Question

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Part 1
Reasoning  Make a model that shows 9divided bythree fourths
. Use pencil and paper. Describe how the model changes if the divisor is half as big or twice as big.
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Part 1
Which of these models shows 9divided bythree fourths
​?

Answer 1

Work it out with a simple rectangular (or number‑line) model. The arithmetic first:
9 ÷ (3/4) = 9 × (4/3) = 12.

Two equivalent pencil‑and‑paper models and what to look for:

1) Whole‑unit bars divided into fourths
- Draw 9 unit bars (or one long bar labeled 0 to 9).
- Divide each unit into 4 equal parts (fourths). Now the 9 units contain 9 × 4 = 36 fourths.
- Group those fourths into groups of 3 fourths (because the divisor is 3/4). Each group of 3 fourths is one “share” of size 3/4.
- Count the groups: 36 ÷ 3 = 12 groups. So the model shows 12 groups of size 3/4 — that is 9 ÷ 3/4 = 12.

2) Number‑line jumps of length 3/4
- Draw a number line from 0 to 9. Mark off equal steps of length 3/4 along the line.
- Count how many steps fit from 0 to 9. You will get 12 steps (because 9 ÷ 0.75 = 12).

How the model changes if the divisor is half as big or twice as big
- If the divisor is half as big: (3/4) ÷ 2 = 3/8. Each group is smaller, so you can fit twice as many groups into the same 9 units. Numerically: 9 ÷ (3/8) = 9 × (8/3) = 24. In the bar model you would group the 36 fourths into groups of (3/8) = 1.5 fourths (or better: divide each unit into 8 eighths so there are 72 eighths, then group into groups of 3 eighths → 72 ÷ 3 = 24 groups). On a number line you would make jumps of length 3/8 and count 24 jumps from 0 to 9.

- If the divisor is twice as big: 2 × (3/4) = 3/2. Each group is larger, so you fit half as many groups. Numerically: 9 ÷ (3/2) = 9 × (2/3) = 6. In the bar model you could group the units into pieces of size 3/2 (one and a half), giving 6 groups. On the number line you make jumps of length 1.5 and count 6 jumps from 0 to 9.

Which model shows 9 ÷ 3/4?
- Any model that results in 12 groups of size 3/4 (e.g., 9 unit bars each divided into fourths and grouped into 12 groups of three fourths, or a number line with 12 jumps of length 3/4) shows 9 ÷ 3/4.