Explaining the concepts of ‘trading’ or ‘renaming Trading (or grouping) – grouping numbers into groups of 1s, 10s, 100s, 1000s (etc) in order to perform simple additions and subtractions For example, you get 10 of a particular unit and then trade it for one of the subsequent unit (i.e 10 ones are traded for 1 ten. 10 tens are traded for 1 hundred). It is important for teachers to make sure the students understand the link between the blocks, the actions and the words – than merely move the blocks as directed by the teacher without making the connections with the concept of trading and the addition process. A simple trading example is: = ? = 55 Renaming – Numbers need to be renamed in a varietyy of ways rather than being understood in terms of counting or even place value (Booker et al, 2010, pg 81). For example, 89 can be viewed as 8 tens and 9 ones, as well as 89 ones. 568 can be interpreted as 5 hundreds, 6 tens and 8 ones; or 56 tens, 8 ones; or 568 ones. **** RENAMING AND TRADING GO HAND-IN-HAND ****

Explain why these understandings are essential for children’s number learning. Renaming and trading numbers are crucial number processes in developing number sense (Booker et al, 2010, pg 82). Renaming of numbers is used everyday. It is important that students understand trading and renaming in order to create meaningful mathematical experiences. Renaming is a crucial way of understanding numbers. Booker et al state that understanding how numbers can be renamed is important for comparison and rounding, counting on and back, and, later, students will need an understanding of renaming in order to understand the algorithms for subtracting and diving large numbers (2012, pg 119). These concepts lay the foundations to a students future mathematical education, for example, once an understanding of renaming and trading has been gained, children can then extend this thought process to counting on and back by hundreds, tens and by ones (especially with larger numbers) Booker et al, 2010, pg 121). Students must understand when renaming is appropriate, for example, in Stage 1 Mathematics (NSW), students may be shown that you only need to rename sometimes. An example is: = ? (show working and explain why you do/do not have to rename a number). Answer: Renaming is the same thing as borrowing. In this particular problem, 32 can be subtracted from 65 without renaming (65-32=33). However, if the problem was 62-35, you would have to rename the 2 to 12, by borrowing from the 6, which would then become

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. Examples Source: State Government of Victoria. (2009). Department of Education and Early Childhood Development. Mathematics Developmental Continuum. Retrieved from ingresources/maths/mathscontinuum/number/N22501 P.htm#a1 ingresources/maths/mathscontinuum/number/N22501 P.htm#a1

Examples (cont’d) 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. Source: State Government of Victoria. (2009). Department of Education and Early Childhood Development. Mathematics Developmental Continuum. Retrieved from ingresources/maths/mathscontinuum/number/N22501 P.htm#a1 ingresources/maths/mathscontinuum/number/N22501 P.htm#a1

Examples (cont’d) Number expanders are a common tool used in the classroom to help explain the concept of renaming. Blank, to show hundreds, tens and ones. 236 = 2 hundreds + 3 tens + 6 ones 236 = 2 hundreds + 36 ones 236 = 23 tens + 6 ones 236 = 236 ones

Examples (cont’d) Trading Adding numbers Numbers of any size can be added together easily. When adding 2, 3 and 4 digit numbers using a written method, write the numbers in a vertical list. You will need to properly line up the place value columns so you get the correct total. Example: Step 1: Make sure each number is carefully listed in neat place value columns to avoid errors. Step 2: Working from right to left, add each column, starting with the units column. You may have to trade to the next column. 9 ones + 5 ones = 14 ones. Trade ten ones for one ten 6 tens + 1 ten = 7 tens 2 hundreds + 7 hundreds = 9 hundreds Answer = 974

Subtracting numbers When we subtract, we take away one of the two numbers from the other. Make sure the same place value columns line up underneath each other. 6 ones cannot be taken away from 5 ones. We need to add ten ones to make 15, and change the 3 at the top of the tens column to ones less 6 ones, equals 9 ones. Step 2: (Tens column) 4 tens cannot be taken away from 2 tens. Trade 1 hundred to make 12, and change the 4 at the top of the next column to a less 4 equals 8. Step 3: (Hundreds column) 3 hundreds less 2 hundreds leave 1 hundred. Answer = 189 Examples (cont’d)