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3rd Grade algebra · Ages 8–9

Letters Are Boxes

The first letter shows up in a maths problem this year. It is not scary — it is a box with a name on it.

nthe unknown

The one idea

A letter like n is a box with a name on it. It stands for a number we haven’t found yet.

Children often freeze the first time they see a letter in maths. Tell them the truth: it is exactly the same as the empty box they have used for years. This year also brings multiplication, so the hidden number can now be a missing factor.

📚 The 4 lessons⚖️ Balance scale✏️ Practice

This page is the right one if they can already…

  • Finds a missing number in a + or − equation
  • Knows what a number machine is
  • Understands multiplication as equal groups

Not yet? Start at 2nd Grade: Number Machines — algebra only works one rung at a time.

The four 3rd Grade algebra lessons

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

1

Swap the box for a letter

n + 6 = 10 is the same puzzle as □ + 6 = 10. The letter is just a name for the box.

n6
10

Worked example

n + 6 = 10

  1. 1

    Read n as “the box”.

    □ + 6 = 10

  2. 2

    Take 6 off both sides.

    n = 10 − 6

  3. 3

    Work it out.

    n = 4

Answer: n = 4

Check: 4 + 6 = 10 ✓

⚠️ The usual mistake

Thinks n has to be 14, because n is the 14th letter.

Some children try to give letters a fixed number value.

Fix: Show the same letter meaning different numbers in two puzzles: n + 1 = 3 and n + 1 = 9. The letter is a box, not a code.

💬 Say this

“The n is just the box with a name on it. What number is hiding in the box?”

🏠 Try it at home

Write puzzles on sticky notes using your initials as the unknown: T + 3 = 8. Let them make one for you.

2

The missing factor

If 4 equal groups make 20, share 20 into 4 groups to find how many are in each.

Worked example

4 × n = 20

  1. 1

    There are 4 equal groups making 20.

    n + n + n + n = 20

  2. 2

    Share 20 into 4 equal groups.

    20 ÷ 4

  3. 3

    Work it out.

    n = 5

Answer: n = 5

Check: 4 × 5 = 20 ✓

⚠️ The usual mistake

Writes 80.

They see a × sign and multiply the numbers they can see.

Fix: Put it back in: is 4 × 80 equal to 20? Way too big. The groups must be smaller than the total.

💬 Say this

“4 equal groups make 20. How many are in each group?”

🏠 Try it at home

Deal 20 cards into 4 piles, one at a time like a card dealer. How many in each pile? You just solved 4 × n = 20.

3

Multiplication machines

A × machine makes every number a certain number of times bigger. Division runs it backwards.

nn × 3
13
26
412
7?

Worked example

The rule is × 3. What comes out for 7? And what went in if 24 came out?

  1. 1

    Forwards: do the rule.

    7 × 3 = 21

  2. 2

    Backwards: do the opposite of × 3.

    24 ÷ 3

  3. 3

    Work it out.

    8

Answer: 7 → 21, and 8 went in to make 24.

Check: 8 × 3 = 24 ✓

⚠️ The usual mistake

Says the rule is + 2, because 1 → 3.

They only checked the first row, where + 2 happens to work.

Fix: Test on the next row: 2 + 2 = 4, but the table says 6. The rule must work every time.

💬 Say this

“Is it adding the same amount, or making it so many times bigger? Test it on every row.”

🏠 Try it at home

Wheels: 1 car has 4 wheels, 2 cars have 8. How many for 5 cars? How many cars have 28 wheels?

4

Turn a story into a number sentence

Every word story can be written as an equation. Use a letter for the thing you don’t know.

Worked example

Mia had some stickers. She got 5 more. Now she has 12. How many did she start with?

  1. 1

    Choose a letter for what you don’t know.

    n = stickers at the start

  2. 2

    Write the story in order.

    n + 5 = 12

  3. 3

    Undo the + 5.

    n = 12 − 5 = 7

Answer: Mia started with 7 stickers.

Check: 7 stickers and 5 more is 12 ✓

⚠️ The usual mistake

Writes 12 + 5 = 17.

They grab the two numbers and the word “more”, and add.

Fix: Write the story in order before doing any maths: “some, then 5 more, makes 12”.

💬 Say this

“What don’t we know yet? Let’s call it n, and write the story in order.”

🏠 Try it at home

Make up a story from your day for them to turn into n + __ = __. Then they make one for you.

⚖️ Keep it fair

Get n 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: n + 2 = 5

n

n + 2 = 5

✏️ 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.

Find n

Multiplication and division puzzles. Remember to put your answer back in and check.

Find n.

n × 9 = 81

Times machines

Forwards, backwards, or find the rule — now with × and ÷.

What comes out?

The words, in plain English

Unknown
The number we need to find.
In n + 6 = 10, n is the unknown.
Variable
A letter standing for a number.
n, x, or any letter.
Factor
A number you multiply.
4 and 5 are factors of 20.
Equation
A number sentence with an = sign.
4 × n = 20
Inverse
The opposite operation that undoes it.
÷ is the inverse of ×.

🌍 Algebra in an ordinary day

  • “Tickets are $4 each and we paid $20. How many tickets?”
  • Sharing a bag of 24 sweets between 3 friends
  • A recipe for 2 people that needs doubling

🏁 By the end of 3rd Grade

  • Uses a letter for an unknown without fuss
  • Solves missing-factor equations like 4 × n = 20
  • Finds × and ÷ rules from tables
  • Writes a word story as an equation and solves it

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

2nd Grade: Number Machines

One rung up →

4th Grade: Number Recipes

All 12 gradesMore maths, grade by grade →

Frequently Asked Questions

What algebra should a 3rd Grade child learn?

A letter like n is a box with a name on it. It stands for a number we haven’t found yet. By the end of the year they should be able to: uses a letter for an unknown without fuss; solves missing-factor equations like 4 × n = 20; finds × and ÷ rules from tables; writes a word story as an equation and solves it.

Isn’t 3rd Grade too early for algebra?

Not the way it is taught here. Algebra is finding a number that is hiding, and at this age the hiding spot is n — something a child can point at. There are no letters until this year, and the ideas match what 3rd grade maths already covers.

What is the most common 3rd Grade algebra mistake?

Writes 80. They see a × sign and multiply the numbers they can see. The fix: Put it back in: is 4 × 80 equal to 20? Way too big. The groups must be smaller than the total.

What if my child is stuck?

Go down one rung. Every lesson here builds on the 2nd 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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