This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that n × (a/b) = (n × a)/b, recognizing this as a multiple of the unit fraction 1/b (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same. This works because 2/3 is really 2 copies of the unit fraction 1/3, so 5 groups of (2 copies of 1/3) equals (5 × 2) copies of 1/3, which equals 10/3. To multiply 5 × (2/3), we multiply 5 × 2 = 10, and keep the denominator 3, giving 10/3. We can also see this as 5 groups of 2 unit fractions (1/3), which equals 10 unit fractions = 10 × (1/3) = 10/3. Choice A is correct because multiplying: 5 × 2 = 10, keeping denominator 3: 10/3. This demonstrates understanding that multiplying by whole number affects numerator only. Choice B represents multiplying both numerator and denominator, which happens when students incorrectly think both parts multiply by 5. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: 2/3 is '2 copies of 1/3,' so 5 groups of that is (5 × 2) copies of 1/3 = 10 × (1/3) = 10/3. Connect to repeated addition: 5 × (2/3) = 2/3 + 2/3 + 2/3 + 2/3 + 2/3 = 10/3. Convert improper fractions to mixed numbers for interpretation: 10/3 = 3 1/3.

This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that n × (a/b) = (n × a)/b, recognizing this as a multiple of the unit fraction 1/b (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same: 6 × (a/b) = (6 × a)/b. For Keisha's trail mix problem, each batch uses 3/10 cup of raisins, and she makes 6 batches, so we need 6 × (3/10) cups total. To multiply 6 × (3/10), we multiply 6 × 3 = 18, and keep the denominator 10, giving 18/10. The visual shows 6 groups of 3/10, totaling 18 tenths. Choice B is correct because multiplying: 6 × 3 = 18, keeping denominator 10: 18/10. This demonstrates understanding that multiplying by whole number affects numerator only. Choice D represents multiplying both numerator and denominator, which happens when students incorrectly think both parts multiply by 6. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: 3/10 is '3 copies of 1/10,' so 6 groups of that is (6 × 3) copies of 1/10 = 18 × (1/10) = 18/10. Use visuals: draw 6 groups of 3/10 (six sets of 3 shaded tenths), count total tenths: 18 tenths = 18/10. Convert improper fractions to mixed numbers for interpretation: 18/10 = 1 8/10 = 1 4/5 cups.

This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that n × (a/b) = (n × a)/b, recognizing this as a multiple of the unit fraction 1/b (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same: n × (a/b) = (n × a)/b. This works because 5/12 is really 5 copies of the unit fraction 1/12, so 8 groups of (5 copies of 1/12) equals (8 × 5) copies of 1/12, which equals 40/12. To multiply 8 × (5/12), we multiply 8 × 5 = 40, and keep the denominator 12, giving 40/12. We can also see this as 8 groups of 5 unit fractions (1/12), which equals 40 unit fractions = 40 × (1/12) = 40/12. Choice A is correct because multiplying: 8 × 5 = 40, keeping denominator 12: 40/12. This demonstrates understanding that multiplying by whole number affects numerator only. Choice C represents multiplying both numerator and denominator, which happens when students incorrectly think both parts multiply by 8. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: 5/12 is '5 copies of 1/12,' so 8 groups of that is (8 × 5) copies of 1/12 = 40 × (1/12) = 40/12. Use visuals: draw 8 groups of 5/12 (eight sets of 5 shaded twelfths), count total twelfths: 40 twelfths = 40/12. Convert improper fractions to mixed numbers for interpretation: 40/12 = 3 4/12 = 3 1/3.

This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that n × (a/b) = (n × a)/b, recognizing this as a multiple of the unit fraction 1/b (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same: n × (a/b) = (n × a)/b. This works because a/b is really a copies of the unit fraction 1/b, so n groups of (a copies of 1/b) equals (n × a) copies of 1/b, which equals (n × a)/b. For example, 3 × (2/5) means 3 groups of two-fifths, which equals 6 fifths total, or 6/5. To multiply 9 × (2/5), we multiply 9 × 2 = 18, and keep the denominator 5, giving 18/5. We can also see this as 9 groups of 2 unit fractions (1/5), which equals 18 unit fractions = 18 × (1/5) = 18/5. Choice B is correct because multiplying: 9 × 2 = 18, keeping denominator 5: 18/5, using unit fractions: 9 × (2 × (1/5)) = (9 × 2) × (1/5) = 18 × (1/5) = 18/5, or repeated addition: 2/5 added 9 times = 18/5. This demonstrates understanding that multiplying by whole number affects numerator only. Choice A represents adding numerators and denominators to get 11/5, which happens when students use wrong operation or make calculation error. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: a/b is 'a copies of 1/b,' so n groups of that is (n × a) copies of 1/b = (n × a)/b. Use visuals: draw 9 groups of 2/5 (nine sets of 2 shaded fifths), count total fifths: 18 fifths = 18/5. Practice with unit fractions first: 9 × (1/5) = 9/5 (easier to see), then extend: 9 × (2/5) = 9 × [2 × (1/5)] = 18 × (1/5) = 18/5. Connect to repeated addition: 9 × (2/5) = 2/5 added 9 times = 18/5. Convert improper fractions to mixed numbers for interpretation: 18/5 = 3 3/5. Watch for: multiplying denominator (wrong), adding instead of multiplying, arithmetic errors in numerator multiplication, and forgetting that denominator stays the same.

This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that $n \times \left(\frac{a}{b}\right) = \frac{n \times a}{b}$, recognizing this as a multiple of the unit fraction $\frac{1}{b}$ (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same: $n \times \left(\frac{a}{b}\right) = \frac{n \times a}{b}$. This works because $\frac{a}{b}$ is really a copies of the unit fraction $\frac{1}{b}$, so n groups of (a copies of $\frac{1}{b}$) equals (n × a) copies of $\frac{1}{b}$, which equals $\frac{n \times a}{b}$. For example, $3 \times \left(\frac{4}{7}\right)$ means 3 groups of four-sevenths, which equals 12 sevenths total, or $\frac{12}{7}$. To multiply $3 \times \left(\frac{4}{7}\right)$, we multiply 3 × 4 = 12, and keep the denominator 7, giving $\frac{12}{7}$. We can also see this as 3 groups of 4 unit fractions ($\frac{1}{7}$), which equals 12 unit fractions = 12 × ($\frac{1}{7}$) = $\frac{12}{7}$. Choice C is correct because multiplying: 3 × 4 = 12, keeping denominator 7: $\frac{12}{7}$; using unit fractions: $3 \times \left(4 \times \left(\frac{1}{7}\right)\right) = \left(3 \times 4\right) \times \left(\frac{1}{7}\right) = 12 \times \left(\frac{1}{7}\right) = \frac{12}{7}$; repeated addition: $\frac{4}{7} + \frac{4}{7} + \frac{4}{7} = \frac{12}{7}$. This demonstrates understanding that multiplying by whole number affects numerator only. Choice A represents multiplying both numerator and denominator by 3, giving $\frac{12}{21}$, which happens when students incorrectly think both parts multiply by n. To help students: Use formula $n \times \left(\frac{a}{b}\right) = \frac{n \times a}{b}$—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: $\frac{a}{b}$ is 'a copies of $\frac{1}{b}$,' so n groups of that is (n × a) copies of $\frac{1}{b}$ = $\frac{n \times a}{b}$. Use visuals: draw 3 groups of $\frac{4}{7}$ (three sets of 4 shaded sevenths), count total sevenths: 12 sevenths = $\frac{12}{7}$. Practice with unit fractions first: $3 \times \left(\frac{1}{7}\right) = \frac{3}{7}$ (easier to see), then extend: $3 \times \left(\frac{4}{7}\right) = 3 \times \left[4 \times \left(\frac{1}{7}\right)\right] = 12 \times \left(\frac{1}{7}\right) = \frac{12}{7}$. Connect to repeated addition: $3 \times \left(\frac{4}{7}\right) = \frac{4}{7} + \frac{4}{7} + \frac{4}{7} = \frac{12}{7}$. Convert improper fractions to mixed numbers for interpretation: $\frac{12}{7} = 1 \frac{5}{7}$. Watch for: multiplying denominator (wrong), adding instead of multiplying, arithmetic errors in numerator multiplication, and forgetting that denominator stays the same.

This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that n × (a/b) = (n × a)/b, recognizing this as a multiple of the unit fraction 1/b (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same. This works because 4/7 is really 4 copies of the unit fraction 1/7, so 9 groups of (4 copies of 1/7) equals (9 × 4) copies of 1/7, which equals 36/7. To multiply 9 × (4/7), we multiply 9 × 4 = 36, and keep the denominator 7, giving 36/7. We can also see this as 9 groups of 4 unit fractions (1/7), which equals 36 unit fractions = 36 × (1/7) = 36/7. Choice C is correct because multiplying: 9 × 4 = 36, keeping denominator 7: 36/7. This demonstrates understanding that multiplying by whole number affects numerator only. Choice B represents multiplying both numerator and denominator, which happens when students incorrectly think both parts multiply by 9. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: 4/7 is '4 copies of 1/7,' so 9 groups of that is (9 × 4) copies of 1/7 = 36 × (1/7) = 36/7. Practice with unit fractions first: 9 × (1/7) = 9/7 (easier to see), then extend: 9 × (4/7) = 9 × [4 × (1/7)] = 36 × (1/7) = 36/7. Convert improper fractions to mixed numbers for interpretation: 36/7 = 5 1/7.

This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that n × (a/b) = (n × a)/b, recognizing this as a multiple of the unit fraction 1/b (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same. Since we're multiplying 7 × (1/4), and the numerator is already 1, we get 7 × 1 = 7 in the numerator, keeping denominator 4. To multiply 7 × (1/4), we multiply 7 × 1 = 7, and keep the denominator 4, giving 7/4. We can also see this as 7 groups of 1 unit fraction (1/4), which equals 7 unit fractions = 7 × (1/4) = 7/4. Choice A is correct because multiplying: 7 × 1 = 7, keeping denominator 4: 7/4. This demonstrates understanding that multiplying by whole number affects numerator only. Choice C represents multiplying both numerator and denominator, which happens when students incorrectly think both parts multiply by 7. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: 1/4 is '1 copy of 1/4,' so 7 groups of that is 7 copies of 1/4 = 7 × (1/4) = 7/4. Practice with unit fractions first: 7 × (1/4) = 7/4 shows the pattern clearly. Connect to repeated addition: 7 × (1/4) = 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 7/4. Convert improper fractions to mixed numbers for interpretation: 7/4 = 1 3/4.

This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that n × (a/b) = (n × a)/b, recognizing this as a multiple of the unit fraction 1/b (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same. For Carlos's juice problem, each serving is 2/11 liter, and he pours 4 servings, so we need 4 × (2/11) liters total. To multiply 4 × (2/11), we multiply 4 × 2 = 8, and keep the denominator 11, giving 8/11. We can also see this as 4 groups of 2 unit fractions (1/11), which equals 8 unit fractions = 8 × (1/11) = 8/11. Choice A is correct because multiplying: 4 × 2 = 8, keeping denominator 11: 8/11. This demonstrates understanding that multiplying by whole number affects numerator only. Choice C represents multiplying both numerator and denominator, which happens when students incorrectly think both parts multiply by 4. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: 2/11 is '2 copies of 1/11,' so 4 groups of that is (4 × 2) copies of 1/11 = 8 × (1/11) = 8/11. Connect to repeated addition: 4 × (2/11) = 2/11 + 2/11 + 2/11 + 2/11 = 8/11. Watch for: multiplying denominator (wrong), adding instead of multiplying, arithmetic errors in numerator multiplication.

This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that n × (a/b) = (n × a)/b, recognizing this as a multiple of the unit fraction 1/b (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same: n × (a/b) = (n × a)/b. This works because a/b is really a copies of the unit fraction 1/b, so n groups of (a copies of 1/b) equals (n × a) copies of 1/b, which equals (n × a)/b. For example, 3 × (2/5) means 3 groups of two-fifths, which equals 6 fifths total, or 6/5. To multiply 3 × (2/5), we multiply 3 × 2 = 6, and keep the denominator 5, giving 6/5. We can also see this as 3 groups of 2 unit fractions (1/5), which equals 6 unit fractions = 6 × (1/5) = 6/5. Choice B is correct because multiplying: 3 × 2 = 6, keeping denominator 5: 6/5; using unit fractions: 3 × (2 × (1/5)) = (3 × 2) × (1/5) = 6 × (1/5) = 6/5; repeated addition: 2/5 + 2/5 + 2/5 = 6/5. This demonstrates understanding that multiplying by whole number affects numerator only. Choice A represents multiplying both numerator and denominator by 3, giving 6/15, which happens when students incorrectly think both parts multiply by n. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: a/b is 'a copies of 1/b,' so n groups of that is (n × a) copies of 1/b = (n × a)/b. Use visuals: draw 3 groups of 2/5 (three sets of 2 shaded fifths), count total fifths: 6 fifths = 6/5. Practice with unit fractions first: 3 × (1/5) = 3/5 (easier to see), then extend: 3 × (2/5) = 3 × [2 × (1/5)] = 6 × (1/5) = 6/5. Connect to repeated addition: 3 × (2/5) = 2/5 + 2/5 + 2/5 = 6/5. Convert improper fractions to mixed numbers for interpretation: 6/5 = 1 1/5. Watch for: multiplying denominator (wrong), adding instead of multiplying, arithmetic errors in numerator multiplication, and forgetting that denominator stays the same.

This question tests 4th grade understanding of multiplying a fraction by a whole number using the principle that n × (a/b) = (n × a)/b, recognizing this as a multiple of the unit fraction 1/b (CCSS.4.NF.4.b). To multiply a whole number by a fraction, we multiply the whole number by the numerator and keep the denominator the same: n × (a/b) = (n × a)/b. This works because a/b is really a copies of the unit fraction 1/b, so n groups of (a copies of 1/b) equals (n × a) copies of 1/b, which equals (n × a)/b. For example, 9 × (5/11) means 9 groups of five-elevenths, which equals 45 elevenths total, or 45/11. To multiply 9 × (5/11), we multiply 9 × 5 = 45, and keep the denominator 11, giving 45/11. We can also see this as 9 groups of 5 unit fractions (1/11), which equals 45 unit fractions = 45 × (1/11) = 45/11. Choice D is correct because multiplying: 9 × 5 = 45, keeping denominator 11: 45/11; using unit fractions: 9 × (5 × (1/11)) = (9 × 5) × (1/11) = 45 × (1/11) = 45/11; repeated addition: 5/11 + 5/11 + ... (9 times) = 45/11. This demonstrates understanding that multiplying by whole number affects numerator only. Choice A represents multiplying both numerator and denominator by 9, giving 45/99, which happens when students incorrectly think both parts multiply by n. To help students: Use formula n × (a/b) = (n × a)/b—multiply the NUMERATOR by whole number, KEEP DENOMINATOR same. Show why: a/b is 'a copies of 1/b,' so n groups of that is (n × a) copies of 1/b = (n × a)/b. Use visuals: draw 9 groups of 5/11 (nine sets of 5 shaded elevenths), count total elevenths: 45 elevenths = 45/11. Practice with unit fractions first: 9 × (1/11) = 9/11 (easier to see), then extend: 9 × (5/11) = 9 × [5 × (1/11)] = 45 × (1/11) = 45/11. Connect to repeated addition: 9 × (5/11) = 5/11 + 5/11 + ... (9 times) = 45/11. Convert improper fractions to mixed numbers for interpretation: 45/11 = 4 1/11. Watch for: multiplying denominator (wrong), adding instead of multiplying, arithmetic errors in numerator multiplication, and forgetting that denominator stays the same.