pieces to represent the . See how many purple pieces () can be made with the 6 brown pieces. The answer is 3.

(A)  (B) one black and one orange

He thought of one and one-half and 20. The product of these he thought of as 20 + 10 or 30.

There are many possible story problems. Examples:

“Mary made 8 batches of a recipe that calls for one-quarter cup of mulk for each batch. How many cups of milk did Mary use?”

“John runs one-quarter of the track in a minute. How many rounds of the track would John run in 8 minutes?”

(A)  17 (Divide 32 by 4 and 36 by 4 and add the 8 and 9; or add 32 and 36 and divide 68 by 4

(B) (Add and subtract the whole numbers to get 10, and then add and subtract the fifths to get .

Whatever concrete model is used, the student should have traced 3 of the quarter pieces (such as 3 purple pieces in Fraction Factory) and then showed these quarters traded for 6 eighths (such as 6 of the brown pieces in the Fraction Factory). In some way the picture should show the removal of 5 of these eighths leaving one-eighth as the answer.

The second mixed number could be any mixed number greater than three and one-fifth and less than 4. (A common error made by students is to think that mixed numbers are multiplied by the whole number parts and the fractional parts and adding these two products together.

(A) The picture should clearly show a half piece, a sixth piece, and a fourth piece. The easiest way to get the answer of one-twelfth is to find the piece that will add to the others to make 1.

(B) One-twelfth

(C) They should agree with Mary because 24 is divisible by 2, 4, 6, and 12 while 32 is not; a class of 32, in other words, could not have voting fractional results as reported.

While not the only possible strategy, a common one would be first to subtract the 3 full cups from the 4 cups, and then the third of a cup from the two-thirds and the five-sixths from the 1 cup, so you are left with one-sixth and one-third, or three-sixths and one-half.

The rectangle should be accurately drawn. Then one-half of it should be shaded (for example, 2 cm x 12 cm or 4 cm x 6 cm). The one-half should then be “quartered” so that one-quarter of one-half is clear. While there are a variety of ways that this could be drawn, each one should clearly show “one-quarter of one-half,” not just the answer one-eighth and not its commutative pair, “one-half of one-quarter.”

(A) Point F (B) Point D (C) Point E (D) Point H

This would be a good portfolio question where you would expect each submission to be a different design, with each design having 56 squares in gold, 14 squares in white, and 70 squares in black.