Free math worksheets

- Identify the outermost operation first. - Describe the complete divisor as one quantity. - Use terms such as sum, difference, product, and quotient.

1. The outermost operation is division, so the entire expression is a quotient. 2. The dividend is \(120\), and the divisor is the product of \(4\) and \(6\). 3. One complete description is: “The quotient of \(120\) and the product of \(4\) and \(6\).”

The quotient of \(120\) and the product of \(4\) and \(6\).

- Identify the outermost operation first. - Treat each grouped quantity as one factor. - Describe the main structure before the parts inside the grouping symbols.

1. The outermost operation is multiplication, so the entire expression is a product. 2. The first factor is the sum of \(75\) and \(25\). The second factor is the difference between \(12\) and \(8\). 3. One complete description is: “The product of the sum of \(75\) and \(25\) and the difference between \(12\) and \(8\).”

The product of the sum of \(75\) and \(25\) and the difference between \(12\) and \(8\).

- Identify the outermost operation first. - Describe the complete quantity on each side of the subtraction sign. - Use product and quotient for the grouped calculations.

1. The outermost operation is subtraction, so the entire expression is a difference. 2. The first quantity is the product of \(15\) and \(4\). The second quantity is the quotient of \(80\) and \(2\). 3. One complete description is: “The difference between the product of \(15\) and \(4\) and the quotient of \(80\) and \(2\).”

The difference between the product of \(15\) and \(4\) and the quotient of \(80\) and \(2\).

- Treat \(14+9\) as one complete quantity. - Compare how that same quantity appears in the two expressions. - Describe the relationship without finding the value of the grouped sum.

1. Expression B is the grouped quantity \(14+9\). 2. Expression A consists of \(6\) copies of that same grouped quantity, so its value is \(6\) times the value of Expression B.

Expression A is \(6\) times the value of Expression B.

- Identify the operation performed last. - Name each part of the expression based on its operation. - Evaluate inside the grouping symbols before finding the final sum.

1. The expression is a sum. Its first addend is \(4200\), and its second addend is the difference \(150 - 75\). 2. Evaluate the difference: \(150 - 75 = 75\). 3. Evaluate the sum: \(4200 + 75 = 4275\).

Structure: the sum of \(4200\) and the difference of \(150\) and \(75\) Value: \(4275\)

- 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) \(\div\)

- Apply the order of operations when there are no grouping symbols. - Notice how grouping symbols change which operation is performed last. - Classify the entire expression by its outermost operation.

1. For part a), multiply first: \(5\times8=40\). Then \(25+40=65\). The outermost operation is addition, so the entire expression is a sum. 2. For part b), evaluate the grouping symbols first: \(25+5=30\). Then \(30\times8=240\). The outermost operation is multiplication, so the entire expression is a product.

a) Value: \(65\); classification: sum b) Value: \(240\); classification: product

- Compare the order of operations with and without grouping symbols. - Identify the operation that combines the largest parts of the expression. - Classify the entire expression by its outermost operation.

1. For part a), divide first: \(40\div4=10\). Then \(120-10=110\). The outermost operation is subtraction, so the entire expression is a difference. 2. For part b), evaluate the grouping symbols first: \(120-40=80\). Then \(80\div4=20\). The outermost operation is division, so the entire expression is a quotient.

a) Value: \(110\); classification: difference b) Value: \(20\); classification: quotient

- Apply the order of operations in each expression. - Compare the effect of the grouping symbols. - Classify the entire expression by its outermost operation.

1. For part a), evaluate the grouping symbols first: \(6+9=15\). Then \(90\div15=6\). The outermost operation is division, so the entire expression is a quotient. 2. For part b), divide before adding: \(90\div6=15\). Then \(15+9=24\). The outermost operation is addition, so the entire expression is a sum.

a) Value: \(6\); classification: quotient b) Value: \(24\); classification: sum

- Identify which quantity is evaluated first. - Determine the complete divisor. - Make the wording clear about what is divided by what.

1. For part a), the grouped sum is the divisor, so one description is “Divide \(80\) by the sum of \(12\) and \(8\).” 2. For part b), evaluate the grouped sum: \(12+8=20\). Then \(80\div20=4\).

a) Divide \(80\) by the sum of \(12\) and \(8\). b) \(4\)

- Match the positive changes with water added and the negative changes with water removed. - Compare the terms being added with the terms being subtracted in the two expressions. - In Expression 2, interpret each grouped sum as one combined quantity before evaluating.

1. For part a), Expression 1 gives \(120+30-10+40-20+50=210\,\text{L}\). Expression 2 gives \(120+120-30=210\,\text{L}\). Both expressions have the same value. 2. For part b), \((30+40+50)\) represents the total amount of water added, and \((10+20)\) represents the total amount of water removed.

a) Both expressions equal \(210\,\text{L}\). b) \((30+40+50)\) is the total water added, and \((10+20)\) is the total water removed.

- Identify which numbers are directly connected by addition. - Use grouping symbols to show which operation occurs first. - Compare each choice with the tree structure.

1. The tree first adds \(30\) and \(20\), and then subtracts that sum from \(100\). 2. Expression B matches the tree: \(100 - (30 + 20)\). 3. Evaluate: \(100 - 50 = 50\).

B) \(100 - (30 + 20) = 50\)

- Read the tree from its upper branch toward the final operation. - Describe which operation produces an intermediate amount before naming the final operation. - Keep the money units consistent when evaluating the tree.

1. For part a), the tree multiplies \(3\) by \(\$4.50\), then adds \(\$6.50\) to that product. 2. For part b), the product is \(3\times\$4.50=\$13.50\). Then \(\$13.50+\$6.50=\$20.00\).

a) Multiply \(3\) by \(\$4.50\), then add \(\$6.50\). b) \(\$20.00\)

- Regard all of Expression P as one quantity. - Locate that same complete quantity inside Expression Q. - Describe what is done to it in Expression Q without calculating its value.

1. Expression P is the complete grouped quantity \(82-(27+15)\). 2. Expression Q takes that entire quantity and adds \(9\), so Expression Q is \(9\) greater than Expression P.

Expression Q is \(9\) greater than Expression P.

- Focus on what the parentheses group together. - Identify the complete divisor in the expression. - Compare that structure with the order of operations in Leo’s description.

1. The grouping symbols show that \(8+1\) is one complete quantity. 2. That entire grouped quantity is the divisor, so the expression means to divide \(72\) by the sum of \(8\) and \(1\). 3. Leo is not correct because his description treats the \(+1\) as an operation performed after the division.

Leo is not correct. The expression means “divide \(72\) by the sum of \(8\) and \(1\).”

- Match each number in the expression to a quantity in the situation. - Interpret the grouped part before interpreting the multiplication by \(5\). - Do not calculate; describe what each part counts.

1. The grouped sum \(18+12\) represents the total number of books in one box. 2. Multiplying that quantity by \(5\) represents the total number of books in all \(5\) boxes.

\(18+12\) represents the number of books in one box. The entire expression represents the total number of books in all \(5\) boxes.

- Identify the operation performed last. - Name the expression on each side of the final subtraction. - Evaluate the grouped expressions before the final difference.

1. The entire expression is a difference. The minuend is the sum \(780 + 220\), and the subtrahend is the difference \(150 - 60\). 2. Evaluate the grouped expressions: \(780 + 220 = 1000\) and \(150 - 60 = 90\). 3. Evaluate the final difference: \(1000 - 90 = 910\).

Structure: the difference between the sum of \(780\) and \(220\) and the difference of \(150\) and \(60\) Value: \(910\)

- Identify the operation performed last. - Name the minuend and subtrahend as expressions. - Evaluate each grouped difference before the final subtraction.

1. The entire expression is a difference. The minuend is the difference \(3400 - 1200\), and the subtrahend is the difference \(500 - 150\). 2. Evaluate the grouped expressions: \(3400 - 1200 = 2200\) and \(500 - 150 = 350\). 3. Evaluate the final difference: \(2200 - 350 = 1850\).

Structure: the difference between the difference of \(3400\) and \(1200\) and the difference of \(500\) and \(150\) Value: \(1850\)

- Apply the order of operations one expression at a time. - Notice how the grouping symbols in b change the calculation. - Classify the entire expression by its outermost operation.

1. For part a), evaluate both products: \(9\times6=54\) and \(4\times3=12\). Then \(54-12=42\). The outermost operation is subtraction, so the expression is a difference. 2. For part b), evaluate the grouping symbols: \(6-4=2\). Then \(9\times2\times3=54\). The outermost operation is multiplication, so the expression is a product. 3. Compare the values: \(54>42\), so expression b has the greater value.

a) Value: \(42\); classification: difference b) Value: \(54\); classification: product Expression b has the greater value.

- Perform multiplication before addition or subtraction. - When only addition and subtraction remain, work from left to right. - Classify the expression by the operation performed last.

1. Multiply first: \(50 \times 3 = 150\). The expression becomes \(200 - 150 + 12\). 2. Evaluate addition and subtraction from left to right: \(200 - 150 = 50\), and \(50 + 12 = 62\). 3. The outermost operation is addition, so the expression is a sum.

Value: \(62\); classification: sum

- Evaluate both sets of grouping symbols first. - Use the order of operations inside the second set. - Identify the operation that connects the two grouped quantities.

1. Evaluate the first set of grouping symbols: \(24 + 36 = 60\). 2. In the second set, multiply before subtracting: \(15 - 3 \times 3 = 15 - 9 = 6\). 3. Divide: \(60 \div 6 = 10\). 4. The outermost operation is division, so the expression is a quotient.

Value: \(10\); classification: quotient

- Perform multiplication and division before addition and subtraction. - When only addition and subtraction remain, work from left to right. - Classify the expression by the operation performed last.

1. Multiply and divide first: \(5 \times 15 = 75\) and \(32 \div 4 = 8\). The expression becomes \(125 - 75 + 8\). 2. Evaluate from left to right: \(125 - 75 = 50\), and \(50 + 8 = 58\). 3. The outermost operation is addition, so the expression is a sum.

Value: \(58\); classification: sum

- Evaluate the grouping symbols before the operation outside them. - Apply the order of operations inside the grouping symbols. - Classify the expression by its outermost operation.

1. Inside the grouping symbols, multiply first: \(6 \times 7 = 42\). 2. Subtract: \(48 - 42 = 6\). 3. Multiply: \(6 \times 6 = 36\). 4. The outermost operation is multiplication, so the expression is a product.

Value: \(36\); classification: product

- Evaluate the two grouped quantities separately. - Apply the order of operations inside each set of grouping symbols. - Identify the operation that connects the two grouped quantities.

1. Evaluate the first set of grouping symbols: \(14 + 6 \times 11 = 14 + 66 = 80\). 2. Evaluate the second set: \(2 \times 12 - 4 = 24 - 4 = 20\). 3. Divide: \(80 \div 20 = 4\). 4. The outermost operation is division, so the expression is a quotient.

Value: \(4\); classification: quotient

- Identify the outermost operation in each expression. - Describe each grouped or multiplied quantity as one complete part. - Evaluate after writing the description.

1. For part a), one description is “Divide the sum of \(45\) and \(15\) by the difference between \(20\) and \(10\).” The value is \(60\div10=6\). 2. For part b), one description is “Subtract the product of \(3\) and \(9\) from the product of \(8\) and \(7\).” The value is \(56-27=29\).

a) Divide the sum of \(45\) and \(15\) by the difference between \(20\) and \(10\). Value: \(6\) b) Subtract the product of \(3\) and \(9\) from the product of \(8\) and \(7\). Value: \(29\)

- Identify the complete quantity being subtracted from \(150\). - Describe the product inside the grouped sum. - Apply the order of operations when evaluating.

1. One description is: “Subtract the sum of \(40\) and the product of \(5\) and \(8\) from \(150\).” 2. Inside the grouping symbols, multiply first: \(5 \times 8 = 40\). 3. Add: \(40 + 40 = 80\). 4. Subtract: \(150 - 80 = 70\).

Subtract the sum of \(40\) and the product of \(5\) and \(8\) from \(150\). Value: \(70\)

- Identify which two numbers are combined first in each tree. - Use grouping symbols when addition must occur before multiplication. - Compare what is multiplied by \(5\) in the two trees.

1. For part a), Tree A first adds \(12\) and \(8\), then multiplies by \(5\), so its expression is \((12+8)\times5\). Tree B first multiplies \(8\) and \(5\), then adds \(12\), so its expression is \(12+8\times5\). 2. For part b), Tree A has value \(20\times5=100\). Tree B has value \(12+40=52\). 3. For part c), Tree A groups the addition so it happens before multiplication. Tree B multiplies before adding, so the same numbers are combined in different structures.

a) Tree A: \((12+8)\times5\); Tree B: \(12+8\times5\) b) Tree A: \(100\); Tree B: \(52\) c) The trees group the numbers differently, so the operations are performed in a different order.

- Describe what each branch of the trees represents in the shopping situation. - Find the total cost before calculating the change. - Check whether the final operation in each tree matches the situation.

1. For part a), Tree A is correct because it adds both purchase costs before subtracting the total cost from \(\$100.00\). Tree B incorrectly adds the paintbrush cost after subtracting the chalk cost. 2. For part b), the chalk costs \(5\times\$8.00=\$40.00\), and the paintbrush sets cost \(3\times\$12.00=\$36.00\). The total cost is \(\$76.00\), so the change is \(\$100.00-\$76.00=\$24.00\). 3. For part c), the numerical expression is \(100-(5\times8+3\times12)\), and its value is \(24\).

a) Tree A, because it subtracts the total cost from the amount paid. b) \(\$24.00\) c) \(100-(5\times8+3\times12)=24\)