Indicator of progress
Students are able to rename three-digit whole numbers (e.g. 265 as 26 tens + 5 ones) flexibly without materials, appreciating that this process does not change the size of a number. Before this, students may only be able to rename using materials.
The ability to flexibly rename numbers is an essential skill that is developed over many years of schooling.
Illustration 1: Limited renaming
A student is able to rename 235 as 2 hundreds + 3 tens + 5 ones, but is unable to rename this as 23 tens + 5 ones without materials.
Illustration 2: Flexible renaming
A student moving towards this skill will be able to estimate the result of 234 ÷ 10 as 23, by first renaming 234 as 23 tens + 4 ones, and then thinking “I have a number which is a bit more than 23 tens. How many tens are in this number? My estimate is 23.” Estimates produced by such thinking will assist students to reject incorrect answers that might come from trying to apply a half-remembered rule about moving a decimal point.
Illustration 3: Renaming in computation
Renaming is first encountered in the context of whole numbers, and is used in computation. For example, calculating 80 + 40 using the addition algorithm requires renaming 12 tens as 1 hundred + 2 tens. Similarly, calculating 820 − 350 using the decomposition algorithm requires renaming (decomposing) the 8 hundred as 7 hundred + 10 tens.
Students who cannot rename successfully will not be able to carry out such addition and subtraction problems. Errors in these calculations may indicate difficulty with renaming.
Illustration 4: Generalisation to decimals
Later, the skill of renaming is used in the wider context of decimal numbers. For example, students are more likely to understand that 0.4 ÷ 10 = 0.04, if they first rename 4 tenths as 40 hundredths. These 40 objects are then shared amongst 10 people, giving 4 objects to each person (ie 4 hundredths each or 0.04).
Renaming can also assist students with mental estimation; for example, 1.34 ÷ 0.1 can be read as “How many tenths in 1.34? I know 1.34 is 13 tenths + 4 hundredths, so the answer is close to 13.” Students without a firm understanding of whole number renaming will have difficulty flexibly renaming decimal numbers.
Examples of the types of tasks that would be illustrative of renaming three-digit whole numbers flexibly, aligned from the Mathematics Online Interview:
- Question 8 - Reading numerals (to 3-digits)
- Question 9 - Calculator tasks to 3-digit numbers
- Question 10 - Ordering tasks to 3-digit numbers
- Question 11 - Bundling tasks
- Question 13 - 3-digit chart task
NOTE TO TEACHERS:
As students’ number understanding increases they use more places in the place value system. They start with whole numbers and move through tens, hundreds, thousands, and so on, and then also use decimal places, such as tenths, hundredths, thousandths, and so on. Although the curriculum sometimes specifies how many places students should be considering, there are some key principles that apply across all of them.
In our base 10 system, once we have ten of a particular place (e.g. 10 hundreds) we start using the next place value (e.g. 1 thousand). As we go from one place value to a neighbouring place value the magnitude of the numbers changes by a factor of 10. That is, for example, 7 tenths is 10 times bigger than 7 hundredths and 5 hundreds are 1/10 the size of 5 thousands.
Moreover, the places continue beyond thousands and millions, and beyond hundredths, with each change of place involving a factor of 10. This is part of the ‘endless base 10 chain’, part of which is shown below.
For three-digit whole numbers we only deal with a small part of this chain, shown below. What is significant in all the activities is the important role of 10.
The general teaching strategy here is to move from the most direct concrete representation of number to the abstract symbolic representation of number. So, the first activity uses the simplest model, where children make their own bundles of pop-sticks. This draws attention to the importance of ten. In this model, a number is represented by the number of sticks. Activity 2 uses MAB, where the bundling has already been done for students in the construction of the longs and flats. This is convenient, but it relies upon students having already developed a strong notion of how the bundling works. In this model, a number is represented by the total volume of pieces used to make it. Activity 3 leaves the concrete models of numbers behind, and works on the symbolic level with number expanders. Students will take some time to progress from each of these models to the next.
Activity 1: Using pop-sticks to rename involves bundling and unbundling of pop-sticks to rename numbers.
Activity 2: Using MAB to rename involves exchanging MAB to rename numbers.
Activity 3: Using a Number Expander to rename uses a physical manipulative to represent actions with the materials but the numbers are only represented symbolically.
Activity 1: Using pop-sticks to rename
Ensure that students are familiar with using pop-sticks and rubber bands to represent three-digit whole numbers, as an extension of work with 2-digit numbers. For example:
- 43 is made with 4 bundles of ten and 3 singles,
- 143 is made with 1 group of ten bundles of ten (i.e. a hundred group), 4 bundles of ten and 3 singles.
Then ask them to make 143 using only bundles of ten and singles, (i.e. 14 bundles of ten and 3 singles). Give them practice with other three-digit numbers. Students can make challenges for each other to complete. The advantage of using the pop-sticks is that all the individual units are easily seen and can be bundled and unbundled readily; a disadvantage is that many sticks are required for larger three-digit numbers. Students can prepare bundles of ten and of ten tens (100) to keep for use on many occasions.
Activity 2: Using MAB to rename
Ensure that students are familiar with using MAB to represent numbers. For example:
- 43 is made with 4 longs and 3 minis
- 43 can also be made with 43 minis
- 143 is made with 1 flat, 4 longs and 3 minis
Note to Teachers: Emphasise that 43 separate minis is cumbersome compared with the convenience of using 4 longs and 3 minis instead. Highlight, that there are still the same number of blocks (really, the total volume is still the same).
Then ask them to make 143 using only:
- longs and minis (14 longs and 3 minis)
- flats and minis (1 flat and 43 minis)
Make the point that while we could also use 143 separate minis, it is too cumbersome.
Give students practice with other three-digit numbers. Students can also make challenges for each other to complete. MAB are useful because quite large numbers can be represented. A disadvantage is that a long, for example, has to be exchanged for 10 separate minis, rather than broken up into 10 minis. Teachers will need to highlight that the same number of blocks is present after the exchange.
Activity 3: Using a Number Expander to rename
Number Expanders are useful tools because they offer a hands-on way of manipulating the symbolic representation (numeral) of a number. They make a bridge from physical to symbolic models for number. Opening and closing the number expander act as a reminder of the actions with materials (but not as a replacement for this). Because they do not physically model the size of the number, they can be made with any number of place value columns and so can represent very large or small numbers at more advanced levels. Read more about Number Expanders.
Below are some pictures of a number expander with 3 place value columns, showing various ways of ‘expanding’ (or renaming) 236.
||Here is a fully open blank number expander with 3 blank place value columns|
||The number 236 has been written in the blank place value columns. This shows 236 = 2 hundreds + 3 tens + 6 ones|
||This shows 236 = 2 hundreds + 36 ones|
||This shows 236 = 23 tens + 6 ones|
||This shows 236 = 236 ones|
- Print the template of 8 number expanders, then copy and slice into rows to provide each student with their own copy. Template found here (PDF - 11Kb).
- Ask students to write their own three-digit whole number (eg. 517) into the 3 blank rectangles.
- Show students how to fold the expander: the shaded rectangles are folded in half with a 'valley fold' and then a 'mountain fold' is used to put the shaded rectangle behind the white rectangle on its left.
- Ask them to fold and unfold at various places to make as many different expansions as they can.
- Discuss and explore particularly interesting examples such as numbers with zeros. For example, 410 is 41 tens 0 ones; 507 is 50 tens 7 ones; 700 is 7 hundreds, 70 tens and 700 ones.
- Use the photocopier to enlarge a few number expanders and make a wall display for future reference.