Year 8 · Spring 1

Representations: from clay to silicon · Lesson 1 of 5

Tally marks to number bases

TheoryTopic 3 Binary, denary and hex

Year 8 · Representations: from clay to silicon · Lesson 1 of 5

You will learn to

  • Compare tally marks with place value, and explain why place value needs fewer symbols.
  • Explain that each denary column is worth 10 times the one on its right.
  • Explain that each binary column is worth 2 times the one on its right.
  • Use the byte toy to switch bits on and off and read the value.

Year 8 · Representations: from clay to silicon · Lesson 1 of 5

Success criteria

You have got it when you can say:

  • I can write the binary place values 128, 64, 32, 16, 8, 4, 2, 1 from memory.
  • I can explain why 9 is followed by 10 in denary, and 1 is followed by 10 in binary.
  • I can make a given number with the byte toy and read back its value.

Year 8 · Representations: from clay to silicon · Lesson 1 of 5

Retrieval starter

Questions from before. Write your answers, then show them.

  1. Last unit

    Which controls how a page looks?

    A DNSB CSSC HTML

    B. CSS

  2. Last unit

    A box has width 100px, padding 10px, border 2px, margin 5px. How wide is it on screen, from border edge to border edge?

    A 124pxB 134pxC 112px

    A. 124px

  3. Earlier this year

    How many rows in a truth table with 3 inputs?

    A 8B 6C 3

    A. 8

Year 8 · Representations: from clay to silicon · Lesson 1 of 5

Key words

Denary
Base 10. Our everyday numbers.
Binary
Base 2. Digits 0 and 1. Columns 128 64 32 16 8 4 2 1.
Hexadecimal
Base 16: digits 0–9 then A–F. One digit = 4 bits.
Overflow
A result too big for the bits available.

The explanation · step 1 of 3

How normal numbers already work

We use ten digits, 0 to 9. When we run out, we start a new column on the left. That is why 9 is followed by 10.

ThousandsHundredsTensOnes
1000100101
3507

3507 = (3 × 1000) + (5 × 100) + (0 × 10) + (7 × 1). Binary works exactly the same way, with 2 instead of 10.

The explanation · step 2 of 3

Binary → denary

  1. Write the place values above the bits.
  2. Find every column with a 1.
  3. Add those column values together.
e.g.Example

01101011 → the 1s are under 64, 32, 8, 2, 1 → 64 + 32 + 8 + 2 + 1 = 107.

★Remember

A 1 means 'count this column'. A 0 means 'skip it'.

The explanation · step 2 of 3

The maths: Binary to denary, as a formula

The explanation · step 3 of 3

Bonus: negative numbers

Two's complement makes the leftmost bit worth −128. To write −5: write 5 (00000101), flip every bit (11111010), add 1 (11111011).

★Remember

This is A-level content. KS3 and GCSE use positive whole numbers only.

The explanation · step 3 of 3

The maths: Negative numbers (two's complement)

Year 8 · Representations: from clay to silicon · Lesson 1 of 5

Try it: Build a byte

Year 8 · Representations: from clay to silicon · Lesson 1 of 5

Try it: How many patterns?

Year 8 · Representations: from clay to silicon · Lesson 1 of 5

Quick quiz · 1 of 3

00000101 in denary is…

  1. A 101
  2. B 5
  3. C 6

B. 5

The 1s are under 4 and 1. 4 + 1 = 5.

Year 8 · Representations: from clay to silicon · Lesson 1 of 5

Quick quiz · 2 of 3

1111 in hex is…

  1. A E
  2. B F
  3. C 15

B. F

1111 = 8 + 4 + 2 + 1 = 15, and 15 is written F.

Year 8 · Representations: from clay to silicon · Lesson 1 of 5

Quick quiz · 3 of 3

Shifting left by 1 place…

  1. A Halves the number
  2. B Doubles the number
  3. C Adds 1

B. Doubles the number

Every bit moves into a column worth twice as much.

Year 8 · Representations: from clay to silicon · Lesson 1 of 5

Exit ticket

Answer on paper before you go. Then check.

  1. What is 1011 in denary?

    A 13B 11C 7

    B. 11 8 + 2 + 1 = 11. Add the columns that hold a 1.

  2. What is 200 in 8-bit binary?

    A 11001000B 10011000C 11010000

    A. 11001000 128 fits (72 left), 64 fits (8 left), then 8 fits: 11001000.