Chapter 1 of 8

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

M3.8.1 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

  • Write 'a number add 5' as n + 5
  • Write 3 × n as 3n and n × n as n²

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.

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

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