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6th Grade algebra · Ages 11–12

Balance and Tip

Equations are balanced scales and inequalities are tipped ones. Grade 6 learns to handle both — and to tidy up messy expressions.

xthe unknown

The one idea

An equation is a balance; an inequality is a tipped scale. Whatever you do to one side, do to the other.

The biggest Grade 6 trap is adding things that are not alike: 3x + 4 is not 7x. If your child says “apples and oranges” out loud while simplifying, they will make this mistake far less. Inequalities are new too — the answer is a whole range of numbers, not one.

📚 The 4 lessons⚖️ Balance scale✏️ Practice

This page is the right one if they can already…

  • Solves x + 9 = 23 and 6x = 42
  • Knows 2x means 2 × x
  • Can plot points on a grid

Not yet? Start at 5th Grade: Undo It — algebra only works one rung at a time.

The four 6th Grade algebra lessons

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

1

Combine like terms

You can add x’s to x’s and numbers to numbers — but not x’s to numbers. They are different things.

xxxx
4x + 4

Worked example

Simplify 3x + 4 + x.

  1. 1

    Spot the like terms. x on its own means 1x.

    3x and 1x are alike; 4 is different

  2. 2

    Add the x’s.

    3x + 1x = 4x

  3. 3

    Leave the 4 as it is.

    4x + 4

Answer: 4x + 4

Check: Try x = 2: 3(2) + 4 + 2 = 12 and 4(2) + 4 = 12 ✓

⚠️ The usual mistake

Writes 3x + 4 = 7x.

Adding everything in sight looks like simplifying.

Fix: 3 apples + 4 oranges is not 7 apples. Test with a number: if x = 1, 3x + 4 is 7, but if x = 2 it is 10 — and 7x is 14.

💬 Say this

“Which things are the same kind? Apples with apples, oranges with oranges.”

🏠 Try it at home

Empty a pocket of mixed coins. “3 dimes + 4 pennies + 1 dime” — tidy it up. You can’t call it 8 dimes.

2

One-step equations, all four ways

Whatever was done to x, do the opposite. That includes dividing — and undoing a divide means multiplying.

Worked example

x ÷ 4 = 3

  1. 1

    x was divided by 4. What undoes ÷ 4?

    × 4

  2. 2

    Multiply both sides by 4.

    x = 3 × 4

  3. 3

    Work it out.

    x = 12

Answer: x = 12

Check: 12 ÷ 4 = 3 ✓

⚠️ The usual mistake

Solves x ÷ 4 = 3 as x = 3 ÷ 4.

They copy the operation they see instead of reversing it.

Fix: Ask: is x bigger or smaller than 3? It got divided down to 3, so it started bigger.

💬 Say this

“What happened to x? What undoes that?”

🏠 Try it at home

“I cut a pizza into 4 slices and each slice cost $3. What did the pizza cost?” That is x ÷ 4 = 3.

3

Inequalities: a tipped scale

x > 4 means x is any number bigger than 4. The answer is a whole stretch of the number line, not one spot.

−1012345678910

Worked example

x + 3 > 7

  1. 1

    Treat it like an equation: take 3 off both sides.

    x > 7 − 3

  2. 2

    Work it out.

    x > 4

  3. 3

    Draw it: open circle at 4 (4 itself doesn’t count), shade to the right.

    5, 6, 7, … all work

Answer: x > 4

Check: Try x = 5: 5 + 3 = 8, and 8 > 7 ✓. Try x = 4: 7 > 7 is false, so 4 is not included.

⚠️ The usual mistake

Gives one answer: x = 5.

Every equation so far had exactly one answer.

Fix: Ask “what else works?” until they see it never stops. Then draw the shaded ray.

💬 Say this

“Is that the only number that works? What about 6? 100? 4.5?”

🏠 Try it at home

Find inequalities in the wild: “Must be over 120 cm to ride.” Write it: h > 120.

4

Share out the brackets

3(x + 2) means three lots of (x + 2). Multiply everything inside the brackets by 3.

Worked example

Expand 3(x + 2).

  1. 1

    It means three lots of x + 2.

    (x + 2) + (x + 2) + (x + 2)

  2. 2

    Count the x’s and the 2’s.

    3 x’s and three 2’s

  3. 3

    Write it out.

    3x + 6

Answer: 3x + 6

Check: Try x = 4: 3(4 + 2) = 18 and 3(4) + 6 = 18 ✓

⚠️ The usual mistake

Writes 3x + 2.

They multiply the first thing in the brackets and forget the rest.

Fix: Draw the rectangle: 3 tall, x + 2 wide. Both pieces have a height of 3.

💬 Say this

“The 3 has to visit everyone inside the brackets.”

🏠 Try it at home

Packing lunches: 3 lunches, each with a sandwich and 2 apples. How many of each? 3(s + 2a) = 3s + 6a.

⚖️ Keep it fair

Get x on its own using only fair moves. This is exactly how equations are solved at every grade, right up to Grade 10. Try the wrong button too — see what happens.

Puzzle: 2x + 3 = 11

xx
ten

2x + 3 = 11

Tip: clear the loose dots first, then share. Undo the adding before the multiplying.

✏️ 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.

One-step equations

All four operations, bigger numbers. Do the opposite to both sides.

Solve for x.

11x = 176

Combine like terms

Tap the simplified version. x’s with x’s, numbers with numbers.

Which is the simplified version?

3x + 4 + 5x + 3

The words, in plain English

Like terms
Terms with the same letter part.
3x and x are like; 3x and 4 are not.
Simplify
Tidy an expression by combining like terms.
3x + 4 + x = 4x + 4
Inequality
Says one side is bigger or smaller.
x > 4
Expand
Multiply out the brackets.
3(x + 2) = 3x + 6
Term
One piece of an expression, between + and − signs.
4x + 4 has two terms.

🌍 Algebra in an ordinary day

  • “You need at least 60 points to pass” — that is p ≥ 60
  • Tidying a shopping list: 2 milks + 3 eggs + 1 milk
  • Buying 3 of a meal deal: 3(drink + snack)

🏁 By the end of 6th Grade

  • Combines like terms and never adds x’s to numbers
  • Solves one-step equations with all four operations
  • Solves and graphs one-step inequalities
  • Expands brackets with the distributive property

Where this fits

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

  1. Pre-K❓The Hiding Game
  2. K🎁Make It Fair
  3. 1□The Same As
  4. 2?Number Machines
  5. 3nLetters Are Boxes
  6. 4nNumber Recipes
  7. 5xUndo It
  8. 6xBalance and Tip
  9. 7xTwo Steps Back
  10. 8y = mx + bLines and Slope
  11. 9xAlgebra 1
  12. 10f(x)Curves and Functions

← One rung down

5th Grade: Undo It

One rung up →

7th Grade: Two Steps Back

All 12 gradesMore maths, grade by grade →

Frequently Asked Questions

What algebra should a 6th Grade child learn?

An equation is a balance; an inequality is a tipped scale. Whatever you do to one side, do to the other. By the end of the year they should be able to: combines like terms and never adds x’s to numbers; solves one-step equations with all four operations; solves and graphs one-step inequalities; expands brackets with the distributive property.

Isn’t 6th Grade too early for algebra?

No — balance and tip is standard 6th Grade maths in most curricula. This page breaks it into single moves so it never feels like a jump.

What is the most common 6th Grade algebra mistake?

Solves x ÷ 4 = 3 as x = 3 ÷ 4. They copy the operation they see instead of reversing it. The fix: Ask: is x bigger or smaller than 3? It got divided down to 3, so it started bigger.

What if my child is stuck?

Go down one rung. Every lesson here builds on the 5th 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.

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