Evaluating expressions: how to substitute a value for a variable and simplify using the order of operations, with brackets, exponents, negative numbers (the −3² trap), fractions, two variables and formulas, twenty practice questions with answers, the mistakes that lose marks, and a printable sheet

An expression is a string of numbers, variables and operations with no equals sign, like 3x² − 2x + 1. To evaluate it is to find its value when the variables are given numbers, and the whole skill is two steps done carefully: substitute with brackets, then follow the order of operations. It is the Grade 6 entry to algebra (Common Core 6.EE.2) and it comes back in every formula a student ever uses. The method, the traps and twenty practice questions are below, with the printable.

Download the evaluating expressions sheet (PDF) (the method, the order of operations and the worked examples on page 1, twenty practice questions with answers on page 2; two pages, free).

The two steps

1. Substitute in brackets. Rewrite the expression with each variable replaced by its value inside brackets: 3x² − 2x + 1 at x = −2 becomes 3(−2)² − 2(−2) + 1. The brackets are not decoration; they keep a negative sign attached to its number and they make 2x into 2 × (value), which children forget when the x disappears.
2. Order of operations. Brackets, exponents, multiply and divide left to right, add and subtract left to right. 3(−2)² − 2(−2) + 1 = 3(4) − (−4) + 1 = 12 + 4 + 1 = 17.

The two orders people get wrong: multiplication does not come before division (8 ÷ 2 × 4 = 16, left to right), and addition does not come before subtraction (10 − 3 + 2 = 9).

Worked examples

1. 5x + 7 at x = 3. 5(3) + 7 = 15 + 7 = 22.
2. x² − 4x at x = 6. (6)² − 4(6) = 36 − 24 = 12.
3. x² at x = −5. (−5)² = 25. Without the bracket, −5² reads as −(5²) = −25, which is the most common wrong answer in the topic.
4. −x² at x = −5. −(−5)² = −(25) = −25. The square applies to x; the minus in front stays.
5. 2(a + b) − 3b at a = 4, b = −2. 2((4) + (−2)) − 3(−2) = 2(2) + 6 = 4 + 6 = 10.
6. (x + 3)/(x − 1) at x = 5. (5 + 3)/(5 − 1) = 8/4 = 2. The fraction bar is a bracket for the top and for the bottom.
7. 3y/4 at y = 2/3. 3(2/3)/4 = 2/4 = 1/2.
8. 2x³ at x = −1. 2(−1)³ = 2(−1) = −2; an odd power keeps the sign.
9. The area formula A = ½bh at b = 10, h = 7. ½(10)(7) = 35.
10. F = 9C/5 + 32 at C = 25. 9(25)/5 + 32 = 45 + 32 = 77 °F.
11. √(a² + b²) at a = 3, b = 4. √(9 + 16) = √25 = 5. The root is a bracket round everything under it.
12. 4 − 2(x − 1) at x = −3. 4 − 2((−3) − 1) = 4 − 2(−4) = 4 + 8 = 12. Not 2 × (−4) first and then "4 − 2 = 2": the 2 belongs to the bracket.

Twenty practice questions

Evaluate each. Answers follow.

1. 4x − 1 at x = 2
2. 10 − 3x at x = 4
3. x² + 1 at x = 3
4. x² at x = −4
5. −x² at x = −4
6. 2x² − x at x = −3
7. (x − 2)(x + 2) at x = 5
8. 3a + 2b at a = 2, b = 5
9. a² − b² at a = 6, b = 4
10. ab − c at a = −2, b = 3, c = −5
11. (p + q)/2 at p = 7, q = 11
12. 12/x + x at x = 4
13. x/3 + 1 at x = 2/3
14. 5(x − 3) + 2x at x = 3
15. 2x³ at x = 2
16. (−x)² at x = 3
17. 3x − 2y at x = 1/2, y = −1
18. √(x + 9) at x = 16
19. P = 2l + 2w at l = 8, w = 5
20. d = 5t² at t = 3

Answers: 1) 7; 2) −2; 3) 10; 4) 16; 5) −16; 6) 21; 7) 21; 8) 16; 9) 20; 10) −1; 11) 9; 12) 7; 13) 11/9; 14) 6; 15) 16; 16) 9; 17) 7/2; 18) 5; 19) 26; 20) 45.

The mistakes that lose marks

• Squaring a negative without the bracket: x² at x = −3 is 9.
• Reading 2x as "2 and x": at x = 5 it is 10, not 25.
• Doing multiplication before division, or addition before subtraction, instead of left to right.
• Forgetting that a fraction bar and a root sign are brackets.
• Substituting into only one of the places a variable appears.
• Confusing evaluate (find the value) with solve (find x).

Where to get help

If the method is not sticking from a page, a 1-on-1 online class fixes it faster than another worksheet: one tutor with a master's degree, one child, on a shared whiteboard, in your time zone (7 am before school, after 4 pm and weekend mornings, US time; after-school slots for the UK, Canada, Australia and the UAE), for about US$10 to $13 an hour. The first 30-minute class is free: how 1-on-1 online tutoring works.

Related pages

• Equivalent expressions: how to tell and the properties used
• The greater than or equal to sign and the inequality symbols
• Graphing linear equations, four methods

Questions parents ask

See how we teach this, 1-on-1 online →