Conceptual Obstacles with Numbers Less Than 1: 5.0
Supporting materials
- Related Progression Points
- Developmental Overview of Numbers and Operations (PDF - 2.8Mb)
- Developmental Overview of Proportional Reasoning and Multiplicative Thinking (PDF - 41Kb)
Indicator of Progress
Success depends on students' use of number sense and an understanding of the meaning of operations to estimate the effect of multiplying and dividing by decimals and fractions.
Numbers between 0 and 1 present special difficulties, because the effect of the operation is the opposite of what students expect and the first meanings of these operations (i.e. multiplication as repeated addition for equal groups, division as partition or quotition) do not apply.
See: More about meanings for multiplication.
Illustration 1: Dividing by a fraction
3 ÷ 1/2 = 6
Students may be surprised that 3 ÷ 1/2 = 6 (i.e. how many halves in three - there are six). Many expect that the answer will be less than 3, because 3 has been divided up.
Overgeneralising from their whole number knowledge, they expect falsely that multiplication always makes bigger and that division makes smaller.
Illustration 2: Decimal Fraction
I buy 0.4 kg of meat at $12.40 per kg. What does it cost?
Teachers have always known that students find such a problem very difficult, but they often assume it is mainly because students do not know how to multiply decimals. However, many students get this question wrong even with a calculator, because they choose the wrong operation. Knowing they want an answer smaller than $12.40, they divide instead of multiplying by 0.4. They get the wrong answer $31, or the more realistic $3.10, instead of the correct answer of $4.96.
Teaching Strategies
The teaching challenge here is to extend a limited conception, which has developed naturally from previous school experiences.
Activity 1: Extending the meaning of multiplication and division extends students' understanding to meanings that are not limited to whole numbers.
Activity 2: Multiply Estimo helps give students an intuitive feel for the effect of multiplication and division on the size of numbers, which can guide them in the choice of operation.
Activity 3: Identifying the operation (calculator use puts the focus on choice of operation) and
Activity 4: Identifying the operation (multiple choice format puts the focus on choice of operation) puts the focus on choosing operations, in one case by using calculators to prevent students being swamped by calculation, in the other by using multiple choice items.
Activity 1: Extending the meaning of multiplication and division
The equal groups interpretation of multiplication and division works with whole number multipliers and divisors. For example,
3 × 5 is 3 groups of 5 and so is equal to 5 + 5 + 5 = 15
However, the equal groups model does not make sense for calculations like this:
(1/3) × 37, 0.35 × 7, 0.35 × 7.2, 0.35 × 0.72, (1/3) × (1/5)
4.3 × 37, 4.35 × 7, 4.35 × 7.2, 4.35 × 0.72
Modelling multiplication (and associated division operations) cannot be done with an equal groups interpretation and discrete materials, but needs to employ continuous materials (e.g. lengths) and the idea of multiplicative comparison.
Ask students to model the above multiplications using concrete materials or computer generated images, as illustrated below.
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The green column is one fifth the length of the orange column. The brown column is one third the length of the green column. One third of one fifth of the orange column is one fifteenth of the orange column. 1/3 × 1/5 = 1/15 |
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The green column is 0.8 the length of the orange column. The brown column is 0.65 the length of the green column. 0.65 of 0.8 of the orange column is 0.52 of the orange column. 0.65 × 0.8 = 0.52
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Activity 2: Multiply Estimo
A game for pairs of students with one shared calculator and a die. Through discussion within groups, teachers encourage students to improve their estimates in the light of growing experience.
Instructions
Student A throws the die twice. The numbers that are thrown make the two digit starting number (e.g. if a 6 and a 3 are thrown, the starting number is 63).
Student B throws the die twice. The numbers that are thrown make the two digit target number (e.g. if a 3 and a 2 are thrown, the target number is 32).
Student A enters the starting number (for example 63) into the calculator and hands it to Student B.
Student B presses the multiplication button and any number he wishes, then presses =. The aim is to get the target number. For example, if he enters 0.5, then the screen will now read 31.5 (i.e. 63 × 0.5). Since this is not '32 point something', he passes the calculator back to Student A.
Student A presses the multiplication button and any number she wishes, then presses =. For example, if she enters 1.1, then the screen will now read 34.65 (i.e. 31.5 × 1.1). Since this is not '32 point something', she passes the calculator back to Student B.
Student B multiplies the current number (in this case 34.65) by anything he wishes. Play continues, alternating until the winner gets a number that is 32 point something.
Variations, in increasing level of difficulty
- Play DIVIDE ESTIMO instead of MULTIPLY ESTIMO.
- Throw the die three times to get starting and target numbers in the hundreds.
- Use decimal starts and targets.
- Make the target range smaller (e.g. have to get the target between 32 and 32.01)
- Make the target 3.14159265358979323846 (or as many of these digits fit on your calculator) and call the game Estimo Pi.
Activity 3: Identifying the operation (calculator use puts the focus on choice of operation)
Give students a range of short word problems to solve on their calculators. This focuses their attention on what operation to use, without calculations to distract them. Students should work out the answers on the calculators and write down the buttons that they use. Again this focuses attention on the process.
The samples are arranged here in increasing order of difficulty, by changing from whole numbers, to decimals, to decimals less than one. Research shows that it is the nature of the number that the individual perceives to be doing the multiplying or the dividing that matters most – not the nature of the number being operated on.
In the first example below, I perceive that a quantity of $12.30 is being multiplied by 4 (i.e. that the cost is four lots of $12.30). In this case I perceive 4 to be the multiplier. In (a) the multiplier (i.e. 4) is a whole number and this question is easier for me than question (c) where the perceived multiplier is a decimal less than one.
Three multiplication questions in increasing order of difficulty:
| (a) I buy 4 kg of meat at $12.30 per kg. What is the cost? |
| (b) I buy 4.2 kg of meat at $12.30 per kg. What is the cost? |
| (c) I buy 0.4 kg of meat at $12.30 per kg. What is the cost? |
Here (a) can be solved by repeated addition and so can be done by students at a lower developmental level than (b), which requires multiplication. Many students want to divide for (c) because they want an answer less than $12.30. This puts it at the highest level.
Three division questions in increasing order of difficulty:
| (a) I travelled 160 km in 4 hours. How fast was I going? |
| (b) I travelled 160 km in 4.2 hours. How fast was I going? |
| (c) I travelled 160 km in 0.4 hours. How fast was I going? |
In this set, (a) is easiest because of the whole number of hours. (b) is of intermediate difficulty. Many students will know that the answer for (c) will be more than 160km/hour, but will want to multiply. This question is only done successfully by students at the highest level.
Activity 4 - Identifying the operation (multiple choice format puts the focus on choice of operation)
Multiple choice items can also focus attention on operations, not calculations. More people use the wrong operation when the perceived 'multiplier' or 'divisor' is less than one. (The correct answers are highlighted in blue.)
(a) A spaceship travelled at a speed of 160.9 km/h for 0.32 hours. How far did it go?
| Circle the answer | 160.9 × 0.32 | 160.9 ÷ 0.32 | 0.32 ÷ 160.9 |
(b) A spaceship travelled at a speed of 0.32 km/h for 160.9 hours. How far did it go?
| Circle the answer | 160.9 × 0.32 | 160.9 ÷ 0.32 | 0.32 ÷ 160.9 |
(c) Stephen will be paid $60000 for three fifths of the year. At the same rate, how much would he be paid for a whole year?
Circle all the correct answers
| $60000 × (3/5) | $60000 ÷ (3/5) | (3/5) × $60000 × (3/5) | (3/5) ÷ $60000 |
(d) The Aussie dollar is worth 0.71USD. How much is AU$20 in USD?
Circle all the correct answers
| 20 × 0.71 | 0.71 × 20 | 20 ÷ 0.71 | 0.71 ÷ 20 | 20 - 0.71 |
(e) The Aussie dollar is worth 0.71USD. How much is US$20 in AUD?
Circle all the correct answers
| 20 × 0.71 | 0.71 × 20 | 20 ÷ 0.71 | 0.71 ÷ 20 | 20 - 0.71 |