Appendix A — Answer Key, Chapter 5: Multiplying and Dividing Fractions

SOL 6.CE.1 a–c · Covers textbook Chapter 5 and the companion workbook. Item numbers match the textbook; workbook items that repeat the textbook are answered once here, and workbook-only items are keyed in the final section. Every answer is in simplest form. Reasoning answers show an acceptable response, not the only wording. Estimates accept any reasonable rounding.

Lesson 5.1 — Modeling Fraction Multiplication

Guided practice

1. $\tfrac{1}{2} \times \tfrac{1}{4} = \tfrac{1}{8}$. The square has $2 \times 4 = 8$ cells and 1 is double-shaded.
2. $\tfrac{2}{5} \times \tfrac{1}{3} = \tfrac{2}{15}$. The square has 15 cells and 2 are double-shaded.
3. $\tfrac{1}{3}$ of $9$: three equal groups of 3, take one group, so the answer is $3$.
4. $2 \times \tfrac{3}{8} = \tfrac{3}{8} + \tfrac{3}{8} = \tfrac{6}{8} = \tfrac{3}{4}$
5. $\tfrac{3}{4} \times \tfrac{2}{3} = \tfrac{6}{12} = \tfrac{1}{2}$. The model has 12 cells (3 rows by 4 columns) with 6 double-shaded.

Independent practice

6. a) $\tfrac{1}{10}$ b) $\tfrac{3}{8}$ c) $\tfrac{4}{15}$
7. $5 \times \tfrac{2}{3} = \tfrac{10}{3} = 3\tfrac{1}{3}$
8. $\tfrac{2}{3}$ of $12$: three equal groups of 4, take two groups, so the answer is $8$.
9. $\tfrac{3}{5} \times \tfrac{5}{6} = \tfrac{15}{30} = \tfrac{1}{2}$. The model has 30 cells with 15 double-shaded.
10. $\tfrac{3}{8} \times \tfrac{4}{5} = \tfrac{12}{40} = \tfrac{3}{10}$. Cut a square into 8 rows and 5 columns for 40 cells; shade 3 of the 8 rows one way and 4 of the 5 columns the other way; 12 cells are double-shaded.
11. $\tfrac{1}{2} \times \tfrac{2}{3} = \tfrac{2}{6} = \tfrac{1}{3}$ cup. Model: shade $\tfrac{2}{3}$ of a square one way and $\tfrac{1}{2}$ of it the other way; 2 of the 6 cells are double-shaded.
12. Shading $\tfrac{1}{3}$ one direction and $\tfrac{1}{4}$ the other direction cuts the square into 3 rows and 4 columns, which makes $3 \times 4 = 12$ equal cells. The denominators tell how many cuts each direction makes, so the number of cells — the denominator of the product — is their product. One cell is double-shaded, so $\tfrac{1}{3} \times \tfrac{1}{4} = \tfrac{1}{12}$.

Exit ticket 5.1

1. $\tfrac{1}{2} \times \tfrac{2}{5} = \tfrac{2}{10} = \tfrac{1}{5}$
2. $12$
3. $4 \times \tfrac{3}{10} = \tfrac{12}{10} = \tfrac{6}{5} = 1\tfrac{1}{5}$
4. Shading $\tfrac{1}{3}$ of the square in one direction and $\tfrac{1}{4}$ in the other divides it into 12 equal cells. Exactly one cell is shaded both ways, and one cell out of 12 is $\tfrac{1}{12}$.

Lesson 5.2 — Multiplying Fractions and Mixed Numbers

Guided practice

1. $\tfrac{6}{15} = \tfrac{2}{5}$
2. $\tfrac{28}{56} = \tfrac{1}{2}$
3. $\tfrac{3}{2} \times \tfrac{2}{3} = \tfrac{6}{6} = 1$
4. $\tfrac{3}{1} \times \tfrac{5}{6} = \tfrac{15}{6} = \tfrac{5}{2} = 2\tfrac{1}{2}$
5. $\tfrac{12}{5} \times \tfrac{5}{4} = \tfrac{60}{20} = 3$

Independent practice

6. a) $\tfrac{15}{120} = \tfrac{1}{8}$ b) $\tfrac{45}{60} = \tfrac{3}{4}$ c) $\tfrac{6}{36} = \tfrac{1}{6}$
7. $\tfrac{4}{3} \times \tfrac{9}{4} = \tfrac{36}{12} = 3$
8. $\tfrac{7}{2} \times \tfrac{8}{7} = \tfrac{56}{14} = 4$
9. $\tfrac{25}{36}$ (already in simplest form, since 25 and 36 share no factor but 1)
10. $\tfrac{5}{2} \times \tfrac{16}{5} = \tfrac{80}{10} = 8$
11. Estimate: about $3 \times 1 = 3$ miles. Exact: $\tfrac{3}{4} \times \tfrac{8}{3} = \tfrac{24}{12} = 2$ miles.
12. Prediction: less than $\tfrac{4}{5}$, because $\tfrac{7}{8}$ is between 0 and 1, so it takes only part of $\tfrac{4}{5}$. Computation: $\tfrac{7}{8} \times \tfrac{4}{5} = \tfrac{28}{40} = \tfrac{7}{10}$. Since $\tfrac{7}{10} = \tfrac{28}{40}$ and $\tfrac{4}{5} = \tfrac{32}{40}$, the product is indeed less. The prediction was right.

Exit ticket 5.2

1. $\tfrac{12}{72} = \tfrac{1}{6}$
2. $\tfrac{3}{2} \times \tfrac{8}{3} = \tfrac{24}{6} = 4$
3. $\tfrac{6}{1} \times \tfrac{5}{12} = \tfrac{30}{12} = \tfrac{5}{2} = 2\tfrac{1}{2}$
4. A mixed number is a sum of a whole number and a fraction, and multiplying the whole parts and fraction parts separately drops some of the products you need. Rewriting it as a single improper fraction makes it one numerator over one denominator, so the multiply-across rule applies directly.

Lesson 5.3 — Modeling Fraction Division

Guided practice

1. Six thirds fit into 2, so $2 \div \tfrac{1}{3} = 6$.
2. $1 \div \tfrac{1}{8} = 8$
3. $\tfrac{1}{2} = \tfrac{4}{8}$, so $\tfrac{4}{8} \div \tfrac{1}{8} = 4$.
4. $\tfrac{3}{4} \div \tfrac{1}{4} = 3$
5. $\tfrac{2}{3} = \tfrac{4}{6}$, so $\tfrac{4}{6} \div \tfrac{1}{6} = 4$.

Independent practice

6. a) $8$ b) $15$ c) $3 \div \tfrac{2}{3}$: cut 3 wholes into 9 thirds and group them 2 at a time, giving $\tfrac{9}{2} = 4\tfrac{1}{2}$
7. $7$
8. $5$
9. $\tfrac{3}{4} \div \tfrac{1}{2} = \tfrac{3}{4} \div \tfrac{2}{4} = 3 \div 2 = \tfrac{3}{2} = 1\tfrac{1}{2}$. Model: a strip of $\tfrac{3}{4}$ holds one whole half ($\tfrac{2}{4}$) with $\tfrac{1}{4}$ left over, and $\tfrac{1}{4}$ is half of a half — so one and a half halves fit.
10. $1\tfrac{1}{2} \div \tfrac{1}{4} = \tfrac{6}{4} \div \tfrac{1}{4} = 6$
11. $6$ pieces. Model: lay out $4\tfrac{1}{2}$ feet as 18 quarter-feet; each $\tfrac{3}{4}$-foot piece uses 3 of them, and $18 \div 3 = 6$. (Or: $\tfrac{9}{2} \div \tfrac{3}{4} = \tfrac{9}{2} \times \tfrac{4}{3} = 6$.)
12. Because division asks how many copies of the divisor fit inside the dividend, and $\tfrac{1}{4}$ is much smaller than 1. Each whole holds 4 fourths, so 3 wholes hold 12 of them. The answer counts small pieces, and there are more small pieces than there were wholes. "Dividing makes things smaller" is only true when you divide by a number greater than 1.

Exit ticket 5.3

1. $10$
2. $\tfrac{3}{4} = \tfrac{6}{8}$, so $\tfrac{6}{8} \div \tfrac{1}{8} = 6$.
3. $\tfrac{5}{6} \div \tfrac{2}{6} = 5 \div 2 = \tfrac{5}{2} = 2\tfrac{1}{2}$
4. The question asks how many groups of $\tfrac{1}{3}$ cup fit into 2 cups, which is exactly what $2 \div \tfrac{1}{3}$ means. Each cup holds 3 scoops, so 2 cups hold $6$ scoops.

Lesson 5.4 — Dividing Fractions and Mixed Numbers

Guided practice

1. $\tfrac{1}{2} \times \tfrac{4}{3} = \tfrac{4}{6} = \tfrac{2}{3}$
2. $\tfrac{3}{5} \times \tfrac{5}{2} = \tfrac{15}{10} = \tfrac{3}{2} = 1\tfrac{1}{2}$
3. $\tfrac{2}{3} \times \tfrac{1}{4} = \tfrac{2}{12} = \tfrac{1}{6}$
4. $\tfrac{4}{3} \times \tfrac{3}{2} = \tfrac{12}{6} = 2$
5. $\tfrac{3}{1} \times \tfrac{2}{3} = \tfrac{6}{3} = 2$

Independent practice

6. a) $\tfrac{5}{6} \times \tfrac{3}{2} = \tfrac{15}{12} = \tfrac{5}{4} = 1\tfrac{1}{4}$ b) $\tfrac{7}{10} \times \tfrac{8}{7} = \tfrac{56}{70} = \tfrac{4}{5}$ c) $\tfrac{9}{10} \times \tfrac{5}{3} = \tfrac{45}{30} = \tfrac{3}{2} = 1\tfrac{1}{2}$
7. $\tfrac{9}{4} \times \tfrac{2}{1} = \tfrac{18}{4} = \tfrac{9}{2} = 4\tfrac{1}{2}$
8. $\tfrac{10}{3} \times \tfrac{2}{5} = \tfrac{20}{15} = \tfrac{4}{3} = 1\tfrac{1}{3}$
9. $\tfrac{5}{12} \times \tfrac{6}{5} = \tfrac{30}{60} = \tfrac{1}{2}$
10. $\tfrac{6}{1} \times \tfrac{8}{3} = \tfrac{48}{3} = 16$
11. $7\tfrac{1}{2} \div 1\tfrac{1}{4} = \tfrac{15}{2} \times \tfrac{4}{5} = \tfrac{60}{10} = 6$ full batches.
12. Asking how many $\tfrac{1}{8}$ fit into $\tfrac{3}{4}$ is the same as asking for $\tfrac{3}{4}$ of the 8 eighths in one whole, which is $\tfrac{3}{4} \times 8 = 6$. Multiplying by 8 is multiplying by the reciprocal of $\tfrac{1}{8}$, so the reciprocal rule is just a shortcut for counting how many pieces fit. Check of 6a: $\tfrac{5}{4} \times \tfrac{2}{3} = \tfrac{10}{12} = \tfrac{5}{6}$, which is the original dividend.

Exit ticket 5.4

1. $\tfrac{4}{5} \times \tfrac{3}{2} = \tfrac{12}{10} = \tfrac{6}{5} = 1\tfrac{1}{5}$
2. $\tfrac{7}{2} \times \tfrac{8}{7} = \tfrac{56}{14} = 4$
3. $\tfrac{5}{8} \times \tfrac{1}{5} = \tfrac{5}{40} = \tfrac{1}{8}$
4. Reciprocal of $\tfrac{3}{7}$ is $\tfrac{7}{3}$; of $5$ is $\tfrac{1}{5}$; of $1\tfrac{1}{2} = \tfrac{3}{2}$ is $\tfrac{2}{3}$.

Lesson 5.5 — Multiplying and Dividing by a Number Between 0 and 1

Guided practice

1. Less, since $\tfrac{2}{5}$ is between 0 and 1. $20 \times \tfrac{2}{5} = \tfrac{40}{5} = 8$.
2. More, since the divisor is between 0 and 1. $20 \div \tfrac{2}{5} = \tfrac{20}{1} \times \tfrac{5}{2} = \tfrac{100}{2} = 50$.
3. Less. $\tfrac{3}{4} \times \tfrac{4}{5} = \tfrac{12}{20} = \tfrac{3}{5}$, and $\tfrac{3}{5} = \tfrac{12}{20}$ while $\tfrac{3}{4} = \tfrac{15}{20}$.
4. $6 \div \tfrac{1}{2} = 12$, which is more than 6.
5. $8 \times 1 = 8$. Multiplying by 1 leaves a number unchanged, so 1 is the boundary: factors below 1 take only part of the number and shrink it, and factors above 1 take more than all of it and grow it.

Independent practice

6. a) less b) more c) more d) less
7. a) $10 \times \tfrac{3}{5} = \tfrac{30}{5} = 6$, less than 10 b) $10 \div \tfrac{3}{5} = \tfrac{10}{1} \times \tfrac{5}{3} = \tfrac{50}{3} = 16\tfrac{2}{3}$, more than 10
8. $\tfrac{5}{2} \times \tfrac{4}{5} = \tfrac{20}{10} = 2$, which is less than $2\tfrac{1}{2}$
9. $\tfrac{5}{2} \times \tfrac{5}{4} = \tfrac{25}{8} = 3\tfrac{1}{8}$, which is more than $2\tfrac{1}{2}$
10. Products less than 9: $9 \times \tfrac{1}{3} = 3$ and $9 \times \tfrac{5}{6} = \tfrac{45}{6} = \tfrac{15}{2} = 7\tfrac{1}{2}$. Not less than 9: $9 \times \tfrac{3}{2} = \tfrac{27}{2} = 13\tfrac{1}{2}$, which is greater, and $9 \times 1 = 9$, which is equal.
11. $8 \times \tfrac{3}{4} = 6$ inches; $8 \times \tfrac{5}{4} = 10$ inches. The factor $\tfrac{5}{4}$ enlarges the photo, because $\tfrac{5}{4}$ is greater than 1 — it asks for the whole width plus another quarter of it. The factor $\tfrac{3}{4}$ is between 0 and 1, so it takes only part of the width.
12. Multiplication counterexample: $8 \times \tfrac{1}{2} = 4$, which is smaller than 8. Division counterexample: $8 \div \tfrac{1}{2} = 16$, which is larger than 8. He left out the condition that the second number must be greater than 1. His rules hold only when multiplying or dividing by a number greater than 1.

Exit ticket 5.5

1. Less. $16 \times \tfrac{5}{8} = \tfrac{80}{8} = 10$.
2. More. $16 \div \tfrac{5}{8} = \tfrac{16}{1} \times \tfrac{8}{5} = \tfrac{128}{5} = 25\tfrac{3}{5}$.
3. $\tfrac{3}{4} \times \tfrac{2}{3} = \tfrac{6}{12} = \tfrac{1}{2}$, which is less than $\tfrac{3}{4}$.
4. $\tfrac{5}{8}$ is between 0 and 1, so multiplying by it takes only 5 of the 8 equal parts of the number — less than all of it. $\tfrac{8}{5}$ is greater than 1, so multiplying by it takes 8 of only 5 equal parts — more than all of it. The two factors are reciprocals, so they move a number in opposite directions.

Chapter 5 Review

Part A — Modeling multiplication and division (6.CE.1a)

1. $\tfrac{2}{3} \times \tfrac{1}{2} = \tfrac{2}{6} = \tfrac{1}{3}$. The model has $3 \times 2 = 6$ cells with 2 double-shaded.
2. $\tfrac{3}{4} \div \tfrac{1}{8} = 6$. Model: a strip of $\tfrac{3}{4}$ renamed as $\tfrac{6}{8}$ holds six $\tfrac{1}{8}$ pieces.
3. $4 \times \tfrac{2}{3} = \tfrac{2}{3} + \tfrac{2}{3} + \tfrac{2}{3} + \tfrac{2}{3} = \tfrac{8}{3} = 2\tfrac{2}{3}$
4. $\tfrac{3}{5}$ of $10$: five equal groups of 2, take three groups, so the answer is $6$.

Part B — Multiplying and dividing (6.CE.1b)

5. a) $\tfrac{6}{24} = \tfrac{1}{4}$ b) $\tfrac{15}{90} = \tfrac{1}{6}$ c) $\tfrac{42}{84} = \tfrac{1}{2}$
6. a) $\tfrac{3}{2} \times \tfrac{12}{5} = \tfrac{36}{10} = \tfrac{18}{5} = 3\tfrac{3}{5}$ b) $\tfrac{7}{3} \times \tfrac{3}{2} = \tfrac{21}{6} = \tfrac{7}{2} = 3\tfrac{1}{2}$
7. a) $\tfrac{3}{4} \times \tfrac{8}{3} = \tfrac{24}{12} = 2$ b) $\tfrac{5}{6} \times \tfrac{12}{5} = \tfrac{60}{30} = 2$ c) $\tfrac{2}{5} \times \tfrac{1}{4} = \tfrac{2}{20} = \tfrac{1}{10}$
8. a) $\tfrac{9}{2} \times \tfrac{4}{3} = \tfrac{36}{6} = 6$ b) $\tfrac{10}{3} \times \tfrac{3}{5} = \tfrac{30}{15} = 2$
9. $\tfrac{8}{1} \times \tfrac{3}{2} = \tfrac{24}{2} = 12$

Part C — Multiplying and dividing by a number between 0 and 1 (6.CE.1c)

10. a) Less; $24 \times \tfrac{5}{6} = \tfrac{120}{6} = 20$. b) More; $24 \div \tfrac{5}{6} = \tfrac{24}{1} \times \tfrac{6}{5} = \tfrac{144}{5} = 28\tfrac{4}{5}$.
11. Multiplying by $\tfrac{1}{2}$ means taking half of the number, and half of something is less than all of it. Computing gives $\tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4}$, and $\tfrac{1}{4} < \tfrac{1}{2}$ because fourths are smaller pieces than halves.
12. $\tfrac{7}{8} \times \tfrac{4}{3} = \tfrac{28}{24} = \tfrac{7}{6} = 1\tfrac{1}{6}$, which is greater than $\tfrac{7}{8}$. $\tfrac{7}{8} \div \tfrac{4}{3} = \tfrac{7}{8} \times \tfrac{3}{4} = \tfrac{21}{32}$, which is less, since $\tfrac{7}{8} = \tfrac{28}{32}$. $\tfrac{7}{8} \times \tfrac{2}{3} = \tfrac{14}{24} = \tfrac{7}{12}$, which is less, since $\tfrac{7}{8} = \tfrac{21}{24}$ and $\tfrac{14}{24} < \tfrac{21}{24}$. Only $\tfrac{7}{8} \times \tfrac{4}{3}$ is greater than $\tfrac{7}{8}$, because $\tfrac{4}{3}$ is the only one of the three operations that scales up.

Part D — Mixed application and reasoning

13. $\tfrac{3}{4} \times \tfrac{5}{2} = \tfrac{15}{8} = 1\tfrac{7}{8}$ pounds
14. $\tfrac{21}{4} \div \tfrac{3}{4} = \tfrac{21}{4} \times \tfrac{4}{3} = \tfrac{84}{12} = 7$ pieces
15. They answer different questions. The division asks how many $\tfrac{1}{6}$ pieces fit into $\tfrac{5}{6}$, and five of them fit, so the answer is a count: 5. The multiplication asks for $\tfrac{1}{6}$ of $\tfrac{5}{6}$, which is a small part of an already small amount: $\tfrac{5}{36}$. Dividing by a number between 0 and 1 grows the result, and multiplying by one shrinks it.
16. $\tfrac{3}{4} \div \tfrac{2}{5} = \tfrac{3}{4} \times \tfrac{5}{2} = \tfrac{15}{8} = 1\tfrac{7}{8}$. Check: $\tfrac{15}{8} \times \tfrac{2}{5} = \tfrac{30}{40} = \tfrac{3}{4}$, the original dividend. The check works because multiplication undoes division: if the quotient tells how many copies of the divisor make the dividend, then multiplying the quotient by the divisor must rebuild the dividend.

Workbook-only items

Page 2, fill in the blanks. The number of equal cells equals the denominator times the denominator. The number of double-shaded cells equals the numerator times the numerator.

Page 2, shade and solve.

Page 3, fraction of a set. $\tfrac{3}{4}$ of 8: groups of 2, take 3, answer 6. $\tfrac{1}{3}$ of 9: groups of 3, take 1, answer 3. $\tfrac{2}{3}$ of 12: groups of 4, take 2, answer 8. $\tfrac{3}{4}$ of 16: groups of 4, take 3, answer 12.

Page 3, repeated addition. $3 \times \tfrac{2}{5} = \tfrac{2}{5} + \tfrac{2}{5} + \tfrac{2}{5} = \tfrac{6}{5} = 1\tfrac{1}{5}$ $2 \times \tfrac{3}{8} = \tfrac{3}{8} + \tfrac{3}{8} = \tfrac{6}{8} = \tfrac{3}{4}$ $5 \times \tfrac{2}{3} = \tfrac{2}{3} + \tfrac{2}{3} + \tfrac{2}{3} + \tfrac{2}{3} + \tfrac{2}{3} = \tfrac{10}{3} = 3\tfrac{1}{3}$ $4 \times \tfrac{3}{10} = \tfrac{3}{10} + \tfrac{3}{10} + \tfrac{3}{10} + \tfrac{3}{10} = \tfrac{12}{10} = \tfrac{6}{5} = 1\tfrac{1}{5}$

Page 5, multiply table.

Page 5, convert to improper fractions. $1\tfrac{1}{2} = \tfrac{3}{2}$; $2\tfrac{1}{4} = \tfrac{9}{4}$; $2\tfrac{5}{8} = \tfrac{21}{8}$; $3\tfrac{1}{2} = \tfrac{7}{2}$

Page 6, estimate and compute. Estimates round each factor to the nearest whole number; other reasonable roundings are acceptable.

Page 6, apply. Estimate about $3 \times 1 = 3$ miles; exact $\tfrac{3}{4} \times \tfrac{8}{3} = 2$ miles.

Page 6, predict. Less. $\tfrac{7}{8} \times \tfrac{4}{5} = \tfrac{28}{40} = \tfrac{7}{10}$.

Page 8, count the pieces. Twelve $\tfrac{1}{4}$ fit into 3, so $3 \div \tfrac{1}{4} = 12$. Six, so $2 \div \tfrac{1}{3} = 6$. Eight, so $1 \div \tfrac{1}{8} = 8$. Eight, so $4 \div \tfrac{1}{2} = 8$. Fifteen, so $5 \div \tfrac{1}{3} = 15$.

Page 8, common denominator. $\tfrac{1}{2} \div \tfrac{1}{8} = \tfrac{4}{8} \div \tfrac{1}{8} = 4$. $\tfrac{2}{3} \div \tfrac{1}{6} = \tfrac{4}{6} \div \tfrac{1}{6} = 4$. $\tfrac{3}{4} \div \tfrac{1}{8} = \tfrac{6}{8} \div \tfrac{1}{8} = 6$.

Page 9, complete. Two whole thirds fit, using up $\tfrac{4}{6}$. The leftover is $\tfrac{1}{6}$, which is half of a third. So $\tfrac{5}{6} \div \tfrac{1}{3} = 2\tfrac{1}{2}$.

Page 9, divide. a) $7$ b) $5$ c) $3$ d) $1\tfrac{1}{2}$ e) $4\tfrac{1}{2}$ f) $6$

Page 9, apply. $6$ pieces.

Page 9, explain. Because $\tfrac{1}{4}$ is smaller than one whole, so more than one fourth fits inside each whole. Three wholes hold twelve fourths, and the quotient counts those small pieces.

Page 11, reciprocal table. $\tfrac{3}{4} \to \tfrac{4}{3}$; $\tfrac{1}{5} \to 5$; $7 \to \tfrac{1}{7}$; $1\tfrac{1}{2} \to \tfrac{2}{3}$; $\tfrac{3}{7} \to \tfrac{7}{3}$; $5 \to \tfrac{1}{5}$; $\tfrac{2}{9} \to \tfrac{9}{2}$

Page 11, check one. $\tfrac{3}{4} \times \tfrac{4}{3} = \tfrac{12}{12} = 1$

Page 11, no reciprocal. Zero. No number multiplied by 0 gives 1, since every product with 0 is 0.

Page 11, divide. $\tfrac{3}{4} \div \tfrac{2}{5} = \tfrac{3}{4} \times \tfrac{5}{2} = \tfrac{15}{8} = 1\tfrac{7}{8}$. $\tfrac{5}{8} \div \tfrac{5}{6} = \tfrac{5}{8} \times \tfrac{6}{5} = \tfrac{30}{40} = \tfrac{3}{4}$. $\tfrac{1}{2} \div \tfrac{3}{4} = \tfrac{1}{2} \times \tfrac{4}{3} = \tfrac{4}{6} = \tfrac{2}{3}$. $\tfrac{2}{3} \div 4 = \tfrac{2}{3} \times \tfrac{1}{4} = \tfrac{2}{12} = \tfrac{1}{6}$.

Page 12, divide and check. a) $1\tfrac{1}{2}$ b) $1\tfrac{1}{4}$ c) $\tfrac{4}{5}$ d) $1\tfrac{1}{2}$ e) $\tfrac{1}{2}$ f) $16$ g) $4\tfrac{1}{2}$ h) $1\tfrac{1}{3}$ i) $4$ j) $2$ k) $2$ l) $\tfrac{1}{12}$

Page 12, check by multiplying. Yes. $\tfrac{3}{2} \times \tfrac{2}{5} = \tfrac{6}{10} = \tfrac{3}{5}$, the original dividend.

Page 12, apply. $6$ full batches.

Page 14, rule table. Multiply by a number between 0 and 1: less than $n$. Multiply by exactly 1: equal to $n$. Multiply by a number greater than 1: greater than $n$. Divide by a number between 0 and 1: greater than $n$. Divide by a number greater than 1: less than $n$.

Page 14, predict and compute.

Page 15, sort. Less than 9: $9 \times \tfrac{1}{3} = 3$, $9 \times \tfrac{5}{6} = 7\tfrac{1}{2}$, $9 \div 3 = 3$. Equal to 9: $9 \times 1 = 9$. Greater than 9: $9 \times \tfrac{3}{2} = 13\tfrac{1}{2}$, $9 \div \tfrac{3}{4} = 12$.

Page 15, compute and compare. $2\tfrac{1}{2} \times \tfrac{4}{5} = 2$, which is less than $2\tfrac{1}{2}$. $2\tfrac{1}{2} \div \tfrac{4}{5} = \tfrac{25}{8} = 3\tfrac{1}{8}$, which is more than $2\tfrac{1}{2}$.

Page 15, apply. Scaled by $\tfrac{3}{4}$: $6$ inches. Scaled by $\tfrac{5}{4}$: $10$ inches. The factor $\tfrac{5}{4}$ enlarges the photo, because it is greater than 1 — it keeps the whole width and adds another quarter of it.

Page 15, disprove it. Multiplication: $8 \times \tfrac{1}{2} = 4$, smaller than 8. Division: $8 \div \tfrac{1}{2} = 16$, larger than 8. He left out the condition that the second number must be greater than 1.