This unit requires students to apply their number sense about the size of decimals to estimate and calculate the product of decimal fractions. In doing so they generalise about the effect of multiplying and dividing by ten and one hundred.
- Express a multiplication of two decimals as a product of fractions, e.g. 0.4 x 0.7 as 4/10 x 7/10 = 28/100.
- Connect the product of the two fractions to the decimal answer, e.g. 28/100 = 0.28.
- Know the effect of multiplying and dividing a decimal number by ten or one hundred.
- Use multiplication and division by ten or one hundred to find the size of a product of decimals, e.g. 0.4 x 0.7 is the product of 4 x 7 divided by one hundred.
- Use known benchmarks, especially one half and one, to estimate the products of two decimals, e.g. 0.4 x 0.7 is a bit less than 0.5 x 0.7.
In this unit the students will develop number sense related to multiplying decimals. Making sensible estimates for the products of decimals requires a flexible connection of place number understanding with whole numbers, the meaning of multiplication, and multiplication with fractions.
In the age of calculators, it is a dubious use of instruction time to concentrate extensively on pencil and paper algorithms. Rather, emphasis should go on giving students strategies to recognise the reasonableness of answers and to estimate the products of decimals.
The decimal system includes a restricted set of equivalent fractions for situations where one units are inadequate for purpose. Commonly situations where tenths, hundredths, thousandths, etc. of units are needed involve measurement. The prefixes deci (tenth), centi (hundredth) and milli (thousandth) are applied to base units, such as metres and litres, to obtain a necessary degree of precision. Tenths, hundredths, thousandths, etc. are the result of division by ten of the previous larger unit. Therefore, understanding the effect of multiplying and dividing a decimal amount by ten, one hundred, and one thousand is foundational to estimating the products of decimals. The effect of division by ten is to make each unit one tenths of its previous size, represented as a shift of the digits one place to the right. For example, 3.9 ÷ 10 = 0.39, 3.9 ÷ 100 = 0.039, 3.9 ÷ 1000 = 0.0039.
Applying multiplication to measurement situations often involves a multiplier and a rate. A rate is a relationship between two measures, such as 60 kilometres per hour (velocity), 456 kilograms per cubic metre (density), or 30 people per square kilometre (also density). The multiplier is applied to the rate resulting in a measure, e.g. 3 kilograms of meat at $12.50 per kilogram costs $37.50.
Decimals might more correctly be termed decimal fractions. Deci is the prefix meaning one tenth, to indicate that decimal fractions are powers of one tenth, or negative powers of ten, e.g. 1/1000 = (1/10)³ or 10⁻³. Estimation of the size of products requires understanding of the multiplication of decimal fractions. Knowing 1/10 x 1/10 = 1/100 and 1/10 x 1/100 = 1/1000, etc., makes it possible to know the size of the products for factors like 0.6 x 0.4 and 0.3 x 0.08. Expressing both multiplications as fractions gives 6/10 x 4/10= 24/100 and 3/10 x 8/100 = 24/1000, so the products are 0.24 and 0.024 respectively. A sound understanding of multiplication of fractions is therefore a prerequisite for multiplying decimals with a sense of the size of product.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
- Using physical objects to connect decimals, as numbers, with physicial quantities. An area model is used extensively in the unit. You may need to revisit the area(array) model in simple whole number contexts to remind students about how the factors are side lengths and the products are areas.
- Directly model the recording of equations, highlighting the fraction meaning of decimals and the symbols, x as ‘of’, and = as ‘the same as.’ For example, 0.3 x 0.9 = means “What is three tenths of nine tenths?”
- Use calculators to confirm predictions about the results of multiplying decimals. Where differences between predictions and outcomes occur, ask students to work out where their predictions went awry. This strategy helps to cause cognitive conflict and address common misconceptions. The area model is particularly good in identifying where the size of units is misinterpreted.
- Encourage students to work collaboratively in partnerships. Students need time to develop mathematical arguments and to rehearse those arguments with a peer is important for developing clarity and risk taking.
- At first use rates that students are more familiar with, such as items in a packet, or speed in kilometres per hour. Progress to more unfamiliar rates such as density and fuel consumption.
- Fold in an out of different levels of abstraction, i.e. materials, images (diagrams), equations. Use symbols as a means to connect across situations. Thinking with patterns in equations, and reasoning why patterns occur, is at the heart of mathematics.
The contexts for this unit are about rates, which is unavoidable with multiplication. Situations involve electricity and fuel consumption, density, and cost. Adapt the contexts if needed to meet the interests and backgrounds of your students. For example, using lengths of fabric at a decimal amount per metre, the cost of petrol per litre, or costs of vegetables, seafood, or other ingredient per kilogram, might be more familiar to your students. For students who have whānau in other nations, such as India or Samoa, currency exchange rates might be of interest.
In this session students are introduced to multiplying decimals in everyday contexts. They refresh their understanding of multiplying fractions and the size of decimals using an area model. Students should work through the units Getting Partial to Fractions and Getting Partial to Decimals before attempting this unit.
- Begin with the first three slides of PowerPoint One. For each scenario:
What operation do we need to carry out to solve this problem?
Treat the significant digits as whole numbers to calculate the product:
451 x 100 = 45 100 45 x 208 = 9 360 15 x 537 = 8 055
- Ask your students to choose which of the four options is correct. Expect them to justify their choices using sense of the magnitude of the numbers. For example:
- Slide One: If you get 4.5 Yuan for $1.00 then you should get 45 Yuan for $10.00 and 450 Yuan for $100.
- Slide Two: 4.5 equals four and one half. 4½ x 2 = 10 so the answer must be close to $10.00.
- Slide Three: 0.537kg is a bit more than half of one kilogram. The answer should be about half of $15.00 which is $7.50.
- Point out that the unit is about multiplying decimals and that number sense will be important in determining whether an answer is reasonable or not. Write the calculation 0.4 x 0.6 = ? on the board.
- Ask for the meaning of the symbols:
What does 0.4 mean? (four tenths)
What does 0.6 mean? (six tenths)
What does x mean? (of)
What does equals mean? (the same amount as)
- Ask your students to predict the answer to 0.4 x 0.6 = ? Many may opt for 2.4 since both factors have the decimal point in the ‘middle’, as does 2.4.
- Slide Four of PowerPoint One shows how an area model can be used to find the answer. The model is important for understanding the metric units for length, area, and later volume. Four tenths of six tenths is shown by the orange rectangle. One unit of the 24 that fill the rectangle is shown.
How big is the area that represents 0.4 x 0.6? (24 hundredths)
What is the answer to 0.4 x 0.6? (0.24)
- You may have to put the digits of 0.24 on a place value chart to remind students of the nested nature of place value. In this case 20 hundredths are nested in 2 tenths. Slides 5 and 6 present two other examples of simple multiplication of decimals. In both cases challenge your students to predict the answers first before using the area model to confirm and make sense of the correct answer. You might provide Copymaster One to enable students to draw their own model of the multiplication.
- Provide Copymaster Two for students to practise solving simple decimal multiplication problems. Look for your students to:
- Use the appropriate whole number multiplication fact.
- Convert the decimals to fractions to mark the correct side lengths.
- Understand that when tenths of tenths are found the answer is in hundredths.
- Retain reference to one to find the answer.
- Correctly express the answer as a decimal.
In the next two sessions student learn more about the place value structure of decimals. They connect their knowledge of multiplication with fractions to finding decimals place value units, especially tenths of tenths, tenths of hundredths, and hundredths of hundredths. The learn about the effect of multiplying and dividing a number by ten and one hundred.
- Make several square metres from butchers’ paper.
Who can mark one tenth of one tenth on this square metre? (You might use a place value block flat to mark the area)
What will you call one tenth of one tenth? (one hundredth of a square metre(m²))
Note that one hundredth of a square metre is a square decimetre (dm²) though the unit is rarely used in New Zealand.
How might we record finding one tenth of one tenth? (1/10 x 1/10 = 1/100 or 0.1 x 0.1 = 0.01)
- Recall that last session you worked out calculations like 0.6 x 0.5 = ?
How do I write that calculation using fractions? (6/10 x 5/10 = 30/100 )
Why is the answer not 0.030? (Thirty hundredths equals three tenths)
- Proceed in the same way to find:
- One tenth of one hundredth equals one thousandth (1/10 x 1/100 = 1/1000 or 0.1 x 0.01 = 0.001) that can be marked with a place value long unit (1cm x 10cm).
- One hundredth of one hundredth equals one ten thousandth (1/100 x 1/100 = 1/10000, or 0.01 x 0.01 = 0.0001) that can be marked with the area of a unit place value cube (1cm x 1cm). Note that one ten thousandth of one square metre is one square centimetre (1 cm²).
- Ask your students to recreate these units using the squares on Copymaster One. They should label the diagrams with the operation and the unit itself, e.g. 1/10 x 1/100 = 1/1000 ).
- Once students return, ask problems like:
- If 0.6 x 0.05 = 6/10 x 5/100, why is the answer not 0.0030?
- If 0.12 x 0.12 = 12/100 x 12/100, what is the answer? Why?
- If 2.5 x 0.4 = 25/10 x 4/10, What is the answer? Why?
- You might take considerable time over the previous examples, asking students to draw diagrams to support their thinking.
- Provide students with Copymaster Three that deals with the patterns created when a number is multiplied or divided by ten. Students can attempt the problems in pairs with a calculator shared between them.
- Discuss a couple of the patterns to see if students correctly identify changes and consistencies between consecutive equations. See if they have generalised the effect of multiplying and dividing by ten.
- Watch Videos One, Two, Three, and Four to see why numbers behave the way they do when multiplied and divided by ten. For each video, ask your students to anticipate what will happen.
What is the effect of multiplying/dividing by ten?
- Pose these problems for students to solve in pairs:
3.7 x 10 = 3.7 x 100 = 3.7 ÷ 10 = 3.7 ÷ 10 = 1/10 x 3.7= 1/100 x 3.7=
Be aware that students may not see that 3.7 divided by ten has the same result as one tenth of 3.7. That is an important connection if they are to use the effect to estimate the results of decimal multiplication.
- Begin with PowerPoint Three as a guide to applying decimal multiplication to a real-life context. You might discuss density which is the ratio of mass to volume. Bringing in several objects with varying densities will help illustrate the attribute. For example, a basketball and a block of wood may look to have the same volume but vary considerably in mass.
- Work your way through the slides, inviting students to solve the problems. Do your students:
- Have a reasonable expectation of the number size of each answer, using benchmarks like 0.5 as one half?
- Understand that decimals can be thought of as fractions, to establish the size of the answer?
- Recognise that division by ten can be used to find the answer in decimal form?
- Copymaster Four contains a set of problems that require multiplication of decimals. Let your students solve the problems in small teams. Most of the problems involve a rate, such as kilowatts per hour or litres per 100 kilometres. Rates in real life tend to be less familiar to students than the simple rates they encounter in multiplication normally, e.g. marbles per bag. After students attempt the problems it is worthwhile to discuss the rates that appear.
This session is devoted to applying multiplication of decimals. The beginning explicitly confronts common misconceptions about decimals as a window to the understandings that students have developed so far. The final part of the session involves a game where multiplication of decimals is required.
- Work your way through the four slides of PowerPoint Four. On each slide challenge the students:
- Is this student correct or incorrect?
- Explain why they are correct or incorrect.
- Suggest something to the student that will improve their understanding of decimals.
- Focus on the following issues in your discussion:
- Slide One: ‘Adding zeros’ is a rule that students commonly learn. When a number is multiplied by ten, all the place value units become ten times the size. So effectively the digits move one place to the left, relative to the decimal point. The correct answer is 10 x 2.8 = 28.
- Slide Two: Thinking that the decimal point is a separator can result in students operating of both sides of the decimal point independently. The role of the decimal point is to mark the ones place. The decimal places are connected. Felix could think that the answer is one tenth of the product of 6 x 48 = 288, so is 28.8. He could also use fraction multiplication 6 x 48/10 = 288/10 = 28 8/10 .
- Slide Three: It is common for students to match the number of decimal places in the product with the number of places in the factors. Len could think that the answer is one tenth of one tenth (1/100) of the product of 8 x 3 = 24, so is 0.24. He could also use fraction multiplication 8/10 x 3/10 = 24/100, or 0.24.
- Slide Four: Emma has generalised that zeros to the right do not affect the size of the decimal. For example, she thinks 3.70 has the same value as 3.7. In most cases that is true except when the zero on 3.70 represents a degree of accuracy, such as giving the number of centimetres, rounded to the nearest centimetre. Her answer is correct, possibly by accidently using Len’s incorrect idea. Emma should know that 1.5 is the same as one and one half. The answer should be one and one half lots of eight tenths, which is twelve tenths. So the correct answer should be 1.2 .
- After the class discussion introduce the game Decimal Pathways (Copymaster Five). The rules are as follows:
- Each player needs a pen of a different colour to their opponent.
- One copy of the game board is needed for each game. The gameboard could be laminated and whiteboard pens used instead of felt pens.
- To Play
Players take turns to:
- Connect one factor on the left of their panel with one factor on the right, e.g. 0.3 and 0.8.
- Calculate the product of the factors, e.g. 0.3 x 0.8 = 0.24.
- Draw around the circle on the gameboard that has that product. If that product is not on the board the player misses that turn.
- Once a circle is coloured it cannot be claimed by the other player.
- The object of the game is to create a pathway of circles from a Player’s Start to their Finish. The first player to do that wins. If no player can make a pathway then the game is a draw. Note the pathway can be constructed in any order.
- Look for your students to:
- Calculate the products fluently using a combination of basic facts and place value knowledge.
- Play strategically by capturing the circles in the centre of the board first.
- Think ahead to capture circles that are most advantageous.
- To Play