Lesson 4.3 · Factors, Multiples and Patterns
Number and shape patterns
Patterns are everywhere: the stripes on a flag, the numbers on houses along a street, the way a plant grows new leaves. In math, a pattern follows a rule. If you know the rule, you can figure out what comes next, even 20 steps later.
Number patterns
Look at this list of numbers:
Each number is 3 more than the one before it. The rule is start at 4, add 3. Each number in the pattern is called a term. The first term is 4, the second term is 7, and so on.
Definition
Pattern rule
A rule tells you how to get from one term to the next, such as "add 5" or "multiply by 2." A full rule also says where to start.
Rules can use any operation:
- Start at 50, subtract 6: 50, 44, 38, 32, 26, …
- Start at 3, multiply by 2: 3, 6, 12, 24, 48, …
Finding the rule
To find the rule of a number pattern, compare each term with the next one. Ask: what did I do to get from this number to the next?
For 5, 9, 13, 17: , , . The rule is add 4.
For 2, 6, 18, 54: the jumps are 4, 12, 36. Those are not the same, so it is not an adding rule. Try multiplying: , , . The rule is multiply by 3.
Check every step
A rule must work for every pair of terms in the pattern, not just the first two. Test it all the way to the end before you use it.
Features the rule doesn't tell you
A pattern often has hidden features. Look at the rule start at 1, add 4:
Every term is odd. The rule never says "odd," but it happens anyway. Why? You start with an odd number, and adding an even number (4) to an odd number always gives another odd number.
Now try start at 1, add 3: 1, 4, 7, 10, 13, 16, … The terms go odd, even, odd, even. Adding an odd number flips odd to even and even to odd each time.
Shape patterns
A shape pattern repeats or grows. This one repeats:
🔺🟦🟦🔺🟦🟦🔺🟦🟦
The part that repeats, 🔺🟦🟦, is called the core. It has 3 shapes.
To find a shape far down the line, use the core. Which shape is 20th? Every 3rd shape is the end of a core, so shapes 3, 6, 9, 12, 15, 18 are all 🟦 (the last one in the core). Shape 19 starts a new core, so it is 🔺, and shape 20 is 🟦.
Tip
You can also divide. R2. That means 6 full cores, then 2 more shapes. The 2nd shape in the core is 🟦.
Shape patterns can also grow. Here each step adds 2 squares:
Step 1: 🟩🟩
Step 2: 🟩🟩🟩🟩
Step 3: 🟩🟩🟩🟩🟩🟩
The number of squares is 2, 4, 6, … The rule is start at 2, add 2. Step 10 would have 20 squares.
Worked example: Continue the pattern
The rule is start at 7, add 5. Write the first six terms.
Notice the ones digits: they go 7, 2, 7, 2, … That happens because adding 5 twice adds 10, which brings the ones digit back to where it was.
Worked example: Find the missing term
Find the missing number: 64, 56, 48, ___, 32.
The terms go down: and . The rule is subtract 8.
. Check: ✔. The missing number is 40.
Worked example: A shape far away
A pattern repeats ⭐🌙🌙☀️. What is the 15th shape?
The core has 4 shapes. R3. That is 3 full cores and then 3 more shapes. The 3rd shape in the core is 🌙.
The 15th shape is 🌙.
Common mistake
Don't find the rule from the first two terms only. In 2, 4, 8, 16, the first jump is "add 2," but is 6, not 8. The real rule is multiply by 2. Always check the rule on every term.
Practice
The rule is start at 5, add 6. What is the fifth term?
Type a number.
The pattern 3, 10, 17, 24, 31 follows the rule "add ___." What number goes in the blank?
Type a number.
Find the missing number: 90, 81, 72, 63, 54, ___.
Type a number.
The rule is start at 5, multiply by 2. What is the fifth term?
Type a number.
A pattern repeats: 🟥🟨🟦. What is the 14th shape?
The rule is start at 2, add 4. Which is true about every term in the pattern?
Step 1 of a pattern has 4 dots. Each step adds 3 more dots. How many dots are in step 10?
Type a number.
Which rule makes a pattern whose terms go odd, even, odd, even, …?