Q) I would like to bring history of maths into my Year 6 classroom as a way to enhance a concept in the curriculum. Have you any ideas?
A) I suggest you teach pupils about traditional Chinese numbers and their relationship to our system of numbers, which is Hindu-Arabic. Place value is as important in traditional Chinese numbers as it is in our system, but the numbers are written vertically. In the first example you can see what I mean: the 't' shape is seven and the cross (not a plus sign) is 10. Written vertically, this means seven lots of 10. The character beneath these two is four (four units as it is not modified by another character).
The second example shows how 723 is written.
This exercise helps in the understanding of the construction of our base-10 system and is a nice way to practise breaking numbers down into their different components. I have used it with Years 6 and 7 and also with people at a maths event in a shopping mall. I found that the best way to do this was to photocopy the each individual Chinese number onto card and laminate it, so that each number could be built vertically and we could discuss the numbers as we built them. Making large characters for use would be good for having a whole-class discussion, setting problems and asking children to build or translate the number.
I had great fun using Chinese painting to create the numbers and I think this would be a nice cross-curricular activity with a multi-cultural theme that pupils will enjoy doing while learning a great deal about number structure.
I was interested to learn that the Chinese have four major types of numerals: the traditional or literary form, a business or legal form, a calculating rod form and an abbreviated form.
Enter "Chinese numbers" into a search engine to find some interesting sites. Chinese numbers can be found at www.mandarintools.comnumbers.html
Q) How would you approach the teaching of multiplying an expression in a bracket by a number, a letter - or both - outside the bracket? A common mistake is to multiply only one of the terms by the number outside - eg for 3(2a + 4) pupils write 6a + 4 or even 32a + 34 (the answer should be 6a + 12).
A) The distributive law is a(b + cab + ac where multiplication is distributive over addition and subtraction. Some students don't understand that the "3" outside the brackets indicates that there are three lots of the expression in the brackets. The stage that is missing is that 3(2a+4) actually means (2a+4)+(2a+4)+(2a+4), which we rearrange as 2a + 2a + 2a + 4 + 4 + 4 = 3 x 2a + 3 x 4 = 6a + 12.
Writing this out a few times soon makes us want to use the shortcut: the distributive law. There are several ways to approach teaching this law, depending on ability. Pupils will have met it in primary school when looking at mental methods for arithmetic, eg in working out 6 x 99. This can be rewritten as 6(100 - 1600 - 6 = 594. You could begin with a revision of this method to show an application of the law. Set some problems and include questions containing decimals, eg 3 x 43.25 = 3(40 + 3 + 0.25) =
3 x 40 + 3 x 3 + 3 x 0.25 =
120 + 9 + 0.75= 129.75
An alternative could be to display a 3 x 14 rectangle (= 42 units2) and show how the area can be worked out by splitting it into two pieces of 3 x 10 = 30 units2 and 3 x 4 = 12 units2.
A calculator with an algebra button is useful for showing what happens in algebraic terms. Pupils are asked to investigate a(b + c) and ab + ac for different sets of a, b and c. With an algebraic calculator they can type in the algebra once they have entered the values for the variables. Make sure they include negative values.
You could use this poem as a starting point and discuss what it means.
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
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