Question

1. consider $\frac{3}{5} div \frac{2}{3}$.a. draw a tape diagram of $\frac{3}{5} div \frac{2}{3}$. use it to calculate the quotient.b. confirm your solution to part (a) by using the invert and multiply strategy.for problems 2-4, determine the unknown number that makes the number sentence true.2. $\frac{1}{2} div 3 = \frac{1}{2} \times \frac{square}{3}$3. $\frac{2}{5} div \frac{1}{4} = \frac{2}{square} \times \frac{4}{1}$4. $2\frac{1}{4} div \frac{5}{6} = \frac{9}{4} \times \frac{square}{5}$5. consider $\frac{2}{3} div \frac{4}{7}$. which number sentences are true? choose all that apply.a. $\frac{2}{3} div \frac{4}{7} = \frac{2}{3} \times 7 div 4$b. $\frac{2}{3} div \frac{4}{7} = \frac{2}{3} \times 4 div 7$c. $\frac{2}{3} div \frac{4}{7} = \frac{2}{3} \times \frac{7}{4}$d. $\frac{2}{3} div \frac{4}{7} = \frac{2}{3} \times \frac{4}{7}$

Explanation:

Step1: Tape diagram for $\frac{3}{5} \div \frac{2}{3}$

1. Draw a tape representing 1 unit, divided into 15 equal parts (LCM of 5 and 3).
2. Shade $\frac{3}{5}$ of the tape: $\frac{3}{5} = \frac{9}{15}$, so shade 9 parts.
3. Group the shaded 9 parts into sets of $\frac{2}{3} = \frac{10}{15}$? No, instead: find how many $\frac{2}{3}$ fit into $\frac{3}{5}$.

• Rewrite both with denominator 15: $\frac{3}{5}=\frac{9}{15}$, $\frac{2}{3}=\frac{10}{15}$.
• We see $\frac{9}{15} \div \frac{10}{15} = \frac{9}{10}$.

Tape diagram summary: A 15-segment tape, 9 segments shaded. Each "group" of $\frac{2}{3}$ is 10 segments. The 9 shaded segments are $\frac{9}{10}$ of that 10-segment group.

Step2: Invert and multiply for $\frac{3}{5} \div \frac{2}{3}$

$\frac{3}{5} \div \frac{2}{3} = \frac{3}{5} \times \frac{3}{2} = \frac{9}{10}$

Step3: Solve problem 2

Apply invert & multiply: $\frac{1}{2} \div 3 = \frac{1}{2} \times \frac{1}{3}$. The box is 1.

Step4: Solve problem 3

Apply invert & multiply: $\frac{2}{5} \div \frac{1}{4} = \frac{2}{5} \times \frac{4}{1}$. The box is 5.

Step5: Solve problem 4

Convert mixed number: $2\frac{1}{4} = \frac{9}{4}$. Apply invert & multiply: $\frac{9}{4} \div \frac{5}{6} = \frac{9}{4} \times \frac{6}{5}$. The box is 6.

Step6: Evaluate problem 5 options

• Option A: $\frac{2}{3} \div \frac{4}{7} = \frac{2}{3} \times \frac{7}{4} = \frac{2}{3} \times 7 \div 4$ (True)
• Option B: $\frac{2}{3} \times 4 \div 7 = \frac{8}{21}$, but $\frac{2}{3} \div \frac{4}{7} = \frac{7}{6}$ (False)
• Option C: $\frac{2}{3} \div \frac{4}{7} = \frac{2}{3} \times \frac{7}{4}$ (True, invert & multiply)
• Option D: $\frac{2}{3} \times \frac{4}{7} = \frac{8}{21}

eq \frac{7}{6}$ (False)

Answer:

1. a. Quotient: $\frac{9}{10}$

b. Confirmed quotient: $\frac{9}{10}$

1. $\boxed{1}$
2. $\boxed{5}$
3. $\boxed{6}$
4. A. $\frac{2}{3}\div\frac{4}{7}=\frac{2}{3}\times7\div4$, C. $\frac{2}{3}\div\frac{4}{7}=\frac{2}{3}\times\frac{7}{4}$