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Diagnostic test for subtraction discussion
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Beth's error is that she always borrows from the first column. She simply takes 2 from the hundreds and puts a 1 in front of the tens and another 1 in front of the ones.
The strategies described below refer to some examples of Beth's written work. To see these examples, click here. Use your browser 'Back' button to return to this page.
Teaching strategies:
A teacher could help Beth by first of all helping her to see that her answers are not correct. For example, she could check the last subtraction by adding, and see that 158 + 556 is not 704. Then she needs to think about what she is not doing correctly. If Beth learned subtraction using the decomposition algorithm with MultiBase Arithmetic Blocks, then the teacher could ask her to explain each of the steps that she has taken with the concrete material. She should make 774 and then decompose one of the hundreds (flats) into ten tens (longs). She will have 6 hundreds and 17 tens. To get enough ones for the subtraction to take place, she will probably automatically decompose a ten (long), not another hundred (flat). This is the point where her written work is going wrong.
Back to Example completed diagnostic test for subtraction
Don borrows from a 0 treating it as a 10, but fails to decrement it to 9. He continues to use it as a 10 when subtracting from that digit.
The strategies described below refer to some examples of Don's written work. To see these examples, click here. Use your browser 'Back' button to return to this page.
Teaching strategies:
Don seems lost in the symbols. He has nearly learned how to do the procedure, but not quite. Don might be better off being taught a slightly different version of decomposition. Instead of decomposing the hundreds and then the tens separately (or vice versa), he might be better off thinking of a number such as 704 as 70 tens and 4 ones. To get more ones, we decompose one ten giving 69 tens and 14 ones. A number expander could help here. Don also needs to learn to check subtraction by adding the difference to the subtrahend.
Back to Example completed diagnostic test for subtraction
Lynn may be confused by the fact that 0 x anything = 0. She might be using the multiplicative property of 0 accidentally. Alternatively, she may have reasoned that 0 - anything = 0. Children sometimes think this, because if you have zero lollies and you try to take some anyway, you can't - there are still none there.
Lynn is able to use decomposition fairly successfully, however when faced with a zero she has difficulty. For example when there is a zero in the tens column she ignores it and borrows directly from the hundreds column.
The strategies described below refer to some examples of Lynn's written work. To see these examples, click here. Use your browser 'Back' button to return to this page.
Teaching strategies:
Lynn might be helped by thinking about how her method leads to some silly answers on some easy questions. For example, 600 - 122 will be less than 500, not 522. Also using place value material such as MAB at this stage will help Lynn consolidate her knowledge about the properties of zero. It might also be worth gently probing to see if Lynn has any idea of negative numbers e.g. in a temperature context. She may be quite excited to hear that 0 - 3 is negative 3, rather than "it can't be done" and that she will learn more about these numbers in secondary school.
Back to Example completed diagnostic test for subtraction
Meng borrows one from a 0 when necessary and decrements it to 9 without taking one hundred from the column to the left of the 0.
The strategies described below refer to some examples of Meng's written work. To see these examples, click here. Use your browser 'Back' button to return to this page.
Teaching strategies:
To help Meng, she first needs to see that her answers are not correct and that it is not just a "slip" that is causing her to get the questions wrong. For example, 206 - 49 can't be larger than 206. Then Meng needs to go back to concrete material, working through the subtraction step by step decomposing numbers made with MultiBase Arithmetic Blocks.
Back to Example completed diagnostic test for subtraction
Whenever a larger number has to be taken away from a smaller number, Pepu puts down the result as 0. This is a reasonable belief derived from some real situations: if I have 4 objects and try to take away 8, the best I can do is to take 4 of them, leaving none.
The strategies described below refer to some examples of Pepu's written work. To see these examples, click here. Use your browser 'Back' button to return to this page.
Teaching strategies:
Pepu might be helped by thinking about how his method leads to some silly answers on some easy questions. For example, 701 - 699 = 100. Once Pepu realises that her method is flawed then she needs to be lead to consider possible alternative strategies for when there is not enough in one column to complete the subtraction. It might also be worth gently probing to see if Pepu has any idea of negative numbers e.g. in a temperature context. She may be quite excited to hear that 0 - 3 is negative 3, rather than "it can't be done" and that she will learn more about these numbers in secondary school. Some children automatically invent a strategy for subtraction based on negative numbers. You can see an example in the section "Mental Methods for Subtraction".
Back to Example completed diagnostic test for subtraction
Richard subtracts the ones, the tens, the hundreds etc one at a time, but always subtracts the smaller digit from the larger one even when the smaller digit belongs to the upper number (minuend).
Richard's method gives the correct answer on questions which do not require decomposition or "borrowing".
The strategies described below refer to some examples of Richard's written work. To see these examples, click here. Use your browser 'Back' button to return to this page.
Teaching
strategies:
A teacher could reinforce the concept of place value using teaching aids like blocks or an abacus and use money as a concrete example. Help Richard to see subtraction also as 'take away' and not only as a difference.
Ask Richard to make a number such as 74 with MAB blocks. He will have 7 longs and 4 minis. Then ask him to take away 38. He will be able to take away the 30 (3 longs) but he will not be able to take away the 8. He must understand the need for decomposition or borrowing before he can be expected to do it.
Back to Example completed diagnostic test for subtraction
