Week 3 of 10

Pattern Machine

Students find and extend patterns, skip count, and crack the hidden rule of an input-output machine, learning that a rule you can name lets you predict what comes next.

Estimated time: 45-60 minutes
Grade range: Grades 2-5
Skill focus: Patterns, Sequences, Skip counting, Input-output rules
Theme: The Pattern Machine: feed it a number, discover its rule

The story

In the back of the workshop sits the Pattern Machine, all gears and dials. Drop a 2 in the top and a 6 rolls out the bottom. Drop in 5, and out comes 15. The machine won't tell you its secret - but if you watch carefully, you can figure out the rule it's using. Once you know the rule, you can predict every number it will ever make.

What you'll learn

Extend a growing or repeating number pattern and describe its rule in words (for example, 'add 5 each time').
Skip count by 2s, 5s, 10s, 3s, and 4s, and connect skip counting to multiplication.
Figure out the rule of an input-output machine by comparing what goes in with what comes out.

Key ideas

A pattern follows a rule, and naming the rule lets you predict the next term.
Skip counting is repeated addition and the foundation of multiplication.
An input-output rule does the same thing to every number you feed it.

Words to know

Pattern: A sequence that follows a rule you can name.
Rule: The instruction that tells you how to get from one term to the next.
Sequence: An ordered list of numbers or shapes.
Skip counting: Counting forward by the same amount each time, like 5, 10, 15.
Input: The number you put into a machine or rule.
Output: The number that comes out after the rule is applied.

The lesson

A pattern is a sequence that follows a rule. In 3, 6, 9, 12, ... the rule is 'add 3 each time.' Once you name the rule, you can jump ahead: the next term is 15, and you didn't have to draw every step to know it.

Skip counting is patterning you already know. Counting by 5s - 5, 10, 15, 20 - is adding 5 over and over. This is exactly why skip counting and multiplication are cousins: counting by 5s four times gets you to 20, and 4 x 5 = 20.

Patterns can repeat (red, blue, red, blue) or grow (2, 4, 8, 16, where the rule is 'double'). The trick is to check what changes from one term to the next. If the same amount is added each time, the rule is 'add that amount.'

An input-output machine applies one rule to every input. If 2 becomes 6 and 5 becomes 15, the machine multiplies by 3. Test your rule on a third pair to be sure - a good rule works for every input, not just one.

Worked examples

What comes next: 4, 8, 12, 16, ___?

Each term goes up by 4 (the rule is 'add 4'). 16 + 4 = 20. This is also skip counting by 4s.

A machine turns 3 into 8, 5 into 10, and 6 into 11. What is its rule?

Add 5

Compare each input to its output: 3 to 8 is +5, 5 to 10 is +5, 6 to 11 is +5. The same rule works for all three, so the rule is 'add 5'.

Skip count by 5s to find 6 x 5.

5, 10, 15, 20, 25, 30 - six jumps of five lands on 30, which is exactly 6 x 5.

Try it: interactive practice

This is the week's practice game. Work through it right here - try each step yourself, then check your thinking.

Activity

Crack the Machine's Rule

Guessing the rule from input-output pairs is the same reasoning as finding the pattern in a sequence - you look for what stays the same from one step to the next.

Find the missing number

Hands-on challenge

Build a Pattern Chain

What you need

Colored paper strips or beads
Pencil and paper for tables

Steps

1
Use two colors of paper strips or beads to build a repeating pattern at least 12 long.
2
Next to it, build a growing number pattern using tally marks or dots (start with a rule like 'add 3').
3
Ask a partner to name both rules and predict the next three terms.

You've got it when: Your partner can state the rule of each pattern in words and correctly continue it three more steps.

Checkpoint

Quick checks before you move on. Try each one first, then reveal the answer.

1
Continue the pattern: 10, 20, 30, 40, ___, ___.
Show answer
50, 60 (the rule is 'add 10', or skip counting by 10s).
2
A machine turns 4 into 12 and 6 into 18. What is the rule?
Show answer
Multiply by 3 (4 x 3 = 12, 6 x 3 = 18).
3
How is skip counting by 2s related to multiplication?
Show answer
Each skip adds 2, so counting by 2s five times equals 5 x 2 = 10.

Think about it

Why is it useful to know the rule of a pattern instead of just the next number? What can a rule do that a single answer cannot?

Challenge problem

A machine turns 1 into 3, 2 into 5, and 3 into 7. What will it do to 10?

Show hint

This rule has two steps. Notice the output goes up by 2 each time the input goes up by 1 - so multiplying is involved, plus a little extra.

Show answer

21. The rule is 'multiply by 2, then add 1' (1x2+1=3, 2x2+1=5, 3x2+1=7). For 10: 10 x 2 + 1 = 21.

Extension · go further

Invent a two-step machine rule (like 'x 3 then - 1'), make a table of four inputs and outputs, and challenge someone to crack it.

Finished the lesson, activity, and challenge? Mark this week complete to track your progress through the 10-week course.