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6th Grade · 6.EE.A.2

Algebraic expressions in 6th grade

Algebraic expressions are math phrases that mix numbers, letters and operation signs, such as 4n+74n + 7. In 6th grade, each letter stands for a number. Your child learns to write expressions from words, name their parts and find their value when the letter is replaced by a number.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers.

  • 6.EE.A.2.a Write expressions that record operations with numbers and with letters standing for numbers.
  • 6.EE.A.2.b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.
  • 6.EE.A.2.c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

What it means, in plain words

An expression has no equals sign. The letter in it is a variable, a letter that stands for a number. 4n+74n + 7 means "4 times a number, plus 7". Algebraic expressions in 6th grade use a short way of writing: 4n4n means 4 times nn, and n2\frac{n}{2} means nn divided by 2.

Writing expressions means turning words into this short form. "The product of 6 and the difference of yy and 1" becomes 6(y−1)6(y - 1). Your child also learns the names of the parts. A term is a part joined to the rest by a plus or minus sign, so 4n+74n + 7 has the terms 4n4n and 7. The coefficient is the number multiplied by the letter, here 4. Sum, difference, product and quotient are the results of adding, subtracting, multiplying and dividing. A part in parentheses, like (y−1)(y - 1), can be read as one amount.

Evaluating expressions means putting a number in place of the letter and working out the value. The order of operations decides what comes first: anything in parentheses, then exponents, then multiplying and dividing, then adding and subtracting. Formulas work the same way. The area of a square is s2s^2, so a square with sides of 5 centimeters has an area of 52=255^2 = 25 square centimeters.

Your child should be able to:

  • Write an expression from words such as "5 more than twice a number"
  • Name the terms and the coefficient of an expression, and read a part in parentheses as one amount
  • Evaluate an expression for a given value of the letter, in the right order of operations
  • Use a formula, such as s2s^2 for the area of a square, to find a real amount

Example

Write an expression for "7 less than 4 times a number".

  1. Call the number nn.
  2. "4 times a number" is 4n4n.
  3. "7 less than" means 7 is taken away from 4n4n, so the subtraction comes after it.
  4. The expression is 4n−74n - 7, not 7−4n7 - 4n.

Example

Evaluate 3x2+23x^2 + 2 when x=4x = 4.

  1. Put 4 in place of xx: 3×42+23 \times 4^2 + 2.
  2. Exponent first: 42=164^2 = 16.
  3. Then multiply: 3×16=483 \times 16 = 48.
  4. Then add: 48+2=5048 + 2 = 50. The value of the expression is 50.

Where children usually get stuck

  • Your child writes "3 less than a number" as 3−n3 - n.

    The words are copied in the order they appear. In "3 less than a number", the 3 is taken away from the number, so it comes second.

    What helps. Try it with a real number first. "3 less than 10" is 7. Ask which gives 7, 10−310 - 3 or 3−103 - 10. Then put nn where the 10 was: n−3n - 3.

  • To evaluate 2x22x^2 when x=5x = 5, your child works out 2×5=102 \times 5 = 10, then 102=10010^2 = 100.

    Your child works from left to right. The exponent belongs only to the xx, so it has to come before the multiplying.

    What helps. Ask what is being squared. Circle x2x^2 and do it first: 52=255^2 = 25, then 2×25=502 \times 25 = 50.

  • Your child evaluates 4n4n when n=6n = 6 and writes 46.

    In earlier grades two digits side by side made one number. With 6 in place of nn, 4n4n looks like 46.

    What helps. Have your child write the multiplication sign back in when replacing the letter: 4×6=244 \times 6 = 24. The sign can be dropped again once the habit sticks.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

  1. 1Warm-up
    Marcus wrote the phrase "2 times the sum of x and 5" on the board. The table below shows Sofia's and Noah's expressions for it.
    StudentExpression
    Sofia
    2(x+5)\displaystyle 2(x+5)
    Noah
    2(5+x)\displaystyle 2(5+x)
    Select the student or students who wrote a correct expression.
    A
    Sofia
    B
    Noah
    C
    Both students
    D
    Neither student
    Show the answer and how to solve it

    Both students are correct because x + 5 and 5 + x are the same sum, and both expressions multiply that grouped sum by 2.

    1. 1The inner part is 'the sum of x and 5.' Because addition is commutative, x + 5 and 5 + x represent exactly the same sum.
    2. 2Multiply that sum by 2, keeping the entire sum grouped in parentheses.
    3. 3Compare both expressions with the table. Sofia wrote 2(x + 5) and Noah wrote 2(5 + x). Both multiply the whole grouped sum by 2; the only difference is addend order inside the parentheses, which is allowed by commutativity.
    4. 4Both students wrote a correct expression.

    Common mistake. A common mistake is choosing only Sofia and overlooking Noah. Since addition is commutative, 5 + x is just as valid as x + 5, so 2(5 + x) also correctly matches the phrase.

    The idea underneath. In a product involving a sum or difference, the order named in the phrase matters, and any inner sum or difference must stay together as one quantity inside parentheses.

  2. 2On level
    Draw lines to match each phrase to the algebraic expression that means the same thing. One expression will not be used.
    Connect each item on the left to its match on the right.
    Phrase
    Expression
    the sum of x and 5
    the product of 3 and t
    the difference of n and 8
    n−8\displaystyle n - 8
    8−n\displaystyle 8 - n
    3t\displaystyle 3t
    x+5\displaystyle x + 5

    Use Tab to navigate between items. Click or tap a card to select it, then click or tap a card on the opposite side to create a match. You can also drag a card or its connector dot to a card on the opposite side.

    Show the answer and how to solve it

    The sum phrase matches x + 5, the product phrase matches 3t, and the difference phrase matches n - 8. The extra expression 8 - n is unused because it reverses the order of the difference phrase.

    1. 1Each operation word points to a specific symbol.
    2. 2Match the sum and product phrases to their expressions.
    3. 3For difference phrases, the order in the words must be kept exactly. "The difference of n and 8" means n comes first and 8 is subtracted from it.
    4. 4The expression 8−n8 - n reverses that order, putting 8 first. Since neither phrase calls for that arrangement, it is the unused expression.

    Common mistake. A common mistake is choosing 8 - n for "the difference of n and 8" because both use the same numbers, but the order in the phrase requires n - 8, not 8 - n.

    The idea underneath. Each phrase names exactly one operation, so look for the expression that uses that same operation.

  3. 3On level
    Each expression on the left is written in a nonstandard way.
    Drag to match each one to the same expression written in standard algebra notation on the right.
    Connect each item on the left to its match on the right.
    Nonstandard form
    Standard algebra notation
    x3\displaystyle x3
    x÷5\displaystyle x \div 5
    4×m−2\displaystyle 4 \times m - 2
    4m−2\displaystyle 4m - 2
    x/5\displaystyle x/5
    3x\displaystyle 3x

    Use Tab to navigate between items. Click or tap a card to select it, then click or tap a card on the opposite side to create a match. You can also drag a card or its connector dot to a card on the opposite side.

    Show the answer and how to solve it

    The correct matches are x3 → 3x, x ÷ 5 → x/5, and 4 × m − 2 → 4m − 2.

    1. 1Look at the left column. Two items share the variable x: x3 uses multiplication, and x ÷ 5 uses division. Because x appears in both, you cannot match by letter alone — you must read the operation to choose the correct right-side expression.
    2. 2For the two multiplication items: move the coefficient in front of the variable and drop the multiplication sign. x3 becomes 3x, and 4 × m − 2 becomes 4m − 2 (only the multiplication sign is removed; the constant term −2 is unchanged).
    3. 3For the division item: keep the variable as the numerator and write the divisor as the denominator. x ÷ 5 becomes x/5.
    4. 4All three left items have exactly one match on the right. Every right-side expression is used exactly once.

    Common mistake. A common mistake is matching x ÷ 5 with 3x because both contain x. Reading the operation reveals x ÷ 5 is a division, so it matches x/5, while x3 is a multiplication that matches 3x.

    The idea underneath. Standard algebra notation keeps the meaning the same while following clear patterns: coefficients go before variables, multiplication signs are usually omitted, and quotients are written with a slash. When two left expressions share a variable, the operation tells you which standard form is the match.

  4. 4Stretch
    Noor wrote an expression for the phrase below.
    Phrase: the fifth power of xNoor’s expression: x5\displaystyle \begin{array}{l} \text{Phrase: the fifth power of } x \\ \text{Noor's expression: } x^5 \end{array}
    Select the best evaluation of Noor's work.
    A
    Noor used multiplication instead of an exponent.
    B
    The expression is correct.
    C
    Noor used the wrong exponent.
    D
    Noor swapped the base and exponent.
    Show the answer and how to solve it

    The best evaluation is: The expression is correct. The fifth power of x is x5x^5, and Noor wrote exactly that.

    1. 1Look at the phrase and Noor's expression.
    2. 2The fifth power of x means x is used as a factor 5 times.
    3. 3Noor's expression is x5x^5, which also shows x used as a factor 5 times.
    4. 4The phrase and Noor's expression match exactly, so the expression is correct.

    Common mistake. A common mistake is assuming an error must exist because the item seems designed to test error detection. Always compare the target exponent with the written exponent — here both are 5, so the expression is correct.

    The idea underneath. In exponent notation, the exponent tells how many times the base is used as a factor, so changing the exponent changes the meaning of the whole expression.