Free math worksheets

- 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.

- 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.

- Determine how much water remains before making either comparison. - Pay attention to the different meanings of “how many liters greater” and “how many times as great.” - Use the same two water amounts to answer both comparison questions.

1. The amount left is \(48-8=40\,\text{L}\). 2. The additive difference is \(40-8=32\,\text{L}\). 3. Since \(8 \times 5=40\), the amount left is \(5\) times as great as the amount used.

a) The amount left is \(32\,\text{L}\) greater. b) The amount left is \(5\) times as great.

- 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.

- Ask what operation the equation \(5\times8=40\) represents. - Find the additive difference independently. - Use precise comparison language for each result.

1. The equation \(5\times8=40\) shows a multiplicative comparison, not an additive difference. 2. The additive difference is \(40-8=32\), so \(40\) is \(32\) more than \(8\). 3. Since \(5\times8=40\), \(40\) is \(5\) times as many as \(8\).

Mason confuses “times as many” with “more than.” Correct statements are: \(40\) is \(32\) more than \(8\), and \(40\) is \(5\) times as many as \(8\).

- Translate the addition equation into “more than” language. - Translate the multiplication equation into “times as many” language. - Compare what the numbers \(21\) and \(4\) mean in the two statements.

1. The equation \(7+21=28\) shows that \(28\) is \(21\) more than \(7\). 2. The equation \(4\times7=28\) shows that \(28\) is \(4\) times as many as \(7\). 3. The first relationship measures an additive difference; the second measures a multiplicative factor.

Additive: \(28\) is \(21\) more than \(7\). Multiplicative: \(28\) is \(4\) times as many as \(7\). The first compares by difference, while the second compares by factor.