Number · Number and Numeration Systems · Lesson 1

Modelling Numbers up to and beyond One Billion

Ghana's national budget, the population of Africa and the money in our banks are all counted in billions. In this lesson you will see exactly how big one billion is, build large numbers with multi-base blocks, and make big cedi amounts with GH₵20, GH₵50, GH₵100 and GH₵200 notes.

NaCCA B7.1.1.1.1 ⏱ about 45 min

By the end you will be able to

✔Name the place value of every digit in a number up to the hundred-billions.
✔Model large numbers with multi-base pieces: cube = 100,000, rod = 1,000,000, flat = 10,000,000 and block = 100,000,000.
✔Work out how many notes of a given cedi value make a large amount of money.
✔Write a number in more than one way using multiples of 10, 50, 100 and 200.

Key words

Place value: The value a digit has because of its position in a number. In 3,000,000,000 the 3 is worth three billion.
Billion: One thousand million: 1,000,000,000, which is $1 0^{9}$ (nine zeros).
Multi-base blocks: Pieces of different sizes, each 10 times the one before, used to model place value.
Expanded form: A number written as the sum of the values of its digits, e.g. 2,050,000 = 2,000,000 + 50,000.
Denomination: The face value of a note or coin, such as GH₵50 or GH₵200.

Curriculum reference

Content standard: B7.1.1.1 Demonstrate understanding and the use of place value for expressing quantities recorded as base ten numerals as well as rounding these to given decimal places and significant figures.

Indicator: Model number quantities more than 1,000,000,000 using graph sheets, isometric papers and multi-base blocks.

Starter

How much is a billion cedis?

Imagine you could count one cedi every second, day and night, without sleeping. Counting to one million would take you about 11½ days. Counting to one billion would take you about 31 years! Big numbers are hard to imagine, so in this lesson we will build them with blocks and bundles of notes until they make sense.

1. Place value up to the billions

Our number system is a base ten system. Every place is 10 times the place on its right. Ten ones make one ten, ten tens make one hundred, and the pattern carries on forever: ten hundred-millions make one billion.

To read a big number easily, we split the digits into groups of three from the right, using commas: ones, thousands, millions, billions. Each group has its own hundreds, tens and ones.

Billions	Hundred millions	Ten millions	Millions	Hundred thousands	Ten thousands	Thousands	Hundreds	Tens	Ones
4	2	5	6	1	0	3	0	0	0

The number in the table is 4,256,103,000. We read it as four billion, two hundred and fifty-six million, one hundred and three thousand. Step through the animation to see the value of each digit.

▶ Watch how each digit's value depends on its place, then see the number in expanded form.

2. Modelling big numbers with multi-base blocks

In earlier classes you used small cubes for ones. For really big numbers we let one small cube stand for a much larger amount. In Basic 7 we use these pieces:

Piece	Made of	Value
Cube	the smallest piece	100,000
Rod	10 cubes in a line	1,000,000
Flat	10 rods side by side	10,000,000
Block	10 flats stacked	100,000,000

▶ Each new piece is ten of the piece before it. Count how many blocks you would need for one billion.

To model a number, start with the largest piece that fits, then move to smaller pieces. For example, 2,345,600,000 is 23 blocks (2,300,000,000), 4 flats (40,000,000), 5 rods (5,000,000) and 6 cubes (600,000).

You can also model numbers on graph sheets or isometric paper: shade one small square for a cube, a strip of ten squares for a rod and a 10 by 10 square for a flat. On isometric paper you can draw the block as a real-looking 3D shape.

3. Making big amounts with cedi notes

Banks, mobile money agents and market queens deal with large amounts of money every day. Our notes come in values such as GH₵20, GH₵50, GH₵100 and GH₵200. To find how many notes make an amount, divide the amount by the value of one note.

1Write down the amount and the value of one note.
2Divide: number of notes = amount ÷ value of one note.
3Check by multiplying back: number of notes × value of one note should give the amount.

Sometimes we must use several kinds of note together. Then we choose how many of some notes to use, find the money still needed, and divide that by the last note value. There are usually many correct answers, so always check that your total is exactly right.

4. Writing a number in many ways

Just as a pile of notes can be made in different ways, a number can be written as a sum of multiples of 200, 100, 50 and 10 in many different ways. For 5,560:

5,560 = 20 × 200 + 10 × 100 + 11 × 50 + 1 × 10 (that is 4,000 + 1,000 + 550 + 10)
5,560 = 15 × 200 + 20 × 100 + 10 × 50 + 6 × 10 (that is 3,000 + 2,000 + 500 + 60)

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

How many blocks make one billion?

Each block stands for 100,000,000. How many blocks are needed to model 1,000,000,000?

Show the solution

Step 1Number of blocks = 1,000,000,000 ÷ 100,000,000.
Step 2Cancel the same number of zeros (eight) from both numbers: 10 ÷ 1 = 10.
Step 3Check: 10 × 100,000,000 = 1,000,000,000 ✓

Answer: 10 blocks

Example 2

Modelling 2,345,600,000

Model 2,345,600,000 with blocks, flats, rods and cubes, using as few pieces as possible.

Show the solution

Step 1Blocks (100,000,000 each): 2,345,600,000 contains 23 hundred-millions, so use 23 blocks = 2,300,000,000. Left: 45,600,000.
Step 2Flats (10,000,000 each): 45,600,000 contains 4 ten-millions, so use 4 flats = 40,000,000. Left: 5,600,000.
Step 3Rods (1,000,000 each): use 5 rods = 5,000,000. Left: 600,000.
Step 4Cubes (100,000 each): use 6 cubes = 600,000. Left: 0 ✓

Answer: 23 blocks, 4 flats, 5 rods and 6 cubes

Example 3

Counting GH₵200 notes

A bank needs GH₵185,000,000 in GH₵200 notes. How many notes is that?

Show the solution

Step 1Number of notes = 185,000,000 ÷ 200.
Step 2Divide by 100: 185,000,000 ÷ 100 = 1,850,000.
Step 3Divide by 2: 1,850,000 ÷ 2 = 925,000.
Step 4Check: 925,000 × 200 = 185,000,000 ✓

Answer: 925,000 notes

Example 4

Using three kinds of note

Make GH₵1,000,000 using GH₵50, GH₵100 and GH₵200 notes. Each kind must be used.

Show the solution

Step 1Choose some GH₵200 notes, e.g. 3,000 of them: 3,000 × 200 = GH₵600,000.
Step 2Choose some GH₵100 notes, e.g. 2,000 of them: 2,000 × 100 = GH₵200,000.
Step 3Money still needed: 1,000,000 − 600,000 − 200,000 = GH₵200,000.
Step 4GH₵50 notes needed: 200,000 ÷ 50 = 4,000 notes.
Step 5Check: 600,000 + 200,000 + 4,000 × 50 = 600,000 + 200,000 + 200,000 = 1,000,000 ✓

Answer: 3,000 × GH₵200 + 2,000 × GH₵100 + 4,000 × GH₵50 (other answers are possible)

Summary

Each place value is 10 times the place on its right; 1 billion = 1,000,000,000 = $1 0^{9}$ .
Cube = 100,000, rod = 1,000,000, flat = 10,000,000, block = 100,000,000, and 10 blocks = 1 billion.
To model a number, use the largest pieces first, then smaller ones.
Number of notes = amount ÷ value of one note; check by multiplying back.
A number can be written in many ways as multiples of 200, 100, 50 and 10. Always check the total.

How much is a billion cedis?

1. Place value up to the billions

2. Modelling big numbers with multi-base blocks

3. Making big amounts with cedi notes

4. Writing a number in many ways

Worked examples

How many blocks make one billion?

Modelling 2,345,600,000

Counting GH₵200 notes

Using three kinds of note

Summary

Practice: 15 questions