Level 3 · Mathematics M3.8

Letters for numbers and simple equations

Use letters to stand for numbers, tidy up expressions, and solve simple equations.

Start with a box of beans and write what you do not know as a letter. Put numbers in, collect like terms, use brackets, and keep a balance level to solve equations. Then write your own equations and find the rule of a sequence.

You are ready for this book if

You can work out 3 + 4 × 2 in the right order.

Written and drawn by attnom. Free to read, print and share.

M3.8.1 Chapter 1

Letters for unknown numbers

Ravi has a closed box of beans and 5 loose beans. How can he write down all his beans?

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

Ravi has a closed box of beans. He also has 5 loose beans. He cannot see how many beans are in the box.

In algebra, a letter stands for a number we do not know. We call it an unknown. Ravi writes n for the number of beans in the box.

Draw it first

The box and 5 more beans

  1. Step 1: a box labelled n, with the words box: n beans under it. Step 2: a plus sign and 5 loose beans, with the words 5 loose beans. Step 3: the words n + 5 under both.nbox: n beans

    1 Draw the closed box. Write n on it. It holds n beans.

  2. Step 1: a box labelled n, with the words box: n beans under it. Step 2: a plus sign and 5 loose beans, with the words 5 loose beans. Step 3: the words n + 5 under both.nbox: n beans+5 loose beans

    2 Draw the 5 loose beans beside it, with a plus sign.

  3. Step 1: a box labelled n, with the words box: n beans under it. Step 2: a plus sign and 5 loose beans, with the words 5 loose beans. Step 3: the words n + 5 under both.nbox: n beans+5 loose beansn + 5

    3 Write what you see: n + 5.

Numbers, letters and signs written together are an expression. Ravi has the expression n + 5.

The letter is not a puzzle. The box holds a real number. It may be 3 one day and 9 the next. The letter n stands for whichever it is.

Two rows. In the first row a box marked 3 and 5 loose beans make 8, with the words n is 3, so n + 5 is 8. In the second row a box marked 9 and 5 loose beans make 14, with the words n is 9, so n + 5 is 14.3+=8n is 3, so n + 5 is 89+=14n is 9, so n + 5 is 14
The same expression, n + 5, fits both boxes.

Worked example

Ana has n pencils in a tin. Her brother has 3 more than Ana. How many pencils does her brother have?

  1. Ana has n pencils.
  2. 3 more means we add 3.
  3. Her brother has n + 3 pencils.
  4. Try n = 10. Ana has 10, and her brother has 10 + 3 = 13.

Answer: n + 3

To take beans away we use a minus sign. Amina has n stickers and gives 3 away. She has n − 3 left.

A long bar labelled n, then 3 small squares drawn dashed and crossed out. Under them is the expression n − 3.nn − 3
The bar is n. Its length does not tell you n. The crossed squares are the 3 taken away.

Now Ravi has 3 boxes. Each box holds the same number of beans. He could write n + n + n.

There is a shorter way. We write 3n. There is no sign between the 3 and the n. The 3 in front is the coefficient. It counts the boxes.

See it three ways

With things

Put 3 cups on a table. Put the same number of beans in each cup, and do not count them.

As a picture

Three boxes, each labelled n, joined by plus signs.n+n+n

In symbols

n + n + n = 3 × n = 3n.

What is n times n? It is n multiplied by itself. We write n² and say n squared.

On top, a square with n rows of n squares. Its side is n on the top and on the left, and inside it is n². Under it are the words n² is 16 squares. At the bottom, two bars of n with 4 squares each, and the words 2n is 8 squares. In this picture n is 4.nnn²n² is 16 squaresnn2n is 8 squares
Here n is 4. The square n² has 16 small squares. The two bars 2n have only 8.

Worked example

A box holds n beans. Write the beans in 3 boxes. Then write n × n. Try n = 4.

  1. 3 boxes of n beans is 3 × n. We write 3n.
  2. n × n means n multiplied by itself. We write n².
  3. Try n = 4. Then 3n = 3 × 4 = 12.
  4. And n² = 4 × 4 = 16.

Answer: 3n and n². When n = 4, they are 12 and 16.

Try it

A cup you cannot see into

You need: a cup or a small tin, dry beans, stones or seeds, a friend, paper and a pencil.

  1. Ask your friend to put some beans in the cup and cover it. Do not look.
  2. Put 5 more beans on the table, next to the cup.
  3. Write n + 5 on the paper. Here n means the beans in the cup.
  4. Now count the beans in the cup, then count them all. Is the total n plus 5?

What you will see: Whatever the cup holds, the total is that number plus 5. The letter n is a plan that works for any number.

If you cannot try it, this is what happens.
A long bar labelled n and 5 small squares side by side. Under them is the expression n + 5.nn + 5

A wrong idea many people have

n² means the same as 2n.

Why it feels right: Both have a 2 in them, and both seem to say "twice".

The test that breaks it: Let n be 4. Then 2n is 4 + 4 = 8. But n² is 4 × 4 = 16. The square picture has 16 small squares.

What is true: n² is n × n. The small 2 says n is used two times in a product. 2n is n + n.

Key words

unknown
a number we do not know yet, written as a letter.
expression
numbers, letters and signs written together, such as n + 5.
coefficient
the number in front of a letter. In 3n it is 3.
squared
multiplied by itself. n squared is n × n, written n².

Back to your guess

The answer is B: He writes n + 5, where n is the number of beans in the box.

We can write a number we do not know as a letter. Ravi has n + 5 beans. This is true whatever the box holds.

Check yourself

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

  1. Ravi has one box of beans and 4 loose beans. The bar is the box and the squares are the loose beans. The length of the bar does not tell you n. Which expression shows all his beans?

    One long bar labelled n and 4 small squares beside it.n
    Tap your answer.

    1. Yes, that is right. The box is n and the loose beans are 4 more. Adding gives n + 4.

    2. Not this one. This says 4 boxes. Ravi has one box and 4 loose beans. Go to M3.8.1. Try another.

    3. Not this one. We write a number before a letter, not after it. A plus sign must also join the box and the beans. Go to M3.8.1. Try another.

    4. Not this one. A minus sign takes beans away. Ravi has 4 beans more, not fewer. Go to M3.8.1. Try another.

    See the answer

    n + 4

    The box is n and the loose beans are 4 more. Adding gives n + 4.

    • If you chose B: This says 4 boxes. Ravi has one box and 4 loose beans. Go to M3.8.1.
    • If you chose C: We write a number before a letter, not after it. A plus sign must also join the box and the beans. Go to M3.8.1.
    • If you chose D: A minus sign takes beans away. Ravi has 4 beans more, not fewer. Go to M3.8.1.

  2. Tariq has 3 boxes. Every box holds the same number of beans, n. How many beans does he have?

    Three boxes in a row, each labelled n.nnn
    Tap your answer.

    1. Not this one. This is one box and 3 more beans. Tariq has 3 boxes. Go to M3.8.1. Try another.

    2. Not this one. We put the number first and the letter after it. n3 is not written. Go to M3.8.1. Try another.

    3. Yes, that is right. 3 boxes of n beans is n + n + n. We write this as 3 × n, or 3n.

    4. Not this one. This is 3 loose beans and one box. Tariq has 3 boxes, which is 3 times n. Go to M3.8.1. Try another.

    See the answer

    3n

    3 boxes of n beans is n + n + n. We write this as 3 × n, or 3n.

    • If you chose A: This is one box and 3 more beans. Tariq has 3 boxes. Go to M3.8.1.
    • If you chose B: We put the number first and the letter after it. n3 is not written. Go to M3.8.1.
    • If you chose D: This is 3 loose beans and one box. Tariq has 3 boxes, which is 3 times n. Go to M3.8.1.

  3. What does n² mean? The picture shows a square with side n.

    A square with side n on the top and on the left. Inside it is n².nnn²
    Tap your answer.

    1. Not this one. The small 2 does not say to multiply by 2. If n is 5, then n × 2 is 10, but n² is 25. Go to M3.8.1. Try another.

    2. Not this one. This is 2n, two boxes of n. The square has n rows of n. Go to M3.8.1. Try another.

    3. Not this one. The small 2 is not added. It says n is multiplied by itself. Go to M3.8.1. Try another.

    4. Yes, that is right. n² is n multiplied by itself. The area of a square with side n is n × n.

    See the answer

    n × n

    n² is n multiplied by itself. The area of a square with side n is n × n.

    • If you chose A: The small 2 does not say to multiply by 2. If n is 5, then n × 2 is 10, but n² is 25. Go to M3.8.1.
    • If you chose B: This is 2n, two boxes of n. The square has n rows of n. Go to M3.8.1.
    • If you chose C: The small 2 is not added. It says n is multiplied by itself. Go to M3.8.1.

  4. Mei says that n² and 2n are the same. Test her with n = 5. Which sentence is true?

    Tap your answer.

    1. Not this one. You worked n² as 2 × 5. But n² is 5 × 5. Go to M3.1.7. Try another.

    2. Yes, that is right. 2n is 2 × 5 = 10. n² is 5 × 5 = 25. They are not the same, so Mei is wrong.

    3. Not this one. These are the wrong way round. 2n is 5 + 5, which is small. n² is 5 × 5. Go to M3.8.1. Try another.

    4. Not this one. You worked 2n as 5 × 5. But 2n is 2 × 5, which is 10. Go to M3.8.1. Try another.

    See the answer

    2n is 10 and n² is 25

    2n is 2 × 5 = 10. n² is 5 × 5 = 25. They are not the same, so Mei is wrong.

    • If you chose A: You worked n² as 2 × 5. But n² is 5 × 5. Go to M3.1.7.
    • If you chose C: These are the wrong way round. 2n is 5 + 5, which is small. n² is 5 × 5. Go to M3.8.1.
    • If you chose D: You worked 2n as 5 × 5. But 2n is 2 × 5, which is 10. Go to M3.8.1.

  5. Amina has n stickers. She gives 3 away. The bar is her stickers and the crossed squares are the 3 she gives away. How many stickers are left?

    A long bar labelled n and 3 small squares drawn dashed and crossed out.n
    Tap your answer.

    1. Yes, that is right. She starts with n and takes 3 away, so she has n − 3 left.

    2. Not this one. Adding 3 gives her more stickers. She gives 3 away, so she has fewer. Go to M3.8.1. Try another.

    3. Not this one. This takes n away from 3. Amina takes 3 away from n. The order matters. Go to M3.8.1. Try another.

    4. Not this one. 3n means 3 boxes of n. Nobody is multiplying here. Amina gives away 3. Go to M3.8.1. Try another.

    See the answer

    n − 3

    She starts with n and takes 3 away, so she has n − 3 left.

    • If you chose B: Adding 3 gives her more stickers. She gives 3 away, so she has fewer. Go to M3.8.1.
    • If you chose C: This takes n away from 3. Amina takes 3 away from n. The order matters. Go to M3.8.1.
    • If you chose D: 3n means 3 boxes of n. Nobody is multiplying here. Amina gives away 3. Go to M3.8.1.

  6. Look at the picture. Each bar is a box with n beans. Each square is one loose bean. Which expression shows all the beans?

    Two long bars, each labelled n, and 3 small squares.nn
    Tap your answer.

    1. Not this one. You added the 2 and the 3. But the 3 squares are loose beans, not boxes. Only the bars are n. Go to M3.8.1. Try another.

    2. Not this one. There are 2 bars, so 2 boxes, not 1. The 2 boxes make 2n. Go to M3.8.1. Try another.

    3. Yes, that is right. There are 2 bars of n, which is 2n. The 3 squares are 3 more. Together they are 2n + 3.

    4. Not this one. This is 2 loose beans and 3 boxes. The picture has 2 boxes and 3 loose beans. Go to M3.8.1. Try another.

    See the answer

    2n + 3

    There are 2 bars of n, which is 2n. The 3 squares are 3 more. Together they are 2n + 3.

    • If you chose A: You added the 2 and the 3. But the 3 squares are loose beans, not boxes. Only the bars are n. Go to M3.8.1.
    • If you chose B: There are 2 bars, so 2 boxes, not 1. The 2 boxes make 2n. Go to M3.8.1.
    • If you chose D: This is 2 loose beans and 3 boxes. The picture has 2 boxes and 3 loose beans. Go to M3.8.1.

  7. A box holds n beans. There are also 5 loose beans. If n is 9, how many beans are there in all?

    See the answer

    14 beans

    n + 5 with n = 9 is 9 + 5 = 14.

  8. What is n² when n is 6?

    See the answer

    36

    n² is n × n. With n = 6 it is 6 × 6 = 36.

Keep it

Copy this into your exercise book.

A letter stands for an unknown number. n + 5 is a box and 5 more. 3n is 3 × n. n² is n × n, and it is not 2n.

Draw: Draw a box marked n beside 5 beans, and write n + 5. Draw a square with side n and write n² in it.

M3.8.2 Chapter 2

Putting numbers in

Leila has 3 boxes of beans and 2 loose beans. Each box holds 4 beans. How many beans does she have?

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

In the last chapter, n stood for the beans in a box. Leila has 3 boxes and 2 loose beans. She has 3n + 2 beans.

Now we are told that each box holds 4 beans. Then we can find the number of beans. We put 4 in place of n. This is called to substitute. The number we get is the value of the expression.

Draw it first

Cut each bar into 4 squares

  1. Step 1: three bars labelled n and 2 small squares, with the words 3n + 2. Step 2: the same bars, now each cut into 4 squares, with the words n = 4. Step 3: the words 3 × 4 + 2 = 14 squares.nnn3n + 2

    1 Draw 3 bars of n and 2 squares. This is 3n + 2. The length of a bar does not tell you n.

  2. Step 1: three bars labelled n and 2 small squares, with the words 3n + 2. Step 2: the same bars, now each cut into 4 squares, with the words n = 4. Step 3: the words 3 × 4 + 2 = 14 squares.nnnn = 4

    2 Now we know n is 4, so each bar is drawn 4 squares long. Cut every bar into 4 squares.

  3. Step 1: three bars labelled n and 2 small squares, with the words 3n + 2. Step 2: the same bars, now each cut into 4 squares, with the words n = 4. Step 3: the words 3 × 4 + 2 = 14 squares.nnnn = 43 × 4 + 2 = 14 squares

    3 Count the squares: 3 × 4 + 2 = 14.

Take care with 3n. It means 3 × n. It does not put 3 and 4 side by side. With n = 4, 3n is 3 × 4 = 12, not 34.

Worked example

Find the value of 3n + 2 when n = 4.

  1. Write 3n as 3 × n.
  2. Put in n = 4. This gives 3 × 4 + 2.
  3. Multiply first: 3 × 4 = 12.
  4. Then add: 12 + 2 = 14.

Answer: 14

A number machine. The number n = 4 goes in. The first step is × 3 and gives 12. The second step is + 2 and gives 14. The expression 3n + 2 is written over the last box.4n× 312+ 2143n + 2
Put in 4. The machine multiplies by 3, then adds 2.

Worked example

Find the value of n² + 1 when n = 5.

  1. Write n² as n × n.
  2. Put in n = 5. This gives 5 × 5 + 1.
  3. Multiply first: 5 × 5 = 25.
  4. Then add: 25 + 1 = 26.

Answer: 26

A table helps when you want many values. Put in n = 1, then 2, then 3, and so on.

A table with two columns, n and 3n + 2. The rows are 1 and 5, 2 and 8, 3 and 11, 4 and 14, 5 and 17.n3n + 21528311414517
Each time n goes up by 1, the value goes up by 3.

A formula is a rule written with letters. The distance all the way round a rectangle is its perimeter. A rectangle has 2 lengths and 2 widths. So P = 2l + 2w.

A rectangle with the words length l on the top and on the bottom, and the words width w on the left and on the right. Small marks show that opposite sides are equal.length llength lwidth wwidth w
P is the perimeter, l is the length and w is the width.

Worked example

A rectangle is 8 cm long and 5 cm wide. Find its perimeter.

  1. The formula is P = 2l + 2w.
  2. Two lengths make 2 × 8 = 16 cm.
  3. Two widths make 2 × 5 = 10 cm.
  4. Add them: 16 + 10 = 26 cm.

Answer: 26 cm

A rectangle 8 cm long and 5 cm wide. The top and the bottom are marked 8 cm. The left side and the right side are marked 5 cm.8 cm8 cm5 cm5 cm
Go round: 8 + 5 + 8 + 5 = 26 cm. The formula gives the same.

See it three ways

With things

Put 4 beans in each of 3 cups. Add 2 loose beans. Count every bean.

As a picture

A number machine. The number 4 goes in. The first step is × 3 and gives 12. The second step is + 2 and gives 14.4× 312+ 214

In symbols

3n + 2 with n = 4 is 3 × 4 + 2 = 14.

Try it

Measure the way round a book

You need: a book or a sheet of paper, a ruler or a piece of string, paper and a pencil.

  1. Measure the length of the book in cm. Call it l.
  2. Measure the width in cm. Call it w.
  3. Work out 2l + 2w.
  4. Now lay string round the edge. Measure the string. Is it the same?

What you will see: A book 20 cm long and 15 cm wide gives 2 × 20 + 2 × 15 = 70 cm. Your numbers will differ. The string agrees with the formula, to within a few millimetres.

If you cannot try it, this is what happens.
A rectangle 20 cm long and 15 cm wide. The top and the bottom are marked 20 cm. The left side and the right side are marked 15 cm.20 cm20 cm15 cm15 cm

A wrong idea many people have

When n = 4, the expression 3n means 34.

Why it feels right: In a number like 34, two digits stand side by side. A letter beside a number looks the same.

The test that breaks it: Take 3 boxes with 4 beans in each. Count them: 4 + 4 + 4 = 12 beans, not 34.

What is true: A number next to a letter means multiply. 3n is 3 × n, so 3n is 12 when n is 4.

Key words

substitute
to put a number in place of a letter.
value
the number an expression gives when you substitute.
formula
a rule written with letters, such as P = 2l + 2w.

Back to your guess

The answer is B: She has 14. The boxes are 3 × 4 = 12, because 3n means 3 times n. Then add 2.

3n means 3 times n, so the boxes hold 3 × 4 = 12 beans. With 2 loose beans that is 14. It would be 34 only if the 3 and the 4 were a two digit number.

Check yourself

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

  1. Find the value of 3n + 2 when n = 5.

    Tap your answer.

    1. Not this one. You put 3 and 5 side by side as 35. But 3n means 3 × 5 = 15. Go to M3.8.2. Try another.

    2. Not this one. You added 5 and 2 first. Multiply before you add: 3 × 5 = 15, then 15 + 2. Go to M3.2.6. Try another.

    3. Not this one. You added 3 + 5 + 2. A number next to a letter means multiply: 3 × 5. Go to M3.8.2. Try another.

    4. Yes, that is right. 3n is 3 × 5 = 15. Then 15 + 2 = 17.

    See the answer

    17

    3n is 3 × 5 = 15. Then 15 + 2 = 17.

    • If you chose A: You put 3 and 5 side by side as 35. But 3n means 3 × 5 = 15. Go to M3.8.2.
    • If you chose B: You added 5 and 2 first. Multiply before you add: 3 × 5 = 15, then 15 + 2. Go to M3.2.6.
    • If you chose C: You added 3 + 5 + 2. A number next to a letter means multiply: 3 × 5. Go to M3.8.2.

  2. Find the value of 20 − 2n when n = 4.

    Tap your answer.

    1. Yes, that is right. 2n is 2 × 4 = 8. Then 20 − 8 = 12.

    2. Not this one. You left out the 2. The term 2n is 2 × 4 = 8, not 4. Go to M3.8.2. Try another.

    3. Not this one. You did 20 − 2 first and then multiplied by 4. Multiply first: 2 × 4 = 8. Go to M3.2.6. Try another.

    See the answer

    12

    2n is 2 × 4 = 8. Then 20 − 8 = 12.

    • If you chose B: You left out the 2. The term 2n is 2 × 4 = 8, not 4. Go to M3.8.2.
    • If you chose C: You did 20 − 2 first and then multiplied by 4. Multiply first: 2 × 4 = 8. Go to M3.2.6.

  3. Use P = 2l + 2w. The rectangle is 9 cm long and 4 cm wide. What is its perimeter?

    A rectangle with the top marked 9 cm and the left side marked 4 cm.9 cm4 cm
    Tap your answer.

    1. Not this one. You multiplied 9 × 4. That is the area inside, not the distance round the edge. Go to M3.8.2. Try another.

    2. Not this one. You added one length and one width. Going round, you pass 2 lengths and 2 widths. Go to M3.8.2. Try another.

    3. Yes, that is right. 2 × 9 = 18 and 2 × 4 = 8. Then 18 + 8 = 26 cm.

    4. Not this one. You doubled the length but not the width. Both sides come twice. Go to M3.8.2. Try another.

    See the answer

    26 cm

    2 × 9 = 18 and 2 × 4 = 8. Then 18 + 8 = 26 cm.

    • If you chose A: You multiplied 9 × 4. That is the area inside, not the distance round the edge. Go to M3.8.2.
    • If you chose B: You added one length and one width. Going round, you pass 2 lengths and 2 widths. Go to M3.8.2.
    • If you chose D: You doubled the length but not the width. Both sides come twice. Go to M3.8.2.

  4. Kofi uses P = 2l + 2w with l = 7 and w = 3. Which working is right?

    Tap your answer.

    1. Not this one. 2l means 2 times l, not 2 and 7 side by side. Go to M3.8.2. Try another.

    2. Not this one. This is only half the way round. Multiply each side by 2. Go to M3.8.2. Try another.

    3. Not this one. The formula has a plus sign. It adds the two lengths and the two widths. Go to M3.8.2. Try another.

    4. Yes, that is right. 2l is 2 × 7 = 14 and 2w is 2 × 3 = 6. Then 14 + 6 = 20.

    See the answer

    2 × 7 + 2 × 3 = 20

    2l is 2 × 7 = 14 and 2w is 2 × 3 = 6. Then 14 + 6 = 20.

    • If you chose A: 2l means 2 times l, not 2 and 7 side by side. Go to M3.8.2.
    • If you chose B: This is only half the way round. Multiply each side by 2. Go to M3.8.2.
    • If you chose C: The formula has a plus sign. It adds the two lengths and the two widths. Go to M3.8.2.

  5. Let n = 3. Which is bigger, 2n or n + 4?

    Tap your answer.

    1. Not this one. Not always. With n = 3, 2n is 6 but n + 4 is 7. Put the number in and check. Go to M3.8.2. Try another.

    2. Yes, that is right. With n = 3, n + 4 is 3 + 4 = 7. And 2n is 2 × 3 = 6. So n + 4 is bigger.

    3. Not this one. Put in n = 3. Then 2n is 6 and n + 4 is 7. They are not equal. Go to M3.8.2. Try another.

    See the answer

    n + 4 is bigger. It is 7, and 2n is 6.

    With n = 3, n + 4 is 3 + 4 = 7. And 2n is 2 × 3 = 6. So n + 4 is bigger.

    • If you chose A: Not always. With n = 3, 2n is 6 but n + 4 is 7. Put the number in and check. Go to M3.8.2.
    • If you chose C: Put in n = 3. Then 2n is 6 and n + 4 is 7. They are not equal. Go to M3.8.2.

  6. From an earlier chapter

    The picture shows 5 boxes of n beans and 2 loose beans. Which expression is it?

    Five bars, each labelled n, and 2 small squares.nnnnn
    Tap your answer.

    1. Not this one. There are 5 bars, so 5 boxes of n. You cannot add the 5 and the 2 together. Go to M3.8.1. Try another.

    2. Not this one. This is 5 loose beans and 2 boxes. The picture has 5 boxes and 2 loose beans. Go to M3.8.1. Try another.

    3. Yes, that is right. 5 bars of n make 5n. The 2 loose squares are 2 more. Together they are 5n + 2.

    4. Not this one. A number in front of a letter says how many of it. It is not a two digit number. Go to M3.8.1. Try another.

    See the answer

    5n + 2

    5 bars of n make 5n. The 2 loose squares are 2 more. Together they are 5n + 2.

    • If you chose A: There are 5 bars, so 5 boxes of n. You cannot add the 5 and the 2 together. Go to M3.8.1.
    • If you chose B: This is 5 loose beans and 2 boxes. The picture has 5 boxes and 2 loose beans. Go to M3.8.1.
    • If you chose D: A number in front of a letter says how many of it. It is not a two digit number. Go to M3.8.1.

  7. Find the value of 4n + 1 when n = 6.

    See the answer

    25

    4n is 4 × 6 = 24. Then 24 + 1 = 25.

  8. A rectangle is 12 cm long and 7 cm wide. Use P = 2l + 2w to find its perimeter.

    See the answer

    38 cm

    2 × 12 = 24 and 2 × 7 = 14. Then 24 + 14 = 38 cm.

Keep it

Copy this into your exercise book.

Put the number in place of the letter. 3n means 3 × n, so n = 4 gives 12, not 34. Multiply before you add. P = 2l + 2w.

Draw: Draw a machine: 4, then × 3, then 12, then + 2, then 14. Draw a rectangle and write l, w, l, w on its sides.

M3.8.3 Chapter 3

Collecting like terms

Sipho has 3 boxes of mangoes and 2 boxes of limes. A mango box holds a mangoes. A lime box holds b limes. He gets 5 more mango boxes and gives away 1 lime box. How many pieces of fruit does he have?

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

Sipho has 3 boxes of mangoes and 2 boxes of limes. Each mango box holds a mangoes. Each lime box holds b limes. He has 3a + 2b pieces of fruit.

Each part of an expression, with the sign in front of it, is a term. The expression 3a + 2b has two terms. They are 3a and 2b.

Three rows of bars. In the first row, 2 bars of a and 3 bars of a, with the words 2a and 3a are like terms. In the second row, 2 bars of a and 3 hatched bars of b, with the words 2a and 3b: not like terms. In the third row, 2 bars of a and 3 small squares, with the words 2a and 3: not like terms.aaaaa2a and 3a are like termsaabbb2a and 3b: not like termsaa2a and 3: not like terms
The shading tells the letters apart. The length of a bar is only for the picture.

The numbers a and b may be different, so 3a and 2b cannot be joined. But 3a and 5a are both a number of a. Terms with the same letter part, such as 3a and 5a, are like terms. The letter part of a² is not the same as a.

We can count like terms together. We cannot count terms with different letters together. A plain number is not like a term with a letter.

Draw it first

Group the bars that are alike

  1. Step 1: four rows of bars: 3 bars of a, 2 bars of b, 5 bars of a, and 1 crossed out bar of b, with the words 3a + 2b + 5a − b. Step 2: the a bars together and the b bars together, with the words like terms together. Step 3: 8 bars of a and 1 bar of b, with the words 8a + b.aaabbaaaaab3a + 2b + 5a − b

    1 Draw one row for each term: 3a + 2b + 5a − b. The crossed bar is taken away.

  2. Step 1: four rows of bars: 3 bars of a, 2 bars of b, 5 bars of a, and 1 crossed out bar of b, with the words 3a + 2b + 5a − b. Step 2: the a bars together and the b bars together, with the words like terms together. Step 3: 8 bars of a and 1 bar of b, with the words 8a + b.aaaaaaaabbblike terms together

    2 Move the bars that are alike next to each other.

  3. Step 1: four rows of bars: 3 bars of a, 2 bars of b, 5 bars of a, and 1 crossed out bar of b, with the words 3a + 2b + 5a − b. Step 2: the a bars together and the b bars together, with the words like terms together. Step 3: 8 bars of a and 1 bar of b, with the words 8a + b.aaaaaaaab8a + b

    3 Count each kind. There are 8 bars of a. The crossed bar cancels 1 of the 2 bars of b, so 1 bar of b is left.

We add the numbers in front of the letters and keep the letter. 3a + 5a = 8a. And 2b − b = b. We write b, not 1b.

Worked example

Simplify 3a + 2b + 5a − b.

  1. The a terms are 3a and 5a. The b terms are 2b and −b.
  2. Count the a bars: 3 + 5 = 8.
  3. Count the b bars: 2 − 1 = 1.
  4. So 3a + 2b + 5a − b = 8a + b.

Answer: 8a + b

To simplify an expression is to write it in its shortest form. We collect the like terms to do this.

Worked example

Simplify 4x + 3 + 2x − 1.

  1. The x terms are 4x and 2x. The plain numbers are 3 and −1.
  2. Count the x bars: 4 + 2 = 6.
  3. Add the plain numbers: 3 − 1 = 2.
  4. So 4x + 3 + 2x − 1 = 6x + 2.

Answer: 6x + 2

The term a² is not like the term a. The term a² is a square. The term a is a bar. The two shapes are different.

On the left, a hatched square with side a on the top and on the left, and a² inside. On the right, one long bar labelled a.aaa²a
a² and a are not like terms.

Worked example

Check that 8a + b is the same as 3a + 2b + 5a − b when a = 2 and b = 3.

  1. Put a = 2 and b = 3 into 3a + 2b + 5a − b.
  2. The a terms: 3 × 2 + 5 × 2 = 16.
  3. The b terms: 2 × 3 − 3 = 3.
  4. Add them: 16 + 3 = 19.
  5. Now 8a + b gives 8 × 2 + 3 = 19. The two agree.

Answer: 19 and 19. They agree.

See it three ways

With things

Put 2 cups of beans and 3 more cups of beans in a row. Every cup holds the same number of beans, a. Count the cups. 5 cups of a beans hold 5a beans.

As a picture

Five bars of a in a row, with the words 2a + 3a = 5a under them.aaaaa2a + 3a = 5a

In symbols

2a + 3a = 5a. Add the numbers in front. The letter stays.

Try it

Cups and bowls

You need: 3 cups and 2 bowls, beans or stones to put in them, paper and a pencil.

  1. Put the same number of beans in each cup. Put the same number of stones in each bowl. Do not count them.
  2. Let a be the beans in a cup and b the stones in a bowl. Put 3 cups and 2 bowls on the table.
  3. Add 4 more cups. Take away 1 bowl.
  4. Count the cups and the bowls. Write the number of beans and stones with a and b.

What you will see: There are 7 cups of a beans and 1 bowl of b stones. This is 3a + 2b + 4a − b = 7a + b in all. The numbers a and b may differ, so they stay apart.

If you cannot try it, this is what happens.
Seven bars of a in the first row and one hatched bar of b in the second row, with the words 7a + b under them.aaaaaaab7a + b

A wrong idea many people have

Always add the numbers in front. So 3a + 2b = 5ab.

Why it feels right: 3 + 2 is 5, and it seems tidy to keep both letters.

The test that breaks it: Let a be 10 and b be 1. Then 3a + 2b is 30 + 2 = 32. But 5ab is 5 × 10 × 1 = 50. They are not equal.

What is true: Only like terms are joined. 3a + 2b has two different kinds of term, so it stays 3a + 2b.

Key words

term
one part of an expression, with the sign in front of it.
like terms
terms with the same letter part, such as 3a and 5a.
simplify
to write an expression in its shortest form.

Back to your guess

The answer is B: 8a + b. The mango boxes are counted together, and the lime boxes are counted together.

The numbers a and b may be different, so 3a and 2b cannot be joined. He has 3 + 5 = 8 boxes of a mangoes and 2 − 1 = 1 box of b limes. That is 8a + b pieces of fruit.

Check yourself

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

  1. Which pair is a pair of like terms?

    Tap your answer.

    1. Not this one. The letters are different. 4x is a number of x, and 7y is a number of y. Go to M3.8.3. Try another.

    2. Yes, that is right. Both terms have the letter x and nothing else. The numbers in front may differ.

    3. Not this one. The 7 is a plain number. It has no x, so it is not like 4x. Go to M3.8.3. Try another.

    4. Not this one. x² is a square, but x is a bar. They are different shapes, so they are not like terms. Go to M3.8.3. Try another.

    See the answer

    4x and 7x

    Both terms have the letter x and nothing else. The numbers in front may differ.

    • If you chose A: The letters are different. 4x is a number of x, and 7y is a number of y. Go to M3.8.3.
    • If you chose C: The 7 is a plain number. It has no x, so it is not like 4x. Go to M3.8.3.
    • If you chose D: x² is a square, but x is a bar. They are different shapes, so they are not like terms. Go to M3.8.3.

  2. Simplify 5a + 3a.

    Tap your answer.

    1. Yes, that is right. 5 bars of a and 3 bars of a make 8 bars of a. So 5a + 3a = 8a.

    2. Not this one. You multiplied 5 × 3. When you collect like terms, you add the numbers in front. Go to M3.8.3. Try another.

    3. Not this one. The letter stays. 5 bars of a and 3 more bars of a are 8 bars of a, not just 8. Go to M3.8.3. Try another.

    4. Not this one. a + a is 2a, not a square. When we add like terms, the letter part does not change. Go to M3.8.1. Try another.

    See the answer

    8a

    5 bars of a and 3 bars of a make 8 bars of a. So 5a + 3a = 8a.

    • If you chose B: You multiplied 5 × 3. When you collect like terms, you add the numbers in front. Go to M3.8.3.
    • If you chose C: The letter stays. 5 bars of a and 3 more bars of a are 8 bars of a, not just 8. Go to M3.8.3.
    • If you chose D: a + a is 2a, not a square. When we add like terms, the letter part does not change. Go to M3.8.1.

  3. Simplify 4x + 3y + 2x − 2y.

    Tap your answer.

    1. Not this one. You added 3y and 2y. But the sign in front of 2y is minus, so it is taken away. Go to M3.8.3. Try another.

    2. Yes, that is right. The x terms: 4x + 2x = 6x. The y terms: 3y − 2y = y. So the answer is 6x + y.

    3. Not this one. You joined everything into one term. The x terms and the y terms stay apart. Go to M3.8.3. Try another.

    4. Not this one. You took the 3y away from the 2y. But we start with 3y, then take 2y away. That leaves 1y. Go to M3.8.3. Try another.

    See the answer

    6x + y

    The x terms: 4x + 2x = 6x. The y terms: 3y − 2y = y. So the answer is 6x + y.

    • If you chose A: You added 3y and 2y. But the sign in front of 2y is minus, so it is taken away. Go to M3.8.3.
    • If you chose C: You joined everything into one term. The x terms and the y terms stay apart. Go to M3.8.3.
    • If you chose D: You took the 3y away from the 2y. But we start with 3y, then take 2y away. That leaves 1y. Go to M3.8.3.

  4. Simplify 7x + 4 − 3x + 1.

    Tap your answer.

    1. Not this one. The last term is plus 1, not minus 1. The plain numbers are 4 and 1, which make 5. Go to M3.8.3. Try another.

    2. Not this one. The sign in front of 3x is minus. So 7x − 3x is 4x. 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: 7x − 3x = 4x. The plain numbers: 4 + 1 = 5. So the answer is 4x + 5.

    See the answer

    4x + 5

    The x terms: 7x − 3x = 4x. The plain numbers: 4 + 1 = 5. So the answer is 4x + 5.

    • If you chose A: The last term is plus 1, not minus 1. The plain numbers are 4 and 1, which make 5. Go to M3.8.3.
    • If you chose B: The sign in front of 3x is minus. So 7x − 3x is 4x. 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.

  5. The first row is 2a + 3b. The second row is 3a − 2b, and the crossed bars are taken away. What is the whole picture, simplified?

    The first row has 2 bars of a and 3 hatched bars of b. The second row has 3 bars of a and 2 hatched bars of b that are dashed and crossed out.aabbbaaabb
    Tap your answer.

    1. Not this one. You counted the crossed bars of b. They are taken away, so they cancel 2 of the 3 bars. Go to M3.8.3. Try another.

    2. Not this one. You joined the letters. Bars of a and bars of b are different, so they stay apart. Go to M3.8.3. Try another.

    3. Yes, that is right. The a bars: 2 + 3 = 5. The b bars: 3 − 2 = 1. So the answer is 5a + b.

    See the answer

    5a + b

    The a bars: 2 + 3 = 5. The b bars: 3 − 2 = 1. So the answer is 5a + b.

    • If you chose A: You counted the crossed bars of b. They are taken away, so they cancel 2 of the 3 bars. Go to M3.8.3.
    • If you chose B: You joined the letters. Bars of a and bars of b are different, so they stay apart. Go to M3.8.3.

  6. Look at the picture. Which sentence is true about a and a²?

    On the left, a hatched square with side a on the top and on the left, and a² inside. On the right, one long bar labelled a.aaa²a
    Tap your answer.

    1. Not this one. a² is a times a, a square. The term a is a bar. The shapes differ, so they are not like terms. Go to M3.8.3. Try another.

    2. Yes, that is right. Like terms have the same letter part. The terms a and a² have different letter parts. They cannot join into one term.

    3. Not this one. They are different shapes. If a is 3, then a is 3 but a² is 9. Go to M3.8.1. Try another.

    See the answer

    They are not like terms. One is a bar and the other is a square.

    Like terms have the same letter part. The terms a and a² have different letter parts. They cannot join into one term.

    • If you chose A: a² is a times a, a square. The term a is a bar. The shapes differ, so they are not like terms. Go to M3.8.3.
    • If you chose C: They are different shapes. If a is 3, then a is 3 but a² is 9. Go to M3.8.1.

  7. Collect the like terms in 3a + 2b + 5a. What is the number in front of a?

    See the answer

    8

    The a terms are 3a and 5a. 3 + 5 = 8, so the answer is 8a + 2b.

  8. In 2b + 6b − 3b, how many bars of b are left after collecting?

    See the answer

    5

    2 + 6 = 8, and 8 − 3 = 5. So 2b + 6b − 3b = 5b.

  9. From an earlier chapter

    Find the value of 3a + 2b when a = 4 and b = 5.

    Tap your answer.

    1. Not this one. You added every number: 3 + 4 + 2 + 5. A number next to a letter means multiply. Go to M3.8.2. Try another.

    2. Not this one. You put the numbers side by side: 34 and 25. 3a means 3 × 4, and 2b means 2 × 5. Go to M3.8.2. Try another.

    3. Yes, that is right. 3a is 3 × 4 = 12 and 2b is 2 × 5 = 10. Then 12 + 10 = 22.

    See the answer

    22

    3a is 3 × 4 = 12 and 2b is 2 × 5 = 10. Then 12 + 10 = 22.

    • If you chose A: You added every number: 3 + 4 + 2 + 5. A number next to a letter means multiply. Go to M3.8.2.
    • If you chose B: You put the numbers side by side: 34 and 25. 3a means 3 × 4, and 2b means 2 × 5. Go to M3.8.2.

Keep it

Copy this into your exercise book.

Like terms have the same letter part. Collect them: 3a + 5a = 8a. A letter alone, such as b, means 1b. Different letters stay apart.

Draw: Draw bars for 3a + 2b + 5a − b. Group the a bars and the b bars. Write 8a + b.

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

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.

M3.8.5 Chapter 5

Solving equations in one step

A balance is level. The left pan holds a bag of x stones and 7 loose stones. The right pan holds 12 loose stones. How many stones are in the bag?

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

A balance is level when both sides weigh the same. Every stone weighs the same. A bag holds x stones, and we do not know x yet. The bag itself weighs nothing.

The left pan holds the bag and 7 loose stones, so it weighs x + 7. The right pan holds 12 loose stones. We write x + 7 = 12.

This is an equation. It says that two things are equal. To solve it is to find the number that x stands for.

Draw it first

Take the same from both sides

  1. Step 1: a level balance. The left side has a bag of x stones and 7 loose stones, with the words x + 7. The right side has 12 loose stones, with the words 12. Step 2: the same balance with 7 stones taken from each side. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Step 3: the words x = 5.xx + 712

    1 The balance is level. The left side is x + 7 and the right side is 12.

  2. Step 1: a level balance. The left side has a bag of x stones and 7 loose stones, with the words x + 7. The right side has 12 loose stones, with the words 12. Step 2: the same balance with 7 stones taken from each side. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Step 3: the words x = 5.xx5

    2 Take 7 stones from each side. The balance stays level, and the bag is on its own.

  3. Step 1: a level balance. The left side has a bag of x stones and 7 loose stones, with the words x + 7. The right side has 12 loose stones, with the words 12. Step 2: the same balance with 7 stones taken from each side. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Step 3: the words x = 5.xx5x = 5

    3 The bag balances 5 stones, so x = 5.

Here is the rule. Whatever you do to one side, do to the other side. Then the two sides stay equal.

Worked example

Solve x + 7 = 12.

  1. The 7 is added to x. The opposite of adding 7 is taking away 7.
  2. Take 7 from both sides: x = 12 − 7 = 5.
  3. So x = 5.
  4. Check: put 5 back into x + 7. We get 5 + 7 = 12. It is true.

Answer: x = 5

Adding and taking away undo each other. Multiplying and dividing undo each other. Each one is the inverse operation of the other.

Draw it first

Share both sides into 4 equal groups

  1. Step 1: a level balance. The left side has 4 bags of x stones, with the words 4x. The right side has 20 loose stones, with the words 20. Step 2: each side shared into 4 equal groups. One group on the left is one bag, with the words x. One group on the right is 5 stones, with the words 5. Under the balance, the 20 stones are drawn as 4 groups of 5, with the words 4 groups of 5. Step 3: the words x = 5.xxxx4x20

    1 Four bags balance 20 stones. This is 4x = 20.

  2. Step 1: a level balance. The left side has 4 bags of x stones, with the words 4x. The right side has 20 loose stones, with the words 20. Step 2: each side shared into 4 equal groups. One group on the left is one bag, with the words x. One group on the right is 5 stones, with the words 5. Under the balance, the 20 stones are drawn as 4 groups of 5, with the words 4 groups of 5. Step 3: the words x = 5.xx54 groups of 5

    2 Share each side into 4 equal groups. The 20 stones make 4 groups of 5. One bag balances 5 stones.

  3. Step 1: a level balance. The left side has 4 bags of x stones, with the words 4x. The right side has 20 loose stones, with the words 20. Step 2: each side shared into 4 equal groups. One group on the left is one bag, with the words x. One group on the right is 5 stones, with the words 5. Under the balance, the 20 stones are drawn as 4 groups of 5, with the words 4 groups of 5. Step 3: the words x = 5.xx54 groups of 5x = 5

    3 So x = 5.

Worked example

Solve 4x = 20.

  1. 4x means 4 × x. The opposite of multiplying by 4 is dividing by 4.
  2. Divide both sides by 4: 4x ÷ 4 = 20 ÷ 4.
  3. Work out 20 ÷ 4 = 5. So x = 5.
  4. Check: put 5 back into 4x. We get 4 × 5 = 20. It is true.

Answer: x = 5

What if something is taken away from x? Look at x − 3 = 8. The whole x is cut into two parts. One part is 3, and the other part is 8.

One long bar with the word x over it. It is cut into two parts. The left part is 8 and the right part is 3.83x
x − 3 = 8. Put the 3 back, and the whole bar is x.

Worked example

Solve x − 3 = 8.

  1. The 3 is taken away from x. The opposite of taking away 3 is adding 3.
  2. Add 3 to both sides of x − 3 = 8.
  3. On the right side: 8 + 3 = 11.
  4. So x = 11.
  5. Check: put 11 back into x − 3. We get 11 − 3 = 8. It is true.

Answer: x = 11

See it three ways

With things

Count out 12 stones. Hide 7 of them in your hand. Count the stones left on the table.

As a picture

One long bar with the number 12 over it. It is cut into two parts. The left part is x, shaded, and the right part is 7.x712

In symbols

x + 7 = 12, so x = 12 − 7 = 5.

Try it

Make your own balance

You need: a ruler or a flat stick, a pencil or a round bottle to rest it on, 2 cups that are the same, about 50 small stones or washers, all the same size.

  1. Rest the stick across the pencil so that it can tip. Ask a friend to hide some stones in one cup.
  2. Put that cup and 7 loose stones on the left. Put the empty cup on the right, so the two cups cancel. Add loose stones to the right until the stick is level.
  3. Take 7 stones from the left. Take 7 stones from the right. Is the stick still level?
  4. Now the stones in the cup balance the stones on the right. Count them. That is x.

What you will see: The two cups cancel. The stick stays level because you take the same from both sides. The stones left on the right tell you how many were in the cup.

If you cannot try it, this is what happens.
A level balance. The left side is one bag of x stones, with the words x. The right side is 5 loose stones, with the words 5.xx5

A wrong idea many people have

x + 7 = 12, so x = 12 + 7 = 19.

Why it feels right: There is a 7 and a 12 and a plus sign, and you are used to adding.

The test that breaks it: Put 19 back into x + 7. You get 19 + 7 = 26, not 12. So 19 is not the answer.

What is true: Use the opposite operation. The 7 is added to x, so take 7 from both sides.

Key words

equation
two things that are equal, with a letter for an unknown number.
solve
to find the number that makes an equation true.
inverse operation
the operation that undoes another, such as taking away and adding.

Back to your guess

The answer is A: x = 5. Take 7 stones from both sides, and the bag balances 5 stones.

The pans stay level if we take the same from both sides. Take 7 stones from each pan. The bag is left on its own, and it balances 12 − 7 = 5 stones.

Check yourself

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

  1. The balance is level. Solve x + 3 = 9.

    A level balance. The left side has a bag of x stones and 3 loose stones, with the words x + 3. The right side has 9 loose stones, with the words 9.xx + 39
    Tap your answer.

    1. Not this one. You added 3 and 9. The 3 is added to x, so take 3 from both sides. Go to M3.8.5. Try another.

    2. Yes, that is right. Take 3 from both sides. The bag balances 9 − 3 = 6 stones. Check: 6 + 3 = 9.

    3. Not this one. That is the 3 loose stones on the left. The number x is what the bag holds. Go to M3.8.5. Try another.

    See the answer

    6

    Take 3 from both sides. The bag balances 9 − 3 = 6 stones. Check: 6 + 3 = 9.

    • If you chose A: You added 3 and 9. The 3 is added to x, so take 3 from both sides. Go to M3.8.5.
    • If you chose C: That is the 3 loose stones on the left. The number x is what the bag holds. Go to M3.8.5.

  2. Solve 3x = 18.

    Tap your answer.

    1. Not this one. You took 3 away from 18. But 3x means 3 times x. The opposite of times is divide. Go to M3.8.5. Try another.

    2. Not this one. You added 3 to 18. But 3x means 3 times x. The opposite of times is divide. Go to M3.8.5. Try another.

    3. Not this one. You multiplied 3 by 18. The 3 is already multiplying x, so divide both sides by 3. Go to M3.8.5. Try another.

    4. Yes, that is right. Divide both sides by 3. x = 18 ÷ 3 = 6. Check: 3 × 6 = 18.

    See the answer

    6

    Divide both sides by 3. x = 18 ÷ 3 = 6. Check: 3 × 6 = 18.

    • If you chose A: You took 3 away from 18. But 3x means 3 times x. The opposite of times is divide. Go to M3.8.5.
    • If you chose B: You added 3 to 18. But 3x means 3 times x. The opposite of times is divide. Go to M3.8.5.
    • If you chose C: You multiplied 3 by 18. The 3 is already multiplying x, so divide both sides by 3. Go to M3.8.5.

  3. Solve x − 5 = 9.

    Tap your answer.

    1. Not this one. You took 5 from 9. But 5 was already taken away from x. The opposite is to add 5. Go to M3.8.5. Try another.

    2. Not this one. That is the answer to x − 5, not the number x. Add 5 to both sides to find x. Go to M3.8.5. Try another.

    3. Yes, that is right. Add 5 to both sides. x = 9 + 5 = 14. Check: 14 − 5 = 9.

    See the answer

    14

    Add 5 to both sides. x = 9 + 5 = 14. Check: 14 − 5 = 9.

    • If you chose A: You took 5 from 9. But 5 was already taken away from x. The opposite is to add 5. Go to M3.8.5.
    • If you chose B: That is the answer to x − 5, not the number x. Add 5 to both sides to find x. Go to M3.8.5.

  4. Solve x ÷ 4 = 3.

    Tap your answer.

    1. Not this one. You added 4 and 3. The x is divided by 4, so the opposite is to multiply by 4. Go to M3.8.5. Try another.

    2. Yes, that is right. Multiply both sides by 4. x = 3 × 4 = 12. Check: 12 ÷ 4 = 3.

    3. Not this one. You divided 3 by 4. But x was divided by 4, so multiply both sides by 4. Go to M3.8.5. Try another.

    See the answer

    12

    Multiply both sides by 4. x = 3 × 4 = 12. Check: 12 ÷ 4 = 3.

    • If you chose A: You added 4 and 3. The x is divided by 4, so the opposite is to multiply by 4. Go to M3.8.5.
    • If you chose C: You divided 3 by 4. But x was divided by 4, so multiply both sides by 4. Go to M3.8.5.

  5. The balance is level. Which move keeps it level?

    A level balance. The left side has a bag of x stones and 4 loose stones, with the words x + 4. The right side has 10 loose stones, with the words 10.xx + 410
    Tap your answer.

    1. Not this one. Then the left side is lighter and the balance tips. Take the same from the other side too. Go to M3.8.5. Try another.

    2. Not this one. Then the left side is lighter and the balance tips. Whatever you do on one side, do on the other. Go to M3.8.5. Try another.

    3. Not this one. This makes the sides further apart. Do the same to both sides. Go to M3.8.5. Try another.

    4. Yes, that is right. Equal things stay equal when you take the same from each. This is the one rule of solving equations.

    See the answer

    Take the same number of stones from both sides.

    Equal things stay equal when you take the same from each. This is the one rule of solving equations.

    • If you chose A: Then the left side is lighter and the balance tips. Take the same from the other side too. Go to M3.8.5.
    • If you chose B: Then the left side is lighter and the balance tips. Whatever you do on one side, do on the other. Go to M3.8.5.
    • If you chose C: This makes the sides further apart. Do the same to both sides. Go to M3.8.5.

  6. Tariq solves x + 8 = 15 and says x = 23. Test his answer by putting it back into x + 8. What do you find?

    Tap your answer.

    1. Yes, that is right. The correct answer takes 8 from 15: x = 15 − 8 = 7. Check: 7 + 8 = 15. Tariq added the 8 instead.

    2. Not this one. The equation says x + 8, not x − 8. Put 23 into x + 8. Go to M3.8.5. Try another.

    3. Not this one. The right side of the equation is 15. A true answer must give 15, not 31. Go to M3.8.5. Try another.

    See the answer

    23 + 8 = 31, not 15. His answer is wrong.

    The correct answer takes 8 from 15: x = 15 − 8 = 7. Check: 7 + 8 = 15. Tariq added the 8 instead.

    • If you chose B: The equation says x + 8, not x − 8. Put 23 into x + 8. Go to M3.8.5.
    • If you chose C: The right side of the equation is 15. A true answer must give 15, not 31. Go to M3.8.5.

  7. From an earlier chapter

    Expand 2(x + 3).

    Tap your answer.

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

    2. Yes, that is right. 2 × x = 2x and 2 × 3 = 6. So 2(x + 3) = 2x + 6.

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

    See the answer

    2x + 6

    2 × x = 2x and 2 × 3 = 6. So 2(x + 3) = 2x + 6.

    • If you chose A: You multiplied only the first term. The 2 multiplies the 3 as well. Go to M3.8.4.
    • If you chose C: You multiplied only the second term. The 2 multiplies the x as well. Go to M3.8.4.

  8. Solve x + 9 = 20.

    See the answer

    x = 11

    Take 9 from both sides. x = 20 − 9 = 11. Check: 11 + 9 = 20.

  9. Solve 6x = 42.

    See the answer

    x = 7

    Divide both sides by 6. x = 42 ÷ 6 = 7. Check: 6 × 7 = 42.

Keep it

Copy this into your exercise book.

An equation is a level balance. Do the same to both sides. Use the inverse operation: take away to undo add, divide to undo multiply. Always put the answer back.

Draw: Draw a balance. Put x and 7 stones on the left and 12 stones on the right. Cross out 7 from each side. Write x = 5.

M3.8.6 Chapter 6

Solving equations in two steps

A balance is level. The left pan holds 3 bags of x stones and 5 loose stones. The right pan holds 20 loose stones. How many stones are in each bag?

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

In the last chapter each equation took one step. Now the bag is multiplied and a number is added too. The equation 3x + 5 = 20 needs two steps. It is a two-step equation.

Draw it first

Two steps on the balance

  1. Step 1: a level balance. The left side has 3 bags of x stones and 5 loose stones, with the words 3x + 5. The right side has 20 loose stones, with the words 20. Step 2: 5 stones taken from each side. The left side is 3 bags, with the words 3x. The right side is 15 stones, with the words 15. Step 3: each side shared into 3 equal groups. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Under the balance, the 15 stones are drawn as 3 groups of 5, with the words 3 groups of 5. Step 4: the words x = 5.xxx3x + 520

    1 Three bags and 5 stones balance 20 stones.

  2. Step 1: a level balance. The left side has 3 bags of x stones and 5 loose stones, with the words 3x + 5. The right side has 20 loose stones, with the words 20. Step 2: 5 stones taken from each side. The left side is 3 bags, with the words 3x. The right side is 15 stones, with the words 15. Step 3: each side shared into 3 equal groups. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Under the balance, the 15 stones are drawn as 3 groups of 5, with the words 3 groups of 5. Step 4: the words x = 5.xxx3x15

    2 Take 5 stones from each side. Three bags balance 15 stones.

  3. Step 1: a level balance. The left side has 3 bags of x stones and 5 loose stones, with the words 3x + 5. The right side has 20 loose stones, with the words 20. Step 2: 5 stones taken from each side. The left side is 3 bags, with the words 3x. The right side is 15 stones, with the words 15. Step 3: each side shared into 3 equal groups. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Under the balance, the 15 stones are drawn as 3 groups of 5, with the words 3 groups of 5. Step 4: the words x = 5.xx53 groups of 5

    3 Share each side into 3 equal groups. The 15 stones make 3 groups of 5. One bag balances 5 stones.

  4. Step 1: a level balance. The left side has 3 bags of x stones and 5 loose stones, with the words 3x + 5. The right side has 20 loose stones, with the words 20. Step 2: 5 stones taken from each side. The left side is 3 bags, with the words 3x. The right side is 15 stones, with the words 15. Step 3: each side shared into 3 equal groups. The left side is one bag, with the words x. The right side is 5 stones, with the words 5. Under the balance, the 15 stones are drawn as 3 groups of 5, with the words 3 groups of 5. Step 4: the words x = 5.xx53 groups of 5x = 5

    4 So x = 5.

The order matters. In 3x + 5, x was first multiplied by 3, and then 5 was added. We undo the steps backwards. Take away 5 first, then divide by 3.

A number machine. The number x = 5 goes in. The first step is × 3 and gives 15. The second step is + 5 and gives 20.5x× 315+ 520
Building 3x + 5: multiply by 3, then add 5.
A number machine that runs backwards. The number 20 goes in. The first step is − 5 and gives 15. The second step is ÷ 3 and gives 5. The letter x is written over the last box.20− 515÷ 35x
Solving it: start at 20, take away 5, then divide by 3.

Worked example

Solve 3x + 5 = 20.

  1. Take 5 from both sides: 3x = 20 − 5 = 15.
  2. Divide both sides by 3: x = 15 ÷ 3 = 5.
  3. Check: put 5 back. 3 × 5 + 5 = 15 + 5 = 20. It is true.

Answer: x = 5

Some equations have brackets. In 2(x + 3) = 14, the 2 multiplies both x and 3. One way is to expand the bracket first.

A rectangle 2 high. Its width is x + 3, with x on the left part and 3 on the right part. The left part has the area 2x and the right part has the area 6.x32x + 32x6
2(x + 3) = 2x + 6. The whole rectangle has the area 14.

Worked example

Solve 2(x + 3) = 14.

  1. Expand the bracket: 2(x + 3) = 2x + 6, so 2x + 6 = 14.
  2. Take 6 from both sides: 2x = 14 − 6 = 8.
  3. Divide both sides by 2: x = 8 ÷ 2 = 4.
  4. Check: put 4 back. 2 × (4 + 3) = 2 × 7 = 14. It is true.

Answer: x = 4

There is another way. Divide both sides by 2 first. This gives x + 3 = 7, and then x = 4. Both ways give the same answer.

See it three ways

With things

Picture a see-saw. On one side are 2 bags of stones and 6 loose stones. On the other side are 14 loose stones.

As a picture

A level balance. The left side has 2 bags of x stones and 6 loose stones, with the words 2x + 6. The right side has 14 loose stones, with the words 14.xx2x + 614

In symbols

2(x + 3) = 14 becomes 2x + 6 = 14. Then 2x = 8 and x = 4.

Sometimes x is on both sides. First take the same number of bags from both sides. This leaves the bags on one side only.

Draw it first

Bags on both sides

  1. Step 1: a level balance. The left side has 5 bags of x stones, with the words 5x. The right side has 3 bags of x stones and 8 loose stones, with the words 3x + 8. Step 2: 3 bags taken from each side. The left side is 2 bags, with the words 2x. The right side is 8 stones, with the words 8. Step 3: each side shared into 2 equal groups. The left side is one bag, with the words x. The right side is 4 stones, with the words 4. Under the balance, the 8 stones are drawn as 2 groups of 4, with the words 2 groups of 4. Step 4: the words x = 4.xxxxx5xxxx3x + 8

    1 Five bags balance 3 bags and 8 stones. This is 5x = 3x + 8.

  2. Step 1: a level balance. The left side has 5 bags of x stones, with the words 5x. The right side has 3 bags of x stones and 8 loose stones, with the words 3x + 8. Step 2: 3 bags taken from each side. The left side is 2 bags, with the words 2x. The right side is 8 stones, with the words 8. Step 3: each side shared into 2 equal groups. The left side is one bag, with the words x. The right side is 4 stones, with the words 4. Under the balance, the 8 stones are drawn as 2 groups of 4, with the words 2 groups of 4. Step 4: the words x = 4.xx2x8

    2 Take 3 bags from each side. Two bags balance 8 stones.

  3. Step 1: a level balance. The left side has 5 bags of x stones, with the words 5x. The right side has 3 bags of x stones and 8 loose stones, with the words 3x + 8. Step 2: 3 bags taken from each side. The left side is 2 bags, with the words 2x. The right side is 8 stones, with the words 8. Step 3: each side shared into 2 equal groups. The left side is one bag, with the words x. The right side is 4 stones, with the words 4. Under the balance, the 8 stones are drawn as 2 groups of 4, with the words 2 groups of 4. Step 4: the words x = 4.xx42 groups of 4

    3 Share each side into 2 equal groups. The 8 stones make 2 groups of 4. One bag balances 4 stones.

  4. Step 1: a level balance. The left side has 5 bags of x stones, with the words 5x. The right side has 3 bags of x stones and 8 loose stones, with the words 3x + 8. Step 2: 3 bags taken from each side. The left side is 2 bags, with the words 2x. The right side is 8 stones, with the words 8. Step 3: each side shared into 2 equal groups. The left side is one bag, with the words x. The right side is 4 stones, with the words 4. Under the balance, the 8 stones are drawn as 2 groups of 4, with the words 2 groups of 4. Step 4: the words x = 4.xx42 groups of 4x = 4

    4 So x = 4.

Worked example

Solve 5x − 2 = 3x + 6.

  1. Add 2 to both sides: 5x = 3x + 6 + 2 = 3x + 8.
  2. Take 3x from both sides: 5x − 3x = 8, so 2x = 8.
  3. Divide both sides by 2: x = 8 ÷ 2 = 4.
  4. Check the left side: 5 × 4 − 2 = 20 − 2 = 18.
  5. Check the right side: 3 × 4 + 6 = 12 + 6 = 18. Both sides match.

Answer: x = 4

Try it

Two steps on a stick balance

You need: a ruler or flat stick, a pencil to rest it on, 6 cups that are the same, 50 small stones or washers, all the same size.

  1. Ask a friend to put the same number of stones in each of 3 cups. Use 2 to 6 stones. Cover them. Rest the stick on the pencil.
  2. Put the 3 full cups and 5 loose stones on the left. Put 3 empty cups on the right, so the cups cancel. Add loose stones to the right until the stick is level.
  3. Take 5 stones from each side. Is the stick still level?
  4. Share the stones on the right into 3 equal groups. One group is the number of stones in one cup.

What you will see: First you take away the loose number. Then you share. The empty cups cancel the full ones, and the stick stays level at every step.

If you cannot try it, this is what happens.
A level balance. The left side has 3 bags of x stones, with the words 3x. The right side has 15 loose stones, with the words 15.xxx3x15

A wrong idea many people have

In 3x + 5 = 20, divide only the 3x by 3, and leave the rest.

Why it feels right: The 3 is stuck to the x, so it seems the only thing to divide.

The test that breaks it: Try it. You get x + 5 = 20, so x = 15. Put 15 back: 3 × 15 + 5 = 50, not 20.

What is true: Take away the 5 first. Or divide every term on both sides by 3.

Key words

two-step equation
an equation that needs two steps to solve, such as 3x + 5 = 20.

Back to your guess

The answer is A: x = 5. Take 5 stones from each side, then share the 15 stones into 3 equal groups.

Two things happen to the bags: there are 3 of them, and 5 stones are added. Undo both. Take 5 from each side to get 3x = 15. Then divide by 3 to get x = 5.

Check yourself

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

  1. The balance is level. Solve 2x + 4 = 14.

    A level balance. The left side has 2 bags of x stones and 4 loose stones, with the words 2x + 4. The right side has 14 loose stones, with the words 14.xx2x + 414
    Tap your answer.

    1. Not this one. You added 4 to 14. But the 4 is added to the 2x, so take 4 away first. Go to M3.8.6. Try another.

    2. Not this one. You divided 14 by 2 and took away 4. The 4 is not divided by itself. Take 4 from 14 first, then divide by 2. Go to M3.8.6. Try another.

    3. Yes, that is right. Take 4 from both sides: 2x = 10. Divide by 2: x = 5. Check: 2 × 5 + 4 = 14.

    See the answer

    5

    Take 4 from both sides: 2x = 10. Divide by 2: x = 5. Check: 2 × 5 + 4 = 14.

    • If you chose A: You added 4 to 14. But the 4 is added to the 2x, so take 4 away first. Go to M3.8.6.
    • If you chose B: You divided 14 by 2 and took away 4. The 4 is not divided by itself. Take 4 from 14 first, then divide by 2. Go to M3.8.6.

  2. Solve 4x − 3 = 17.

    Tap your answer.

    1. Not this one. You took 3 from 17. But 3 was already taken away from 4x, so add 3 to both sides. Go to M3.8.6. Try another.

    2. Yes, that is right. Add 3 to both sides: 4x = 20. Divide by 4: x = 5. Check: 4 × 5 − 3 = 17.

    3. Not this one. You added 3, which is right, but you stopped. This is 4x = 20. Divide both sides by 4. Go to M3.8.6. Try another.

    See the answer

    5

    Add 3 to both sides: 4x = 20. Divide by 4: x = 5. Check: 4 × 5 − 3 = 17.

    • If you chose A: You took 3 from 17. But 3 was already taken away from 4x, so add 3 to both sides. Go to M3.8.6.
    • If you chose C: You added 3, which is right, but you stopped. This is 4x = 20. Divide both sides by 4. Go to M3.8.6.

  3. Solve 2(x + 3) = 14.

    Tap your answer.

    1. Yes, that is right. Divide both sides by 2: x + 3 = 7. Take 3 from both sides: x = 4. Check: 2 × (4 + 3) = 14.

    2. Not this one. This is the value of 2x, not x. You still need to divide both sides by 2. Go to M3.8.6. Try another.

    3. Not this one. You took 3 from 14. But the 2 multiplies the whole bracket. Divide 14 by 2 first, or expand the bracket. Go to M3.8.6. Try another.

    See the answer

    4

    Divide both sides by 2: x + 3 = 7. Take 3 from both sides: x = 4. Check: 2 × (4 + 3) = 14.

    • If you chose B: This is the value of 2x, not x. You still need to divide both sides by 2. Go to M3.8.6.
    • If you chose C: You took 3 from 14. But the 2 multiplies the whole bracket. Divide 14 by 2 first, or expand the bracket. Go to M3.8.6.

  4. The balance is level. The left side is 5 bags of x. The right side is 3 bags of x and 8 loose stones. Solve 5x = 3x + 8.

    A level balance. The left side has 5 bags of x stones, with the words 5x. The right side has 3 bags of x stones and 8 loose stones, with the words 3x + 8.xxxxx5xxxx3x + 8
    Tap your answer.

    1. Not this one. After you take 3 bags from each side, 2 bags balance 8 stones. That is 2x = 8. Now divide by 2. Go to M3.8.6. Try another.

    2. Not this one. You added 5 and 3, then divided 8 by that. The bags are taken from both sides, 5 − 3 = 2. Go to M3.8.6. Try another.

    3. Yes, that is right. Take 3 bags from both sides: 2x = 8. Divide by 2: x = 4. Check: 5 × 4 = 20 and 3 × 4 + 8 = 20.

    See the answer

    4

    Take 3 bags from both sides: 2x = 8. Divide by 2: x = 4. Check: 5 × 4 = 20 and 3 × 4 + 8 = 20.

    • If you chose A: After you take 3 bags from each side, 2 bags balance 8 stones. That is 2x = 8. Now divide by 2. Go to M3.8.6.
    • If you chose B: You added 5 and 3, then divided 8 by that. The bags are taken from both sides, 5 − 3 = 2. Go to M3.8.6.

  5. What is the right first step to solve 3x + 5 = 20?

    Tap your answer.

    1. Not this one. Then the sides are no longer equal. Whatever you do, do it to every term on both sides. Go to M3.8.6. Try another.

    2. Not this one. The loose number is already added to the bags. To undo adding, take it away. Go to M3.8.6. Try another.

    3. Not this one. Multiplying makes the numbers bigger and does not free x. Undo the adding first. Go to M3.8.6. Try another.

    4. Yes, that is right. We undo the steps backwards. The loose number was added last, so we take it away first. Then we divide.

    See the answer

    Take the loose number away from both sides.

    We undo the steps backwards. The loose number was added last, so we take it away first. Then we divide.

    • If you chose A: Then the sides are no longer equal. Whatever you do, do it to every term on both sides. Go to M3.8.6.
    • If you chose B: The loose number is already added to the bags. To undo adding, take it away. Go to M3.8.6.
    • If you chose C: Multiplying makes the numbers bigger and does not free x. Undo the adding first. Go to M3.8.6.

  6. Leila solves 2x + 1 = 11 and says x = 4. Put 4 back into the equation. What do you find?

    Tap your answer.

    1. Not this one. The right side of the equation is 11. A true answer must make the left side 11, not 9. Go to M3.8.6. Try another.

    2. Yes, that is right. The right answer is x = 5, because 2 × 5 + 1 = 11. Leila forgot to divide 10 by 2.

    3. Not this one. 2x means 2 times x. Put 4 in place of x, then multiply: 2 × 4. Go to M3.8.2. Try another.

    See the answer

    2 × 4 + 1 = 9, not 11. Her answer is wrong.

    The right answer is x = 5, because 2 × 5 + 1 = 11. Leila forgot to divide 10 by 2.

    • If you chose A: The right side of the equation is 11. A true answer must make the left side 11, not 9. Go to M3.8.6.
    • If you chose C: 2x means 2 times x. Put 4 in place of x, then multiply: 2 × 4. Go to M3.8.2.

  7. From an earlier chapter

    Solve x − 6 = 10.

    Tap your answer.

    1. Yes, that is right. Add 6 to both sides. x = 10 + 6 = 16. Check: 16 − 6 = 10.

    2. Not this one. You took 6 from 10. But 6 was already taken away from x, so add 6 to both sides. Go to M3.8.5. Try another.

    3. Not this one. You multiplied 6 by 10. The opposite of taking away is adding. Go to M3.8.5. Try another.

    See the answer

    16

    Add 6 to both sides. x = 10 + 6 = 16. Check: 16 − 6 = 10.

    • If you chose B: You took 6 from 10. But 6 was already taken away from x, so add 6 to both sides. Go to M3.8.5.
    • If you chose C: You multiplied 6 by 10. The opposite of taking away is adding. Go to M3.8.5.

  8. Solve 3x + 5 = 20.

    See the answer

    x = 5

    Take 5 from both sides: 3x = 15. Divide by 3: x = 5. Check: 3 × 5 + 5 = 20.

  9. Solve 3(x + 2) = 24.

    See the answer

    x = 6

    Divide both sides by 3: x + 2 = 8. Take 2 from both sides: x = 6. Check: 3 × (6 + 2) = 24.

Keep it

Copy this into your exercise book.

Undo the steps backwards. Take away the loose number first, then divide. Expand a bracket first, or divide by the number outside. Always put the answer back.

Draw: Draw a balance with 3 bags and 5 stones on the left and 20 stones on the right. Draw each step and write x = 5.

M3.8.7 Chapter 7

Writing your own equations

Three pieces of rope of the same length and a 5 m piece are joined. The whole rope is 26 m long. How long is each of the 3 pieces?

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

A word problem tells a story with numbers. To solve it, turn the story into an equation. Then solve the equation, and check the answer.

Words and symbols mean the same thing. This table shows how some words are written with a letter.

A table with two columns, words and symbols. The rows are: add 5 to n, n + 5. 3 times n, 3n. Take 7 from n, n − 7. Double n, add 6, 2n + 6. Is the same as, =.wordssymbolsadd 5 to nn + 53 times n3ntake 7 from nn − 7double n, add 62n + 6is the same as=
Take care with the order: "take 7 from n" is n − 7, not 7 − n.

Here are four steps. First, pick a letter for the unknown, and say what it means. Second, write the equation. Third, solve it. Fourth, check the answer.

Worked example

Joseph has some beans. He gives 7 away and has 12 left. How many beans did he have at the start?

  1. Let n be the beans Joseph had at the start.
  2. He gave away 7 and has 12 left, so n − 7 = 12.
  3. Add 7 to both sides: n = 12 + 7 = 19.
  4. Check: 19 − 7 = 12. It is true. He had 19 beans.

Answer: 19 beans

One long bar with the letter n over it. It is cut into two parts. The left part is 12 beans, the beans left. The right part is 7 beans, the beans given away.12 beans7 beansn
The whole bar is n. The 7 given away and the 12 left make n.

Draw it first

From a story to an equation

  1. Step 1: a bar with 3 equal shaded parts, each labelled x, and one part labelled 5 m. Step 2: a bracket over the whole bar with the words 26 m. Step 3: the words 3x + 5 = 26. Step 4: the words x = (26 − 5) ÷ 3 = 7.xxx5 m

    1 Let x be the length of one equal piece, in metres. Draw 3 pieces of x and the 5 m piece.

  2. Step 1: a bar with 3 equal shaded parts, each labelled x, and one part labelled 5 m. Step 2: a bracket over the whole bar with the words 26 m. Step 3: the words 3x + 5 = 26. Step 4: the words x = (26 − 5) ÷ 3 = 7.xxx5 m26 m

    2 The whole rope is 26 m. Write that over the bar.

  3. Step 1: a bar with 3 equal shaded parts, each labelled x, and one part labelled 5 m. Step 2: a bracket over the whole bar with the words 26 m. Step 3: the words 3x + 5 = 26. Step 4: the words x = (26 − 5) ÷ 3 = 7.xxx5 m26 m3x + 5 = 26

    3 Write the equation from the bar: 3x + 5 = 26.

  4. Step 1: a bar with 3 equal shaded parts, each labelled x, and one part labelled 5 m. Step 2: a bracket over the whole bar with the words 26 m. Step 3: the words 3x + 5 = 26. Step 4: the words x = (26 − 5) ÷ 3 = 7.xxx5 m26 m3x + 5 = 26x = (26 − 5) ÷ 3 = 7

    4 Solve it. Take 5 from 26 and divide by 3. This gives x = 7.

Worked example

Three pieces of rope of the same length and a 5 m piece are joined. The rope is 26 m long. How long is each of the 3 pieces?

  1. Let x be the length of one piece, in metres.
  2. Three pieces and 5 m make 26 m, so 3x + 5 = 26.
  3. Take 5 from both sides: 3x = 26 − 5 = 21.
  4. Divide both sides by 3: x = 21 ÷ 3 = 7.
  5. Check in the equation: 3 × 7 + 5 = 21 + 5 = 26. It is true.
  6. Check in the story: three pieces of 7 m and one of 5 m make 26 m. Each piece is 7 m.

Answer: 7 m

Formulas are equations too. If you know the perimeter and one side of a rectangle, you can find the other side.

A rectangle 4 cm wide. The top and the bottom are marked length l. The left side and the right side are marked 4 cm. Under it are the words perimeter 22 cm.length llength l4 cm4 cmperimeter 22 cm
The perimeter is 22 cm. We find the length l.

Worked example

A rectangle is 4 cm wide. Its perimeter is 22 cm. How long is it?

  1. The formula is P = 2l + 2w. Put in P = 22 and w = 4: 22 = 2l + 2 × 4.
  2. Work out the last part: 2 × 4 = 8.
  3. So 2l + 8 = 22. Take 8 from both sides: 2l = 22 − 8 = 14.
  4. Divide both sides by 2: l = 14 ÷ 2 = 7.
  5. Check: 2 × 7 + 2 × 4 = 14 + 8 = 22. It is true.

Answer: 7 cm

See it three ways

With things

Think of a number. Double it. Add 6. Tell a friend the answer, which is 20. Your friend must find your number.

As a picture

A number machine. The number n = 7 goes in. The first step is × 2 and gives 14. The second step is + 6 and gives 20.7n× 214+ 620

In symbols

2n + 6 = 20, so 2n = 14 and n = 7.

Try it

Think of a number

You need: a friend, paper and a pencil.

  1. Think of a number, and write it down where your friend cannot see it.
  2. Multiply it by 3, then add 4. Tell your friend only the answer.
  3. Your friend writes the equation 3n + 4 = your answer, and solves it.
  4. Show your number. Is it the same as the one your friend found?

What you will see: The equation always finds the number you thought of. If you thought of 5, the answer you told was 19, and 3n + 4 = 19 gives n = 5.

If you cannot try it, this is what happens.
A number machine. The number n = 5 goes in. The first step is × 3 and gives 15. The second step is + 4 and gives 19.5n× 315+ 419

A wrong idea many people have

If my answer fits my equation, my answer must be right.

Why it feels right: The numbers match, so everything seems fine.

The test that breaks it: Suppose you write 3x − 5 = 16 for the rope story by mistake. You get x = 7. Check: 3 × 7 − 5 = 16. It fits. But 3 pieces of 7 m and 5 m make 26 m, not 16 m.

What is true: Check in the story as well as in the equation. A wrong equation can still fit its own answer.

Key words

word problem
a story with numbers, to be turned into an equation.
solution
the number that makes an equation true.

Back to your guess

The answer is C: Each piece is 7 m. The equation 3x + 5 = 26 gives x = 7.

Let x be the length of one piece. Three pieces and 5 m make 26 m, so 3x + 5 = 26. Take 5 from both sides, then divide by 3. Each piece is 7 m.

Check yourself

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

  1. Joseph has n beans. He gets 4 more and then has 15. Which equation is right, and what is n?

    Tap your answer.

    1. Not this one. A minus sign means he gave 4 away. Joseph gets 4 more, so we add. Go to M3.8.7. Try another.

    2. Yes, that is right. Joseph gets 4 more, so n + 4 = 15. Take 4 from both sides: n = 15 − 4 = 11. Check: 11 + 4 = 15.

    3. Not this one. 4n means 4 times n. Joseph does not have 4 boxes. He gets 4 more beans. Go to M3.8.1. Try another.

    See the answer

    n + 4 = 15, so n = 11

    Joseph gets 4 more, so n + 4 = 15. Take 4 from both sides: n = 15 − 4 = 11. Check: 11 + 4 = 15.

    • If you chose A: A minus sign means he gave 4 away. Joseph gets 4 more, so we add. Go to M3.8.7.
    • If you chose C: 4n means 4 times n. Joseph does not have 4 boxes. He gets 4 more beans. Go to M3.8.1.

  2. Amina thinks of a number. She doubles it and takes away 3. The answer is 11. Which equation is right, and what is her number?

    Tap your answer.

    1. Yes, that is right. Add 3 to both sides: 2n = 14. Divide by 2: n = 7. Check: 2 × 7 − 3 = 11.

    2. Not this one. You missed the doubling. The number is doubled first, so we write 2n. Go to M3.8.7. Try another.

    3. Not this one. She takes 3 away. A plus sign would add 3. Go to M3.8.7. Try another.

    See the answer

    2n − 3 = 11, so n = 7

    Add 3 to both sides: 2n = 14. Divide by 2: n = 7. Check: 2 × 7 − 3 = 11.

    • If you chose B: You missed the doubling. The number is doubled first, so we write 2n. Go to M3.8.7.
    • If you chose C: She takes 3 away. A plus sign would add 3. Go to M3.8.7.

  3. The bar shows 4 equal pieces of rope and one piece of 3 m. The whole rope is 19 m. How long is each equal piece?

    One bar with the words 19 m over it. It is cut into 4 equal shaded parts, each labelled x, and one part labelled 3 m.xxxx3 m19 m
    Tap your answer.

    1. Not this one. You took 3 from 19, which is right, but you stopped. 16 m is all 4 pieces. Divide by 4. Go to M3.8.7. Try another.

    2. Not this one. You added 3 to 19 and divided by 4. The 3 m is part of the rope, so take it away from 19. Go to M3.8.7. Try another.

    3. Yes, that is right. The equation is 4x + 3 = 19. Take 3 from both sides: 4x = 16. Divide by 4: x = 4. Check: 4 × 4 + 3 = 19.

    See the answer

    4 m

    The equation is 4x + 3 = 19. Take 3 from both sides: 4x = 16. Divide by 4: x = 4. Check: 4 × 4 + 3 = 19.

    • If you chose A: You took 3 from 19, which is right, but you stopped. 16 m is all 4 pieces. Divide by 4. Go to M3.8.7.
    • If you chose B: You added 3 to 19 and divided by 4. The 3 m is part of the rope, so take it away from 19. Go to M3.8.7.

  4. Lucía says the solution of 5x + 2 = 32 is x = 7. Put 7 back into the equation. What do you find?

    Tap your answer.

    1. Not this one. The right side is 32. A true answer must make the left side 32, not 37. Go to M3.8.7. Try another.

    2. Yes, that is right. The right answer is x = 6, because 5 × 6 + 2 = 32. The 7 does not fit, so Lucía made a slip.

    3. Not this one. 5x means 5 times x. Put 7 in place of x, then multiply: 5 × 7. Go to M3.8.2. Try another.

    See the answer

    5 × 7 + 2 = 37, not 32. She is wrong.

    The right answer is x = 6, because 5 × 6 + 2 = 32. The 7 does not fit, so Lucía made a slip.

    • If you chose A: The right side is 32. A true answer must make the left side 32, not 37. Go to M3.8.7.
    • If you chose C: 5x means 5 times x. Put 7 in place of x, then multiply: 5 × 7. Go to M3.8.2.

  5. A rectangle is 6 cm wide. Its perimeter is 30 cm. Which equation finds the length l?

    A rectangle 6 cm wide. The top is marked length l. The left side is marked 6 cm.length l6 cm
    Tap your answer.

    1. Not this one. This adds only one length and one width. The perimeter goes round all four sides. Go to M3.8.2. Try another.

    2. Not this one. You doubled the length but not the width. There are two widths, so 2 × 6. Go to M3.8.2. Try another.

    3. Not this one. This multiplies the sides, which is the area. The perimeter adds the four sides. Go to M3.8.2. Try another.

    4. Yes, that is right. P = 2l + 2w with w = 6 gives 2l + 2 × 6 = 30, so 2l + 12 = 30. Then 2l = 18 and l = 9.

    See the answer

    2l + 12 = 30

    P = 2l + 2w with w = 6 gives 2l + 2 × 6 = 30, so 2l + 12 = 30. Then 2l = 18 and l = 9.

    • If you chose A: This adds only one length and one width. The perimeter goes round all four sides. Go to M3.8.2.
    • If you chose B: You doubled the length but not the width. There are two widths, so 2 × 6. Go to M3.8.2.
    • If you chose C: This multiplies the sides, which is the area. The perimeter adds the four sides. Go to M3.8.2.

  6. From an earlier chapter

    Solve 3x + 4 = 19.

    Tap your answer.

    1. Yes, that is right. Take 4 from both sides: 3x = 15. Divide by 3: x = 5. Check: 3 × 5 + 4 = 19.

    2. Not this one. You took 4 from 19, which is right, but you stopped. This is 3x = 15. Divide by 3. Go to M3.8.6. Try another.

    3. Not this one. You added 4 to 19 and stopped. The 4 is added to 3x, so take it away. Go to M3.8.6. Try another.

    See the answer

    5

    Take 4 from both sides: 3x = 15. Divide by 3: x = 5. Check: 3 × 5 + 4 = 19.

    • If you chose B: You took 4 from 19, which is right, but you stopped. This is 3x = 15. Divide by 3. Go to M3.8.6.
    • If you chose C: You added 4 to 19 and stopped. The 4 is added to 3x, so take it away. Go to M3.8.6.

  7. I think of a number. I multiply it by 3 and add 4. The answer is 19. What is my number?

    See the answer

    5

    The equation is 3n + 4 = 19. Take 4 from both sides: 3n = 15. Divide by 3: n = 5. Check: 3 × 5 + 4 = 19.

  8. Joseph has some beans. He gives 7 away and has 12 left. How many beans did he have at the start?

    See the answer

    19 beans

    The equation is n − 7 = 12. Add 7 to both sides: n = 19. Check: 19 − 7 = 12.

Keep it

Copy this into your exercise book.

Pick a letter and say what it means. Write the equation from the story. Solve it. Check in the equation and in the story.

Draw: Draw a bar for the rope story. It has 3 parts of x and a part of 5. Write 3x + 5 = 26.

M3.8.8 Chapter 8

Sequences and the nth term

A dot pattern has 5, 8, 11 and 14 dots in its first four pictures. How many dots will the 100th picture have?

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

A sequence is a list of numbers that follows a rule. Each number in the list is a term. Here the sequence is 5, 8, 11, 14.

Draw it first

Count the dots in each picture

  1. Four dot pictures: term 1 with 5 dots, term 2 with 8 dots, term 3 with 11 dots and term 4 with 14 dots. Each picture has 2 ring dots and columns of 3 black dots. Between the pictures, the marks +3 show the difference. Step 2: the words black: 3 more. ring: always 2. Step 3: the words dots = 3 × picture number + 2.term 15 dots+3term 28 dots+3term 311 dots+3term 414 dots

    1 Draw the pictures and count the dots: 5, 8, 11, 14.

  2. Four dot pictures: term 1 with 5 dots, term 2 with 8 dots, term 3 with 11 dots and term 4 with 14 dots. Each picture has 2 ring dots and columns of 3 black dots. Between the pictures, the marks +3 show the difference. Step 2: the words black: 3 more. ring: always 2. Step 3: the words dots = 3 × picture number + 2.term 15 dots+3term 28 dots+3term 311 dots+3term 414 dotsblack: 3 more. ring: always 2

    2 Each picture adds one column of 3 black dots. The 2 ring dots never change.

  3. Four dot pictures: term 1 with 5 dots, term 2 with 8 dots, term 3 with 11 dots and term 4 with 14 dots. Each picture has 2 ring dots and columns of 3 black dots. Between the pictures, the marks +3 show the difference. Step 2: the words black: 3 more. ring: always 2. Step 3: the words dots = 3 × picture number + 2.term 15 dots+3term 28 dots+3term 311 dots+3term 414 dotsblack: 3 more. ring: always 2dots = 3 × picture number + 2

    3 Picture number n has 3 × n + 2 dots.

Each term has a term number. The first term has term number 1, the second has term number 2, and so on. We call the term number n.

The rule that gives any term from its term number is the nth term. For this sequence it is 3n + 2. We will see how to find it.

Look at how far each term is from the next. The terms go up by 3 each time. This is the common difference. It is the number in front of n.

A table with three columns: n, 3n and term. The rows are 1, 3, 5. Then 2, 6, 8. Then 3, 9, 11. Then 4, 12, 14.n3nterm135268391141214
Every term is 2 more than 3n.

Worked example

Find the nth term of 5, 8, 11, 14.

  1. The terms go up by 3 each time. The common difference is 3, so the nth term starts with 3n.
  2. Write 3n for each term number: 3, 6, 9, 12.
  3. Compare with the sequence: 5 − 3 = 2, 8 − 6 = 2, 11 − 9 = 2.
  4. Each term is 2 more than 3n. So the nth term is 3n + 2.
  5. Check with n = 4: 3 × 4 + 2 = 14. It matches the fourth term.

Answer: 3n + 2

Now we can find any term. We do not need to draw all the pictures. We put the term number into the nth term.

Worked example

The nth term of a sequence is 3n + 2. Find the 100th term.

  1. The 100th term has n = 100.
  2. Put n = 100 into 3n + 2. This gives 3 × 100 + 2.
  3. Multiply first: 3 × 100 = 300.
  4. Then add: 300 + 2 = 302.

Answer: 302

Four dot pictures: term 1 with 7 dots, term 2 with 11 dots, term 3 with 15 dots and term 4 with 19 dots. Each picture has 3 ring dots and columns of 4 black dots. Between the pictures, the marks +4 show the difference.term 17 dots+4term 211 dots+4term 315 dots+4term 419 dots
The sequence 7, 11, 15, 19 goes up by 4, so its nth term starts with 4n.

Worked example

Find the nth term of 7, 11, 15, 19. Then find the 50th term.

  1. The terms go up by 4 each time, so the nth term starts with 4n.
  2. The first term of 4n is 4. The first term of the sequence is 7. And 7 − 4 = 3.
  3. So the nth term is 4n + 3.
  4. For n = 50: 4 × 50 = 200.
  5. Then 200 + 3 = 203.

Answer: 4n + 3 and 203

See it three ways

With things

Put 2 stones in a ring. Add a column of 3 stones, and count all of them. Add another column, and count again.

As a picture

A number machine. The term number n = 4 goes in. The first step is × 3 and gives 12. The second step is + 2 and gives 14.4n× 312+ 214

In symbols

The nth term is 3n + 2. With n = 4 it is 3 × 4 + 2 = 14.

Try it

Grow a pattern with stones

You need: about 40 small stones or beans.

  1. Put 2 stones in a ring. Put a column of 3 stones beside it. Count them all.
  2. Add another column of 3 stones. Count them all again.
  3. Add one more column of 3. Write the three counts.
  4. Guess how many stones there are with 10 columns. Then check with 3 × 10 + 2.

What you will see: The counts are 5, 8 and 11. With 10 columns there are 3 × 10 + 2 = 32 stones. The ring of 2 is the "plus 2" in the rule.

If you cannot try it, this is what happens.
Three dot pictures: term 1 with 5 dots, term 2 with 8 dots and term 3 with 11 dots. Each picture has 2 ring dots and columns of 3 black dots. Between the pictures, the marks +3 show the difference.term 15 dots+3term 28 dots+3term 311 dots

A wrong idea many people have

The nth term of 5, 8, 11, 14 is n + 3, because the terms go up by 3.

Why it feels right: The sequence adds 3 each time, so adding 3 seems to be the rule.

The test that breaks it: Put in n = 1. Then n + 3 is 4, but the first term is 5. Put in n = 4. Then n + 3 is 7, but the fourth term is 14.

What is true: The common difference is the number in front of n: it is 3n. Then fix the start: 3n + 2.

Key words

sequence
a list of numbers that follows a rule.
term number
the place of a term in the list: 1 for the first, 2 for the second.
nth term
a rule that gives any term from its term number, such as 3n + 2.
common difference
the amount a sequence goes up by each time.

Back to your guess

The answer is C: 302 dots, because each picture has 3 times its number, plus 2.

Each new picture adds a column of 3 dots, and 2 ring dots never change. Picture number n has 3n + 2 dots. For n = 100 that is 3 × 100 + 2 = 302.

Check yourself

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

  1. The dot pictures have 4, 7, 10 and 13 dots. What is the nth term?

    Four dot pictures: term 1 with 4 dots, term 2 with 7 dots, term 3 with 10 dots and term 4 with 13 dots. Each picture has 1 ring dot and columns of 3 black dots. Between the pictures, the marks +3 show the difference.term 14 dots+3term 27 dots+3term 310 dots+3term 413 dots
    Tap your answer.

    1. Not this one. Test n = 2. Then n + 3 is 5, but the second term is 7. The difference 3 goes in front of n. Go to M3.8.8. Try another.

    2. Not this one. The terms of 4n are 4, 8, 12, 16. This goes up by 4, but the sequence goes up by 3. Go to M3.8.8. Try another.

    3. Not this one. The terms of 3n are 3, 6, 9, 12. Each term of the sequence is 1 more than that. Go to M3.8.8. Try another.

    4. Yes, that is right. The terms go up by 3, so start with 3n. The terms of 3n are 3, 6, 9, 12. Each term is 1 more. So the nth term is 3n + 1.

    See the answer

    3n + 1

    The terms go up by 3, so start with 3n. The terms of 3n are 3, 6, 9, 12. Each term is 1 more. So the nth term is 3n + 1.

    • If you chose A: Test n = 2. Then n + 3 is 5, but the second term is 7. The difference 3 goes in front of n. Go to M3.8.8.
    • If you chose B: The terms of 4n are 4, 8, 12, 16. This goes up by 4, but the sequence goes up by 3. Go to M3.8.8.
    • If you chose C: The terms of 3n are 3, 6, 9, 12. Each term of the sequence is 1 more than that. Go to M3.8.8.

  2. The nth term of a sequence is 3n + 2. What is the 50th term?

    Tap your answer.

    1. Yes, that is right. Put n = 50 into 3n + 2. 3 × 50 = 150, and 150 + 2 = 152.

    2. Not this one. You put 3 and 50 side by side as 350. But 3n means 3 × n, so 3 × 50 = 150. Go to M3.8.2. Try another.

    3. Not this one. You added 3 + 50 + 2. A number next to a letter means multiply: 3 × 50. Go to M3.8.2. Try another.

    See the answer

    152

    Put n = 50 into 3n + 2. 3 × 50 = 150, and 150 + 2 = 152.

    • If you chose B: You put 3 and 50 side by side as 350. But 3n means 3 × n, so 3 × 50 = 150. Go to M3.8.2.
    • If you chose C: You added 3 + 50 + 2. A number next to a letter means multiply: 3 × 50. Go to M3.8.2.

  3. Is 62 a term of the sequence 5, 8, 11, 14, ...?

    Tap your answer.

    1. Not this one. The list shows only the first four terms. The three dots mean it goes on for ever, so use the nth term. Go to M3.8.8. Try another.

    2. Yes, that is right. Solve 3n + 2 = 62. Take 2 from both sides: 3n = 60. Divide by 3: n = 20. It is a whole number, so 62 is the 20th term.

    3. Not this one. The rule is 3n + 2, not 3n. Take 2 from 62 first. Then 60 ÷ 3 = 20. Go to M3.8.6. Try another.

    See the answer

    Yes. 3n + 2 = 62 gives n = 20, a whole number.

    Solve 3n + 2 = 62. Take 2 from both sides: 3n = 60. Divide by 3: n = 20. It is a whole number, so 62 is the 20th term.

    • If you chose A: The list shows only the first four terms. The three dots mean it goes on for ever, so use the nth term. Go to M3.8.8.
    • If you chose C: The rule is 3n + 2, not 3n. Take 2 from 62 first. Then 60 ÷ 3 = 20. Go to M3.8.6.

  4. Find the nth term of 7, 11, 15, 19, ...

    Tap your answer.

    1. Not this one. The number added to 4n is 7 − 4 = 3, not 7. The terms of 4n are 4, 8, 12, 16, and each term is 3 more. Go to M3.8.8. Try another.

    2. Not this one. Test n = 2. Then n + 4 is 6, but the second term is 11. The difference 4 goes in front of n. Go to M3.8.8. Try another.

    3. Yes, that is right. The terms go up by 4, so start with 4n: 4, 8, 12, 16. Each term is 3 more. So the nth term is 4n + 3.

    See the answer

    4n + 3

    The terms go up by 4, so start with 4n: 4, 8, 12, 16. Each term is 3 more. So the nth term is 4n + 3.

    • If you chose A: The number added to 4n is 7 − 4 = 3, not 7. The terms of 4n are 4, 8, 12, 16, and each term is 3 more. Go to M3.8.8.
    • If you chose B: Test n = 2. Then n + 4 is 6, but the second term is 11. The difference 4 goes in front of n. Go to M3.8.8.

  5. What is the common difference of the sequence 5, 8, 11, 14?

    Tap your answer.

    1. Yes, that is right. 8 − 5 = 3, and 11 − 8 = 3, and 14 − 11 = 3. The terms go up by 3 each time.

    2. Not this one. That is the number added in 3n + 2. The difference is the step between terms. Go to M3.8.8. Try another.

    3. Not this one. That is the first term. The difference is how much the terms go up by each time. Go to M3.8.8. Try another.

    See the answer

    3

    8 − 5 = 3, and 11 − 8 = 3, and 14 − 11 = 3. The terms go up by 3 each time.

    • If you chose B: That is the number added in 3n + 2. The difference is the step between terms. Go to M3.8.8.
    • If you chose C: That is the first term. The difference is how much the terms go up by each time. Go to M3.8.8.

  6. Joseph says the nth term of 5, 8, 11, 14 is n + 3. Test it with n = 2. What do you find?

    Tap your answer.

    1. Not this one. The second term of the sequence is 8, not 5. A correct nth term must give 8. Go to M3.8.8. Try another.

    2. Yes, that is right. The right rule is 3n + 2, which gives 3 × 2 + 2 = 8 for n = 2. So n + 3 is wrong.

    3. Not this one. Put in n = 2 again. 2 + 3 is 5, not 8. Go to M3.8.2. Try another.

    See the answer

    n + 3 gives 5 for n = 2, but the second term is 8. He is wrong.

    The right rule is 3n + 2, which gives 3 × 2 + 2 = 8 for n = 2. So n + 3 is wrong.

    • If you chose A: The second term of the sequence is 8, not 5. A correct nth term must give 8. Go to M3.8.8.
    • If you chose C: Put in n = 2 again. 2 + 3 is 5, not 8. Go to M3.8.2.

  7. From an earlier chapter

    Solve 2x + 7 = 19.

    Tap your answer.

    1. Not this one. You took 7 from 19, which is right, but you stopped. This is 2x = 12. Divide by 2. Go to M3.8.6. Try another.

    2. Not this one. You added 7 to 6, or took 6 from 19. Take 7 from both sides first, then divide by 2. Go to M3.8.6. Try another.

    3. Yes, that is right. Take 7 from both sides: 2x = 12. Divide by 2: x = 6. Check: 2 × 6 + 7 = 19.

    See the answer

    6

    Take 7 from both sides: 2x = 12. Divide by 2: x = 6. Check: 2 × 6 + 7 = 19.

    • If you chose A: You took 7 from 19, which is right, but you stopped. This is 2x = 12. Divide by 2. Go to M3.8.6.
    • If you chose B: You added 7 to 6, or took 6 from 19. Take 7 from both sides first, then divide by 2. Go to M3.8.6.

  8. The nth term of a sequence is 4n + 3. What is the 20th term?

    See the answer

    83

    Put n = 20 into 4n + 3. 4 × 20 = 80, and 80 + 3 = 83.

  9. What is the common difference of the sequence 6, 11, 16, 21?

    See the answer

    5

    11 − 6 = 5, and 16 − 11 = 5, and 21 − 16 = 5. The terms go up by 5 each time.

Keep it

Copy this into your exercise book.

Find the common difference. It is the number in front of n. Compare with that many n, and add what is left. Then put in n = 100 to jump far ahead.

Draw: Write 5, 8, 11, 14 and under it 3, 6, 9, 12. Write 2 more under each. Write the nth term 3n + 2.

Key words, A to Z

Every key word in this book, what it means, and the chapter that teaches it.

brackets
the signs ( ) that group terms. A number outside multiplies all of them. M3.8.4
coefficient
the number in front of a letter. In 3n it is 3. M3.8.1
common difference
the amount a sequence goes up by each time. M3.8.8
common factor
a number that divides every term. M3.8.4
equation
two things that are equal, with a letter for an unknown number. M3.8.5
expand
to multiply out brackets, such as 3(x + 4) = 3x + 12. M3.8.4
expression
numbers, letters and signs written together, such as n + 5. M3.8.1
factorise
to take a common factor outside brackets, such as 6x + 9 = 3(2x + 3). M3.8.4
formula
a rule written with letters, such as P = 2l + 2w. M3.8.2
inverse operation
the operation that undoes another, such as taking away and adding. M3.8.5
like terms
terms with the same letter part, such as 3a and 5a. M3.8.3
nth term
a rule that gives any term from its term number, such as 3n + 2. M3.8.8
sequence
a list of numbers that follows a rule. M3.8.8
simplify
to write an expression in its shortest form. M3.8.3
solution
the number that makes an equation true. M3.8.7
solve
to find the number that makes an equation true. M3.8.5
squared
multiplied by itself. n squared is n × n, written n². M3.8.1
substitute
to put a number in place of a letter. M3.8.2
term
one part of an expression, with the sign in front of it. M3.8.3
term number
the place of a term in the list: 1 for the first, 2 for the second. M3.8.8
two-step equation
an equation that needs two steps to solve, such as 3x + 5 = 20. M3.8.6
unknown
a number we do not know yet, written as a letter. M3.8.1
value
the number an expression gives when you substitute. M3.8.2
word problem
a story with numbers, to be turned into an equation. M3.8.7