Calculating with Large Numbers: 3.5

• Indicator of Progress
• Teaching Strategies

Supporting materials

• Related Progression Points
• Developmental Overview of Number (PDF - 108Kb)
• Professional Reading: Mental computation and estimation (PDF - 110Kb)

Indicator of Progress

At this level students are able to generalise the properties of the place value system to larger place values. Success depends on students understanding how multiplication and division by powers of ten affects place value. They realise that products such as 50 × 700, for example, are determined by using the basic number fact 5 × 7 together with use of place value knowledge to determine 10 × 100.

Before achieving this, students may lose track of place value when multiplying. Students may think that, for example, hundreds times hundreds is only in the thousands. Achievement at this level relies on computational fluency with basic number facts and understanding the place value system.

Illustration 1: Inappropriate use of the multiplication algorithm

Students who are not aware of the relationship between place value and multiplying by powers of ten will often use the multiplication algorithm to calculate obvious products such as 23 x 1000, painstakingly multiplying all the zeros as shown below.

Or

These students will also have difficulty with products like 40 x 900, and will not realise that this can be done mentally.

Illustration 2: Capacity to self-correct

Students with a good understanding of number properties and the place value system can use estimation to check answers and correct computational errors. This applies to both multiplication and division.

When doing division, having a zero in one of the places of the dividend can cause difficulties. As an example, when using the division algorithm to determine 7209 ÷ 3, many students will respond with 243 because, after dealing with the 72, they may say “0 can’t be shared amongst 3, so now we do 9 shared among 3.” A student with a good understanding of place value will know that the answer should be more than 6000 ÷ 3 which is 2000, which will alert them to an error in their answer of 243.

Illustration 3: Links to the Mathematics Online Interview

Examples of the types of tasks that would be illustrative of successful use of computation with large numbers aligned from the Mathematics Online Interview:

• Question 32 – Multiplication problems
• Question 33 – Division problems
• Question 34 – Off to the circus
• Question 35 – Sharing our money
• Question 36 – In your head
• Question 37 – Missing number

Teaching Strategies

Good teaching should build upon the use of concrete materials for small powers of ten but should develop the capacity to generalise to larger powers of ten. Careful explanation and demonstration, with a reliance on number properties, is necessary to build up the understanding of how multiplication and division work with different place values, and then students need practice to gain fluency with the principles. Teachers can also model for students the ways in which this understanding can be used to check calculations and make sense of the world around them.

Activity 1: Understanding multiplying by powers of 10 highlights some key properties of multiplication with powers of ten that students need to know.

Activity 2: Big number mental multiplication game provides students with the opportunity to reinforce their understanding of multiplying numbers that are multiples of 10, 100 and 1000.

Activity 3: Check it out approximately reminds students that mental computation can be used to estimate and/or check the answer.

Activity 4: Making it meaningful suggests ways that real-life situations can be used to help students calculate and make sense of large number situations.

Activity 1: Understanding multiplying by powers of 10

One of the key understandings required for calculating with large numbers is how place value works with multiplication. The PowerPoint presentation [LINK TO MultiplicationPowersOf10.ppt] highlights ways of modelling the properties using concrete materials such as MAB and using key number facts. It highlights issues associated with multiplying and dividing a number by powers of ten (e.g. x 100, x 10000, ÷ 1000, etc.) and makes reference to a number slide which can be used to illustrate the changes in the place value of digits that occur. Here are Instructions for making a number slide

Its use highlights to students how multiplying by each additional power of ten causes the place value of the digits to increase by one place (or move to the left), whereas dividing by each power of ten decreases the place value of the digits (or moves them to the left). In particular, it points out that it is NOT helpful to talk about ’moving the decimal point’.

The PowerPoint also highlights the importance of establishing fact families for place value. Students can investigate families of examples such as

• 8 × 3 = 24
• 8 × 30 = 240
• 8 × 300 = 2400
• 80 × 30 = 2400
• 80 × 300 = 24000.

These fact families highlight that multiplication of numbers like 800 × 3000 depends on the basic number fact 8 × 3 and then an understanding of the place value associated with 100 × 1000 (or hundreds × thousands), to get 8 × 3 × 100 × 1000 or 24 × 100 000 or 2 400 000. The PowerPoint shows how some of this can be modelled with materials; a calculator could also be used to establish the patterns in the families of examples. Explanation and demonstration could be followed with Activity 2 which provides students with practice using these properties.

Activity 2: Big number mental multiplication game

The big number mental multiplication game is a game for two players that allows students to practise mentally multiplying large numbers with an emphasis on powers of ten and place value (as per the discussion in Activity 1). A printable file of the spinners for the game are available here [link to SpinnerMultiplication.pdf]. The instructions are given below and are also included with the game.

Players take it in turns to play. When it is Player 1’s turn, Player 2 gets to choose which two spinners will be used. The two spinners are spun (players can do one each) and Player 1 has to determine mentally the product of the two numbers that are spun. Recording on paper is permitted in order to keep track of place value. The answer is then checked on a calculator. If the answer is correct, Player 1 gets a point. [Note that there are a few combinations that may be too big for a standard small calculator. These cases actually make good discussion points, but may need to be resolved by the teacher.]

It is then Player 2’s turn, with Player 1 getting to choose which spinners are used. Play alternates between the players. The opponent always chooses which two spinners are to be used, unless a player’s last answer was incorrect. In this case, the player whose turn it is gets to choose for themself which two spinners will be used. This means that the player can choose an easy combination of spinners, to build up confidence and expertise again. The winner is the first player to reach 10 points, provided both players have had the same number of turns.

Special note about making spinners: It is easy to make reliable robust spinners with a pencil and a paperclip. Put the paperclip onto the spinner and put the pencil tip through one end of the paper clip so that the pencil tip rests on the centre of the spinner, with the pencil held vertically. The paperclip should be able to spin around the pencil tip, and act as a pointer/arrow for a spinner. It can be spun by giving the paperclip a flick while the pencil is held firmly in place on the spinner’s centre.

Activity 3: Check it out approximately

Students should be encouraged to estimate and/or check the answers to computational problems. This will also require skill with rounding numbers to the nearest 10, 100, 1000, and so on, and making appropriate choices as to which is most useful.

This may arise when students are working on the long multiplication algorithm. For example, if they are working out 287 × 37, they should be encouraged to estimate the answer to act as a check. In this case, 287 is close to 300 and 37 is close to 40, so the answer should be close to 300 × 40 which is 12000. In this particular example, students should also be able to deduce that the estimate of 12000 is larger than the correct answer.

Similar comments apply if students are doing division problems. If working out something like 8072 ÷ 4 students could estimate by working out 8000 ÷ 4, which is 2000, so that they know their answer should be in the thousands rather than the hundreds or the tens. This is a particularly telling example, because some students will have difficulty with the 0 in 8072 when doing the division algorithm and may come up with an answer that is in the hundreds (typically 218).

Students can also be encouraged to approximate the calculations they need in real life. Suppose we want to know about how many times a student’s heart beats in a year. If the resting pulse is about 80 beats per minute, then in an hour the heart will beat about 60 × 80 or 4800 times. We can approximate this to 5000 to make calculation easier. The implications of and reasons for making such an approximation should be discussed. Then, approximate the number of hours in the day as 20 because it is easier to do 20 x 5000 (and again discuss the reasons and implications), and so get 100 000 beats in a day. Therefore in a year there are 36 500 000 heartbeats. It is worth comparing this with the actual calculated answer of 42 048 000, obtained without using any approximations. It is important to help students realise that although 36 500 000 and 42 048 000 are, in many ways, quite different and a 'long way apart', they are also quite close, and that having the right 'order of magnitude' is probably good enough (i.e. both answers are in the tens of millions, which makes our estimate sufficiently accurate as an estimate).

See also Activity 4.

Activity 4: Making it meaningful

Be alert for examples where large numbers are reported in the media or arise in events associated with students’ lives. Help students to calculate with and question these examples. In many cases it will be appropriate to use a calculator for the actual computations, but the more important principle is to know what operation to choose (e.g. multiplication or division). Students should always be encouraged to estimate as well, using mental strategies and the properties discussed in Activities 1 and 2.

Here are just a few examples:

• A news report at the beginning of 2009 reported the story of an American teenage girl who had sent 14,528 text messages in one month. Is this actually possible? How many messages is that in a day? How many is that in an hour? How frequently did she have to send a message?
  
  An estimated answer that can be dealt with mentally can be obtained by saying that the number of text messages was about 15000, and that there are about 30 days in a month, so this would mean that the girl would have had to send about 500 messages each day. If there were 25 hours in a day (as this is easier to work with than 24, but is still ‘close enough’) then 4 messages in a hour would give 100 messages in a day, so to get 500 messages in a day she needs 5 times that in an hour, or 20 messages an hour. This is one message every 3 minutes. Students may realise that perhaps she sent group messages, and so they could discuss how this might affect the calculations.
  
  
• It is recommended that primary school aged children watch no more than two hours of television a day. If you did this for a whole year, how many days are used up sitting in front of the TV?
  
  We could work this one out mentally by realising that to do 365 × 2 we just have to double 365, and doubling 300 is 600, and doubling 65 is 130, so we get 730 hours of television in a year. At this point, you might get out a calculator and work out 730 ÷ 24 to work out how many days is equal to 730 hours. Alternatively you may realise that 72 is 3 × 24, so 30 × 24 is 720, which means 730 hours is a bit more than 30 days or a month. This makes sense, because 2 hours is 1/12 of a day, so children are watching television for ¹/₁₂ of the time. For a year this “¹/₁₂ of the time” will be a month.
  
  
• At the beginning of May 2009, Melbourne’s water storages had 500 GL (gigalitres) of water. If people have been asked to use only 155 L a day, how long will this water last if it doesn’t rain? (You will need some other assumptions – the population of Melbourne and maybe estimating that industry, not people, uses half the water).
  
• In 2006 there was a rupture in the ground in Indonesia that started erupting with warm mud, like a mud volcano. It was spewing out 150 000 cubic metres of mud every day. How much space would that take up? It kept doing this for at least a year, how big might the mud puddle become? (You might have to assume some information, such as the depth of the puddle).
  
• Is it possible to live to be 1000000 seconds old? What about 1000000 minutes? What about 1000000 hours? What about 1000000 days?