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15 Algebra Questions And Practice Problems For Middle Schoolers

Algebra questions involve using letters or symbols to represent unknown values or values that can change. Here you will find 15 algebra questions to test your knowledge and show you the different ways that algebra can be used to solve a problem to find an unknown value or to make generalizations.

Algebra in middle school

There are many topics and techniques within algebra. In middle school we learn to write and manipulate basic algebraic expressions and linear equations. Later, we develop these techniques to allow us to deal with more complicated polynomials and math problems such as ones that involve quadratic equations or a system of equations, so it is important to establish a solid foundation.


How to solve algebraic equations

When you are presented with an algebraic problem it is important to establish what you are being asked to do, especially if it is on a math test. Here are some of the key terms along with what they mean:

  • Solve the equation – find out the value of the unknown
  • Substitute – put the values you have been given into the algebraic expression
  • Coefficients – the amount a term has been multiplied by. For example, in the expression 4c + 2, 4 is the coefficient.
  • Constants – values which are not variable and stay the same. For example, for 4c + 2, 2 is the constant.
  • Binomial – a binomial or binomial expression is one which has two parts, like our example 4c + 2.
  • Simplify – collect together like terms to make the expression or equation look simpler
  • Expand – multiply out the expressions inside parentheses
  • Factorize – put into parentheses
  • In terms of x – rewrite the equation in the form x =…..

Remember, when working with algebra, we must still apply the order of operations, PEMDAS. i.e. Parenthesis, Exponents (powers, square roots), Multiplication, Division, Addition, Subtraction.

When working with algebraic expressions and equations we must consider carefully which operations to deal with first.


Algebra in elementary school

Students as early as kindergarten begin to solve problems with algebraic thinking. Algebraic thinking starts with students understanding the properties of the operations, the relationships between them, and the composition of numbers.

They begin to relate concrete models to abstract expressions and equations, and continue this work throughout elementary school. They also look at patterns and the order of operations that will contribute to their understanding of algebra in later years.

This flexible understanding with the composition and representation of numbers, as well as the operations sets them up for more complex and abstract equations once they get to middle school. 


Algebra in middle school

The ideas of writing and simplifying expressions, solving equations and substitution are introduced in elementary school. Here are some example elementary level algebrsa questions:


Algebra practice questions for middle school

1. A chocolate bar costs c pence and a drink costs d pence. Write down an expression for the cost of 2 chocolate bars and 2 drinks.

c+d
GCSE Quiz False

2c+d
GCSE Quiz False

2c+2d
GCSE Quiz True

2c-2d
GCSE Quiz False

2 chocolate bars would cost 2 lots of c, or 2c, and 2 drinks would cost 2 lots of d, or 2d.

2. Simplify the expression 4m+5+2m-1

10m
GCSE Quiz False

6m+4
GCSE Quiz True

6m+6
GCSE Quiz False

6m-4
GCSE Quiz False

We need to collect together like terms here so 4m + 2m = 6m and 5 – 1 = 4 (watch out for the negative).

In middle school we learn a variety of different algebra techniques to answer algebra questions and to practice problem solving with algebra. These include:

  • Simplifying algebraic expressions
  • Expanding brackets and factoring
  • Forming algebraic equations from word problems
  • Solving algebraic equations and inequalities
  • Substituting into expressions
  • Changing the subject of an equation
  • Working with real life graphs and straight line graphs
  • Sequences
gcse line graph equation
A slide guiding pupils through the equation of a straight line from Third Space Learning’s online intervention.


Algebra questions for middle school: basic algebra

1. In this pyramid, you add two adjacent blocks to find the value of the block above.

 

basic algebra question 1a

 

What expression will be in the top box?

8a-2b
GCSE Quiz False

12a-b
GCSE Quiz True

12a+5b
GCSE Quiz False

10a+b
GCSE Quiz False

basic algebra question 1b

2.  Brian is a window cleaner. He uses the following formula to calculate the amount to charge his customers:

 

Charge = £20 + 4n

Where n is the number of windows a house has.

 

If a house has 7 windows, how much would Brian charge?

£24
GCSE Quiz False

£67
GCSE Quiz False

£48
GCSE Quiz True

£27
GCSE Quiz False

In this question, n is 7 so we can substitute 7 into the formula.

 

Charge = £20 + 4 × 7

Charge = £48

3. The area of a rectangle is 4x-6.

 

basic algebra question 3

 

Which of the following pairs could be the length and width of the rectangle?

4x and 6x

GCSE Quiz False

4 and x-6

GCSE Quiz False

2 and 2x-3

GCSE Quiz True

2x and 2x-3

GCSE Quiz False

There are two ways of attempting this question. We know that area \;of \;a \;rectangle = length × width so we could multiply each pair together to see which pair makes 4x − 6.

 

\begin{aligned} &4x × 6x \quad \quad\quad\; \;24x^2 \\ &4 × (x − 6) \quad \quad \quad4x − 24\\ &2 × (2x − 3) \quad \quad \;\;4x − 6\\ &2x × (2x − 3) \quad \quad 4x^2 − 6x\\ \end{aligned}

 

Alternatively, if we factorise 4x − 6 we get 2(2x − 3) meaning the sides could be 2 and 2x − 3.

4.  The formula for changing degrees Celsius to degrees Fahrenheit is

 

F=\frac{9C}{5}+32

 

Rearrange this formula to make C the subject.

C=\frac{5(F-32)}{9}
GCSE Quiz True

C=\frac{5F-32}{9}
GCSE Quiz False

C=\frac{5F}{9}-32
GCSE Quiz False

C=5F-\frac{32}{9}
GCSE Quiz False
\begin{aligned} F&=\frac{9C}{5}+32 \hspace{3cm} &\text{subtract 32}\\\\ F-32&=\frac{9C}{5} &\text{multiply by 5}\\\\ 5(F-32)&=9C &\text{divide by 9}\\\\ \frac{5(F-32)}{9}&=C \end{aligned}


Algebra questions for middle school: forming and solving equations

5. Work out the size of the smallest angle.

algebra question ks3

20^{\circ}
GCSE Quiz False

26^{\circ}
GCSE Quiz False

8^{\circ}
GCSE Quiz False

34^{\circ}
GCSE Quiz True

The angles in a triangle add up to 180^{\circ} therefore we can write

 

4x+2x-10+3x-8=180

 

Now we have an equation we can solve.

 

\begin{aligned} 9x-18&=180 \hspace{3cm} &\text{add } 18\\ 9x&=198 &\text{divide by } 9\\ x&=22^{\circ} \end{aligned}

 

The angles are :

 

\begin{aligned} 4\times22&=88^{\circ}\\ 2\times22 -10&=34^{\circ}\\ 3\times22 -8&=58^{\circ} \end{aligned}

 

The smallest angle is 34^{\circ} .

6.  Jamie’s dad is 4 times older than Jamie. In 14 years time, Jamie’s dad will be twice the age of Jamie.

 

What is the sum of Jamie’s age now and Jamie’s dad’s age now?

70
GCSE Quiz False

42
GCSE Quiz False

22
GCSE Quiz False

35
GCSE Quiz True

To solve this we need to write an equation.

 

Let Jamie’s age now be x . Then Jamie’s dad’s age is 4x .

 

In 14 years time Jamie’s age will be x + 14 and Jamie’s dad’s age will be 4x + 14 .

 

Since we know Jamie’s dad’s age will be two times Jamie’s age, we can write

 

4x+14=2(x+14)

 

Now we have an equation we can solve.

 

\begin{aligned} 4x+14&=2(x+14) \hspace{3cm} &\text{expand the brackets}\\ 4x+14&=2x+28 \hspace{3cm} &\text{subtract 2x}\\ 2x+14&=28 \hspace{3cm} &\text{subtract 14}\\ 2x&=14 \hspace{3cm} &\text{expand the brackets}\\ x&=7 \hspace{3cm} \end{aligned}

 

Jamie is currently 7 years old meaning his dad is 28 years old. The sum of their ages is 35 .

Note: when algebraic equations contain denominators on either side, we can use the cross multiplication method to help us work out the answer. For example with the following expressions:

\frac{a}{b} = \frac{c}{d}

This can then become

ad = bc

And so on.


Algebra questions for middle school: graphs

7.  Which of the following lines passes through the point (2, 5)?

y=2x-1
GCSE Quiz False

y=2x+1
GCSE Quiz True

y=4x-2
GCSE Quiz False

y=2x+5
GCSE Quiz False

At the point (2, 5), x is 2 and y is 5. We can check which equation works when we substitute in these values:

 

\begin{aligned} y&=2x−1 \quad \quad \quad 5=2×2−1 \quad \quad \text{False}\\ y&=2x+1 \quad \quad 5=2×2+1 \quad \quad \text{True}\\ y&=4x−2 \quad \quad \quad 5=4×2−2 \quad \quad \text{False}\\ y&=2x+5 \quad \quad 5=2×2+5 \quad \quad \text{False} \end{aligned}


Looking for more algebra math questions?

Try these:

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The content in this article was originally written by secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Jaclyn Wassell.

Beki Christian
Beki Christian
Maths Teacher and Author
Beki has 11 years experience teaching secondary Maths. She is currently working as a Maths tutor as well as writing resources and blog posts for Third Space Learning.
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