Constructing mathematical arguments

Question 1

Sam wrote the following work to divide 3/4 by 2/5: Step 1: 3/4 ÷ 2/5 Step 2: Multiply the numerators and denominators: (3×2)/(4×5) = 6/20 Step 3: Simplify 6/20 to 3/10 Which statement correctly describes the error in Sam’s work?

Accepted Answer

Sam multiplied the fractions directly; to divide fractions you must multiply by the reciprocal of the divisor.

Answer

Sam should have added the fractions instead of dividing them.

Answer

Sam simplified 6/20 too early; he should have kept it as 6/20.

Answer

Sam used the wrong numbers; the problem actually asked for 3/4 ÷ 5/2.

Question 2

In triangle ABC the three sides are equal: AB = BC = CA. Which of the following reasoning chains correctly proves that angle ABC must be 60°?

Accepted Answer

Since the sides are all equal, the three interior angles are all equal. Let each angle be x. The angles of a triangle add to 180°, so 3x = 180 and x = 60°. Therefore angle ABC = 60°.

Answer

Because AB = BC, angles opposite those sides (angle C and angle A) are equal. Let each be y. Then angle B = 180° – 2y. If y = 60°, angle B is 60°.

Answer

All three sides are equal, so each angle is 180°. Therefore angle ABC is 60° because 180° ÷ 3 = 60°.

Answer

Equal sides mean the triangle is right‑angled. Since a right angle is 90°, the other two angles must each be 45°, so angle ABC is 60°.

Question 3

Sam tried to solve (3/4) ÷ (2/5) and wrote the following steps: 1. Change division to multiplication: 3/4 × 2/5 2. Multiply the numerators and denominators: 3×2 = 6 and 4×5 = 20 3. Simplify 6/20 to 3/10 Which statement correctly identifies Sam’s error?

Accepted Answer

Sam multiplied the fractions directly instead of multiplying by the reciprocal of 2/5.

Answer

Sam correctly used the reciprocal of the divisor.

Answer

Sam should have added the numerators and denominators before simplifying.

Answer

Sam simplified the fraction before multiplying, which is not allowed.

Try three questions

Sam wrote the following work to divide 3/4 by 2/5: Step 1: 3/4 ÷ 2/5 Step 2: Multiply the numerators and denominators: (3×2)/(4×5) = 6/20 Step 3: Simplify 6/20 to 3/10 Which statement correctly describes the error in Sam’s work?

In triangle ABC the three sides are equal: AB = BC = CA. Which of the following reasoning chains correctly proves that angle ABC must be 60°?

Sam tried to solve (3/4) ÷ (2/5) and wrote the following steps: 1. Change division to multiplication: 3/4 × 2/5 2. Multiply the numerators and denominators: 3×2 = 6 and 4×5 = 20 3. Simplify 6/20 to 3/10 Which statement correctly identifies Sam’s error?

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