My emphasis for the last week has been on establishing an idea of "one" with my year 9 academic class. We examined how our idea of one influences how we deal with fractions and algebra.
Firstly we looked at common denominator problems and examined in more detail the method for adding fractions with different denominators.
A common idea is to find common multiples or factors of the denominator and then multiply both the numerator and denominator of the fractions until common denominators are found.
eg. 1/2 + 1/3 -> common denominator of 6 (LCM of 2 and 3)
We then need to find equivalent fractions with denominators of six.
eg 1/2 x 3/3 = 3/6
1/3 x 2/2 = 2/6
Now we have common denominators we can add the fractions..
eg 2/6 + 3/6 = 5/6
But.. why does multiplying by 2/2 and 3/3 work??? Understanding "One" is the answer!!!
1/2 x 1 = 1/2
3/3 = 1
Therefore by substitution 1/2 x 3/3 is just multiplying 1/2 by one. Any number multiplied by one is equal to the original value thus any resulting fraction must be equal to 1/2!
This illustrates two different ideas related to one.. "Multiplying by One" and "Dividing a number by itself".
We also looked at cancelling and why it works..
2m / 3m, we commonly use the skill cancel the m's and 2/3 is what is left.
By re-examining how multiplication works with fractions we find that we can rewrite
2m/3m
as
2/3 x m/m
..but we know that anything divided by itself is 1 (other than zero of course!)
Therefore we can simplify to
2/3 x 1
and we know that anything multiplied by one is equal to the original value.... thus we can see why cancelling works..
Quite a fun little lesson.
Russ.
Showing posts with label fractions. Show all posts
Showing posts with label fractions. Show all posts
Thursday, February 17, 2011
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).
PARENTS NOTE: TEACH YOUR CHILDREN TABLES.
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.
Russ.
PARENTS NOTE: TEACH YOUR CHILDREN TABLES.
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.
Russ.
Friday, April 17, 2009
Fractions
Primary parents are always asking about ways of teaching fractions. The main thing I tell them is to spend time with their kids and work through their own thinking. One of their main concerns is that they do it differently to the teacher and don't want to get their child confused.
I relate to this as many times as teachers we have to think if we have broken it down far enough to promote thinking. Sometimes having a framework is handy with steps to teaching a concept or skill. Here's an ebook that does that (it's not perfect but it could help).
There are many other ebooks on mathematics found here
I relate to this as many times as teachers we have to think if we have broken it down far enough to promote thinking. Sometimes having a framework is handy with steps to teaching a concept or skill. Here's an ebook that does that (it's not perfect but it could help).
There are many other ebooks on mathematics found here
Location:Perth, WA, Australia
Perth WA, Australia
Saturday, October 25, 2008
Revisiting fractions
My 10D class has revisited fractions over the last week. For many fractions is like another language others have managed it in the past but have forgotten basic principles. The sequence I have used leading up to percentages of amounts is as follows
Drawing and identifying numerators and denominators
First exercise was identifying a variety of numerical fractions from pictorial form and then constructing pictorial fractions from numerical forms. We spent a lot of time looking at mixed numerals and converting between mixed numerals and improper fractions using pictorial means.
eg. for 3 2/3: draw 3 lots of 3 boxes with all boxes coloured and 1 lot of 3 boxes with two boxes coloured. When students counted the coloured boxes they had 11/3.
Investigating fractions of amounts
It seemed strange to do this here, but funnily enough it worked well as it established relevancy of the topic for many students. We started with a problem 3/4 of $24 is to be given to John and 1/4 to Mary.
I explained it as:
3/4 of 24 is: $6 per part (24/4)
I drew a box and split it into 4 equal parts (drawing attention to the denominator)
I put $6 in each box.
I coloured in three sections that represented John's portion
then counted $6 x 3 parts = $18 for John
I then repeated the same steps for Mary
1/4 of 24 is: $6 per part (24/4) then $6 x 1 part = $6 for Mary
We checked our answer to ensure all the money had been accounted for ($18+$6=$24). Students then completed a number of examples.
Investigating multiples and factors & Equivalent fractions
Next day we looked at multiples and factors. I explained this through examples, showing them examples of multiples and factors, then getting them to find the first five multiples for 2,3,4,7 and then the first five multiples for 2,3,5,7 over 100. After this they found factors of 10, 15, 24 and 42. We investigated patterns in factors (none greater than 1/2 the original valure other than itself, how it helped knowing your tables, factor pairs, 2 is always a factor for even numbers)
Students were then given a fraction wall and identified equivalent fractions in preparation for adding and subtracting fractions. The idea was put forward that fractions rely on parts to be equal otherwise the idea of equivalency would not be able to be used.
Adding and subtracting fractions
In the third lesson we looked at the problem of 1/3 + 1/2 using paper strips. The aim was to establish why equal parts is essential to an understanding of fractions. We used our fraction wall to look for equivalent fractions that allow us to add equal parts. After a few pictorial examples I started to show students how to use multiples and factors to assist in finding common denominators.
Next lesson we look at multiplying fractions...
Drawing and identifying numerators and denominators
First exercise was identifying a variety of numerical fractions from pictorial form and then constructing pictorial fractions from numerical forms. We spent a lot of time looking at mixed numerals and converting between mixed numerals and improper fractions using pictorial means.
eg. for 3 2/3: draw 3 lots of 3 boxes with all boxes coloured and 1 lot of 3 boxes with two boxes coloured. When students counted the coloured boxes they had 11/3.
Investigating fractions of amounts
It seemed strange to do this here, but funnily enough it worked well as it established relevancy of the topic for many students. We started with a problem 3/4 of $24 is to be given to John and 1/4 to Mary.
I explained it as:
3/4 of 24 is: $6 per part (24/4)
I drew a box and split it into 4 equal parts (drawing attention to the denominator)
I put $6 in each box.
I coloured in three sections that represented John's portion
then counted $6 x 3 parts = $18 for John
I then repeated the same steps for Mary
1/4 of 24 is: $6 per part (24/4) then $6 x 1 part = $6 for Mary
We checked our answer to ensure all the money had been accounted for ($18+$6=$24). Students then completed a number of examples.
Investigating multiples and factors & Equivalent fractions
Next day we looked at multiples and factors. I explained this through examples, showing them examples of multiples and factors, then getting them to find the first five multiples for 2,3,4,7 and then the first five multiples for 2,3,5,7 over 100. After this they found factors of 10, 15, 24 and 42. We investigated patterns in factors (none greater than 1/2 the original valure other than itself, how it helped knowing your tables, factor pairs, 2 is always a factor for even numbers)
Students were then given a fraction wall and identified equivalent fractions in preparation for adding and subtracting fractions. The idea was put forward that fractions rely on parts to be equal otherwise the idea of equivalency would not be able to be used.
Adding and subtracting fractions
In the third lesson we looked at the problem of 1/3 + 1/2 using paper strips. The aim was to establish why equal parts is essential to an understanding of fractions. We used our fraction wall to look for equivalent fractions that allow us to add equal parts. After a few pictorial examples I started to show students how to use multiples and factors to assist in finding common denominators.
Next lesson we look at multiplying fractions...
Subscribe to:
Posts (Atom)