1 · Repeating patterns
A repeating pattern has a core that happens again and again. Circle, square, circle, square... the core is "circle, square."
Find the core, then keep going. After circle, square, triangle, circle, square, triangle comes... circle.
2 · Number patterns that add
3, 6, 9, 12... Each number is 3 more than the one before. The rule is "add 3." To find the rule, subtract neighbors: 6 − 3 = 3, 9 − 6 = 3.
Next comes 12 + 3 = 15.
3 · Patterns that count back
Patterns can shrink too. 50, 45, 40, 35... the rule is subtract 5. Next: 30.
If the numbers get smaller, the rule subtracts (or divides).
4 · Growing patterns
Some patterns grow faster and faster. 3, 6, 12, 24: each number is double the last. Next: 48.
A picture pattern can grow too. These rows go 1, 3, 5: two more squares each time. Stage 4 has 7.
5 · Patterns in the times tables
Multiples of 5 always end in 0 or 5. Multiples of 10 end in 0. Multiples of 2 are the even numbers (they end in 0, 2, 4, 6, or 8).
So 35 is a multiple of 5, and 48 is even. Spotting these saves time on a test.
6 · In and out: find the rule
A rule machine changes every number the same way. In 2 → Out 6. In 5 → Out 15. In 8 → Out 24. What is the rule? Each Out is the In times 3.
So In 10 → Out 30. Test your rule on every row before you trust it.
7 · Missing numbers in equations
An equation is a balance: both sides weigh the same. 8 + □ = 15. What goes with 8 to make 15? Think 15 − 8 = 7.
□ − 6 = 9: the number was 9 before taking 6 away, so add back: 9 + 6 = 15. 4 × □ = 24: 24 ÷ 4 = 6.
8 · Working backwards
"I think of a number, add 5, and get 12. What was my number?" Go backwards and do the opposite: 12 − 5 = 7.
"I double a number, then subtract 4, and get 10." Backwards: 10 + 4 = 14, then 14 ÷ 2 = 7.
9 · Two-step problems
"Leo has 3 boxes of 8 crayons. He gives away 5. How many are left?" Two steps: first 3 × 8 = 24, then 24 − 5 = 19.
Write the first answer down before you start the second step.
10 · Which must be true?
"Sam is older than Ali. Ali is older than Kim." Line them up: Sam, Ali, Kim. So Sam is older than Kim: that must be true.
"Kim is 6 years old" might be true, but the clues do not say so. On a test, pick only what the clues force.
11 · Guess and check with a table
"Two numbers add to 10 and multiply to 21." Try pairs that add to 10 and check the product: 1 and 9 → 9. 2 and 8 → 16. 3 and 7 → 21. That is the pair.
A table keeps your guesses tidy so you do not repeat one.
12 · Read like a detective
Some problems hide extra information. "Mia has 4 red pens, 3 blue pens, and 2 notebooks. How many pens?" The notebooks are a distraction: 4 + 3 = 7.
Watch the words: "fewer" means subtract, "each" often means multiply, "altogether" means add.
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