Chapter 6 of 7

All 7 chapters
  1. 1Number sequences, readunderstood
  2. 2Growing shape patterns, readunderstood
  3. 3Number machines, readunderstood
  4. 4Missing numbers, readunderstood
  5. 5Odd and even rules, readunderstood
  6. 6Square and triangle numbers, readunderstood
  7. 7Jumping far ahead, readunderstood

M2.8.6 Chapter 6

Square and triangle numbers

Why are 4 and 9 called square numbers? Are they really squares?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Draw dots to show the square numbers 1, 4, 9 and 16
  • Find the next triangle numbers after 1, 3, 6 and 10

Put 4 stones in 2 rows of 2. They make a square. Put 9 stones in 3 rows of 3. That is a square too.

A square number is the number of dots in a square. There are as many rows as columns.

Draw it first

Build squares of dots

  1. Four squares of dots, built one at a time, labelled 1, 4, 9 and 16. The filled dots in each square are the new ones: one row and one column added to the square before.1

    1 One dot is a square with 1 row of 1. It is 1 × 1 = 1.

  2. Four squares of dots, built one at a time, labelled 1, 4, 9 and 16. The filled dots in each square are the new ones: one row and one column added to the square before.14

    2 Add a row and a column. You get 2 rows of 2. It is 2 × 2 = 4.

  3. Four squares of dots, built one at a time, labelled 1, 4, 9 and 16. The filled dots in each square are the new ones: one row and one column added to the square before.149

    3 Add another row and column. You get 3 rows of 3. It is 3 × 3 = 9.

  4. Four squares of dots, built one at a time, labelled 1, 4, 9 and 16. The filled dots in each square are the new ones: one row and one column added to the square before.14916

    4 Once more. You get 4 rows of 4. It is 4 × 4 = 16.

A table with the heads Square, Total and Added. Row 1: 1 × 1, 1, 1. Row 2: 2 × 2, 4, 3. Row 3: 3 × 3, 9, 5. Row 4: 4 × 4, 16, 7. Row 5: 5 × 5, 25, 9.SquareTotalAdded1 × 1112 × 2433 × 3954 × 41675 × 5259
The Added column counts the filled dots in the picture.

Look at the filled dots. The new dots are 3, then 5, then 7, then 9. They are the odd numbers.

Now try a different shape. Put 1 stone. Below it put a row of 2. Then a row of 3.

Each new row has one more dot. You make a triangle. A triangle number is the number of dots in a triangle.

Four triangles of dots labelled 1, 3, 6 and 10. Each has one more row than the one before. The filled dots are the new row.13610
The new row of each triangle is filled in.

From 1 to 3 you add 2. From 3 to 6 you add 3. From 6 to 10 you add 4. The row you add grows by 1 each time.

A table with the heads Rows, New row and Total. Row 1: 1, 1, 1. Row 2: 2, 2, 3. Row 3: 3, 3, 6. Row 4: 4, 4, 10. Row 5: 5, 5, 15. Row 6: 6, 6, 21. Row 7: 7, 7, 28.RowsNew rowTotal1112233364410551566217728
Add the new row to the total above it.

Worked example

Find the next three triangle numbers after 10.

  1. The next row is the 5th row. It has 5 dots. 10 + 5 = 15.
  2. The 6th row has 6 dots. 15 + 6 = 21.
  3. The 7th row has 7 dots. 21 + 7 = 28.

Answer: 15, 21, 28

See it three ways

With things

Put 1 stone in the top row. Then 2, then 3, then 4. Count all the stones.

As a picture

A triangle of 10 dots in 4 rows. The rows have 1, 2, 3 and 4 dots, labelled 1, 2, 3 and 4. Below it is the sum 1 + 2 + 3 + 4 = 10.12341 + 2 + 3 + 4 = 10

In symbols

1 + 2 + 3 + 4 = 10. The 4th triangle number is 10.

Triangle and square numbers are linked. Two triangles can fit together to make a square.

Try it

Fit two triangles into a square

You need: 16 stones, beans or bottle tops, a pencil and paper.

  1. Make a triangle of 10. Put rows of 1, 2, 3 and 4 stones. Line up the left ends.
  2. Make a triangle of 6 in another place. Put rows of 3, 2 and 1 stones.
  3. Slide the triangle of 6 into the gaps. Line up the right ends.
  4. Count the rows and the columns. Write 10 + 6 = 16.

What you will see: The two triangles fit with no gap. They make a square of 4 rows of 4. So 10 + 6 = 16, a square number.

If you cannot try it, this is what happens.
A square of 16 dots, 4 rows of 4. The open dots make a triangle of 10. The filled dots make a triangle of 6. Below it is 10 + 6 = 16.10 + 6 = 16

It works again with 3 + 6 = 9. And with 10 + 15 = 25. Any two triangle numbers next to each other make a square number.

A wrong idea many people have

Square numbers go up in equal steps, like 1, 4, 7, 10.

Why it feels right: From 1 to 4 you add 3. It seems you add 3 again.

The test that breaks it: Put 3 rows of 3 stones. Count them. There are 9, not 7. From 4 to 9 you add 5.

What is true: The steps get bigger: 3, 5, 7, 9. Each step is the next odd number.

Key words

square number
the number of dots in a square of dots.
triangle number
the number of dots in a triangle of dots.

Back to your guess

The answer is B: That many dots make a square, with as many rows as columns.

A square number is the dots of a square. 4 is 2 rows of 2. 9 is 3 rows of 3. They do not go up by the same step.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A square has 5 rows of 5 dots. How many dots is that?

    See the answer

    25

    5 rows of 5 dots is 5 × 5 = 25. Check with the table: the 5th square number is 25.

  2. Which number is a square number?

    Tap your answer.

    1. Not this one. 10 is a triangle number: 1 + 2 + 3 + 4. It makes a triangle of dots, not a square. Try another.

    2. Not this one. 12 is 3 × 4. The rows and columns differ. The dots make a rectangle. Try another.

    3. Yes, that is right. 16 is 4 × 4. It makes a square of 4 rows of 4.

    See the answer

    16

    16 is 4 × 4. It makes a square of 4 rows of 4.

    • If you chose A: 10 is a triangle number: 1 + 2 + 3 + 4. It makes a triangle of dots, not a square.
    • If you chose B: 12 is 3 × 4. The rows and columns differ. The dots make a rectangle.

  3. The triangle numbers go 1, 3, 6, 10. What are the next two?

    See the answer

    15 and 21

    The 5th row has 5 dots. 10 + 5 = 15. The 6th row has 6 dots. 15 + 6 = 21.

  4. The 5th square number is 25. The new dots are 3, 5, 7, 9, then 11. What is the 6th square number?

    See the answer

    36

    25 + 11 = 36. Check: 6 rows of 6 is 6 × 6 = 36.

  5. Put the triangle numbers 15 and 21 together. Which square number do you get?

    See the answer

    36

    Two triangle numbers next to each other make a square number. 15 + 21 = 36, which is 6 × 6.

  6. From an earlier chapter

    Find the step and the missing number: 6, 11, ?, 21.

    See the answer

    The step is 5. The missing number is 16.

    11 − 6 = 5. So the missing number is 11 + 5 = 16. Check: 16 + 5 = 21. This pattern goes up by the same step each time. Square numbers do not.

  7. Mei makes a square with 4 rows of 4 dots. How many dots does it have?

    The top row and the left column of a square of dots, 4 dots each way, inside a dashed square. A label under it says 4 rows of 4 dots.4 rows of 4 dots
    Tap your answer.

    1. Yes, that is right. 4 rows of 4 dots is 4 × 4 = 16. The 4th square number is 16.

    2. Not this one. You added 4 + 4. A square has 4 rows, so you need 4 × 4. Try another.

    3. Not this one. You counted only the dots drawn. The dashed square has 4 rows, and every row has 4 dots. Try another.

    See the answer

    16

    4 rows of 4 dots is 4 × 4 = 16. The 4th square number is 16.

    • If you chose B: You added 4 + 4. A square has 4 rows, so you need 4 × 4.
    • If you chose C: You counted only the dots drawn. The dashed square has 4 rows, and every row has 4 dots.

  8. The triangle numbers are 1, 3, 6, 10, 15, 21. Which number comes next?

    A triangle of dots with 6 rows. The rows have 1, 2, 3, 4, 5 and 6 dots. A label under it says 6 rows: 21 dots.6 rows: 21 dots
    Tap your answer.

    1. Not this one. You added 6 again, like the last jump. The 7th row has 7 dots. Try another.

    2. Yes, that is right. The next row is the 7th row. It has 7 dots. 21 + 7 = 28.

    3. Not this one. You added 1 dot. The 7th row has 7 dots, so add 7. Try another.

    See the answer

    28

    The next row is the 7th row. It has 7 dots. 21 + 7 = 28.

    • If you chose A: You added 6 again, like the last jump. The 7th row has 7 dots.
    • If you chose C: You added 1 dot. The 7th row has 7 dots, so add 7.

  9. A square has 3 rows of 3 dots. You add dots to make 4 rows of 4. How many dots do you add?

    A square of dots with 3 rows and 3 columns. A label under it says 3 rows of 3 dots.3 rows of 3 dots
    Tap your answer.

    1. Not this one. That is how many dots the new square has. You already have 9 of them. Try another.

    2. Not this one. You added one row of 4. You must add a column of 3 as well. Try another.

    3. Yes, that is right. The new square has 4 × 4 = 16 dots. You have 3 × 3 = 9. So you add 16 − 9 = 7 dots.

    See the answer

    7

    The new square has 4 × 4 = 16 dots. You have 3 × 3 = 9. So you add 16 − 9 = 7 dots.

    • If you chose A: That is how many dots the new square has. You already have 9 of them.
    • If you chose B: You added one row of 4. You must add a column of 3 as well.

  10. Two triangles fit to make a square. One has 3 dots and one has 6. How many dots are in the square?

    Two triangles of dots side by side. The first has 2 rows and is labelled 3. The second has 3 rows and is labelled 6.36
    Tap your answer.

    1. Yes, that is right. Add the dots of both triangles. 3 + 6 = 9. It is 3 rows of 3. 9 is a square number.

    2. Not this one. You multiplied 3 × 6. Two triangles that fit together keep all their dots, so add them. Try another.

    3. Not this one. You counted only the bigger triangle. The square holds the dots of both triangles. Try another.

    See the answer

    9

    Add the dots of both triangles. 3 + 6 = 9. It is 3 rows of 3. 9 is a square number.

    • If you chose B: You multiplied 3 × 6. Two triangles that fit together keep all their dots, so add them.
    • If you chose C: You counted only the bigger triangle. The square holds the dots of both triangles.

Keep it

Copy this into your exercise book.

Square numbers: 1, 4, 9, 16, 25. Triangle numbers: 1, 3, 6, 10, 15, 21. Each new row has one more dot than the last. Triangles of 6 and 10 fit into a square of 16.

Draw: Draw a square of 16 dots. Circle a triangle of 10 inside it. The dots left make a triangle of 6.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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