Q) My students get terribly confused when rounding numbers to different significant figures and decimal places. In fact, sometimes I get confused and am not sure that I have done it correctly. For example, something like 9.999 to two significant figures - is it 10.00?
A) The first confusion students often have is in understanding the difference between rounding to a certain number of significant figures and rounding to a certain number of decimal places.
In your question, 9.999 to two significant figures is in fact 1.0. Your answer of 10.00 would be 9.999 rounded to two decimal places.
The number of significant figures is the total number of digits you have after the number is rounded, whereas the number of decimal places is the number of digits after the decimal point after the number has been rounded.
For readers who aren't sure of the process, here are some examples.
* To round 32.735 to two decimal places (2dp): count two digits to the right of the decimal point, then look at the next figure. If it is a 5 or higher then the number rounds up to 32.74, otherwise it rounds down to 32.73. So 32.735 E 32.74 (2dp).
I would advocate that diagrams should be used when introducing students to rounding numbers. This helps them understand the process. 32.735 lies between 32.73 and 32.74. Every number in the shaded region is rounded up.
Numbers below 32.735, for example 32.733 are rounded down to 32.73.
* To round 32.735 two significant figures (2sf): count two figures from the left to the right and look at the third figure. If it is 5 or higher, the number is rounded up to 33; otherwise the number approximates to 32.
Diagrammatically we have:
Students should be encouraged to draw their own diagrams in the initial stages. A lack of understanding of the basic principles is often why students studying higher level GCSE maths find it difficult to understand problems of upper and lower bounds.
There is a discussion on using significant figures on The TES maths forum (go to www.tes.co.uksectionstaffroom, choose mathematics and type "sig figs" into the search box).
Q) I support a lower set in Year 10 maths. There is a girl in the group who finds it difficult to align the positions of each part of the answer when setting out traditional long-multiplication sums. She often wants to start on the left and doesn't realise the importance of aligning from the right, although she knows her tables really well.
A) It seems she hasn't really understood place value and what happens when multiplying by powers of 10.
I prefer to use the Chinese grid method for long multiplication. It is much more fun and certainly easier.
To work out 329 x 26, show your student how this can be broken down into two parts: (20 x 329) + (6 x 329).
Work through the answers. To do this takes a long time and the more traditional layout is quicker.
The difficulties encountered by students are related to knowing which direction to proceed with the multiplication, so I encourage them to draw an arrow at the top.
The next problem they have is lining up the digits, so I get them to draw pencil guidelines as I have done in the diagram.
They often miss the 0 indicating the multiplication by multiples of 10, so I use a colour code to remind them of this place value.
They are told to align the colours as I have done so the pink in the 10s position begins in the answer in the 10s position.