Mathematics Developmental Continuum P-10

Estimating Large Numbers: 3.75

Supporting materials

 

Indicator of Progress

At this level students comprehend the size of large and small numbers, from thousandths up to millions. They appreciate how the relative size of numbers increase and decrease when multiplying and dividing by 10 and powers of 10. In other words, students understand the different ‘orders of magnitude’ of numbers (e.g. comparing thousands with tens of thousands).

Before achieving this, students may not comprehend the relative changes in size when moving from one place value to the next. For example, they may believe that there is not a great deal of difference in the magnitude of a number in the ‘thousands’ and a number in the ‘tens of thousands’.

Illustration 1: Notation, language and orders of magnitude

One of the reasons that students may have difficulty appreciating the relative size of numbers is because of the way our place value notation works and because our number words do not fully highlight the size of the numbers.

In comparing two numbers like 3467 and 34670, students may well understand that the number of digits has increased by one in the second number (and may inappropriately describe this by saying that zero has been ‘added’ to the end) and they may also be able to describe that the second number now involves an additional place in the place value system. What they may not fully comprehend is the ten-fold change in size: the second number is ten times the size of the first.

The problem is further exacerbated by the fact that number names like thousand and million hide the multiply-by-a-thousand relationship that exists between them. The words ‘million’ and ‘billion’ sound similar but there is a thousand-fold difference in size. Visualising one million is tricky enough; understanding that a billion is one thousand times the size of a million is also difficult.

Illustration 2: Estimating with large numbers

Students with a limited experience of the magnitude of large numbers will often give unrealistic estimates of quantities. For example, they may estimate the number of people in a crowded railway station as being perhaps a hundred or possibly a million, with no sense of the how extreme these two options are.

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 16 – Sorting capital cities
  • Question 17 – Interpreting the number line
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Teaching Strategies

It is important that students are provided with concrete and practical examples of large quantities so that they can visualise them readily. It is also important that the multiplicative relationships are emphasised (e.g. one quantity is a hundred times another).

Activity 1: Revisiting MAB uses familiar concrete materials to help students visualise larger numbers.

Activity 2: A metric number line uses a linear representation to help students see the relationship between smaller and larger numbers.

Activity 3: A 10kg bag of rice gives students a computational challenge and provides them with another point of reference for larger numbers. Some photos of ways in which rice grains can be used to highlight different quantities are also shown, including one showing over 1 billion.

Activity 4: Trillion Dollars is a presentation that helps students to appreciate a trillion by showing how much space would be needed to store a trillion dollars.

Activity 5: Quantity benchmarks suggests ways to visualise large quantities.

Further activities relating to large numbers can be found in other indicators of progress:

 

Activity 1: Revisiting MAB

Most students should be familiar with MAB materials from earlier work on place value and most school storerooms will have supply. For this activity, obtain at least 10 of the big MAB 1000 blocks, some mini/ones, long/tens, flat/hundreds and a cubic metre kit.

Start with the MAB mini/one (the little cubic centimetre) and discuss the relationship between this and the next larger piece, the long/ten, emphasising that there are ten ones in this and that it is ten times bigger than the one. Move to the flat/hundred, and discuss the fact that the flat is ten times bigger than the long, and then highlight that the thousand block (the big MAB cube) is ten times the size of the flat. Demonstrating this by building the block from ten flats is essential. Also compare the original mini and the thousand block. Students should be familiar with this sequence.

The next step is to ask what ten times the thousand looks like and what it is called. Ideally this should be built using ten of the big MAB cubes (see figure below). We could call this a 'superlong'; it contains ten thousand of the tiny cubes.

Now relate this to a cubic metre.  A cubic metre kit, as in the photo below, is ideal; otherwise use metre rulers to mark out a cubic metre, or build a rough-and-ready model from 1m long rolls of newspaper and sticky tape.

Place the MAB ten thousand at the bottom front of the cube, as shown in the diagram. Ask what you would get if you took ten of these ten thousand superlongs. Highlight that this is one hundred thousand, and that this would consist of ten of the ten-thousand superlongs to make a ginormous MAB 'superflat' covering the bottom of the cubic metre.

Finally, ask students to now imagine making ten superflats. This will make ten layers of the hundred-thousand superflats, which will fill up the cube. The total number of tiny cubes here is one million. It is valuable to highlight a single MAB one against the size of a cubic metre (or MAB million).

Activity 3 further emphasises the multiple-of-ten relationship between the different place values and goes further in illustrating a billion and also a trillion.

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Activity 2: A metric number line

Big Numbers On A Number Line (PDF - 64Kb) - Print off, laminate and cut out these number cards. Students work in pairs or small groups for this activity, followed by group discussion with the teacher.

Each group places a 1m ruler (or tape measure) on their table. (A length of string could be used instead, but by using the ruler it is also possible to highlight some nice relationships between place value and the metric system). There are two kinds of cards in the set of number cards: some are numbered between 0 and 1000, and the remaining cards are larger numbers with the heading ‘Don’t include me just yet’. Have students place the cards numbered between 0 and 1000 along the metre ruler, with 0 at one end and 1000 at the other, so that the numbers are correctly positioned.

It is useful to discuss how close 0 and 1 are to each other (and even 0 and 10), together with 99 and 100, and 999 and 1000. Highlight that 100 is only one-tenth of the way along, and so is very small in comparison to 1000. Other benchmarks like 500 (half-way), 667 (about two-thirds of the way), and 750 (three-quarters of the way) can also be highlighted.

Students should then discuss where the remaining ‘Don’t include me just yet’ cards should go. 2000 should be another metre’s length away, 10000 will be at the 10 metre mark (which makes 100 look very tiny indeed), 100000 should be 100m from the start of the ruler (perhaps way out on the school oval), and 1000000 will be 1km away. This discussion should help students appreciate the magnitude of some of these numbers … and note that we haven’t even got up into the billions yet!

Note: Since the distance between 0 and 1 on this scale is 1mm, the number one billion will be one thousand kilometres away.

 

Activity 3: A 10kg bag of rice

How many grains of rice are there in a 10kg bag?

Encourage students to suggest strategies for answering it.

One way to answer it is to use a set of digital kitchen or postal scales that can weigh things to the nearest gram. Add rice to the scales until you have at least 10g on the scales. If you do less than 5g the results are likely to be very inaccurate because the scales are not perfectly accurate for light masses. If you do more than 20g then there will be too many grains of rice to count! Have the students count out the 10g pile of grains (share them out so that everyone contributes to the counting). Discuss whether you are going to count broken grains. Use this result to find out how many grains there in 100g, 1kg and then 10kg. The answer should be somewhere between 400 000 and 600 000, depending on whether you use short grain or long grain rice and on how you count the fragments. This means that two 10kg bags of rice contain about a million grains.

The following photos show rice being used to highlight the magnitude of some big and small numbers. Each grain of rice represents one person. In the first, the big pile of rice depicts the population of China, which is over 1 billion grains of rice.

The second set of rice piles in the next image (below) shows the numbers of staff and students at a Victorian primary school and one in Malawi, Africa. This could lead to a discussion about social and geographical issues, and highlights that numeracy is important to understand the world around us.

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Activity 4: Trillion Dollars

How big is a trillion dollars? (PPT - 230Kb) - helps students to visualise a million dollars, a billion dollars, and a trillion dollars. On each slide, the amount of money shown increases by a factor of 10, starting with a single $100 note. In many respects, one million dollars is quite a small amount of money: just a neat pile at a person’s feet. However, scale this up by three powers of 10, i.e. 1000, and we have a billion dollars and a pile that would cover the floor of a large bedroom, waist deep. Multiply this by 10, and then another 10, and then another 10, and finally we have a trillion dollars.

Of course, this only allows us to visualise how much space the physical money would occupy. Perhaps a more important question is what this money will buy. A valuable activity would be to examine what can be bought with a million dollars, then a billion dollars, and finally a trillion dollars. Suitable points of reference might be chocolate bars (e.g. How many chocolate bars can you buy with $1 000 000?), skateboards, bicycles, computer games, cars, houses, etc.

It is worth noting that the words billion and trillion have not always had consistent meanings. A million has always meant 1000 thousand (i.e. 1 000 000). The number 100 million, or 1 000 000 000, has been called a thousand million in the UK (and Australia in the past), but the US name of one billion is now the agreed meaning. In the UK and a while ago in Australia, a billion used to mean a million million (i.e. 1 000 000 000 000), or what the US and now most people call a trillion. As for the UK trillion, well, that meant a million million million (i.e. 1 000 000 000 000 000 000), which is a number SOOOO big, that to have this much money you would need to cover the area of greater metropolitan Melbourne to a depth of 1m.

 

Activity 5: Quantity benchmarks

It is useful for students to have mental pictures or benchmarks for larger numbers, as well as being aware of what numbers are relevant to their local experience. Some suggestions, mostly involving numbers of people, are given below:

  • The number of students in the class
  • The number of students in the school
  • The population of the suburb
  • The number of people in the Melbourne Cricket Ground when it is sold out (100 000 or so)
  • The population of Tasmania (about 500 000)
  • The population of Melbourne (about 3 900 000)
  • The population of Australia (approaching 22 000 000)
  • The population of the United States of America (about 300 000 000)
  • The population of Indonesia (about 240 000 000)
  • The population of China (about 1 300 000 000)
  • The population of the world (about 6 800 000 000)
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