Learn Without Walls
📄 Save or print this lesson:
← Previous Lesson Lesson 2 of 4 Next Lesson →

Lesson 2: Completing Sequences

About 15-20 minutes -- Screen-free lesson

What You Will Learn

By the end of this lesson, you will be able to:

Fill in the Missing Piece

Sometimes you see a pattern with a piece missing in the middle. This is like a puzzle! You need to figure out the rule first, then use it to find the missing piece.

Try It: What Goes in the Blank?

Puzzle 1:

3, 6, ___, 12, 15
Answer: 9! The rule is "add 3 each time." 3, 6, 9, 12, 15.

Puzzle 2:

A, B, B, A, ___, B, A, B, B
Answer: B! The repeating group is "A, B, B" so the full pattern is: A, B, B, A, B, B, A, B, B.

Puzzle 3:

2, 4, 8, ___, 32, 64
Answer: 16! The rule is "double it each time" (multiply by 2). 2, 4, 8, 16, 32, 64.

Strategy: Work From Both Sides

When there is a blank in the middle, look at the numbers or items on both sides of the blank. This can help you figure out the rule.

For example: 10, ___, 30. Look at the numbers you know: 10 and 30. What is the difference? 30 - 10 = 20. But there are two steps (10 to blank, blank to 30), so each step is 10. The blank must be 20!

Extend the Pattern

Extending a pattern means figuring out what comes next. You have been doing this already! Now let us try some longer ones.

Try It: What Are the Next Three?

Pattern 1: Write the next 3 numbers:

10, 20, 30, 40, ___, ___, ___
Answer: 50, 60, 70! The rule is "add 10 each time." This is counting by tens.

Pattern 2: Write the next 3 items:

up down up up down up up down ___ ___ ___
Answer: up, up, down! The repeating group is "up, down, up" -- wait, let us look more carefully. The groups are: "up down" then "up up down" then "up up down." The repeating group is actually "up, up, down." So the next three are: up, up, down.

Pattern 3: Write the next 3 numbers:

1, 1, 2, 3, 5, 8, ___, ___, ___
Answer: 13, 21, 34! This is a special pattern where each number is the sum of the two numbers before it: 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, 8+13=21, 13+21=34. This is called the Fibonacci sequence, and it appears all over nature!

Visual Patterns

Patterns do not have to use numbers or letters. They can use shapes, positions, and sizes. Here are some visual patterns to solve.

Try It: Growing Patterns

Look at how many squares are in each step. How many will be in Step 4?

Step 1: 1 square

Step 2: 3 squares (a row of 3)

Step 3: 5 squares (a row of 5)

Step 4: ___ squares

Answer: 7 squares! The rule is "add 2 more squares each step." 1, 3, 5, 7. These are the odd numbers!

Try It: Staircase Pattern

Imagine building a staircase with blocks. How many total blocks do you need for each step?

Step 1: 1 block (just one block on the ground)

Step 2: 3 blocks (2 on the bottom, 1 on top)

Step 3: 6 blocks (3 on the bottom, 2 in the middle, 1 on top)

Step 4: ___ blocks

Answer: 10 blocks! The pattern is 1, 3, 6, 10. Each time you add one more row: +2, then +3, then +4. At Step 4: 4 on the bottom, 3 in the next row, 2 in the next, 1 on top = 4+3+2+1 = 10.

Unplugged Activity: Build a Pattern with Objects

Gather some small objects (coins, buttons, crayons, blocks). Create a growing pattern:

  • Step 1: Place 2 objects
  • Step 2: Add 2 more (now you have 4)
  • Step 3: Add 2 more (now you have 6)
  • Keep going! Can you predict how many objects you will have at Step 10?
Answer: At Step 10, you would have 20 objects! The rule is "start with 2, add 2 each step." Another way to figure it out: Step number times 2. So Step 10 = 10 x 2 = 20.

Patterns with Two Things Changing

Some patterns are trickier because more than one thing changes at the same time. Let us look at a few examples.

Size and Color Pattern

Imagine circles that change in both size and color:

big red, small blue, big red, small blue, big red, ___

The next one is: small blue! Two things are alternating: the size (big, small) and the color (red, blue).

Try It: Two Changes at Once

Pattern:

A1, B2, C3, D4, E5, ___
Answer: F6! Two things are changing: the letter goes forward through the alphabet (A, B, C, D, E, F) and the number increases by 1 (1, 2, 3, 4, 5, 6).

Pattern:

1 up, 2 down, 3 up, 4 down, 5 up, ___
Answer: 6 down! The number increases by 1 each time, and the direction alternates between "up" and "down." Odd numbers go up, even numbers go down.

Create Your Own Patterns!

Now it is your turn to be the pattern maker. Creating patterns is actually harder than solving them, because you have to think of a rule first and then follow it carefully.

Unplugged Activity: Pattern Creator

Create three patterns and challenge your parent (or a friend) to figure out the rule and extend them. Here are some ideas to get you started:

  • Number pattern: Pick a starting number and a rule (add 4, subtract 3, double it). Write out 5 numbers and leave the 6th blank.
  • Shape pattern: Draw a repeating pattern using 2-3 different shapes. Draw it 3 times, then leave it for someone to complete.
  • Word pattern: Use words that follow a rule. Example: cat, cats, dog, dogs, fish, ___ (the rule is: animal, then its plural).

Bonus challenge: Can you create a pattern where TWO things change at the same time?

Talk About It

After your child creates a pattern, ask: "What is the rule for your pattern? Could you explain it to someone who has never seen it before?" Being able to describe the rule is important because that is exactly what you do when you write code -- you tell the computer what the rule is.

Decreasing Patterns

Not all patterns go up! Some patterns decrease (go down). These are just as important.

Try It: Counting Down

Pattern:

50, 45, 40, 35, 30, ___
Answer: 25! The rule is "subtract 5 each time."

Pattern:

128, 64, 32, 16, ___
Answer: 8! The rule is "divide by 2 each time" (or "halve it"). 128/2=64, 64/2=32, 32/2=16, 16/2=8.

Real-World Decreasing Patterns

  • Countdown: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, blast off!
  • Eating cookies: If you start with 12 cookies and eat 2 each day: 12, 10, 8, 6, 4, 2, 0!
  • Candles on a menorah: Lit one more each night going up, then eventually they burn down.

Check Your Understanding

1. In the pattern 4, 8, ___, 16, 20, what is the missing number?

Answer: 12! The rule is "add 4 each time." 4, 8, 12, 16, 20.

2. How do you figure out the rule of a pattern when there is a missing piece in the middle?

Answer: Look at the numbers or items on both sides of the blank. Figure out the difference or relationship between the items you can see, then use that rule to fill in the blank.

3. What is special about the Fibonacci sequence?

Answer: In the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13...), each number is the sum of the two numbers before it. It appears all over nature, in things like sunflower seeds and seashell spirals!

Key Takeaways

Ready for More?

Next Lesson

In Lesson 3, you will discover why patterns matter so much to computers and programmers!

Start Lesson 3

Module Progress

You have finished Lesson 2! Two more lessons to go.

Back to Module Home