F3 · Equivalent Fractions: Same Quotient, New Name
“I can rename a fraction by scaling its numerator and denominator by the same factor, and I can scale a fraction down to lowest terms — without changing its value or its place on the number line.”
Rename 2/3 with denominator 6. Step 1: 3 × 2 = 6, so the scaling factor is 2. Step 2: Scale both numbers by 2 — numerator 2 × 2 = 4, denominator 3 × 2 = 6. Step 3: 2/3 = 4/6. Nothing about the value changed — only the name. Model: a bar in thirds with 2 shaded, redrawn in sixths with 4 shaded (the shaded amount is identical), beside a 0–2 number line where 2/3 and 4/6 are the same point. Why it works: the whole is cut into twice as many parts, each part is half as big, so you need twice as many parts to make the same amount.
Why must the numerator be scaled too?
Rename 3/4 with denominator 8. → 3/4 = 6/8
“2/5 renamed with denominator 20: the factor is 20 ÷ 5 = 4, so the numerator is 2 × 4 = ___.” → 8, so 2/5 = 8/20
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.