At a glance. One lesson of about 60 minutes. Students see that the value of a digit comes from its place, extend the rule to tenths, hundredths and thousandths, and compare and order decimals by place.
Calculator: no calculator. Nothing here needs one. The skill is reading the structure of a number, and a calculator would hide it.
What students will learn
Know
- Each place is ten times the place to its right.
- A zero holds a place when nothing else is there.
- Adding a zero to the end of a decimal does not change its value.
Understand
- The value of a digit depends on where it sits, and moving it one place changes it by a factor of ten.
- Comparing decimals is comparing place by place from the left, not comparing lengths.
- A number can be written in other ways: expanded form, words, or a different base.
Do
- State the value of a digit in a whole number or decimal.
- Write numbers from expanded form and from words.
- Compare and order decimals with different numbers of decimal places.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: values of digits and ordering of decimals in familiar and new forms (questions 1 to 6).
- Criterion B. Investigating patterns: the digit-sliding figure. Ask students to state what happens to the value for each step left and each step right, and to say why it works for every digit.
- Criterion C. Communicating: explaining why 0.4 is greater than 0.35 using place language, and writing numbers in expanded form (questions 5 and 9).
- Criterion D. Applying mathematics in real-life contexts: race times and measurement, where a decimal's size matters (the faded example).
The one thing to do with the figure
Ask for a prediction before the first slide. The 7 is worth 70. “If it moves one place right, what is it worth?” Take 7 and 0.7 from the room and let the figure settle it.
Then slide the place all the way left and all the way right and ask what stays the same: the digit. Only the place changes the value, and it changes it by a factor of ten each step. Name that on the board, because multiplying and dividing by 10 and 100 later is this idea again.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea; the second figure. Students set a and b and say the deciding place aloud. |
| 10 min | Worked example on ordering; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose B or D, go back to the first figure before going on. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 4207; 60 and 600; 3 m is longer. |
|---|---|
| Worked example | 0.065 < 0.58 < 0.6 < 0.605. |
| Faded example | 52.9 > 52.3 > 52.09 > 52.089. |
| 1. Value of a digit | C, 0.04. Counting from the point: 7 is tenths, 4 is hundredths, 6 is thousandths. A ignores the place; B counts the 4 as the first digit after the point. |
| 2. Writing a number | 5038. No hundreds, so the 0 holds the place. |
| 3. Digit with a given value | C, 3.07. A is tenths, B is ones, D is tens. |
| 4. Multiply by ten | 40.7. 4 ones becomes 4 tens, 0 tenths becomes 0 ones, 7 hundredths becomes 7 tenths. |
| 5. Expanded form | 209.407. The tens and hundredths are 0. |
| 6. Ordering decimals | B. A puts 0.4 before 0.35; C is reversed; D puts 0.35 before 0.05. |
| 7. Smallest number from digits | 5.27. Ones digit 5 (2 is too small), then tenths 2, hundredths 7. |
| 8. Comparing two places | 100. 80 ÷ 0.8 = 100: tens to ones to tenths is two steps. |
| 9. The longer-is-bigger trap | B. A and D read the digits after the point as a whole number or count the digits. C: unsure what a leading 0 tells you; return to the second figure. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B or D in question 1 | Counted the first digit after the point as the one wanted, or went one place too far. | Draw the chart and write the name of each place above each digit before reading any value. |
| 538 in question 2 | Left out the placeholder zero. | Ask what the number would be if the 5 were in the hundreds place. Then show where the 0 has to sit. |
| 40.07 or 4.7 in question 4 | Moved only some digits, or lost the zero. | Have them write each digit in its own column of a chart and shift the whole chart. |
| 209.47 or 29.407 in question 5 | Closed up the gaps instead of keeping placeholders. | Ask for the value of the digit in each place, listing all seven places. |
| A or D in question 6; 0.35 above 0.4 | Reads the digits after the point as a whole number, or the longer number as the larger. | Write both with the same number of decimal places and compare tenths first. |
| 10 in question 8 | Counted one step between tens and tenths. | Count the steps on the chart: tens, ones, tenths is two steps. |
Questions worth asking
- Is the zero in 407 doing anything? Yes: it keeps the 4 in the hundreds. Listen for “it holds the tens place”.
- Why is 0.5 the same as 0.50? Because 5 tenths is 50 hundredths. Adding a zero on the right changes the name of the last place, not the amount.
- Is a number with more digits always bigger? For whole numbers written without leading zeros it is. For decimals it is not: 0.4 beats 0.35.
- Could we count in a system where each place is worth 2, not 10? Yes: binary. Ask how 10110 is built; it connects the lesson to how computers store numbers.
Differentiation
A way in for anyone stuck
- Give each student a printed place-value chart with the seven place names. Every question can be done on it.
- Use money as a bridge: where 100 small coins make the main unit, 0.40 is 40 small coins and 0.35 is 35, and 40 beats 35.
- Start with the whole-number half of the first figure before the decimal half.
Stretch
- Question 7, then: how many different numbers greater than 5 can be made from the same three digits in the same shape? (4: 5.27, 5.72, 7.25 and 7.52.)
- Write 0.1 + 0.01 + 0.001 as a decimal, then say what it is in thousandths. (0.111, and 111 thousandths.) What would 0.1111 be?
Students learning in English as an additional language
Number names in English change in the teens and again at the decimal point. Say each place name aloud while pointing to the chart.
| Word | Plain meaning | Example |
|---|---|---|
| place value | the value a digit has because of where it sits | In 305 the 3 is worth 300 |
| digit | one of the symbols 0 to 9 | 305 has the digits 3, 0 and 5 |
| decimal point | the dot that separates whole numbers from parts of one | In 3.7 the point sits between 3 and 7 |
| tenth, hundredth | one of ten equal parts, one of a hundred equal parts | 0.1 is one tenth; 0.01 is one hundredth |
| placeholder | a zero that keeps other digits in the right places | the 0 in 407 |
| expanded form | a number written as the sum of the values of its digits | 5038 = 5000 + 30 + 8 |
Sentence starters for explaining aloud or in writing:
- The digit … is in the … place, so it is worth …
- … is greater than … because the … are different: … and …
- Each place is … times the place on its right because …
Two things not to say
“Move the decimal point.” The point does not move. The digits move, and the point is a marker between the ones and the tenths.
“Just add zeros to make them the same length.” That is a method; it needs the reason, that adding a zero on the right of a decimal does not change the number.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Value of a digitWhat is the value of the 5 in 8.052?
0.05, five hundredths. - OrderingPut these in order, smallest first: 0.8, 0.08, 0.88, 0.808.
0.08, 0.8, 0.808, 0.88.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- MoneyKofi's mobile credit is 12.40 and Ama's is 12.04. Who has more, and how much more?
Kofi, by 0.36. - MeasuresA bean plant is 0.30 m tall, another 0.3 m and a third 0.03 m. Which two are the same height?
0.30 m and 0.3 m: 0.30 = 0.3, and 0.03 is a tenth of that.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningWrite a decimal that lies between 0.4 and 0.41, then another between 0.4 and your first. Can this go on?
0.405, then 0.4025 and so on. It never runs out: between any two decimals there is another. - SystemsIn base 2, the places from the right are 1, 2, 4, 8, 16. What does 1101 stand for, and what is the left-hand 1 worth?
13 (8 + 4 + 1), and the left-hand 1 is worth 8.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Draw the place-value chart on the board and move a digit card along it.