Free math worksheets

- Braces are grouping symbols, just like parentheses or brackets. - Evaluate the quantity inside the braces before using the outside operation. - Check that your final step uses the division shown in the expression.

1. Evaluate the grouped sum first: \(24+12=36\). 2. Divide the result by \(6\): \(36\div6=6\).

\(6\)

- Work from the innermost grouping symbols outward. - Keep track of each subtraction operation. - Evaluate one grouped expression at a time.

1. Evaluate the inner differences: \(291 - 144 = 147\) and \(123 - 95 = 28\). 2. Evaluate the bracketed expression: \(147 - 28 = 119\). 3. Complete the subtraction: \(672 - 119 = 553\).

\(553\)

- Treat everything inside the parentheses as one quantity before using the outside multiplication. - Inside the parentheses, follow the order of operations. - Check that the multiplication by \(6\) happens only after the grouped value is known.

1. Inside the parentheses, evaluate the multiplication first: \(3\times5=15\). 2. Complete the grouped difference: \(70-15=55\). 3. Multiply by \(6\): \(6\times55=330\).

\(330\)

- Compare what quantity is divided by \(2\) in the two expressions. - Evaluate the parenthesized expression first in the second row. - Use the differing table values to explain the structural effect of the parentheses.

1. For \(n=50\), \((50+40)\div2=90\div2=45\), while \(50+40\div2=50+20=70\). 2. For \(n=80\), \((80+40)\div2=120\div2=60\), while \(80+40\div2=80+20=100\). 3. For \(n=120\), \((120+40)\div2=160\div2=80\), while \(120+40\div2=120+20=140\). 4. The parentheses change which quantity is divided by \(2\): with parentheses, the entire sum is divided; without them, only \(40\) is divided before the addition.

<table> <tr><td><strong>Starting value \(n\)</strong></td><td>\(50\)</td><td>\(80\)</td><td>\(120\)</td></tr> <tr><td><strong>\((n+40)\div2\)</strong></td><td>\(45\)</td><td>\(60\)</td><td>\(80\)</td></tr> <tr><td><strong>\(n+40\div2\)</strong></td><td>\(70\)</td><td>\(100\)</td><td>\(140\)</td></tr> </table> The parentheses make the entire sum \(n+40\) the dividend. Without the parentheses, the division is performed before the addition.

- Evaluate inside the grouping symbols first. - Rewrite the entire expression after each step. - When only addition and subtraction remain, work from left to right.

1. Evaluate inside the grouping symbols: \(56 + 79 = 135\). 2. Substitute: \(248 - 135 + 42\). 3. Work from left to right: \(113 + 42 = 155\).

\(155\)

- Evaluate inside the grouping symbols first. - Rewrite the expression using the grouped value. - Complete the remaining subtraction.

1. Evaluate inside the grouping symbols: \(367 + 233 = 600\). 2. Subtract: \(1245 - 600 = 645\).

\(645\)

- Evaluate the grouped expression completely before the outside subtraction. - With only subtraction inside one group, work from left to right. - Keep the outside \(456\) unchanged until the grouped value is known.

1. Inside the grouping symbols, work from left to right: \(321-105=216\). 2. Continue inside the group: \(216-66=150\). 3. Subtract the grouped value: \(456-150=306\).

\(306\)

- Evaluate each grouped sum or difference before dividing. - Find the value on each side of the comparison symbol separately. - After evaluating both sides, decide which comparison symbol describes their relationship.

1. For part a), the left side is \(40\div4=10\), and the right side is \(10\), so the values are equal. 2. For part b), the left side is \(36\div9=4\), and the right side is \(6\), so \(4<6\). 3. For part c), the left side is \(35\div5=7\), and the right side is \(6\), so \(7>6\).

a) \(=\) b) \(<\) c) \(>\)

- In every part, identify the grouped quantity before applying the outside operation. - After simplifying a group, return to the remaining operations in their correct order. - Check that ignoring the parentheses would not lead you to the same operation sequence.

1. For part a), evaluate the grouped quantities first: \(24+3=27\) and \(12-4=8\). Then \(27\times8=216\). 2. For part b), \(6+1=7\), so \(42\div7+5=6+5=11\). 3. For part c), \(64-40=24\), so \(24\div8=3\). 4. For part d), \(9-6=3\), so \(81\div3\times2=27\times2=54\).

a) \(216\) b) \(11\) c) \(3\) d) \(54\)

- Evaluate each grouped subtraction first. - Decide which operation connects that grouped value to the target on the right. - Check each completed equation after choosing the operation.

1. For part a), \(82-74=8\), and \(8\times6=48\), so the missing operation is \(\times\). 2. For part b), \(100-64=36\), and \(36\div4=9\), so the missing operation is \(\div\). 3. For part c), \(53-44=9\), and \(9\times9=81\), so the missing operation is \(\times\). 4. For part d), \(95-67=28\), and \(28\div7=4\), so the missing operation is \(\div\).

a) \(\times\) b) \(\div\) c) \(\times\) d) \(\div\)

- Evaluate inside the grouping symbols first. - Decide whether the large first factor must actually be multiplied. - Recall the zero property of multiplication.

1. Evaluate inside the grouping symbols: \(25 \times 4 - 100 = 100 - 100 = 0\). 2. Any number multiplied by \(0\) equals \(0\), so the expression has value \(0\).

\(0\)

- Apply the order of operations to \(A\). - Notice how the grouping symbols change \(B\). - Subtract the lesser value from the greater value.

1. For \(A\), multiply before adding: \(15 + 5 \times 4 = 15 + 20 = 35\). 2. For \(B\), evaluate the grouping symbols first: \((15 + 5) \times 4 = 20 \times 4 = 80\). 3. The positive difference is \(80 - 35 = 45\).

\(A = 35\), \(B = 80\), and the positive difference is \(45\).

- Compare the number inside each grouping symbol with numbers that are easy to multiply mentally. - Consider whether evaluating the grouped quantity creates a convenient factor. - Also consider whether distributing creates easier products before choosing a strategy.

1. For part a), evaluating inside the grouping symbols first gives \((15+5)\times24=20\times24=480\). 2. For part b), distributing is efficient because \(7\times100\) and \(7\times3\) are easy to compute: \(7\times(100-3)=700-21=679\). 3. For part c), evaluating inside the grouping symbols first gives \((125-25)\times9=100\times9=900\).

a) Evaluate inside the grouping symbols first: \(20\times24=480\). b) Use the distributive property: \(700-21=679\). c) Evaluate inside the grouping symbols first: \(100\times9=900\).

- Evaluate multiplication before addition or subtraction. - Use the multiplication properties of \(0\) and \(1\). - In part e, notice the factor outside the grouping symbols.

1. For a), \(0\times13+7=0+7=7\). 2. For b), \(25-0\times4=25-0=25\). 3. For c), \(6\times0+8\times1=0+8=8\). 4. For d), \(1\times9-9\times0=9-0=9\). 5. For e), any product with factor \(0\) is \(0\), so the value is \(0\).

a) \(7\) b) \(25\) c) \(8\) d) \(9\) e) \(0\)

- Identify which numbers are combined in the upper branch. - Use grouping symbols so that addition occurs before subtraction. - The root operation is performed last.

1. The expression is \(150 - (45 + 35)\). 2. Evaluate the grouped sum: \(45 + 35 = 80\). 3. Subtract: \(150 - 80 = 70\).

Expression: \(150 - (45 + 35)\) Value: \(70\)

- Follow the expression tree from the top toward the root. - The sum is used as one complete factor in the expression. - Use grouping symbols because the addition occurs before multiplication.

1. The intermediate value is \(18 + 7 = 25\). 2. The corresponding expression is \((18 + 7) \times 4\). 3. Evaluate the expression: \(25 \times 4 = 100\).

Intermediate value: \(25\) Expression: \((18 + 7) \times 4\) Final value: \(100\)

- Follow one level of the tree at a time. - Treat the subtraction result as the complete dividend in the next operation. - Use grouping symbols to show that the subtraction occurs before division.

1. The intermediate value is \(32 - 14 = 18\). 2. The corresponding expression is \((32 - 14) \div 3\). 3. Evaluate the expression: \(18 \div 3 = 6\).

Intermediate value: \(18\) Expression: \((32 - 14) \div 3\) Final value: \(6\)

- Identify the multiplication branch first. - Add \(26\) to that intermediate value. - Use the tree to check the order of operations.

1. Multiply: \(9\times6=54\). 2. Add: \(54+26=80\). 3. The expression is \(9\times6+26\).

Intermediate value: \(54\) Expression: \(9\times6+26\) Final value: \(80\)

- Evaluate the product on the right branch first. - Subtract that product from \(60\). - The order of operations already matches the tree.

1. Multiply: \(8\times4=32\). 2. Subtract: \(60-32=28\). 3. The expression is \(60-8\times4\).

Intermediate value: \(32\) Expression: \(60-8\times4\) Final value: \(28\)

- Identify the complete divisor in the tree. - Evaluate the difference before dividing. - Use grouping symbols around the divisor.

1. Evaluate the difference: \(25 - 5 = 20\). 2. Divide: \(100 \div 20 = 5\). 3. The expression is \(100 \div (25 - 5)\).

Expression: \(100 \div (25 - 5)\) Final value: \(5\)

- Identify the complete dividend shown by the left branch. - Write the expression before calculating the branch values. - Use grouping symbols around the sum because it is divided as one quantity.

1. The tree represents \((25 + 35) \div 5\). 2. Evaluate the grouped sum: \(25 + 35 = 60\). 3. Divide: \(60 \div 5 = 12\).

Expression: \((25 + 35) \div 5\) Final value: \(12\)

- Identify the operation performed on the left branch before the final subtraction. - Compare the tree order with the usual order of operations before deciding whether grouping symbols are necessary. - After writing the expression, evaluate multiplication before subtraction.

1. The tree is represented by \(12 \times 8 - 46\). Grouping symbols around \(12 \times 8\) are not needed because multiplication is already performed before subtraction. 2. Evaluate the product: \(12 \times 8 = 96\). 3. Subtract: \(96 - 46 = 50\).

Expression: \(12 \times 8 - 46\) Final value: \(50\)

- Identify the operation at each branch of the tree. - Work from the starting numbers toward the final result. - Match each number in the word problem with its position in the tree.

1. The juice costs \(4\times\$14=\$56\). 2. The pretzels cost \(3\times\$8=\$24\). 3. The pretzels and decorations cost \(\$24+\$6=\$30\). 4. The total cost is \(\$56+\$30=\$86\).

a) The intermediate values are \(56\), \(24\), and \(30\). b) The total cost is \(\$86\).

- Evaluate every expression inside parentheses first. - Perform each multiplication before adding. - Keep the two expressions in each pair separate. - Compare the final values within each pair.

1. In Pair 1, evaluate the grouping symbols first: \(9+1=10\) and \(10+1=11\). Then \(12\times 10+13\times 11=120+143=263\). 2. The other expression is \(12\times 11+13\times 10=132+130=262\). 3. In Pair 2, \(29+1=30\) and \(30+1=31\). Then \(20\times 30+21\times 31=600+651=1251\). 4. The other expression is \(20\times 31+21\times 30=620+630=1250\). 5. In each pair, the first value is exactly \(1\) greater than the second value.

Pair 1: \(263\) and \(262\) Pair 2: \(1251\) and \(1250\) In each pair, the first value is \(1\) greater than the second value.

- Evaluate the parentheses first in parts a and c. - In parts b and d, perform multiplication or division before addition or subtraction. - Compare the two values in each pair after evaluating them.

1. Evaluate part a): \((12+8)\times3=20\times3=60\). 2. Evaluate part b): \(12\times3+8\times3=36+24=60\). 3. Evaluate part c): \((24-12)\div4=12\div4=3\). 4. Evaluate part d): \(24\div4-12\div4=6-3=3\). 5. Parts a and b have equal values, and parts c and d have equal values.

a) \(60\) b) \(60\) c) \(3\) d) \(3\) The expressions in each pair have equal values.

- Look for two addends that make a convenient multiple of \(100\) or \(1000\). - For consecutive subtractions, consider whether the amounts being subtracted can be combined into one grouped quantity. - Your final response must show the grouping you used, not only the numerical value.

1. For part a), regroup the convenient addends: \((438+162)+275=600+275=875\). 2. For part b), combine the amounts being subtracted: \(5600-(1200+400)=5600-1600=4000\). 3. For part c), regroup the convenient addends: \((125+875)+999=1000+999=1999\). 4. For part d), combine the amounts being subtracted: \(10{,}000-(2500+3500)=10{,}000-6000=4000\).

a) \((438+162)+275=875\) b) \(5600-(1200+400)=4000\) c) \((125+875)+999=1999\) d) \(10{,}000-(2500+3500)=4000\)

- Evaluate both grouped expressions first. - Keep the subtraction signs outside the grouping symbols. - Complete the remaining subtractions from left to right.

1. Evaluate the grouped expressions: \(340 + 160 = 500\) and \(480 - 120 = 360\). 2. Substitute: \(1500 - 500 - 360\). 3. Work from left to right: \(1000 - 360 = 640\).

\(640\)

- Work from the innermost grouping symbols outward. - Rewrite the expression after each grouped calculation. - Complete the final addition and subtraction from left to right.

1. Evaluate the innermost grouping: \(340 - 160 = 180\). 2. Evaluate the outer grouping: \(120 + 180 = 300\). 3. Substitute and calculate: \(980 - 300 + 45 = 680 + 45 = 725\).

\(725\)

- Round every number to the nearest hundred for the estimate. - Evaluate the exact expression from the innermost grouping outward. - Use the estimate to check whether the exact result is reasonable.

1. Estimate: \(4900 - [(1200 + 800) + (700 - 400)] = 4900 - 2300 = 2600\). 2. Evaluate exactly: \(1214 + 786 = 2000\) and \(654 - 354 = 300\). 3. Then \(4873 - (2000 + 300) = 4873 - 2300 = 2573\). 4. The exact value \(2573\) is close to the estimate \(2600\).

Estimate: \(2600\) Exact value: \(2573\)

- Round to values that make the addition and subtraction easy. - Evaluate inside the grouping symbols first for the exact value. - Compare the exact result with the estimate.

1. One reasonable estimate using hundreds is \(1200 + 3800 - (600 + 1200) = 5000 - 1800 = 3200\). 2. Evaluate exactly: \(567 + 1233 = 1800\), and \(1245 + 3782 = 5027\). 3. Therefore, \(5027 - 1800 = 3227\).

One estimate: \(3200\) Exact value: \(3227\)

- Round each number to the nearest ten for the estimate. - For the exact value, work from the innermost grouping outward. - Check that the exact result is reasonably close to the estimate.

1. Estimate: \(870 - [250 - (130 - 90)] = 870 - 210 = 660\). 2. Evaluate exactly from the inside outward: \(134 - 89 = 45\), then \(245 - 45 = 200\). 3. Therefore, \(872 - 200 = 672\).

Estimate: \(660\) Exact value: \(672\)

- Work from the innermost grouping symbols outward. - Record the value of each inner sum. - Use those values to evaluate the bracketed difference.

1. Evaluate the inner sums: \(1200 + 800 = 2000\) and \(450 + 150 = 600\). 2. Evaluate the bracketed difference: \(2000 - 600 = 1400\). 3. Complete the final subtraction: \(7500 - 1400 = 6100\).

\(6100\)

- Evaluate the grouped expression first. - Complete addition and subtraction from left to right. - Rewrite the outer subtraction after finding the grouped value.

1. Inside the grouping symbols, work from left to right: \(850 + 450 = 1300\). 2. Continue: \(1300 - 300 = 1000\). 3. Subtract: \(2100 - 1000 = 1100\).

\(1100\)

- Round each number to the nearest hundred for the estimate. - Evaluate both grouped expressions exactly. - Compare the exact value with the estimate.

1. Estimate: \((800 + 200) - (500 - 100) = 1000 - 400 = 600\). 2. Evaluate exactly: \(765 + 235 = 1000\) and \(482 - 118 = 364\). 3. Therefore, \(1000 - 364 = 636\).

Estimate: \(600\) Exact value: \(636\)

- Round to values that make the estimate easy to calculate. - Evaluate the inner grouped expressions before the bracketed sum. - Compare the exact result with the estimate.

1. One estimate is \(25{,}000 - [(8000 - 3000) + 2000] = 25{,}000 - 7000 = 18{,}000\). 2. Evaluate exactly: \(8420 - 3120 = 5300\) and \(1550 + 450 = 2000\). 3. The bracketed value is \(7300\), so the exact value is \(25{,}000 - 7300 = 17{,}700\).

Estimate: \(18{,}000\) Exact value: \(17{,}700\)

- Evaluate the grouping symbols before the outside operation. - Determine what grouped value is needed to reach the target. - A subtraction sign before grouping symbols changes how the inner signs affect the total.

1. For part a), the grouped value must be \(290\): \(420-130=290\). Then \(850-290=560\), so \(850-(420-130)=560\). 2. For part b), \(600-250=350\), and \(1500-350=1150\). Therefore, \(1500-(600-250)=1150\). 3. For part c), \(900-400-150=350\), and \(2100-350=1750\). Therefore, \(2100-(900-400-150)=1750\).

a) \(850-(420-130)=560\) b) \(1500-(600-250)=1150\) c) \(2100-(900-400-150)=1750\)

- Evaluate grouping symbols first. - Within one level, multiplication and division are completed from left to right. - Complete multiplication before addition.

1. For part a), evaluate both grouped expressions: \(40\times4=160\). 2. For part b), work from left to right inside the grouping symbols: \(2\times10\div4=20\div4=5\). Then \(100\div5=20\). 3. For part c), multiply before adding: \(56+44=100\).

a) \(160\) b) \(20\) c) \(100\)

- Apply the order of operations to each expression separately. - Notice how grouping symbols change the order of calculation. - Compare the four values after evaluating them.

1. \(A = 36 + 3 = 39\). 2. \(B = 48 \div 4 = 12\). 3. \(C = 144 - 12 = 132\). 4. \(D = 36 \times 8 = 288\). 5. Therefore, \(12 < 39 < 132 < 288\), so the order is \(B, A, C, D\).

\(B = 12\), \(A = 39\), \(C = 132\), \(D = 288\) Order: \(B < A < C < D\)

- Start by evaluating card [B]. - Find the next card whose expression begins with your result. - Record each card letter in order.

1. Card [B]: \((100 - 40) \div 12 = 5\). 2. Card [R] begins with \(5\): \((5 \times 8) + 12 = 52\). 3. Card [E] begins with \(52\): \((52 - 4) \div 8 = 6\). 4. Card [A] begins with \(6\): \(6 \times 15 - 50 = 40\). 5. Card [K] begins with \(40\): \(40 \div 4 + 90 = 100\), returning to card [B]. The letters spell BREAK.

BREAK

- Evaluate each set of grouping symbols separately. - In the second set, perform division from left to right. - Recall what happens when a number is divided by \(1\).

1. Evaluate the first set of grouping symbols: \(17{,}400 + 2600 = 20{,}000\). 2. Evaluate the second set from left to right: \(125 \div 5 = 25\), and then \(25 \div 25 = 1\). 3. Divide by \(1\): \(20{,}000 \div 1 = 20{,}000\).

\(20{,}000\)

- Start with the innermost grouping symbols on the right. - Think about how that result affects the entire product in brackets. - Decide whether you actually need to calculate \(123 + 456\).

1. Evaluate the innermost grouping symbols using the order of operations: \(10 - 2 \times 5 = 10 - 10 = 0\). 2. The product in brackets is \((123 + 456) \times 0 = 0\). The sum \(123 + 456\) does not need to be calculated because any number multiplied by \(0\) equals \(0\). 3. Therefore, \(45{,}678 - 0 = 45{,}678\).

\(45{,}678\)

- Identify which operation must be performed first in each expression. - Evaluate grouping symbols before operations outside them. - Write one intermediate result at a time.

1. For part a), multiply before subtracting: \(1200-25\times30=1200-750=450\). 2. For part b), evaluate the grouping symbols first: \((540+180)\div12=720\div12=60\). 3. For part c), evaluate the grouping symbols first: \(14\times(100-3)=14\times97=1358\). 4. For part d), multiply inside the grouping symbols first: \(640\div(16\times4)=640\div64=10\).

a) \(450\) b) \(60\) c) \(1358\) d) \(10\)

- Evaluate grouping symbols first. - Apply multiplication and division before addition and subtraction. - Compare the four values after evaluating them.

1. Expression 1: \(40 + 24 \div 4 - 2 = 40 + 6 - 2 = 44\). 2. Expression 2: \((40 + 24) \div 4 - 2 = 64 \div 4 - 2 = 16 - 2 = 14\). 3. Expression 3: \(40 + 24 \div (4 - 2) = 40 + 24 \div 2 = 52\). 4. Expression 4: \((40 + 24) \div (4 - 2) = 64 \div 2 = 32\). 5. The greatest value is \(52\), from expression 3. The least value is \(14\), from expression 2.

Greatest value: \(52\), expression 3 Least value: \(14\), expression 2

- Work from the innermost grouping symbols outward. - Apply the order of operations inside the parentheses. - Multiply only after the brackets have been evaluated.

1. Evaluate the innermost grouping symbols: \(12 - 3 \times 2 = 12 - 6 = 6\). 2. Evaluate the brackets: \(15 - 6 = 9\). 3. Multiply: \(8 \times 9 = 72\).

\(72\)

- Work from the innermost grouping symbols outward. - Apply division before addition inside the parentheses. - After the brackets, continue using the order of operations.

1. Evaluate the parentheses: \(48 \div 8 + 2 = 6 + 2 = 8\). 2. Evaluate the brackets: \(42 + 8 = 50\). 3. Divide: \(50 \div 5 = 10\). 4. Add: \(10 + 17 = 27\).

\(27\)

- Begin with the innermost grouping symbols. - Finish evaluating the brackets before multiplying by \(3\). - The subtraction from \(140\) is the final step.

1. Evaluate the parentheses: \(7 + 13 = 20\). 2. Continue inside the brackets: \(20 \times 4 = 80\), and \(80 - 55 = 25\). 3. Multiply: \(3 \times 25 = 75\). 4. Subtract: \(140 - 75 = 65\).

\(65\)

- Evaluate each grouped quantity separately. - Apply multiplication or division before addition or subtraction inside the grouping symbols. - Multiply the two grouped values before subtracting \(80\).

1. Evaluate the first grouped quantity: \(14 \times 5 - 62 = 70 - 62 = 8\). 2. Evaluate the second grouped quantity: \(33 \div 3 + 4 = 11 + 4 = 15\). 3. Multiply: \(8 \times 15 = 120\). 4. Subtract: \(120 - 80 = 40\).

\(40\)

- Work from the innermost grouping symbols outward. - Finish the brackets before dividing by \(5\). - Perform the division before subtracting from \(200\).

1. Evaluate the parentheses: \(18 + 7 = 25\). 2. Evaluate the brackets: \(4 \times 25 - 30 = 100 - 30 = 70\). 3. Divide: \(70 \div 5 = 14\). 4. Subtract: \(200 - 14 = 186\).

\(186\)

- Keep the numbers in the given order. - Use multiplication when you need a larger value. - Apply the order of operations when checking each expression. - More than one answer may be possible.

1. For a), one possible expression is \(2+4+6-8=4\). 2. For b), one possible expression is \(2\times4-6+8=10\). 3. For c), one possible expression is \(2+4\times6-8=18\). 4. For d), one possible expression is \(2+4+6+8=20\).

a) \(2+4+6-8=4\) b) \(2\times4-6+8=10\) c) \(2+4\times6-8=18\) d) \(2+4+6+8=20\)

- Convert quantities to compatible units before combining them. - Remember that \(60\) minutes equals \(1\) hour and \(1000\) grams equals \(1\) kilogram. - Follow the order of operations after the units are compatible.

1. For the time expression, \(1\,\text{h}\ 45\,\text{min} + 45\,\text{min} = 2\,\text{h}\ 30\,\text{min} = 2.5\,\text{h}\). 2. Then \(2.5\,\text{h} + 2.5\,\text{h} = 5.0\,\text{h}\). 3. For the mass expression, \(4.8\,\text{kg} \div 8 = 0.6\,\text{kg}\). 4. Also, \(325\,\text{g} \times 4 = 1300\,\text{g} = 1.3\,\text{kg}\). 5. Then \(0.6\,\text{kg} + 1.3\,\text{kg} = 1.9\,\text{kg}\).

a) \(5.0\,\text{h}\) b) \(1.9\,\text{kg}\)

- Evaluate the expression inside the grouping symbols first. - Convert the mixed area to square feet. - Then multiply and subtract in order.

1. Convert the mixed area: \(5\,\text{yd}^2\ 2\,\text{ft}^2=45\,\text{ft}^2+2\,\text{ft}^2=47\,\text{ft}^2\). 2. Evaluate inside the grouping symbols: \(18\,\text{ft}^2\div 2=9\,\text{ft}^2\), and \(9\,\text{ft}^2+3\,\text{ft}^2=12\,\text{ft}^2\). 3. Multiply: \(12\,\text{ft}^2\times 2=24\,\text{ft}^2\). 4. Subtract: \(47\,\text{ft}^2-24\,\text{ft}^2=23\,\text{ft}^2\).

\(23\,\text{ft}^2\), or \(2\,\text{yd}^2\ 5\,\text{ft}^2\)

- Round every number to the nearest hundred for the estimate. - Work from the innermost grouping symbols outward. - Compare the exact result with the estimate.

1. Estimate: \(8100 - [4600 - (2100 - 900)] = 8100 - (4600 - 1200) = 8100 - 3400 = 4700\). 2. Evaluate the innermost grouping symbols exactly: \(2134 - 899 = 1235\). 3. Evaluate the brackets: \(4567 - 1235 = 3332\). 4. Subtract: \(8123 - 3332 = 4791\). 5. The exact value \(4791\) is reasonably close to the estimate \(4700\).

Estimate: \(4700\) Exact value: \(4791\)

- Read the expression tree from the top branches toward the final box. - Use grouping symbols so the addition occurs before multiplication. - Evaluate one node at a time.

1. For part a), the left branch first adds \(15\) and \(25\), then multiplies the sum by \(12\). The root subtracts \(340\), so the expression is \((15+25)\times12-340\). 2. For part b), evaluate the tree: \(15+25=40\), \(40\times12=480\), and \(480-340=140\). The intermediate values are \(40\) and \(480\), and the final value is \(140\).

a) \((15+25)\times12-340\) b) The intermediate values are \(40\) and \(480\). The final value is \(140\).

- Read each branch from its top values toward the root. - Use grouping symbols where a sum or difference must be evaluated before multiplication or division. - Evaluate each branch before adding the results.

1. For part a), the left branch is \(24\times(15-8)\), and the right branch is \((42+18)\div5\). The branches are added, so the expression is \(24\times(15-8)+(42+18)\div5\). 2. For part b), evaluate the grouped parts: \(15-8=7\) and \(42+18=60\). Then \(24\times7=168\) and \(60\div5=12\). Finally, \(168+12=180\).

a) \(24\times(15-8)+(42+18)\div5\) b) \(180\)

- Read the tree from the top values toward the final box. - Use grouping symbols for the addition and subtraction branches. - Evaluate the two branches before multiplying their values.

1. The left branch adds \(28\) and \(14\), then divides the sum by \(7\): \((28 + 14) \div 7\). 2. The right branch subtracts \(9\) from \(15\): \((15 - 9)\). 3. The root multiplies the branch values, so the expression is \((28 + 14) \div 7 \times (15 - 9)\). 4. Evaluate: \(28 + 14 = 42\), \(42 \div 7 = 6\), and \(15 - 9 = 6\). 5. Multiply: \(6 \times 6 = 36\).

Expression: \((28 + 14) \div 7 \times (15 - 9)\) Final value: \(36\)

- Read each tree from its two upper branches toward the root. - Each compound branch feeding the root must remain one complete quantity in the written expression. - Check that your grouping makes the root operation happen last.

1. For part a), the left tree multiplies the sum \(15+8\) by the difference \(130-45\). The expression is \((15+8)\times(130-45)\). Its value is \(23\times85=1955\). 2. For part b), the right tree divides the difference \(240-80\) by the sum \(5+3\). The expression is \((240-80)\div(5+3)\). Its value is \(160\div8=20\).

a) \((15+8)\times(130-45)=1955\) b) \((240-80)\div(5+3)=20\)

- Identify the two complete quantities used in the final division. - Use grouping symbols around the entire dividend and divisor. - Evaluate the tree one node at a time.

1. For part a), the left side of the final division is \(25\times6-30\), and the right side is \(4\times5\). Therefore, the expression is \((25\times6-30)\div(4\times5)\). 2. For part b), evaluate the nodes: \(25\times6=150\), \(150-30=120\), and \(4\times5=20\). Then \(120\div20=6\).

a) \((25\times6-30)\div(4\times5)\) b) The intermediate values are \(150\), \(120\), and \(20\). The final value is \(6\).

- Read the root operation as the operation performed last. - Treat each compound branch feeding the root as one complete quantity when you write the expression. - Check that removing a needed grouping symbol would change which operation is performed last.

1. For tree a), the expression is \((120-45)\times(14+6)\). The branch values are \(75\) and \(20\), and \(75\times20=1500\). 2. For tree b), the expression is \((250+50)\div(18-8)\). The branch values are \(300\) and \(10\), and \(300\div10=30\). 3. For tree c), the expression is \(400-[12\times8+34]\). The inner product is \(96\), the complete right branch is \(130\), and \(400-130=270\).

a) \((120-45)\times(14+6)=1500\); intermediate values: \(75\) and \(20\) b) \((250+50)\div(18-8)=30\); intermediate values: \(300\) and \(10\) c) \(400-[12\times8+34]=270\); intermediate values: \(96\) and \(130\)

- Identify the final operation at the root of the tree. - Translate each branch into part of one numerical expression before evaluating. - Use nested grouping symbols to preserve the tree structure.

1. The tree represents \(540 + [980 - (320 + 450)]\). 2. Evaluate the innermost group: \(320 + 450 = 770\). 3. Evaluate the difference: \(980 - 770 = 210\). 4. Add: \(540 + 210 = 750\).

Expression: \(540 + [980 - (320 + 450)]\) Value: \(750\)

- Identify the final operation at the root of the tree. - Translate the complete tree into one expression before calculating. - Use grouping symbols to preserve the nested branch structure.

1. The tree represents \(15 \times [200 - (80 + 40)]\). 2. Evaluate the innermost sum: \(80 + 40 = 120\). 3. Subtract: \(200 - 120 = 80\). 4. Multiply: \(15 \times 80 = 1200\).

Expression: \(15 \times [200 - (80 + 40)]\) Value: \(1200\)

- Read the two upper branches as complete quantities. - Write one expression that preserves both branches before calculating. - Evaluate the two grouped branch values before the final subtraction.

1. The tree represents \((800 + 200) - (350 + 150)\). 2. Evaluate the left sum: \(800 + 200 = 1000\). 3. Evaluate the right sum: \(350 + 150 = 500\). 4. Subtract: \(1000 - 500 = 500\).

Expression: \((800 + 200) - (350 + 150)\) Value: \(500\)

- Treat each upper branch as one complete quantity in the written expression. - The root operation is performed after both branch values are known. - Check that both grouped branches remain intact as the dividend and divisor.

1. The left branch is the sum \(7+9\), and the right branch is the difference \(25-17\). 2. The root divides the left branch by the right branch, so the expression is \((7+9)\div(25-17)\). 3. Evaluate the branches: \(7+9=16\) and \(25-17=8\). 4. Divide: \(16\div8=2\).

Expression: \((7+9)\div(25-17)\) Value: \(2\)

- Follow the branches from the numbers toward the final operation. - Translate the complete tree into one expression before evaluating. - Use nested grouping symbols to preserve the right branch structure.

1. The tree represents \(2000 - [(600 + 400) - 500]\). 2. Evaluate the inner sum: \(600 + 400 = 1000\). 3. Evaluate the right branch: \(1000 - 500 = 500\). 4. Evaluate the entire expression: \(2000 - 500 = 1500\).

The expression is \(2000 - [(600 + 400) - 500]\), and its value is \(1500\).

- Read the entire right branch as one quantity feeding the root operation. - Use grouping symbols so the root subtraction happens after the right branch is complete. - Evaluate the multiplication inside the right branch before its subtraction.

1. The right branch represents \(25\times4-60\), and the root subtracts that entire branch from \(140\). 2. The expression is \(140-(25\times4-60)\). 3. Evaluate the product: \(25\times4=100\). 4. Evaluate the grouped difference: \(100-60=40\). 5. Subtract: \(140-40=100\).

Expression: \(140-(25\times4-60)\) Value: \(100\)

- Evaluate the upper branches before moving toward the root. - Find the complete dividend and divisor. - Use grouping symbols to preserve the right branch structure.

1. Evaluate the left product: \(12 \times 5 = 60\). 2. Evaluate the inner product on the right: \(2 \times 30 = 60\). 3. Evaluate the right difference: \(80 - 60 = 20\). 4. Divide: \(60 \div 20 = 3\). 5. The expression is \((12 \times 5) \div [80 - (2 \times 30)]\).

Expression: \((12 \times 5) \div [80 - (2 \times 30)]\) Value: \(3\)

- Evaluate the left and right main branches separately. - Use nested grouping symbols on the right. - Check each intermediate value before the final subtraction.

1. The expression is \((200 + 300) - [600 - (150 + 250)]\). 2. Evaluate the left sum: \(200 + 300 = 500\). 3. Evaluate the inner right sum: \(150 + 250 = 400\). 4. Evaluate the right difference: \(600 - 400 = 200\). 5. Subtract: \(500 - 200 = 300\).

Expression: \((200 + 300) - [600 - (150 + 250)]\) Value: \(300\)

- The tree adds two products. - Evaluate each product before adding.

1. Evaluate the left product: \(8\times5=40\). 2. Evaluate the right product: \(6\times7=42\). 3. Add: \(40+42=82\). 4. The expression is \(8\times5+6\times7\).

Intermediate values: \(40\) and \(42\) Expression: \(8\times5+6\times7\) Final value: \(82\)

- Find the complete dividend and divisor. - Evaluate both branch values before dividing.

1. Evaluate the left difference: \(200 - 80 = 120\). 2. Evaluate the right sum: \(4 + 6 = 10\). 3. Divide: \(120 \div 10 = 12\). 4. The expression is \((200 - 80) \div (4 + 6)\).

Expression: \((200 - 80) \div (4 + 6)\) Final value: \(12\)

- Identify the complete quantity that is divided by \(4\). - Translate the nested branches into one expression before calculating. - Use grouping symbols so the sum is found before the subtraction and the subtraction before the division.

1. The structured calculation is \([100 - (15 + 45)] \div 4\). 2. Evaluate the innermost sum: \(15 + 45 = 60\). 3. Subtract: \(100 - 60 = 40\). 4. Divide: \(40 \div 4 = 10\).

Expression: \([100 - (15 + 45)] \div 4\) Final value: \(10\)

- Evaluate the two upper branches first. - Use grouping symbols around the complete dividend and divisor.

1. Evaluate the left sum: \(24 + 36 = 60\). 2. Evaluate the right difference: \(15 - 5 = 10\). 3. Divide: \(60 \div 10 = 6\). 4. The expression is \((24 + 36) \div (15 - 5)\).

Expression: \((24 + 36) \div (15 - 5)\) Final value: \(6\)

- Treat each upper branch as one complete quantity in the written expression. - Use grouping symbols to show that both upper operations happen before the multiplication. - Evaluate the two grouped branch values before multiplying them.

1. The tree represents \((60 + 40) \times (25 - 15)\). 2. Evaluate the left branch: \(60 + 40 = 100\). 3. Evaluate the right branch: \(25 - 15 = 10\). 4. Multiply the branch values: \(100 \times 10 = 1000\).

Expression: \((60 + 40) \times (25 - 15)\) Intermediate values: \(100\) and \(10\) Final value: \(1000\)

- Treat the subtraction branch as one complete factor in the written expression. - Use grouping symbols so the subtraction happens before multiplication. - After finding the difference, use it in the final multiplication.

1. The tree represents \((912 - 456) \times 2\). 2. Find the difference: \(912 - 456 = 456\). 3. Multiply that result by \(2\): \(456 \times 2 = 912\).

Expression: \((912 - 456) \times 2\) Intermediate value: \(456\) Final value: \(912\)

- Treat the left and right branches as two complete quantities in the written expression. - Use grouping symbols to preserve the subtraction on the right branch. - Evaluate both branch values before performing the final subtraction.

1. The tree represents \((460 + 270) - (530 - 380)\). 2. Find the left intermediate result: \(460 + 270 = 730\). 3. Find the right intermediate result: \(530 - 380 = 150\). 4. Subtract the intermediate results: \(730 - 150 = 580\).

Expression: \((460 + 270) - (530 - 380)\) Intermediate values: \(730\) and \(150\) Final value: \(580\)

- Start with the deepest pair of grouping symbols and work outward. - At each level, treat the grouped result as one quantity in the next operation. - Check the operation immediately outside each pair of grouping symbols before moving on.

1. Evaluate the innermost parentheses: \(2+1=3\). 2. Evaluate the brackets: \(8-3=5\). 3. Multiply inside the braces: \(3\times5=15\). 4. Evaluate the braces: \(24+15=39\). 5. Complete the subtraction: \(90-39=51\).

\(51\)

- Identify the complete divisor in the original expression before evaluating anything. - Compare that divisor with the first operation in Mason’s work. - Evaluate the nested groups from the inside outward before performing the division.

1. Mason ignored that the braces make \(4+[2\times(5+1)]\) one complete divisor. 2. Evaluate the innermost parentheses: \(5+1=6\). 3. Evaluate the bracketed product: \(2\times6=12\). 4. Evaluate the divisor inside the braces: \(4+12=16\). 5. Divide: \(96\div16=6\).

Mason incorrectly treated only \(4\) as the divisor. The braces make the entire quantity \(4+[2\times(5+1)]\) the divisor. The correct value is \(6\).

- Keep track of the four given numbers so that each appears exactly once in each expression. - Think about how grouping symbols can change which operation is performed first. - Work backward from each target and consider what final operation could produce that value. - Check a candidate by evaluating the grouping symbols and then following the order of operations.

1. For target \(1\), one possible expression is \((5+3)\div(4\times2)=8\div8=1\). 2. For target \(12\), one possible expression is \((5+4-3)\times2=6\times2=12\). 3. For target \(50\), one possible expression is \(5\times(3\times4-2)=5\times(12-2)=5\times10=50\).

a) \((5+3)\div(4\times2)=1\) b) \((5+4-3)\times2=12\) c) \(5\times(3\times4-2)=50\) Other expressions are possible if they use \(2\), \(3\), \(4\), and \(5\) exactly once and have the required value.

- Keep the numbers in the given order. - For a small target, think about whether a final division could help. - For larger targets, consider how grouping can create useful factors. - Check each expression using the order of operations.

1. For target \(1\), one possible expression is \(((1+2)\times3-4)\div5=(9-4)\div5=1\). 2. For target \(15\), one possible expression is \(1+2+3+4+5=15\). 3. For target \(36\), one possible expression is \((1+2)\times(3+4+5)=3\times12=36\). 4. For target \(50\), one possible expression is \((1\times2\times3+4)\times5=(6+4)\times5=50\).

a) \(((1+2)\times3-4)\div5=1\) b) \(1+2+3+4+5=15\) c) \((1+2)\times(3+4+5)=36\) d) \((1\times2\times3+4)\times5=50\) Other expressions are possible if they keep the numbers in order and meet the target value.

- Keep the numbers in the given order. - Remember that multiplication and division are performed before addition and subtraction. - For \(31\), try making a factor close to \(3\). - Use grouping symbols when you need to change the usual order.

1. For a), one possible expression is \(10-5\times2+1=1\). 2. For b), one possible expression is \(10\div5+2\times1=4\). 3. For c), one possible expression is \(10-5+2\times1=7\). 4. For d), one possible expression is \(10+5-2\times1=13\). 5. For e), one possible expression is \(10\times(5-2)+1=31\).

a) \(10-5\times2+1=1\) b) \(10\div5+2\times1=4\) c) \(10-5+2\times1=7\) d) \(10+5-2\times1=13\) e) \(10\times(5-2)+1=31\)

- Trace the left and right main branches separately. - Translate the deepest branches into nested grouping symbols before calculating. - After evaluating each main branch, use the operation at the root.

1. The tree represents \([600 - (250 - 50)] - [120 + (80 - 30)]\). 2. On the left, \(250 - 50 = 200\), then \(600 - 200 = 400\). 3. On the right, \(80 - 30 = 50\), then \(120 + 50 = 170\). 4. Subtract: \(400 - 170 = 230\).

Expression: \([600 - (250 - 50)] - [120 + (80 - 30)]\) Value: \(230\)

- Work backward from the target and think about what complete divisor the expression needs. - Try putting one grouping pair inside the other so each pair changes which operations are completed first. - After you find a candidate, test it twice: once with only the parentheses removed and once with only the braces removed.

1. Put parentheses around \(3+3\), then use braces to make \(9-(3+3)\) the complete divisor: \(72\div\{9-(3+3)\}\). 2. Evaluate the inner group: \(3+3=6\). 3. Evaluate the divisor: \(9-6=3\). 4. Divide: \(72\div3=24\). 5. Without the parentheses, \(72\div\{9-3+3\}=72\div9=8\), so the parentheses are essential. 6. Without the braces, \(72\div9-(3+3)=8-6=2\), so the braces are essential.

\(72\div\{9-(3+3)\}=24\). Removing the parentheses gives \(72\div\{9-3+3\}=8\). Removing the braces gives \(72\div9-(3+3)=2\). Therefore both pairs are essential.