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2nd Grade algebra · Ages 7–8

Number Machines

Put a number in, the machine does the same thing every time, a new number comes out. That is a function — in Grade 2.

?the unknown

The one idea

A rule is a machine: the same thing happens to every number that goes in.

Number machines are how functions start. The big leap this year is running the machine backwards — if 15 came out of a “+ 6” machine, what went in? That is the idea of undoing, and every equation they ever solve depends on it.

📚 The 4 lessons⚖️ Balance scale✏️ Practice

This page is the right one if they can already…

  • Adds and subtracts within 20 quickly
  • Skip-counts by 2s, 5s and 10s
  • Finds the missing number in 3 + □ = 7

Not yet? Start at 1st Grade: The Same As — algebra only works one rung at a time.

The four 2nd Grade algebra lessons

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

1

In, rule, out

A machine has a rule. Whatever goes in has the rule done to it, and the result comes out.

Worked example

The rule is + 3. What comes out when 9 goes in?

  1. 1

    Put the number in.

    9

  2. 2

    Do the rule to it.

    9 + 3

  3. 3

    Work it out.

    12

Answer: 12 comes out.

Check: Every number going through a + 3 machine comes out 3 bigger. 12 is 3 bigger than 9 ✓

⚠️ The usual mistake

Does the rule to the last output instead of to the new input.

In a table, they chain the answers together as if it were a counting pattern.

Fix: Cover the other rows. Each number goes through the machine on its own.

💬 Say this

“The machine always does the same thing. What will it do to 9?”

🏠 Try it at home

Be the machine. Your child calls out a number; you secretly add 2 and say the answer. Can they guess your rule?

2

Find the rule

If you know what went in and what came out, the gap between them is the rule.

InOut
27
510
813
10?

Worked example

2 → 7, 5 → 10, 8 → 13. What is the rule?

  1. 1

    Look at the first pair. How do you get from 2 to 7?

    2 + 5 = 7

  2. 2

    Try the same rule on the next pair.

    5 + 5 = 10 ✓

  3. 3

    And the next.

    8 + 5 = 13 ✓

Answer: The rule is + 5.

Check: It works for every row, so 10 → 15.

⚠️ The usual mistake

Finds a rule from the first row and stops.

2 → 7 could be + 5 — but a child might also guess “add 3 then add 2” or something that only works once.

Fix: A rule has to work for every row. Test it on all of them before you trust it.

💬 Say this

“That works for the first one. Does it work for all of them?”

🏠 Try it at home

Play “guess my rule” with + and − rules. Whoever guesses in fewest tries wins.

3

Run the machine backwards

To find what went in, do the opposite of the rule to what came out. Subtraction undoes addition.

Worked example

15 came out of a + 6 machine. What went in?

  1. 1

    The machine added 6. The opposite of adding 6 is…

    taking away 6

  2. 2

    Do the opposite to what came out.

    15 − 6

  3. 3

    Work it out.

    9

Answer: 9 went in.

Check: Put 9 back in the machine: 9 + 6 = 15 ✓

⚠️ The usual mistake

Says 21.

They do the rule again instead of undoing it.

Fix: Ask: should the number that went in be bigger or smaller than 15? The machine made it bigger, so the input was smaller.

💬 Say this

“The machine made it 6 bigger. So what was it before?”

🏠 Try it at home

Walk forward 6 steps from a mark, then walk back 6. Where do you end up? Undoing brings you home.

4

Patterns that grow

A growing pattern adds the same amount every step. Know the step and you can find any number in it.

StepTotal
15
210
315
4?

Worked example

5, 10, 15, ?, 25

  1. 1

    How much does it grow each time?

    10 − 5 = 5, 15 − 10 = 5

  2. 2

    Add that to the last number you know.

    15 + 5

  3. 3

    Check it reaches the next number.

    20 + 5 = 25 ✓

Answer: 20

Check: 5, 10, 15, 20, 25 — every gap is 5.

⚠️ The usual mistake

Writes 16.

They count on by one from the last number, out of habit.

Fix: Write the gap between each pair of numbers above the line. Then the pattern is in front of them.

💬 Say this

“How big is the jump each time? Is it always the same?”

🏠 Try it at home

Count fingers on hands round the table: 5, 10, 15… How many for 6 people?

⚖️ Keep it fair

Get ? 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: ? + 2 = 5

?

? + 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.

Number machines

Sometimes the output is hidden, sometimes the input, sometimes the rule. To run a machine backwards, do the opposite.

What is the rule?

47 → 66 2 → 21

Missing numbers to 100

Bigger numbers, same idea: = means “is the same as”.

What number goes in the box?

□ + 17 = 98

The words, in plain English

Rule
What happens to every number.
+ 3 is a rule.
Input
The number that goes in.
4 goes in.
Output
The number that comes out.
7 comes out.
Opposite
The move that undoes another move.
− 6 undoes + 6.
Growing pattern
Gets bigger by the same amount each time.
5, 10, 15, 20

🌍 Algebra in an ordinary day

  • Every person gets 2 cookies — how many for 4 people?
  • Coins: every nickel is 5 cents, so 3 nickels is…
  • Working backwards: “I spent 6 and have 9 left. What did I start with?”

🏁 By the end of 2nd Grade

  • Finds the output of a + or − machine
  • Finds the rule from a table and tests it on every row
  • Runs a machine backwards using the opposite operation
  • Continues and extends growing patterns

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

1st Grade: The Same As

One rung up →

3rd Grade: Letters Are Boxes

All 12 gradesMore maths, grade by grade →

Frequently Asked Questions

What algebra should a 2nd Grade child learn?

A rule is a machine: the same thing happens to every number that goes in. By the end of the year they should be able to: finds the output of a + or − machine; finds the rule from a table and tests it on every row; runs a machine backwards using the opposite operation; continues and extends growing patterns.

Isn’t 2nd 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 ? — something a child can point at. There are no letters, and the ideas match what 2nd grade maths already covers.

What is the most common 2nd Grade algebra mistake?

Finds a rule from the first row and stops. 2 → 7 could be + 5 — but a child might also guess “add 3 then add 2” or something that only works once. The fix: A rule has to work for every row. Test it on all of them before you trust it.

What if my child is stuck?

Go down one rung. Every lesson here builds on the 1st 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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