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9th Grade algebra · Ages 14–15

Algebra 1

Algebra 1 feels like a hundred new rules. It is really three old moves — balance, undo, rewrite — used on bigger things.

xthe unknown

The one idea

Every algebra move is one of three things: keep it balanced, undo in reverse, or rewrite it in a friendlier shape.

If your teenager is stuck, ask which of the three moves they are trying to make. Factoring and expanding are both “rewrite” — the same thing said a different way — and the area rectangle from Grade 6 still works for both.

📚 The 4 lessons ✏️ Practice

This page is the right one if they can already…

Solves two-step equations with negatives
Expands a(x + b)
Finds slope and writes y = mx + b

Not yet? Start at 8th Grade: Lines and Slope — algebra only works one rung at a time.

The four 9th Grade algebra lessons

In order — each one leans on the last. Every example is broken into single moves and ends by checking itself.

Variables on both sides

Get all the x’s together on one side first — subtract the smaller x term from both sides. Then it’s a two-step equation.

xxxxx

xx15

Worked example

5x + 3 = 2x + 15

1
Take 2x off both sides.
3x + 3 = 15
2
Take 3 off both sides.
3x = 12
3
Divide both sides by 3.
x = 4

Answer: x = 4

Check: Left: 5(4) + 3 = 23. Right: 2(4) + 15 = 23 ✓

⚠️ The usual mistake

Writes 5x + 2x + 3 = 15.

“Move it to the other side” turned into moving without changing the sign.

Fix: Never “move” anything. Do the same thing to both sides — subtract 2x from both — and write it down.

💬 Say this

“What can you take off both sides so x only appears once?”

🏠 Try it at home

Two savings plans: one starts with $15 and adds $2 a week, one starts with $3 and adds $5. When are they level?

Exponent rules come from counting

x³ means x × x × x. Every exponent rule is just counting how many x’s are being multiplied.

Written	Means
x² · x³	(x·x)(x·x·x) = x⁵
(x²)³	(x·x)(x·x)(x·x) = x⁶
x⁵ ÷ x²	x·x·x·x·x ÷ x·x = x³
x⁰	?

Worked example

Simplify x² · x³.

1
Write out what each power means.
(x · x) · (x · x · x)
2
Count the x’s being multiplied.
5 of them
3
Write it as a power.
x⁵

Answer: x⁵ — multiplying powers adds the exponents.

Check: Try x = 2: 2² · 2³ = 4 · 8 = 32, and 2⁵ = 32 ✓

⚠️ The usual mistake

Writes x² · x³ = x⁶.

They multiply the exponents because the bases are being multiplied.

Fix: Write the x’s out and count them. If you forget a rule, you can always rebuild it this way.

💬 Say this

“Write it out the long way. How many x’s are there?”

🏠 Try it at home

Fold a sheet of paper in half again and again. 1 fold = 2 layers, 2 folds = 4… layers = 2ⁿ. How many after 5 folds?

Multiply two brackets

Every term in the first bracket multiplies every term in the second. The area rectangle makes sure nothing is missed.

xx²3x

22x6

Worked example

Expand (x + 2)(x + 3).

1
Draw a rectangle x + 2 tall and x + 3 wide.
four boxes
2
Fill each box: row × column.
x², 3x, 2x, 6
3
Add the boxes, combining like terms.
x² + 5x + 6

Answer: x² + 5x + 6

Check: Try x = 1: (3)(4) = 12 and 1 + 5 + 6 = 12 ✓

⚠️ The usual mistake

Writes x² + 6.

They multiply the firsts and the lasts, and miss the middle.

Fix: Four boxes, four products. If you only have two terms, you’ve missed two boxes.

💬 Say this

“Did every piece in the first bracket meet every piece in the second?”

🏠 Try it at home

Cut a paper rectangle into four pieces with lengths x + 2 and x + 3. Label each piece’s area.

Factor it back

Factoring is expanding in reverse. For x² + bx + c, find two numbers that multiply to c and add to b.

xx²4x

33x12

Worked example

Factor x² + 7x + 12.

1
List pairs that multiply to 12.
1 & 12, 2 & 6, 3 & 4
2
Which pair adds to 7?
3 + 4 = 7
3
Write the brackets.
(x + 3)(x + 4)

Answer: (x + 3)(x + 4)

Check: Expand it back: x² + 4x + 3x + 12 = x² + 7x + 12 ✓

⚠️ The usual mistake

Picks 2 and 6 because they multiply to 12.

They stop at the first pair that multiplies right.

Fix: The pair has two jobs: multiply to c AND add to b. Check both before you write the brackets.

💬 Say this

“Which two numbers multiply to 12 and add to 7?”

🏠 Try it at home

Play “product and sum”: say “multiply to 18, add to 9” and race to find 3 and 6.

✏️ Practice that never runs out

Every question is new. A wrong answer gets a hint; a second one shows the worked solution, so no one stays stuck.

Variables on both sides

Get the x’s together first, then finish it as a two-step equation.

Solve for x.

2x − 9 = x − 6

Factor it

Find two numbers that multiply to the last number and add to the middle one.

Factor it.

x² + 6x + 8

The words, in plain English

Exponent: How many times to multiply a number by itself.; x³ = x · x · x
Binomial: An expression with two terms.; x + 3
Expand: Multiply out the brackets.; (x + 2)(x + 3) = x² + 5x + 6
Factor: Rewrite as brackets multiplied together.; x² + 7x + 12 = (x + 3)(x + 4)
Quadratic: Has an x² term as its highest power.; x² + 5x + 6

🌍 Algebra in an ordinary day

Comparing two deals that start at different prices and grow at different rates
Doubling: bacteria, folded paper, compound interest
Area of a garden that grows by 2 m one way and 3 m the other

🏁 By the end of 9th Grade

Solves equations with variables on both sides
Uses exponent rules and can rebuild them by counting
Expands two brackets with the area model
Factors x² + bx + c

Where this fits

One idea, twelve rungs. The hiding spot just gets cleverer.

← One rung down

8th Grade: Lines and Slope

One rung up →

10th Grade: Curves and Functions

All 12 grades More maths, grade by grade →

Frequently Asked Questions

What algebra should a 9th Grade child learn?

Every algebra move is one of three things: keep it balanced, undo in reverse, or rewrite it in a friendlier shape. By the end of the year they should be able to: solves equations with variables on both sides; uses exponent rules and can rebuild them by counting; expands two brackets with the area model; factors x² + bx + c.

Isn’t 9th Grade too early for algebra?

No — algebra 1 is standard 9th Grade maths in most curricula. This page breaks it into single moves so it never feels like a jump.

What is the most common 9th Grade algebra mistake?

Writes x² · x³ = x⁶. They multiply the exponents because the bases are being multiplied. The fix: Write the x’s out and count them. If you forget a rule, you can always rebuild it this way.

What if my child is stuck?

Go down one rung. Every lesson here builds on the 8th Grade page, and the “ready for this page if” list at the top tells you exactly what they need first. In practice, a wrong answer gives a hint, and a second wrong answer shows the full worked solution.

Is it free?

Yes. All 4 lessons, the pictures, the worked examples, the balance scale and the practice questions are free with no signup. The practice never runs out — every question is generated fresh and checked.

Algebra 1

This page is the right one if they can already…

The four 9th Grade algebra lessons

✏️ Practice that never runs out

Variables on both sides

Factor it

The words, in plain English

🌍 Algebra in an ordinary day

🏁 By the end of 9th Grade

Where this fits

Frequently Asked Questions

What algebra should a 9th Grade child learn?

Isn’t 9th Grade too early for algebra?

What is the most common 9th Grade algebra mistake?

What if my child is stuck?

Is it free?

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