Chapter 4 of 8

All 8 chapters
  1. 1Letters for unknown numbers, readunderstood
  2. 2Putting numbers in, readunderstood
  3. 3Collecting like terms, readunderstood
  4. 4Brackets, readunderstood
  5. 5Solving equations in one step, readunderstood
  6. 6Solving equations in two steps, readunderstood
  7. 7Writing your own equations, readunderstood
  8. 8Sequences and the nth term, readunderstood

M3.8.4 Chapter 4

Brackets

Each of 3 children gets a tin with x beans and 4 more loose beans. How many beans is that in all?

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

  • Expand 3(x + 4) to get 3x + 12
  • Factorise 6x + 9 as 3(2x + 3)

Each of 3 children gets a tin with x beans and 4 loose beans. One child has x + 4 beans. The 3 children have 3(x + 4) beans.

The signs ( ) are brackets. The 3 outside the brackets multiplies everything inside them. A rectangle shows what that means.

Draw it first

Draw 3(x + 4) as a rectangle

  1. Step 1: a rectangle 3 high. Its width is x + 4, with x on the left part and 4 on the right part. Step 2: the left part has the area 3x and the right part has the area 12. Step 3: the words 3(x + 4) = 3x + 12.x43x + 4

    1 Draw a rectangle 3 high and x + 4 across.

  2. Step 1: a rectangle 3 high. Its width is x + 4, with x on the left part and 4 on the right part. Step 2: the left part has the area 3x and the right part has the area 12. Step 3: the words 3(x + 4) = 3x + 12.x43x + 43x12

    2 Cut it into two parts. The x part is 3 bars, so it is 3x. The 4 part has 3 × 4 = 12 squares.

  3. Step 1: a rectangle 3 high. Its width is x + 4, with x on the left part and 4 on the right part. Step 2: the left part has the area 3x and the right part has the area 12. Step 3: the words 3(x + 4) = 3x + 12.x43x + 43x123(x + 4) = 3x + 12

    3 The whole rectangle is both parts: 3(x + 4) = 3x + 12.

To expand the brackets, we multiply every term inside by the number outside. Both terms are multiplied, not just the first.

Worked example

Expand 3(x + 4).

  1. Multiply the first term: 3 × x = 3x.
  2. Multiply the second term: 3 × 4 = 12.
  3. Put them together: 3(x + 4) = 3x + 12.

Answer: 3x + 12

See it three ways

With things

Make 3 rows. In each row put one cup of x beans and 4 loose beans. Count the cups. Count the loose beans. The length of a bar does not tell you x.

As a picture

Three rows. Each row has one bar of x and 4 small squares. Under them are the words 3(x + 4).xxx3(x + 4)

In symbols

3(x + 4) = 3x + 12.

Worked example

Expand 5(2x + 3).

  1. Multiply the first term: 5 × 2x = 10x.
  2. Multiply the second term: 5 × 3 = 15.
  3. Put them together: 5(2x + 3) = 10x + 15.

Answer: 10x + 15

A rectangle 5 high. Its width is 2x + 3, with 2x on the left part and 3 on the right part. The left part has the area 10x and the right part has the area 15.2x352x + 310x15
5(2x + 3) = 10x + 15. The 5 multiplies both parts.

Now go the other way. Start with 6x + 9. Both 6 and 9 can be divided by 3. A number that divides every term is a common factor.

To factorise is to take the common factor outside brackets. It undoes expanding. Take out the highest common factor. Then the numbers left inside the bracket have no common factor except 1.

Draw it first

Share 6x + 9 into equal rows

  1. Step 1: 6 bars of x in a row and 9 small squares in the next row, with the words 6x + 9. Step 2: the same pieces in 3 equal rows. Each row has 2 bars of x and 3 small squares. Step 3: the words 6x + 9 = 3(2x + 3).xxxxxx6x + 9

    1 Draw 6 bars of x and 9 squares. This is 6x + 9.

  2. Step 1: 6 bars of x in a row and 9 small squares in the next row, with the words 6x + 9. Step 2: the same pieces in 3 equal rows. Each row has 2 bars of x and 3 small squares. Step 3: the words 6x + 9 = 3(2x + 3).xxxxxx

    2 Share them into 3 equal rows. Each row has 2 bars of x and 3 squares.

  3. Step 1: 6 bars of x in a row and 9 small squares in the next row, with the words 6x + 9. Step 2: the same pieces in 3 equal rows. Each row has 2 bars of x and 3 small squares. Step 3: the words 6x + 9 = 3(2x + 3).xxxxxx6x + 9 = 3(2x + 3)

    3 One row is 2x + 3. There are 3 rows, so 6x + 9 = 3(2x + 3).

A rectangle 3 high. Its width is 2x + 3, with 2x on the left part and 3 on the right part. The left part has the area 6x and the right part has the area 9.2x332x + 36x9
The area is 6x + 9. One side is 3, so the other side is 2x + 3.

Worked example

Factorise 6x + 9.

  1. The highest common factor of 6 and 9 is 3.
  2. Divide each term by 3: 6x ÷ 3 = 2x, and 9 ÷ 3 = 3.
  3. Write 3 outside the bracket: 6x + 9 = 3(2x + 3).
  4. Check by expanding: 3 × 2x + 3 × 3 = 6x + 9.

Answer: 3(2x + 3)

Try it

Three rows on the table

You need: 3 small cups with the same number of beans in each, 12 loose beans, paper and a pencil.

  1. Make 3 rows. In each row put 1 cup and 4 loose beans.
  2. Count the cups. There are 3. Count the loose beans: 3 × 4 = 12.
  3. Write what is on the table: 3x + 12. Here x is the beans in one cup.
  4. Now write one row as x + 4, and all 3 rows as 3(x + 4). Both say the same.

What you will see: The rows can be counted by row, 3(x + 4), or by kind, 3x + 12. Either way you count the same beans.

If you cannot try it, this is what happens.
Three rows. Each row has one bar of x and 4 small squares. Under them are the words 3(x + 4) = 3x + 12.xxx3(x + 4) = 3x + 12

A wrong idea many people have

3(x + 4) = 3x + 4.

Why it feels right: The 3 is next to the x, so it seems to multiply only the x.

The test that breaks it: Let x be 1. Then 3(x + 4) = 3 × 5 = 15. But 3x + 4 = 3 + 4 = 7. The answer 3x + 12 gives 3 + 12 = 15.

What is true: The number outside the bracket multiplies every term inside it.

Key words

brackets
the signs ( ) that group terms. A number outside multiplies all of them.
expand
to multiply out brackets, such as 3(x + 4) = 3x + 12.
factorise
to take a common factor outside brackets, such as 6x + 9 = 3(2x + 3).
common factor
a number that divides every term.

Back to your guess

The answer is A: 3x + 12. Each child gets x beans and 4 beans, so there are 3 lots of both.

Every child has x + 4 beans, and there are 3 children. That is 3 lots of x and 3 lots of 4. So 3(x + 4) = 3x + 12.

Check yourself

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

  1. Expand 2(x + 5).

    Tap your answer.

    1. Not this one. You multiplied only the first term. The 2 multiplies the 5 as well. Go to M3.8.4. Try another.

    2. Not this one. You multiplied only the second term. The 2 multiplies the x as well. Go to M3.8.4. Try another.

    3. Not this one. You added 2 and 5 and joined them to x. The 2 multiplies each term inside the bracket. Go to M3.8.4. Try another.

    4. Yes, that is right. 2 × x = 2x and 2 × 5 = 10. So 2(x + 5) = 2x + 10.

    See the answer

    2x + 10

    2 × x = 2x and 2 × 5 = 10. So 2(x + 5) = 2x + 10.

    • If you chose A: You multiplied only the first term. The 2 multiplies the 5 as well. Go to M3.8.4.
    • If you chose B: You multiplied only the second term. The 2 multiplies the x as well. Go to M3.8.4.
    • If you chose C: You added 2 and 5 and joined them to x. The 2 multiplies each term inside the bracket. Go to M3.8.4.

  2. Expand 4(3n + 2).

    Tap your answer.

    1. Not this one. You multiplied the 3n by 4 but left the 2. Multiply the 2 by 4 too. Go to M3.8.4. Try another.

    2. Not this one. You added 4 to each term. The number outside multiplies, it does not add. Go to M3.8.4. Try another.

    3. Yes, that is right. 4 × 3n = 12n and 4 × 2 = 8. So 4(3n + 2) = 12n + 8.

    4. Not this one. You multiplied only the 2. The 4 multiplies the 3n as well. Go to M3.8.4. Try another.

    See the answer

    12n + 8

    4 × 3n = 12n and 4 × 2 = 8. So 4(3n + 2) = 12n + 8.

    • If you chose A: You multiplied the 3n by 4 but left the 2. Multiply the 2 by 4 too. Go to M3.8.4.
    • If you chose B: You added 4 to each term. The number outside multiplies, it does not add. Go to M3.8.4.
    • If you chose D: You multiplied only the 2. The 4 multiplies the 3n as well. Go to M3.8.4.

  3. Factorise 8x + 12 by taking out the highest common factor.

    Tap your answer.

    1. Not this one. This equals 8x + 12, but 4x + 6 can still be divided by 2. Take out the highest common factor, which is 4. Go to M3.8.4. Try another.

    2. Yes, that is right. The highest common factor of 8 and 12 is 4. 8x ÷ 4 = 2x and 12 ÷ 4 = 3. So 8x + 12 = 4(2x + 3).

    3. Not this one. You divided 8x by 4 but not 12. Divide every term by the common factor. Go to M3.8.4. Try another.

    4. Not this one. 8 does not divide 12 exactly. The common factor must divide both terms. Go to M3.1.3. Try another.

    See the answer

    4(2x + 3)

    The highest common factor of 8 and 12 is 4. 8x ÷ 4 = 2x and 12 ÷ 4 = 3. So 8x + 12 = 4(2x + 3).

    • If you chose A: This equals 8x + 12, but 4x + 6 can still be divided by 2. Take out the highest common factor, which is 4. Go to M3.8.4.
    • If you chose C: You divided 8x by 4 but not 12. Divide every term by the common factor. Go to M3.8.4.
    • If you chose D: 8 does not divide 12 exactly. The common factor must divide both terms. Go to M3.1.3.

  4. The picture has 4 rows. Each row has one bar of x and 3 small squares. Write the whole picture as one expression with a bracket.

    Four rows. Each row has one bar of x and 3 small squares.xxxx
    Tap your answer.

    1. Yes, that is right. One row is x + 3. There are 4 equal rows, so the picture is 4(x + 3).

    2. Not this one. This is 4 bars but only 3 squares. Every row has 3 squares, so there are 4 × 3 of them. Go to M3.8.4. Try another.

    3. Not this one. This has only one bar of x. The picture has 4 bars, one in each row. Go to M3.8.4. Try another.

    4. Not this one. The 4 rows multiply, they are not added. One row is x + 3, and there are 4 rows. Go to M3.8.4. Try another.

    See the answer

    4(x + 3)

    One row is x + 3. There are 4 equal rows, so the picture is 4(x + 3).

    • If you chose B: This is 4 bars but only 3 squares. Every row has 3 squares, so there are 4 × 3 of them. Go to M3.8.4.
    • If you chose C: This has only one bar of x. The picture has 4 bars, one in each row. Go to M3.8.4.
    • If you chose D: The 4 rows multiply, they are not added. One row is x + 3, and there are 4 rows. Go to M3.8.4.

  5. Mei expands 3(x + 4) and gets 3x + 4. Test her answer by putting in x = 1. What do you find?

    Tap your answer.

    1. Not this one. The value of 3(x + 4) at x = 1 is 3 × 5 = 15, not 7. Work out the left side with the bracket first. Go to M3.8.2. Try another.

    2. Not this one. The value of 3x + 4 at x = 1 is 3 + 4 = 7, not 15. She forgot to multiply the 4 by 3. Go to M3.8.4. Try another.

    3. Yes, that is right. 3(x + 4) with x = 1 is 3 × 5 = 15. But 3x + 4 with x = 1 is 3 + 4 = 7. The values differ, so her answer is wrong. The right answer is 3x + 12.

    See the answer

    The values are 15 and 7, so she is wrong.

    3(x + 4) with x = 1 is 3 × 5 = 15. But 3x + 4 with x = 1 is 3 + 4 = 7. The values differ, so her answer is wrong. The right answer is 3x + 12.

    • If you chose A: The value of 3(x + 4) at x = 1 is 3 × 5 = 15, not 7. Work out the left side with the bracket first. Go to M3.8.2.
    • If you chose B: The value of 3x + 4 at x = 1 is 3 + 4 = 7, not 15. She forgot to multiply the 4 by 3. Go to M3.8.4.

  6. From an earlier chapter

    Simplify 3x + 4 + 2x − 1.

    Tap your answer.

    1. Not this one. You added the 1 instead of taking it away. The sign in front of the 1 is minus. Go to M3.8.3. Try another.

    2. Not this one. You added 3x + 2x + 1 and dropped the 4. Collect the plain numbers on their own: 4 − 1 = 3. Go to M3.8.3. Try another.

    3. Not this one. You joined the plain numbers to the x terms. A plain number is not like a term with x. Go to M3.8.3. Try another.

    4. Yes, that is right. The x terms: 3x + 2x = 5x. The plain numbers: 4 − 1 = 3. So the answer is 5x + 3.

    See the answer

    5x + 3

    The x terms: 3x + 2x = 5x. The plain numbers: 4 − 1 = 3. So the answer is 5x + 3.

    • If you chose A: You added the 1 instead of taking it away. The sign in front of the 1 is minus. Go to M3.8.3.
    • If you chose B: You added 3x + 2x + 1 and dropped the 4. Collect the plain numbers on their own: 4 − 1 = 3. Go to M3.8.3.
    • If you chose C: You joined the plain numbers to the x terms. A plain number is not like a term with x. Go to M3.8.3.

  7. Expand 4(x + 5). What is the plain number in your answer?

    See the answer

    20

    4(x + 5) = 4x + 4 × 5 = 4x + 20. The plain number is 20.

  8. You factorise 10x + 15 by taking out the highest common factor. What number goes outside the bracket?

    See the answer

    5

    The highest common factor of 10 and 15 is 5. So 10x + 15 = 5(2x + 3).

Keep it

Copy this into your exercise book.

The number outside brackets multiplies every term inside. 3(x + 4) = 3x + 12. To factorise, take out the highest common factor. Check by expanding.

Draw: Draw a rectangle 3 high and x + 4 across. Cut it into 3x and 12, and write 3(x + 4) = 3x + 12.

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

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