Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- Write the number that is 4 thousands, 2 hundreds and 7 ones.
Answer
4207. There are no tens, so a 0 holds the tens place. - What is 6 × 10? What is 60 × 10?
Answer
60, then 600. Each multiplication by 10 moves a digit one place to the left. - Which is longer: 3 m or 295 cm?
Answer
3 m, because 3 m is 300 cm. Put both in the same unit before you compare them.
Words used in this lesson
- place value
- the value a digit has because of where it sits. For example: In 305 the 3 is worth 300.
- digit
- one of the symbols 0 to 9. For example: 305 has the digits 3, 0 and 5.
- decimal point
- the dot that separates whole numbers from parts of one. For example: In 3.7 the point sits between 3 and 7.
- tenth, hundredth
- one of ten equal parts, one of a hundred equal parts. For example: 0.1 is one tenth; 0.01 is one hundredth.
- placeholder
- a zero that keeps other digits in the right places. For example: the 0 in 407.
- expanded form
- a number written as the sum of the values of its digits. For example: 5038 = 5000 + 30 + 8.
One digit, many values
A 100 m final can be decided by 0.01 of a second, and a dose of 0.5 mg is ten times a dose of 0.05 mg. The digits look alike; the places they sit in are what matter. The 7 in 7, 70 and 0.07 does not change. Where it sits does. Our number system is built on this idea, called place value: each place is worth ten times the place to its right.
Predict first: the 7 is in the tens place, worth 70. If you move it one place to the right, what is it worth? Then slide it and see. Try the thousandths too.
The 7 is in the tens place, so it is worth 7 × 10 = 70. One place to the right it would be worth 7.
Show me the steps
- The digit 7 in the ones place is worth 7.
- Move it one place left, to the tens: 70. Ten times greater.
- Move it to the hundreds: 700. Ten times greater again.
- Move it to the thousands: 7000.
- Back at the ones, then one place right, the tenths: 0.7. One tenth of 7.
- The hundredths: 0.07, and the thousandths: 0.007. Each step right is a tenth of the step before.
A digit is worth the digit times the value of its place. Each place is ten times the place on its right and one tenth of the place on its left. The same rule carries on past the decimal point: tenths, hundredths, thousandths. Whole numbers and decimals are the numbers used in this unit; negative numbers come in Year 8 Unit 3.
Moving every digit one place left multiplies by 10, and two places left multiplies by 100. So 3.6 × 100 = 360, and moving every digit one place right divides by 10: 52 ÷ 10 = 5.2. The point stays where it is; the digits move.
Zero is a placeholder. In 407 the 0 says there are no tens, which keeps the 4 in the hundreds place. Without it you would have 47.
Comparing decimals
To compare two numbers, compare their places from the left, one place at a time. The first place where the digits differ decides it. It helps to write both numbers with the same number of decimal places, because adding a zero to the end of a decimal does not change its value: 0.4 = 0.40. Put a and b anywhere.
0.40 and 0.35: the tenths are 4 and 3, so the tenths decide and 0.40 > 0.35.
Show me the steps
- a = 0.40 and b = 0.35. Tenths: 4 and 3. The tenths decide, so 0.40 > 0.35.
- a = 0.40 and b = 0.04. Tenths: 4 and 0. The tenths decide, so 0.40 > 0.04.
- a = 0.62 and b = 0.68. Tenths: 6 and 6, the same. Hundredths: 2 and 8, so 0.62 < 0.68.
- a = 0.50 and b = 0.50. Every place matches, so 0.50 = 0.50, and 0.5 is the same number with the zero left off.
- a = 1.00 and b = 0.95. The ones are 1 and 0, so the ones decide and 1.00 > 0.95.
A common mistake: “0.35 is bigger than 0.4 because 35 is bigger than 4.” Count the tenths first: 0.35 has 3 tenths and 0.4 has 4. The number of digits after the point says nothing about size.
Worked example: ordering decimals
Put in order, smallest first: 0.6, 0.58, 0.065, 0.605.
- Give each three decimal places: 0.600, 0.580, 0.065, 0.605.
- Compare the tenths: 0.065 has 0, then 0.580 has 5, then 0.600 and 0.605 both have 6.
- Between 0.600 and 0.605 the hundredths are 0 and 0, so the thousandths decide: 0 then 5.
- In order, with the zeros taken off again: 0.065 < 0.58 < 0.6 < 0.605.
Your go. The first two steps are given. Put these swim times, in seconds, in order from the slowest (largest time) to the fastest (smallest time): 52.3, 52.09, 52.9, 52.089.
- Give each three decimal places: 52.300, 52.090, 52.900, 52.089.
- All have 52 whole, so compare the tenths: 3, 0, 9, 0. Largest first: 52.900, then 52.300.
- Now decide between 52.090 and 52.089 yourself, then check.
Check the full order
52.9 > 52.3 > 52.09 > 52.089. 52.09 and 52.089 both have 0 tenths. The hundredths are 9 and 8, so 52.09 is greater.
Check before you go on
1. In the number 3.746, what is the value of the digit 4?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Write as a number: 5 thousands, 3 tens and 8 ones.
3. Which of these numbers has a 7 that is worth 0.07?
Level 2 · Building
4. Multiply 4.07 by 10 by moving every digit one place to the left. What is 4.07 × 10?
5. Write as a number: (2 × 100) + (9 × 1) + (4 × 0.1) + (7 × 0.001).
6. Which list has the numbers in order, smallest first?
Level 3 · Stretch
7. Use the digits 2, 5 and 7 once each to make the smallest number greater than 5 with one digit before the point and two after it. Give your answer to two decimal places.
8. How many times greater is the 8 in 480 than the 8 in 0.85?
Something to think about
Binary uses only 0 and 1, and the places are worth 1, 2, 4, 8, 16 from the right. What number does 10110 stand for? Which part of the decimal idea did you reuse?
One way to think about it
10110 = 16 + 0 + 4 + 2 + 0 = 22. The idea is the same: a digit is worth its value times its place. Only the places change, doubling each step left instead of becoming ten times greater.
Exit ticket
9. Dilnoza says 0.6 is less than 0.58 because 58 is greater than 6. Which reply is right?