- Which group is larger? Think about what the child's sentence claims. - Could you build one group by making equal copies of the other?

1. Group a) has \(4\) dots and b) has \(12\). 2. Three groups of \(4\) make \(12\), so \(4 \times 3 = 12\). The statement has the groups reversed.

No. b) has \(3\) times as many dots as a), since \(4 \times 3 = 12\).

- Use the number of red pencils as one group. - How many equal groups of red pencils make the blue-pencil count? - Check the factor using multiplication.

1. Three equal groups of \(3\) make the \(9\) blue pencils: \(3\times3=9\). 2. There are three times as many blue pencils as red pencils.

There are \(3\) times as many blue pencils as red pencils; \(3\times3=9\).

- The same two quantities can be compared in two different ways. - One comparison asks for a difference; the other asks for a factor. - Label each result by the kind of comparison it represents.

1. The additive difference is \(240{,}000-80{,}000=160{,}000\). 2. Since \(80{,}000 \times 3=240{,}000\), River County has \(3\) times as many residents. 3. Part a is additive comparison; part b is multiplicative comparison.

a) \(160{,}000\) more residents. b) \(3\) times as many residents. c) Part a is additive; part b is multiplicative.

- For a), think about the operation used to find how many more. - For b), determine how many groups of the smaller amount fit in the larger amount. - Notice the difference between “how many more” and “how many times as many.”

1. For the additive comparison, subtract: \(35 - 7 = 28\). 2. For the multiplicative comparison, divide: \(35 \div 7 = 5\).

a) The bakery made \(28\) more pretzels. b) The bakery made \(5\) times as many pretzels as croissants.

- Pay attention to the difference between “times as large” and “greater than.” - Decide whether each part calls for addition or multiplication. - Rewrite each statement as a numerical expression.

1. For part a, multiply: \(12 \times 9 = 108\). 2. For part b, add: \(12 + 9 = 21\). 3. For part c, add: \(40 + 15 = 55\). 4. For part d, multiply: \(40 \times 4 = 160\).

a) \(108\) b) \(21\) c) \(55\) d) \(160\)

- Each pair must be compared in two different ways. - Keep the meaning of “how much greater” separate from “how many times as large.” - Check each multiplicative comparison with a multiplication fact.

1. For \(48\) and \(6\), the additive difference is \(48-6=42\). Since \(6 \times 8=48\), \(48\) is \(8\) times as large as \(6\). 2. For \(9\) and \(45\), the additive difference is \(45-9=36\). Since \(9 \times 5=45\), \(45\) is \(5\) times as large as \(9\). 3. For \(7\) and \(42\), the additive difference is \(42-7=35\). Since \(7 \times 6=42\), \(42\) is \(6\) times as large as \(7\).

\(48\) and \(6\): \(42\) greater; \(8\) times as large \(9\) and \(45\): \(36\) greater; \(5\) times as large \(7\) and \(42\): \(35\) greater; \(6\) times as large

- For part a), determine how many times the shorter length fits into the longer length. - For part b), find the difference between the two lengths. - Which operation finds how much longer one object is than another?

1. Compare the lengths multiplicatively: \(240 \div 40 = 6\). 2. Compare the lengths additively: \(240\,\text{cm} - 40\,\text{cm} = 200\,\text{cm}\).

a) The blue rope is \(6\) times as long as the red rope. b) The blue rope is \(200\,\text{cm}\) longer than the red rope.

- For a), use the operation that finds the difference between two numbers. - For b), think about how many equal groups of \(75\) make \(300\). - Doubling means multiplying by \(2\).

1. For a), find the additive difference: \(300 - 75 = 225\). 2. For b), find the multiplicative comparison: \(300 \div 75 = 4\). 3. For c), double \(75\): \(75 \times 2 = 150\). Then \(300 \div 150 = 2\).

a) \(300\) is \(225\) greater than \(75\). b) \(75\) fits into \(300\) four times. c) The new number is \(150\), and it fits into \(300\) twice.

- Read “increase by” and “times” as two different relationships rather than as interchangeable wording. - Calculate each instruction independently before comparing the results. - The final question asks for how much greater one result is than the other.

1. Increasing \(8\) by \(4\) gives \(8+4=12\). 2. Four times \(8\) is \(8 \times 4=32\). 3. The larger result is \(32-12=20\) greater than the smaller result.

a) \(12\) b) \(32\) The larger result is \(20\) greater than the smaller result.

- One comparison should describe how much greater one capacity is. - The other should describe how many equal copies of the smaller capacity make the larger capacity. - Make sure your two comparison sentences express different kinds of relationships.

1. The additive difference is \(360-9=351\), so the tank holds \(351\,\text{L}\) more than the watering can. 2. For the multiplicative comparison, find the factor that changes \(9\) to \(360\): \(9\times40=360\). Thus the tank holds \(40\) times as much as the watering can. 3. The additive comparison describes a difference of \(351\,\text{L}\); the multiplicative comparison describes a factor of \(40\).

a) \(360-9=351\). The tank holds \(351\,\text{L}\) more than the watering can. b) \(9\times40=360\). The tank holds \(40\) times as much as the watering can. c) The additive comparison gives the difference between the capacities, while the multiplicative comparison gives the factor relating them.

- Read both quantities from the chart before comparing them. - One comparison asks for a difference; the other asks for a factor. - Use different operations for the two meanings.

1. Bed A has \(8\) seedlings and Bed B has \(32\). 2. The additive difference is \(32-8=24\). 3. Since \(8\times4=32\), Bed B has \(4\) times as many seedlings as Bed A.

a) \(24\) more seedlings b) \(4\) times as many c) Bed B has \(24\) more seedlings than Bed A. Bed B has \(4\) times as many seedlings as Bed A.

- Compare the two actual counts first. - Then work out what three full copies of Mila's amount would be. - Does Jay's count match that total?

1. Jay has \(11 - 8 = 3\) more crayons than Mila. 2. Three times Mila's amount is \(3 \times 8 = 24\), not \(11\). 3. Jay is not right.

No. Jay has \(3\) more crayons. Three times Mila's amount would be \(24\) crayons.

- Use Ava's count as the reference in both questions. - What does a difference measure? What does a factor measure? - Check both answers against the given counts.

1. Cora has \(11 - 6 = 5\) more buttons than Ava. 2. Ben has \(24 \div 6 = 4\) times as many buttons as Ava.

Cora has \(5\) more buttons than Ava: \(11-6=5\). Ben has \(4\) times as many as Ava: \(24\div6=4\).

- Check Nico's first claim using the two counts. - Can the number of extra marbles also be the number of equal groups in this case? - Test the factor you find with multiplication.

1. Nico has \(15 - 5 = 10\) more marbles, so the first part is correct. 2. Nico has \(15 \div 5 = 3\) times as many marbles as Maya. The multiplication equation is \(5 \times 3 = 15\), not \(5 \times 10 = 15\).

“\(10\) more” is right: \(15 - 5 = 10\). “\(10\) times as many” is wrong: \(5 \times 3 = 15\), so Nico has \(3\) times as many.

- Which tray is the starting amount for the four-times comparison? - Find the third tray's count before comparing it with the first. - Does the last question ask for a total or a difference?

1. Four times the second tray's \(5\) cookies is \(4 \times 5 = 20\). 2. The difference from the first tray is \(20 - 9 = 11\).

\(20\) cookies; \(11\) more than on the first tray.

- Both comparisons use Omar's number as the starting amount. - Check whether the sister and cousin should have the same count. - Use two different drawings if the comparisons seem alike.

1. His sister has \(8 + 3 = 11\) postcards. 2. His cousin has \(8 \times 3 = 24\) postcards. 3. Adding three extra postcards and making three equal groups of eight describe different comparisons.

His sister has \(11\) postcards, and his cousin has \(24\). “More” adds \(3\); “times as many” multiplies by \(3\).

- Think about what the \(12\) counts in the first statement. - What does the \(3\) count in the other statement?

1. The additive equation is \(6 + 12 = 18\). 2. The multiplicative equation is \(6 \times 3 = 18\). 3. The results match, but the meanings differ: \(12\) is an extra amount, while \(3\) is a number of equal groups.

No. \(6 + 12 = 18\) compares by a difference, while \(6 \times 3 = 18\) compares by a factor.

- “More than” tells you the difference, not the smaller total. - Work backward from the larger number to find the smaller count. - Then compare the two full counts by equal groups.

1. The second club made \(30 - 24 = 6\) bracelets. 2. Compare the totals: \(30 \div 6 = 5\).

The second club made \(6\) bracelets. The first club made \(5\) times as many.

- Read the counts for all three children. - Test the differences from Ava's count for both other children. - Then check whether either count makes three equal groups of Ava's count.

1. Ava has \(4\) beads and Ben has \(7\), so Ben has \(7 - 4 = 3\) more. 2. Cora has \(12\), and \(4 \times 3 = 12\), so Cora has three times as many as Ava.

Ben has \(3\) more: \(7 - 4 = 3\). Cora has \(3\) times as many as Ava: \(4 \times 3 = 12\).

- Having four times the current amount is about the final total. - The team already owns some balls, so it does not need to buy the whole target. - “Four more” asks only for an extra amount.

1. Four times the current count is \(9 \times 4 = 36\) balls. 2. To reach that total, the team must buy \(36 - 9 = 27\) balls. 3. Wanting only four more means buying \(4\) balls, not \(27\).

To have four times as many, buy \(27\) balls. To have four more, buy \(4\) balls.

- Which person's number is already known? - Find the smaller number that could be repeated three times to make Tala's number. - Compare the two actual sticker counts to check Ben's statement.

1. Ben has \(24 \div 3 = 8\) stickers. 2. The difference is \(24 - 8 = 16\) stickers. 3. Ben is not correct: three times as many gives \(16\) more here, not \(3\) more.

No. Ben has \(8\) stickers, and Tala has \(16\) more.

- Count the dots in each picture. - Find the extra dots before thinking about equal groups. - Check whether your two answers describe different ways of comparing the same pictures.

1. There are \(4\) dots in a) and \(16\) dots in b). 2. The difference is \(16-4=12\) dots. 3. The factor is \(16\div4=4\).

b) has \(12\) more dots and \(4\) times as many dots as a): \(16-4=12\), \(16\div4=4\).

- Test each statement from the numbers rather than from its wording alone. - One equation should show a difference. - The other should show one amount as equal copies of the other.

1. \(600{,}000-300{,}000=300{,}000\), so Ava’s statement is true and is additive. 2. \(300{,}000 \times 2=600{,}000\), so Noah’s statement is true and is multiplicative.

Both statements are true. Ava: \(600{,}000-300{,}000=300{,}000\), additive comparison. Noah: \(300{,}000 \times 2=600{,}000\), multiplicative comparison.

- Use the same pair of visit counts in both comparisons. - Ask what a difference measures and what a factor measures. - Your explanation should distinguish the meanings of the two numerical answers.

1. The additive difference is \(750{,}000-250{,}000=500{,}000\). 2. Since \(250{,}000 \times 3=750{,}000\), the larger count is \(3\) times the smaller count. 3. The first result tells how much larger the count is; the second tells the multiplicative factor.

a) The larger count is \(500{,}000\) more. b) The larger count is \(3\) times as many. c) \(500{,}000\) is an additive difference; \(3\) is a multiplicative factor.

- First find how many blue balloons there are. - Read carefully: \(40\) is the difference, not the number of blue balloons. - Once you know both quantities, determine how many groups of the smaller quantity make the larger quantity.

1. Find the number of blue balloons: \(48 - 40 = 8\). 2. Compare the quantities multiplicatively: \(48 \div 8 = 6\).

a) She buys \(8\) blue balloons. b) She buys \(6\) times as many red balloons as blue balloons.

- Find the remaining rope length before making either comparison. - Parts b) and c) ask two different kinds of comparison about the same two pieces. - Check that each answer matches the wording of its comparison question.

1. The remaining length is \(35-7=28\,\text{m}\). 2. The difference between the pieces is \(28-7=21\,\text{m}\). 3. Since \(7 \times 4=28\), the remaining piece is \(4\) times as long as the piece that was cut off.

a) \(28\,\text{m}\) b) \(21\,\text{m}\) longer c) \(4\) times as long

- Part a must ask about an additive difference. - Part b must ask about a multiplicative factor. - Use the same two book counts in both questions, and do not ask for a given number.

1. One additive comparison question is, “How many more books are there now than originally?” The answer is \(72-8=64\) books. 2. One multiplicative comparison question is, “How many times as many books are there now as there were originally?” Since \(8 \times 9=72\), there are \(9\) times as many books now.

a) One valid question is, “How many more books are there now than originally?” Answer: \(64\) more books. b) One valid question is, “How many times as many books are there now as originally?” Answer: \(9\) times as many.

- Find how much water remains before making either comparison. - Part a) asks for a difference in liters. - Part b) asks how many copies of the water Anna used would equal the amount left.

1. The amount left is \(48 - 8 = 40\,\text{L}\). 2. The difference is \(40 - 8 = 32\,\text{L}\). 3. The comparison factor is \(40 \div 8 = 5\).

a) \(32\,\text{L}\) more. b) \(5\) times as much.

- Your two questions must ask about different mathematical relationships between the same numbers. - Label the relationship before calculating. - Check that one answer is a difference and the other is a factor.

1. One additive comparison question is, “By how many cards does Elias’s collection exceed Mia’s?” The calculation is \(32-8=24\). 2. One multiplicative comparison question is, “Mia’s collection would have to be copied how many times to equal Elias’s collection?” Since \(8 \times 4=32\), the factor is \(4\).

Additive: One valid question is, “By how many cards does Elias’s collection exceed Mia’s?” Answer: \(24\) cards. Multiplicative: One valid question is, “Mia’s collection would have to be copied how many times to equal Elias’s collection?” Answer: \(4\) times.

- Use two different operations for the two statements. - Which operation finds a difference? - Which operation finds how many times one amount fits into another? - Check each statement with its own calculation.

1. Subtraction finds the additive difference: \(15 - 3 = 12\,\text{L}\). The bucket holds \(12\,\text{L}\) more. 2. Division finds the multiplicative comparison: \(15 \div 3 = 5\). The bucket holds \(5\) times as much.

Statement A: \(15 - 3 = 12\), which finds the difference in liters. Statement B: \(15 \div 3 = 5\), which finds how many times as much the bucket holds.

- Parts a) and b) ask for two different kinds of comparison between the same numbers. - For part c), change only the number specified in the claim and then compare the new result with your answer to part a). - Check the claim numerically rather than deciding from the wording alone.

1. The original difference is \(360-120=240\). 2. Since \(120 \times 3=360\), \(360\) is \(3\) times as great as \(120\). 3. Doubling the smaller number gives \(120 \times 2=240\). 4. The new difference is \(360-240=120\). 5. Since \(120\) is half of \(240\), the claim is true.

a) \(240\) b) \(360\) is \(3\) times as great as \(120\). c) The claim is true. After \(120\) is doubled to \(240\), the new difference is \(120\), which is half of \(240\).

- Distinguish between “three times as many” and “\(300\) more.” - Find each girl's number of stickers. - Compare the two results.

1. Find Sarah's number of stickers: \(120 \times 3 = 360\). 2. Find Mia's number of stickers: \(120 + 300 = 420\). 3. Since \(420 > 360\), Mia has more stickers.

Mia has more stickers. She has \(420\), while Sarah has \(360\).

- Put both heights in the same unit before comparing them. - Notice that the two questions ask for different kinds of comparison between the same two heights. - Check that each result matches the wording “centimeters taller” or “times as tall.”

1. Convert the toy chest’s height: \(480\,\text{mm}=48\,\text{cm}\). 2. The additive difference is \(48-8=40\,\text{cm}\). 3. Since \(8 \times 6=48\), the toy chest is \(6\) times as tall as the block.

1) The toy chest is \(40\,\text{cm}\) taller. 2) The toy chest is \(6\) times as tall.

- Distinguish statements that compare by a difference from statements that compare by a factor. - Check each statement against the values shown in the graph. - A statement is ruled out if its stated comparison does not match the two quantities.

1. A is true because \(40 - 25 = 15\). 2. B is false because half of \(40\) is \(20\), not \(30\). 3. C is true because \(35 - 30 = 5\).

Statement B cannot be true.

- What kind of operation does Mason's equation show? - Find the difference using the two given numbers. - Describe the multiplication fact with precise comparison words.

1. The equation \(5 \times 8 = 40\) shows a multiplication factor, not an additive difference. 2. The difference is \(40 - 8 = 32\), so \(40\) is \(32\) more than \(8\). 3. Since \(5 \times 8 = 40\), \(40\) is \(5\) times as large as \(8\).

Mason mixed up “more than” and “times as large.” Correct statements: \(40\) is \(32\) more than \(8\), and \(40\) is \(5\) times as large as \(8\).

- Think about what the addition equation tells you has been added. - Think about how many copies of the smaller number the multiplication equation shows. - Use different comparison phrases for the difference and the factor.

1. The addition equation shows that \(28\) is \(21\) more than \(7\). 2. The multiplication equation shows that \(28\) is \(4\) times as large as \(7\). 3. The first compares by a difference; the second compares by a factor.

Additive: \(28\) is \(21\) more than \(7\). Multiplicative: \(28\) is \(4\) times as large as \(7\). The first compares differences; the second compares equal groups.