Friday, February 4, 2011

Fractions and year 10

We're reviewing fractions and my academic 10's sheepishly owned up to not being confident at fractions. The issue was traced back to poor tables (without it students get hopelessly stuck with LCD methods).


I'm shouting because it's seemingly not PC to rote learn anything. It is hard to get this message heard. People are too busy to do the little things. Curriculum is too full to teach tables in lower school (nonsense), parents are working multiple jobs and don't have time (you can't afford to not find the time), students are too lazy (they have always been too lazy, this hasn't changed), students have little discipline. We are setting students up to fail if we don't take minimum effort to assist them learn key content.

Anyhow, the second element of students not knowing fractions is a lack of actual teaching of what fractions are and how they work. After 60 mins of learning time they could add subtract and multiply fractions and there were a lot of happier students in the room. Here's the method I used.

I started by drawing two objects, one in halves, one with two quarters (colouring in the selected parts) and described fractions as a way of describing the proportion of an object selected. Both objects were the same size and were split into equal parts. I wrote 1/2 and 2/4 (vertically) next to the objects and discussed numerators were the parts selected and denominators were the number of equal parts in each object

I then asked students what would happen if I added the two objects. Students responded that I would have a whole of an object. This was good as it indicated that they had some understanding of a fraction. We discussed how we would expect 2/2 and 4/4 for a whole.

I then added the numerators and denominators and students could see that this was wrong (3/6). I drew what 3/6 would look like.

I then split the 1/2 into quarters and relabelled the 1/2 object 2/4. We talked about equivalent fractions and lowest common multiples at some length.

I then added the numerator and denominators again. This time we had 4/6. I drew this. It was still wrong. Students pointed out not to add the denominators. We noted that adding denominators made no sense as the denominator described the number of parts. Good! We now had 4/4.

We then talked about multiplication. They were happy to accept that to multiply fractions, multiply the numerators and multiply the denominators.

Now we discussed the effect of multiplying by one, how 2/2, 3/3, 4/4 was really one; and used this fact and multiplication to construct equivalent fractions. I pointed out that without tables it was difficult to find lowest common multiples or factors (for denominators) and that simplifying large fractions was a poor alternative for knowing multiples and factors. We then looked back at the cross multiplication method that many had been taught and how that aligned with what we were doing.

Students completed 60 questions of increasing difficulty. All completed working and checked their own answers. Note that there was no "fractions" specific method (such as cross multiplication and lowest common denominator) used here. It simply flowed from their own mathematical understandings.

Finally we discussed that order was important with subtraction. Division was left for another lesson. Formal notes were then given. 60 mins. Happy faces. Job done. Tick.

I'm not saying that this would work with students that have no understanding of fractions. I am saying that proper consolidation of teaching done in upper primary and lower secondary is not difficult with average students and this topic.

The trick will be to consolidate this in algebra, indices and trigonometry topics so that key concepts are not lost in future.


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