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One pattern, many forms

Patterns and sequences · level 2, lesson 6 of 6

A question to start

Look at these V shapes made of dots.

shape 1: 3 dots shape 2: 5 dots shape 3: 7 dots shape 4: 9 dots

Now think back to the toothpick triangles in the last lesson: 3, 5, 7, 9 toothpicks.

The dots and the toothpicks look nothing alike. One is dots in a V, the other is sticks in a zigzag. But they are the same pattern. How can two such different things be the same? And how can you tell?

The idea

Four ways to show a pattern

A growing pattern can be shown in many forms. Here are the four most useful ones, all for the V shapes.

1. Objects or a picture. The dots above. You can see each shape and how it grows.

2. A table.

Shape number 1 2 3 4
Dots 3 5 7 9

3. A number list. 3, 5, 7, 9, ...

4. A sentence (the rule in words). Start at 3 and add 2 each time.

All four describe exactly the same pattern. They are like one story told in four languages.

What each form is good for

Why bother with four forms? Because each one is good at a different job.

  • The picture shows why the pattern grows the way it does. In the V shapes, you can see that each new shape gets one more orange dot and one more blue dot: that is why it grows by 2. And there is always just one green dot at the bottom. A list of numbers can never show you that.
  • The table keeps the shape number next to the count. So you can see which shape has which count, and spot a rule that goes straight from the shape number to the count.
  • The number list shows the gaps clearly: 3, 5, 7, 9 goes +2, +2, +2.
  • The sentence is the easiest to share. You can say it on the phone, and the other person can make the whole pattern without seeing it.

Translating between forms

Moving from one form to another is called translating, just as you might translate a word from English to French. Here is how to do each move.

Picture → table. Count each shape, and write the count under its shape number.

Table → number list. Read the bottom row from left to right.

Number list → sentence. The first number is the start. Find the gaps; if they are all the same, the sentence is start at ... and add ... each time. (That is what you did in lesson 1.)

Sentence → picture. This is the creative one. You have to invent a picture that grows by the right amount. For start at 3 and add 2 each time, you need a first shape with 3 things, and each new shape must get 2 more.

More forms: towers and number lines

You already know two more forms from earlier lessons. The pattern 3, 5, 7, 9 as towers:

shape 1 shape 2 shape 3 shape 4

And on a number line, as hops of 2 starting at 3:

0 1 2 3 4 5 6 7 8 9 10 11 12 +2 +2 +2

The towers are like a picture graph: the height of each tower shows its number. The number line shows the start and the hops.

Why the V shapes and the triangles are the same pattern

Now we can answer the starting question. Translate both into tables:

Shape number 1 2 3 4
V-shape dots 3 5 7 9
Triangle toothpicks 3 5 7 9

The tables are identical. So as patterns, they are the same, even though one is dots and the other is toothpicks. It is just like clap, stamp, stamp and orange, blue, blue in Level 1: same pattern, different clothes.

And look at the pictures again. Both have one special piece (the green dot; the first green toothpick) and then two more pieces for every shape (an orange and a blue). The same reason makes both grow the same way. Seeing the same reason in two places is how mathematicians know two patterns really are the same, and not just by luck for the first few shapes.

Same numbers, different picture, same rule

Here is a second way to see why the same rule can have many pictures. Start at 3 and add 2 each time could be:

  • V shapes of dots (above).
  • A row of triangles from toothpicks (lesson 5).
  • A plant that has 3 leaves on day 1 and grows 2 new leaves every day.
  • Savings: a jar with 3 coins that gets 2 more coins each day.

Same table, same list, same sentence. Four different pictures.

Worked examples

Example 1. Sentence → table → picture. Start at 1 and add 3 each time. Make a table for shapes 1 to 4, and draw a picture.

The list is 1, 4, 7, 10, and the table is:

Shape number 1 2 3 4
Count 1 4 7 10

A picture: start with 1 dot in the middle. Each new shape adds 3 dots: one on the left, one on the right and one on top. So shape 2 is a small upside-down T of 4 dots, shape 3 has 7 dots, and so on. (Any picture that starts with 1 and adds 3 each time is right.)

Example 2. Match the forms. Match each picture to its table and its sentence.

A B C

Tables: (i) 1, 4, 7 · (ii) 2, 4, 6 · (iii) 5, 6, 7.

Sentences: (p) start at 5 and add 1 each time · (q) start at 1 and add 3 each time · (r) start at 2 and add 2 each time.

  1. Count picture A: 2 dots, 4 dots, 6 dots. That is table (ii), and the gaps are +2 from a start of 2: sentence (r).
  2. Picture B: 5, 6, 7. Table (iii), sentence (p).
  3. Picture C: 1, 4, 7. Table (i), sentence (q).

Example 3. Table → sentence, with a gap. Find the missing number and the sentence.

Shape number 1 2 3 4 5
Count 6 10 ? 18 22
  1. From 6 to 10 is +4. From 18 to 22 is +4.
  2. From 10 to 18 is +8, which is two gaps of 4. So the missing number is 10 + 4 = 14.
  3. Sentence: start at 6 and add 4 each time.

Example 4. Picture → sentence, using the picture's reason. A garden path has 1 flower pot at the gate, and then each metre of path gets a pair of pots, one on each side. How many pots for 5 metres of path?

  1. 1 metre: 1 + 2 = 3 pots. 2 metres: 5 pots. 3 metres: 7 pots.
  2. Sentence: start at 3 and add 2 each time. (The same pattern as the V shapes!)
  3. 5 metres: the gate pot, and 5 pairs: 1 + 10 = 11 pots. (List: 3, 5, 7, 9, 11. ✓)

Try it

  1. Write the table for shapes 1 to 5 of: start at 2 and add 5 each time.
  2. Write the sentence for the list 10, 13, 16, 19, 22.
  3. Describe a picture for start at 4 and add 4 each time. (Hint: think of things that come in fours.)
  4. A tower pattern has towers of 2, 3, 4, 5 blocks. Write it as a table and as a sentence.
  5. Fill in the missing numbers: shape 1 has 7, shape 2 has ?, shape 3 has 15, shape 4 has 19. Then write the sentence.
  6. Make a table for the V shapes up to shape 6. How many green, orange and blue dots does shape 6 have?
  7. Two patterns: (a) start at 3 and add 2 each time, and (b) the number of toothpicks for a row of squares (4, 7, 10, ...). Are they the same pattern? Use tables to explain.
  8. Invent a story (like the flower pots) for the pattern 1, 3, 5, 7, 9.
Answers
  1. Shapes 1 to 5: 2, 7, 12, 17, 22.
  2. Start at 10 and add 3 each time.
  3. For example: shape 1 is one car with 4 wheels, shape 2 is two cars (8 wheels), shape 3 is three cars (12 wheels): 4, 8, 12, 16. Or a row of squares that do not touch, 4 toothpicks each.
  4. Table: shape 1, 2, 3, 4 has 2, 3, 4, 5 blocks. Sentence: start at 2 and add 1 each time.
  5. From 7 to 15 is +8, two gaps, so each gap is +4. Shape 2 has 11. Check: 15 + 4 = 19 ✓. Sentence: start at 7 and add 4 each time.
  6. 3, 5, 7, 9, 11, 13. Shape 6 has 1 green, 6 orange and 6 blue dots: 1 + 6 + 6 = 13.
  7. No. (a) is 3, 5, 7, 9; (b) is 4, 7, 10, 13. They start at different numbers and grow by different amounts (2 and 3). Their tables are different, so they are different patterns.
  8. For example: "A frog starts on stone 1, and hops 2 stones each time." Or: "A snowman has 1 ball on day 1, and each day I add 2 buttons." Any story that starts at 1 and adds 2 each time is right.

Common mistake

Thinking patterns that look different must be different. V-shaped dots and toothpick triangles look nothing alike, but their tables are the same. Compare the numbers (or the reason they grow), not the look.

Losing the shape number. A list like 3, 5, 7, 9 does not show which shape each number belongs to. When you need a faraway shape (shape 20), use a table, so the shape number stays next to its count.

Where it leads