Chapter 2 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

M3.8.2 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

  • Find the value of 3n + 2 when n = 4
  • Use the formula P = 2l + 2w for a rectangle

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.

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

No internet tomorrow? Save the whole book as one file (850 KB). It opens on any phone.