Constructing mathematical arguments

Question 1

Maya is trying to find 5/6 ÷ 3/4 and writes these steps: 1. Write the problem: 5/6 ÷ 3/4 2. Change division to multiplication: 5/6 × 3/4 3. Multiply the numerators and denominators: (5×3)/(6×4) = 15/24 4. Simplify 15/24 to 5/8 Which statement correctly explains the error in Maya’s work?

Accepted Answer

Maya should have changed 3/4 to its reciprocal 4/3 before multiplying.

Answer

Maya’s first step is wrong; she should have written 3/4 ÷ 5/6 instead.

Answer

Maya multiplied the numerators together but should have added them.

Answer

Maya simplified 15/24 incorrectly; the correct simplest form is 3/5.

Question 2

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

Answer

Sam should have added the fractions instead of dividing them.

Answer

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

Question 3

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

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°.

Answer

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

Try three questions

Maya is trying to find 5/6 ÷ 3/4 and writes these steps: 1. Write the problem: 5/6 ÷ 3/4 2. Change division to multiplication: 5/6 × 3/4 3. Multiply the numerators and denominators: (5×3)/(6×4) = 15/24 4. Simplify 15/24 to 5/8 Which statement correctly explains the error in Maya’s work?

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°?

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