Instructions
- Complete the sections in order. The questions begin with review and gradually become more challenging.
- Show your work for every calculation, especially when solving equations.
- You may use a calculator only when your teacher or learning plan allows it.
- For questions with a real-world situation, include units in your answer.
- Check your answers by estimating, substituting, or explaining your reasoning.
Learning goals: By the end of this worksheet, you should be able to compare real numbers, evaluate and simplify algebraic expressions, solve one-step and two-step equations, and apply these skills to everyday situations.
Section 1: Number Sense Warm-Up
A. Classify and compare
-
Classify each number as natural, whole, integer, rational, or irrational. Choose the most specific classification that applies.
a. 7
b. -4
c. 0
d. 3/5
e. √2 -
Insert
<,>, or=.a. -3 -8
b. 0.75 3/4
c. √49 6
d. -1.2 -1.02 -
Arrange these numbers from least to greatest:
-2.5, 1/2, -3, 0, √9
Answer: ____
B. Estimate and reason
-
Between which two consecutive integers does √30 lie?
Answer: ____
-
Without calculating exactly, estimate the value of 19.8 × 0.49. Explain how you estimated.
Section 2: Algebra Vocabulary and Expressions
Helpful example: If x = 4, then 3x + 2 = 3(4) + 2 = 14.
-
In the expression 8n - 5, identify each part.
a. Variable: ____
b. Coefficient: ____
c. Constant: ____
d. Number of terms: ____ -
Translate each phrase into an algebraic expression.
a. 6 more than a number p: ____
b. The product of 9 and a number t: ____
c. A number y divided by 4: ____
d. 12 less than twice a number m: ____ -
Evaluate each expression using the given value.
a. 5x - 3 when x = 7: ____
b. 2a² + 1 when a = 3: ____
c. 18 - 4b when b = -2: ____
d. (q + 5)/3 when q = 10: ____ -
Simplify by combining like terms.
a. 4x + 7x: ____
b. 9a - 3 + 2a + 8: ____
c. 5m + 2 - 8m + 6: ____
d. 3(2y + 4) - y: ____
Section 3: Solving Equations
Hint: Use inverse operations. Whatever you do to one side of an equation, do to the other side.
-
Solve each one-step equation. Show the operation used.
a. x + 9 = 21
x = ____b. y - 14 = -2
y = ____c. 6z = 42
z = ____d. r/5 = -3
r = ____ -
Solve each two-step equation.
a. 3x + 4 = 19
x = ____b. 5p - 7 = 28
p = ____c. -2n + 6 = 18
n = ____d. (w/4) + 3 = 10
w = ____ -
Check your answer to question 11a by substituting your value for x into the original equation.
Section 4: Real-World Math Mission
A school club is planning a snack sale. Each snack costs $1.50 to buy and will be sold for $2.25. The club also pays a one-time table fee of $12.
-
Write an expression for the club's total cost to buy n snacks.
Answer: ____
-
Write an expression for the club's total revenue from selling n snacks.
Answer: ____
-
How much profit will the club make if it sells 40 snacks?
Show your work:
Answer: ____
-
Write and solve an equation to find how many snacks the club must sell to earn $48 in profit.
Equation: ___
Solution: ____ snacks
-
Explain what the solution means in this situation.
Section 5: Organize Your Thinking
Complete the organizer. The first row is an example.
| Situation | Algebraic representation | Answer or next step |
|---|---|---|
| Example: Add 5 to a number x | x + 5 | Write the expression |
| A number n multiplied by 7 | ||
| 20 less than a number k | ||
| Three times a number plus 4 | ||
| A number divided by 6 equals 8 | ||
| The cost of 4 notebooks at c dollars each |
Section 6: Reflection and Challenge
-
A student says that -0.6 is greater than -0.2 because 6 is greater than 2. Is the student correct? Explain using a number line or another representation.
-
Create a real-world situation represented by the equation 2x + 15 = 41. Then solve the equation and explain what x represents.
Situation: _____
Solution: __
-
Optional Challenge: The perimeter of a rectangle is 54 centimeters. Its length is 3 centimeters more than twice its width. Write and solve an equation to find the width and length.
Let the width be w.
Equation: ___
Width: ____ cm
Length: ____ cm -
Optional Extension: Find two different values of a and b that make the equation 3a + b = 20 true. Explain how you know both pairs work.
Pair 1: (a, b) = (__, __)
Pair 2: (a, b) = (__, __)
Explanation: ___
Answer Key
1.
a. 7: natural
b. -4: integer
c. 0: whole
d. 3/5: rational
e. √2: irrational
2.
a. >
b. =
c. >
d. <
3. -3, -2.5, 0, 1/2, √9
4. √30 lies between 5 and 6 because 25 < 30 < 36.
5. Approximately 10. One reasonable estimate is 20 × 0.5 = 10.
6.
a. Variable: n
b. Coefficient: 8
c. Constant: -5
d. Number of terms: 2
7.
a. p + 6
b. 9t
c. y/4
d. 2m - 12
8.
a. 32
b. 19
c. 26
d. 5
9.
a. 11x
b. 11a + 5
c. -3m + 8
d. 5y + 12
10.
a. x = 12
b. y = 12
c. z = 7
d. r = -15
11.
a. x = 5
b. p = 7
c. n = -6
d. w = 28
12. 3(5) + 4 = 19, so the answer is correct.
13. 1.50n + 12
14. 2.25n
15. Revenue: 2.25(40) = 90. Cost: 1.50(40) + 12 = 72. Profit: 90 - 72 = $18.
16. 2.25n - (1.50n + 12) = 48; 0.75n - 12 = 48; 0.75n = 60; n = 80 snacks.
17. The club must sell 80 snacks to earn $48 in profit.
18. No. -0.6 is less than -0.2 because it is farther left on the number line. For example, -0.6 < -0.2.
19. Answers will vary. Example: A game costs $15 to enter and $2 per round. If the total cost is $41, then 2x + 15 = 41. Solving gives x = 13 rounds.
20. Perimeter equation: 2w + 2(2w + 3) = 54. Simplify: 2w + 4w + 6 = 54; 6w = 48; w = 8. Width: 8 cm. Length: 19 cm.
21. Answers will vary. Possible pairs include (a, b) = (5, 5) and (6, 2). Both work because 3(5) + 5 = 20 and 3(6) + 2 = 20.