- Parentheses show which operation to do first. - Evaluate the quantity inside the parentheses 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 \(6\): \(36\div6=6\).

\(6\)

- Which operation is inside the parentheses? - After that operation is finished, rewrite the expression with its value in place of the parentheses. - Use the size of your answer as a quick check: it should be less than \(40\).

1. Evaluate the grouped sum first: \(7+8=15\). 2. Subtract: \(40-15=25\).

\(25\)

- Which symbol has been given a value? - After substitution, should that symbol still appear? - Compare your new expression with the original and check that the operation signs and grouping symbols stayed in place.

1. Replace \(n\) with \(7\) while keeping the rest of the expression in the same places. 2. The resulting numerical expression is \(3\times(7+2)\).

\(3\times(7+2)\)

- 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 \(0\).

\(0\)

- 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) \(10\)

- 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 \(45\).

- Start with the operation inside the grouping symbols in each expression. - Keep the outside multiplication until the grouped value is known. - Check that you use the original outside factor after evaluating the group.

1. For part a), evaluate the sum first: \((15+5)\times24=20\times24=480\). 2. For part b), evaluate the difference first: \(7\times(100-3)=7\times97=679\). 3. For part c), evaluate the difference first: \((125-25)\times9=100\times9=900\).

a) \(480\) b) \(679\) c) \(900\)

- In a), evaluate each parenthesized sum before the outside subtraction. - In b), after there are only additions and subtractions, work from left to right. - Compare what happens to the \(150\) in the two expressions.

1. For a), evaluate the grouped sums: \(800+200=1000\) and \(350+150=500\). Then \(1000-500=500\). 2. For b), addition and subtraction have the same priority, so work from left to right: \(800+200=1000\), \(1000-350=650\), and \(650+150=800\). 3. In a), the whole sum \(350+150\) is subtracted. In b), only \(350\) is subtracted before \(150\) is added.

a) \(500\) b) \(800\) The parentheses in a) make \(350+150\) one complete amount to subtract.

- 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 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: \(50\)

- 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: \(912\)

- Compare the original expression with Leo's expression before calculating. - What changed besides replacing \(n\) with \(4\)? - After you identify the change, evaluate the corrected expression.

1. Replacing \(n\) with \(4\) should keep the original grouping: \((4+6)\div2\). 2. Leo dropped the parentheses, which changes the order of operations. 3. Evaluate the correct expression: \((4+6)\div2=5\).

Leo dropped the parentheses when he substituted \(4\) for \(n\). The correct expression is \((4+6)\div2\), and its value is \(5\).

- 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, \(45{,}678 - 0 = 45{,}678\).

\(45{,}678\)

- 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: \(200 - 14 = 186\).

\(186\)

- 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 \(130\)

- Start with the grouping symbols that are farthest inside. - Keep the value of the bracketed expression together before using the outside addition. - Check that each step follows the nesting in the original expression.

1. Evaluate the inner sum: \(320+450=770\). 2. Evaluate the bracketed difference: \(980-770=210\). 3. Add the outside amount: \(540+210=750\).

Inner sum: \(770\) Bracketed value: \(210\) Final value: \(750\)

- Work from the innermost grouping symbols outward. - Compare Riley's written expression with the original. Which grouping symbols changed? - Explain how that change affects the order of the operations.

1. The innermost sum is \(80+40=120\). 2. The bracketed difference is \(200-120=80\). 3. Multiply: \(15\times80=1200\). 4. Sam is correct. Riley changed \(15\times[200-(80+40)]\) into \(15\times200-(80+40)\). That removes the brackets, so the subtraction is no longer done before multiplying by \(15\).

Sam is correct. The value is \(1200\). Riley removed the brackets around \(200-(80+40)\), so he multiplied before doing the bracketed subtraction.

- 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: \(2\)

- 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: \(230\)

- Compare the order in each plan with the grouping symbols in the expression. - Ask what complete quantity the outside subtraction acts on. - After choosing a plan, evaluate it and check that no operation moved outside its group.

1. Plan A follows the nested grouping symbols. First, \(600+400=1000\). 2. Finish the bracketed expression: \(1000-500=500\). 3. Subtract the whole bracketed value from \(2000\): \(2000-500=1500\). 4. Plan B changes the grouping by using \(2000\) before the bracketed expression is complete.

Plan A Final value: \(1500\)

- 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: \(300\)

- 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: \(10\)

- 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.

- 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.