Free math worksheets

- Separate the expression at its addition sign. - Look for the numerical factor attached to the variable.

1. The term containing \(x\) is \(7x\), so its coefficient is \(7\). 2. The term without a variable is \(11\), so the constant term is \(11\).

Coefficient: \(7\) Constant term: \(11\)

- Examine the raised number and the quantity it applies to. - For the other term, find the numerical factor multiplying the variable.

1. In \(u^3\), the base is \(u\) and the exponent is \(3\). 2. In \(2u\), the coefficient is \(2\).

Base: \(u\) Exponent: \(3\) Coefficient of \(2u\): \(2\)

- Write the multiplication that is usually left unwritten. - Ask which numerical factor leaves the variable unchanged.

1. The expression can be written as \(1c\). 2. Therefore, the coefficient of \(c\) is \(1\).

The coefficient is \(1\), because \(c=1c\).

- Examine each term independently. - The same numeral can play the same role in more than one term.

1. In \(3x\), the number \(3\) is the coefficient of \(x\). 2. In \(3y\), the number \(3\) is the coefficient of \(y\). 3. Each occurrence is a separate numerical factor in a different term.

The first \(3\) is the coefficient of \(x\), and the second \(3\) is the coefficient of \(y\).

- The outermost operation is multiplication. - Each complete quantity being multiplied is a factor. - Substitute the value inside the parentheses before multiplying.

1. The expression is a product with factors \(4\) and \(x-2.5\). 2. In words, it is four times the difference between \(x\) and \(2.5\). 3. For \(x=5.5\), \(4\times(5.5-2.5)=4\times3=12\).

a) \(4\) and \(x-2.5\) b) Four times the difference between \(x\) and \(2.5\) c) \(12\)

- The coefficient is the number multiplying the variable. - A minus sign in front of a variable represents an implied factor. - Rewrite division by a number as multiplication by its reciprocal. - Multiplying a variable by \(1\) does not change its value.

1. A coefficient is the numerical factor multiplying a variable. 2. In \(-x\), the implied numerical factor is \(-1\), so the coefficient is \(-1\). 3. In \(1.2y\), the coefficient is \(1.2\). 4. Rewrite \(\frac{z}{10}\) as \(\frac{1}{10}z\), so the coefficient is \(\frac{1}{10}\), or \(0.1\). 5. In \(-\frac{4}{5}a\), the coefficient is \(-\frac{4}{5}\), or \(-0.8\). 6. The expression \(b\) means \(1\times b\), so the coefficient is \(1\).

a) \(-1\) b) \(1.2\) c) \(\frac{1}{10}\), or \(0.1\) d) \(-\frac{4}{5}\), or \(-0.8\) e) \(1\)

- Use the order of operations to decide which operation is performed last. - Imagine substituting numbers for the variables and completing the calculation. - The last operation determines the name of the entire expression.

1. In \(x(y+5)\), the addition inside parentheses is performed first and multiplication is performed last. The expression is a product. 2. In \(a^2-10\), exponentiation is performed first and subtraction is performed last. The expression is a difference. 3. In \((p-q)\div4\), subtraction inside parentheses is performed first and division is performed last. The expression is a quotient. 4. In \(7+3n\), multiplication is performed before addition, so addition is performed last. The expression is a sum. 5. In \((m+1)^2\), addition inside parentheses is performed first and exponentiation is performed last. The expression is a power.

1) Multiplication; product 2) Subtraction; difference 3) Division; quotient 4) Addition; sum 5) Exponentiation; power

- A term includes the sign immediately before it. - Rewrite subtraction as addition of a negative term. - Treat \(-v\) as \(-1\) times \(v\).

1. The signed terms in \(7x-4\) are \(7x\) and \(-4\). Written as a sum, the expression is \((7x)+(-4)\). 2. The signed terms in \(3a+5\) are \(3a\) and \(5\). Written as a sum, the expression remains \((3a)+5\). 3. The signed terms in \(2u-v+6\) are \(2u\), \(-v\), and \(6\). Written as a sum, the expression is \((2u)+(-v)+6\).

1) \((7x)+(-4)\) 2) \((3a)+5\) 3) \((2u)+(-v)+6\)

- A quotient has two structural parts: what is being divided and what it is divided by. - In fraction notation, those parts are the numerator and denominator. - Preserve grouped expressions such as \(a+5\) or \(n+4\) as single parts when changing notation.

1. In \(18\div x\), the dividend is \(18\) and the divisor is \(x\). The same quotient is \(\frac{18}{x}\). 2. In \(\frac{a+5}{3}\), the numerator is \(a+5\) and the denominator is \(3\). The same quotient is \((a+5)\div3\). 3. In \(\frac{2m-1}{n+4}\), the numerator is \(2m-1\) and the denominator is \(n+4\). The same quotient is \((2m-1)\div(n+4)\).

a) Dividend: \(18\); divisor: \(x\); \(\frac{18}{x}\) b) Numerator: \(a+5\); denominator: \(3\); \((a+5)\div3\) c) Numerator: \(2m-1\); denominator: \(n+4\); \((2m-1)\div(n+4)\)

- First identify the operation performed last. - Then examine the grouped part as its own expression.

1. The entire expression is the product of \(5\) and \(a-3\). 2. Rewrite the expression inside the parentheses as \(a+(-3)\). 3. Its two signed terms are \(a\) and \(-3\).

Factors: \(5\) and \(a-3\) Terms inside the parentheses: \(a\) and \(-3\)

- Look at the outermost operation, not the operation inside the numerator. - Treat the numerator as one complete quantity.

1. The operation performed last is division. 2. Therefore, the whole expression is a quotient. 3. The two quantities being divided are \(p+8\) and \(6\).

It is a quotient of \(p+8\) and \(6\).

- The top of a calculation tree represents the final operation. - Each complete branch feeding the top node is one part of that operation.

1. The top node shows multiplication, so multiplication is performed last. 2. The left factor is \(r+4\). 3. The right factor is \(r-1\).

Last operation: multiplication Factors: \(r+4\) and \(r-1\)

- First identify the two complete quantities joined by the outer subtraction. - When rewriting subtraction as addition, keep the negative sign with the variable term.

1. The outer subtraction combines \(12\) and \(\frac{x}{5}\). 2. Rewrite the difference as \(12+\left(-\frac{x}{5}\right)\). 3. Since \(-\frac{x}{5}=-\frac{1}{5}x\), the coefficient of \(x\) is \(-\frac{1}{5}\).

Quantities in the subtraction: \(12\) and \(\frac{x}{5}\) Rewritten as a sum: \(12+\left(-\frac{x}{5}\right)\) Coefficient of \(x\): \(-\frac{1}{5}\)

- Separate the outer product from the operation inside the parentheses. - A term inside one factor is not automatically one of the factors displayed in the whole product.

1. The whole expression is displayed as a product of \(2\) and \(3q+5\). 2. Therefore, \(2\) and \(3q+5\) are the two displayed factors. 3. The quantity \(3q\) is a term inside the factor \(3q+5\), not one of the two displayed factors of the whole product.

\(3q\) is not one of the two displayed factors; it is a term inside the factor \(3q+5\).

- Distinguish an operation inside one term from the operation joining the main parts. - Ask which operation would be completed last when evaluating.

1. The term \(3y\) is a product within the expression. 2. The outermost and last operation combines \(8\) and \(3y\) by addition. 3. The whole expression is a sum, so the student is not correct.

No. The whole expression is a sum because addition is performed last.

- Look at exactly what the exponent is attached to. - Then treat the grouped base as its own expression.

1. The exponent \(2\) applies to the entire grouped quantity. 2. The base is \(k+2\). 3. The base is a sum with terms \(k\) and \(2\).

Base: \(k+2\) Terms in the base: \(k\) and \(2\)

- Keep the subtraction sign with the final term. - A constant term has no variable factor.

1. The signed terms are \(6a\), \(2b\), and \(-15\). 2. The variable terms are \(6a\) and \(2b\). 3. The constant term is \(-15\). 4. There are \(3\) terms in all.

Variable terms: \(6a\) and \(2b\) Constant term: \(-15\) Number of terms: \(3\)

- Separate the quotient into its top and bottom expressions. - Analyze each grouped expression independently after that.

1. The numerator is \(5n-1\), with signed terms \(5n\) and \(-1\). 2. The denominator is \(n+4\), with terms \(n\) and \(4\).

Numerator: \(5n-1\); terms \(5n\) and \(-1\) Denominator: \(n+4\); terms \(n\) and \(4\)

- Ask what numerical factor changes a variable to its opposite. - Write the implied multiplication explicitly.

1. A leading negative sign means multiplication by \(-1\). 2. Thus \(-z=(-1)z\). 3. The coefficient is \(-1\).

The coefficient is \(-1\), because \(-z=(-1)z\).

- Look outside the parentheses to find the final operation. - Treat the entire grouped part as one quantity in that operation.

1. The parentheses form the single quantity \(r+5\). 2. The last operation is subtraction from \(9\). 3. The grouped quantity being subtracted is \(r+5\).

Last operation: subtraction Grouped quantity being subtracted: \(r+5\)

- Use the top node to identify the two main quotient parts. - Then move one level down and analyze only the first branch.

1. The top node is division, so the two complete branch values are the quantities being divided. 2. The first quantity is \(3x+10\). 3. The second quantity is \(5\). 4. The final operation inside \(3x+10\) is addition.

Quantities being divided: \(3x+10\) and \(5\) Final operation inside \(3x+10\): addition

- Analyze each term separately as the product shown. - Compare the displayed factor lists to find what appears in both.

1. The displayed factors of \(ab\) are \(a\) and \(b\). 2. The displayed factors of \(4a\) are \(4\) and \(a\). 3. The displayed common factor is \(a\).

Displayed factors of \(ab\): \(a\), \(b\) Displayed factors of \(4a\): \(4\), \(a\) Displayed common factor: \(a\)

- Separate the numerical factor from the variable power. - Identify what is raised and the number that tells how many equal factors are used.

1. The numerical factor is \(\frac{3}{4}\), so it is the coefficient. 2. The power is \(w^2\), with base \(w\) and exponent \(2\).

Coefficient: \(\frac{3}{4}\) Base: \(w\) Exponent: \(2\)

- Identify the operation joining \(2\) and \(t\). - Use the vocabulary associated with that operation.

1. The divisor is \(2+t\). 2. Its two parts, \(2\) and \(t\), are joined by addition. 3. Therefore, they are terms, not factors.

No. The divisor has terms \(2\) and \(t\), not factors.

- Keep the subtraction sign with the constant term. - Classify the sign of the term itself, not the operation symbol alone.

1. The variable term is \(1.2p\), with coefficient \(1.2\). 2. Rewrite the expression as \(1.2p+(-0.7)\). 3. The constant term is \(-0.7\), which is negative.

Coefficient: \(1.2\) Constant term: \(-0.7\), which is negative

- Rewrite the quotient so the variable appears as one factor. - The coefficient is the complete numerical factor, not just the numerator.

1. The quotient can be rewritten as \(\frac{5}{12}x\). 2. The numerical factor multiplying \(x\) is \(\frac{5}{12}\). 3. Therefore, the coefficient is \(\frac{5}{12}\).

The coefficient is \(\frac{5}{12}\), because \(\frac{5x}{12}=\frac{5}{12}x\).

- Analyze the outer operation before the operation in parentheses. - A grouped expression can be one factor while containing its own terms.

1. The outer operation is multiplication. 2. The factors are \(a\) and \(b+c\). 3. The second factor is a sum with terms \(b\) and \(c\).

Factors: \(a\) and \(b+c\) Terms inside the second factor: \(b\) and \(c\)

- Name the two quantities joined by the outer multiplication. - Then look one level inside the grouped sum for the term without a variable.

1. The outer operation is multiplication. 2. The factors are \(0.6\) and \(x+10\). 3. Inside the second factor, the constant term is \(10\).

Factors: \(0.6\) and \(x+10\) Constant term inside the parentheses: \(10\)

- Expand the power as repeated equal factors. - Separate the numerical factor from the variable factors.

1. Since \(x^2=x\times x\), the displayed factors are \(2\), \(x\), \(x\), and \(y\). 2. The numerical factor is \(2\), so the coefficient is \(2\).

Displayed factors: \(2\), \(x\), \(x\), \(y\) Coefficient: \(2\)

- Look for the choice where the grouped expression participates in multiplication. - Do not confuse being grouped with being a factor.

1. In a), \(x+1\) is an addend. 2. In b), \(x+1\) is multiplied by \(7\), so it is a factor. 3. In c), \(x+1\) is the grouped quantity from which \(7\) is subtracted. 4. The correct choice is b).

b) \(7(x+1)\)

- First identify the main addition. - Then analyze only the nonconstant term as a product.

1. The outer addition gives terms \(\frac{1}{2}(h-6)\) and \(9\). 2. The variable term is a product of \(\frac{1}{2}\) and \(h-6\).

Whole-expression terms: \(\frac{1}{2}(h-6)\) and \(9\) Factors in the variable term: \(\frac{1}{2}\) and \(h-6\)

- Separate the whole sum before examining the quotient term. - Analyze the two quantities only within the term where division occurs.

1. The whole expression is a sum of \(10\div x\) and \(4\). 2. The quotient term is \(10\div x\). 3. The two quantities being divided are \(10\) and \(x\). 4. The other term is \(4\).

Quotient term: \(10\div x\) Quantities being divided: \(10\) and \(x\) Other term: \(4\)

- Count terms using the outermost addition only. - Treat each parenthetical expression as one factor before any rewriting.

1. The outer addition separates two terms: \(3(x+4)\) and \(2(x-1)\). 2. The first term has factors \(3\) and \(x+4\). 3. The second term has factors \(2\) and \(x-1\).

The whole expression has \(2\) terms. First term factors: \(3\) and \(x+4\) Second term factors: \(2\) and \(x-1\)

- Identify the number multiplying the variable in each expression. - Rewrite fractions as decimals to compare the coefficients. - Place negative numbers carefully when ordering from least to greatest.

1. The coefficients are \(A: 0.4\), \(B: \frac{1}{5}=0.2\), \(C: -2\), \(D: \frac{3}{4}=0.75\), and \(E: -1\). 2. Compare the coefficients: \(-2<-1<0.2<0.4<0.75\). 3. Match each coefficient to its expression. The order is \(C,E,B,A,D\).

Coefficients: \(A: 0.4\); \(B: 0.2\); \(C: -2\); \(D: 0.75\); \(E: -1\) Order from least to greatest: \(C,E,B,A,D\)

- Determine how parentheses change the order of operations. - The operation performed last names the structure of the entire expression. - Compare the outermost operation in each pair.

1. In \(a^2+b^2\), the powers are evaluated before the addition. The last operation is addition, so the expression is a sum. In \((a+b)^2\), the sum inside parentheses is evaluated before it is squared. The last operation is exponentiation, so the expression is a power. 2. In \(xy-z\), multiplication is performed before subtraction. The last operation is subtraction, so the expression is a difference. In \(x(y-z)\), the difference inside parentheses is evaluated before multiplication. The last operation is multiplication, so the expression is a product.

a) \(a^2+b^2\) is a sum; \((a+b)^2\) is a power. b) \(xy-z\) is a difference; \(x(y-z)\) is a product.

- A variable with no visible numerical factor has an implied coefficient. - Include each term’s sign as part of its numerical factor. - Rewrite subtraction as addition of a negative term.

1. Rewrite each subtraction as addition of a negative term: \((-5x^2)+x+(-0.8)\). 2. The numerical factor of \(-5x^2\) is \(-5\). 3. The term \(x\) means \(1\times x\), so its coefficient is \(1\). 4. The numerical factor of the constant term \(-0.8\) is \(-0.8\).

a) \((-5x^2)+x+(-0.8)\) b) The numerical factors are \(-5\), \(1\), and \(-0.8\). The coefficient of the \(x\)-term is \(1\).

- Keep each sign attached to its term. - The coefficient is the numerical factor of the variable part.

1. Rewrite subtraction as addition of a negative term: \(3m^2+(-4m)+9\). 2. The terms are \(3m^2\), \(-4m\), and \(9\). 3. The coefficient of the middle term is \(-4\).

Terms: \(3m^2\), \(-4m\), \(9\) Coefficient of the middle term: \(-4\)

- Classify each expression by the operation shown at its outermost level. - For an expression written as a sum, count its signed terms and inspect the variable coefficient.

1. Expression a) is written as the sum of the signed terms \(-3x\) and \(7\). 2. Its variable term has coefficient \(-3\). 3. The other choices are displayed as a product, a three-term sum, or a quotient. 4. The correct choice is a).

a) \(-3x+7\)

- Identify the outermost multiplication first. - Do not break apart a grouped quantity before naming the factors of the whole expression.

1. The parentheses make \(2x-7\) one grouped quantity in the outer product. 2. The two factors of the whole expression are \(4\) and \(2x-7\). 3. The quantities \(2x\) and \(-7\) are terms inside the second factor.

The error is splitting a grouped factor into its terms. The factors are \(4\) and \(2x-7\).

- Identify the two main parts of the outer sum. - Then examine the structure inside the nonconstant part.

1. The whole expression is a sum of \(2\) and \(5(x-1)\). 2. Therefore, \(5(x-1)\) is one term of the whole sum. 3. The same quantity is also a product of factors \(5\) and \(x-1\).

\(5(x-1)\) is both one term of the whole sum and a product of \(5\) and \(x-1\).

- Decide whether the main operation is multiplication or subtraction. - Different levels of an expression can have different kinds of parts.

1. The whole expression is a difference, or a sum of signed terms. 2. Its two signed terms are \(7x\) and \(-2y\), so Priya correctly describes the whole expression. 3. Noah listed factors within the two separate terms, not factors of the whole expression.

Priya is correct: the whole expression has two signed terms, \(7x\) and \(-2y\). Noah listed factors within those terms.

- Start at the top node to identify the two complete quantities in the final operation. - In the first branch, look for the same grouped factor appearing twice.

1. The top operation is subtraction. 2. The first quantity is \((x+3)(x+3)\), which can be written as \((x+3)^2\). 3. The second quantity is \(5\). 4. The repeated factor in the first quantity is \(x+3\).

Quantities in the final subtraction: \((x+3)^2\) and \(5\) Repeated factor: \(x+3\)

- Identify the main operation of the whole expression before naming its parts. - Use the explicitly stated three-factor form when listing the outer factors. - Analyze the grouped sum separately from the outer product.

1. The whole expression is a product. In the stated fully separated form, its three factors are \(5\), \(x\), and \(2+y\). 2. The grouped factor \(2+y\) is a sum with terms \(2\) and \(y\). 3. The student mixed factors from the outer product with terms from the inner sum.

The student mixed two structural levels. Fully separated factors of the outer product: \(5\), \(x\), and \(2+y\) Terms inside the parentheses: \(2\) and \(y\)

- Respect the parentheses before listing parts. - Describe the outer operation and the inner sum at separate levels.

1. The outer subtraction combines \(14\) and the grouped quantity \(3x+2\). 2. Inside the grouped quantity, the terms are \(3x\) and \(2\). 3. The student mixed the outer and inner levels, so the description is not structurally precise.

No. The outer subtraction combines \(14\) and the grouped quantity \(3x+2\); inside that grouped sum, the terms are \(3x\) and \(2\).

- Distinguish terms with variable factors from the term without one. - Remember that an unwritten numerical factor can still be identified.

1. The variable terms are \(x^2\) and \(6x\). 2. Their coefficients are \(1\) and \(6\), respectively. 3. The term \(9\) is the constant term, not usually called a coefficient when no variable factor is present. 4. The label is not standard terminology.

No. The standard description is: coefficients \(1\) and \(6\) for the variable terms, and constant term \(9\).

- Identify the full grouped quantity affected by the exponent. - Then analyze the quotient term inside the base.

1. The base is \(\frac{x}{3}+2\). 2. The exponent is \(4\). 3. Inside the base, the quotient term is \(\frac{x}{3}\), which divides \(x\) by \(3\).

Base: \(\frac{x}{3}+2\) Exponent: \(4\) Quotient term: \(\frac{x}{3}\); quantities being divided: \(x\) and \(3\)