Many kids in grade 7 are unable to operate with fractions. Among them a big segment are unable to grasp what fractions are, a second smaller set are unable to proceed on the arithmetic even after they understand why they are supposed to factorize, take LCM, etc.
As I took supplementary classes for the 7th grade kids, it was obvious that they were comfortable operating fractions when the denominators were the same (5th grade). Of course this just meant that they had a system where they added/subtracted numerators without worrying themselves about what fractions are/were and this was a gotcha in 6th. I am also working with the 6th graders to address the issue here itself.
Numbers are abstractions, but fractions are more so given that they are parts of a whole and a fraction can take different avatars depending on what the whole is. E.g. 1/2 of 1 kg is 1/2 kg, but 1/2 of 1/2 kg is 1/4 kg. A nice abstraction of a whole is something circular. It makes it very obvious when pieces cut diagonally are extra or are missing.
Pizza Party
Most children around Auroville actually know what a pizza is (not all like it) and I spent quite some time with a teaching aid called pizza party (Creatives). The kids that don't like pizzas assume that its a dosa. The game is fairly cheap (Rs.165) and is generally well done (though the suggested games need work).
What it has:
Base cards of fractions 1/2, 1/3, 1/4, 1/6, 1/8. Pieces of the same proportions. A die that has these fractions (5) and creative written on it.
Modified/invented games:
1) Getting familiar with the pieces using base cards. The idea was to roll the die and pick up the corresponding base card. Then in turns roll the die till the fractional piece on your base card comes to complete the pizza.
Well the kid getting 1/2 needs 2 pieces and the one with 1/8 needs 8 so the game if far from fair.
I tried to even out the odds by allowing them to check if the piece fits into the pie and taking as many pieces as it fits, so a kid with 1/8 base card can take 4 pieces of 1/8 when he/she gets 1/2. Now, it seems 1/8 has the advantage. But, we added a no overflow rule i.e. if you get a 1/2 and a 1/8, then you get another 1/2 i.e. you can't use it. The only disadvantage is for 1/3 base card who really do need to wait for 3 such cards to complete the pizza.
2) Complete the rest of the pizza: Roll the die make the rest of the pizza with the pieces you have.
3) Selecting pieces to make a pizza in turns, but you pick the piece for the next person to play. The one who completes a pizza gets a point. You can keep the full pizza as part of the game to see when it gets used and who gives it to who :). Nice game to get into the kids psyche.
4) Pizza delivery game: Group game, the next piece of the pizza is determined by the die roll and the team tries to build 5 pizzas for delivery.
Initially, the children who were getting it wanted to play the game with less luck, but given the mix of kids they ended up playing many different ones. Some kids also wanted new games and we introduced ones with subtraction of two fractions and finding pieces that match the difference, or a piece that is just larger or just smaller than the difference.
One often wonders if doing these fraction games is really worth the time and I started a conversation with the kids of what we learnt (not what we did) from the activity. With feedback starting from 4 pieces of 1/4 makes a whole. This helps reiterate 1/4 means you cut the pizza in 4 pieces and take one. Similar ideas continue for 2 pieces of 1/2 and 8 pieces of 1/8. Then we move on to expressing one set of pieces in terms of another, 1/4+1/4=1/2, 3x1/6=1/3 and my personal favorite 1/2+1/3+1/6 is a whole pizza. Wow, made my day.
A note of caution for teachers using mixed fruit to teach fractions. We ask children to treat everything as 'fruit', adding apples to oranges to make up a whole. The whole is not obvious as a fruit can be added or removed and it would still be a collection of fruits. We then ask them to remember the 'fruits' individuality by asking what fraction of the fruits are bananas. The children get comfortable adding grapes to watermelons, but they will also add 1/2 a pizza slice with 1/8 pizza slice to give 2 pizza slices.
Equivalent fractions
An often skipped section to work quickly towards factorization and LCM is the idea that a fraction can be expressed as equivalent fractions.
By now, most kids can tell stories about fractions.
What is the story of 1/4? You take a pizza and cut it into 4 pieces and take one piece.
The idea can be extended into the relm of 1 out of every 4 pieces. This helps build equivalent fractions. What if you had 8 pieces in the pizza then 1/4 would cover 2 pieces (from the pizza game). So
1/4=2/8=3/12=4/16=...
At this point you can reintroduce the idea of adding fractions with the same denominator say
1/8+3/8 = (1+3)/8 = 4/8
and remind them that the denominator indicates the number of pieces you cut the pizza into. You can add the number of pieces as they are the same size.
1/2+1/4 the pieces are not the same size and can't be added directly. This can easily be seen from the pizza game. With equivalent fractions we can talk about what 1/2 will be if the pizza is cut into 4 pieces. One out of every two gives 2/4 pieces. Now adding:
2/4+1/4 = (2+1)/4 = 3/4
Most smaller fractions can be added by writing them in equivalent fractions and looking for a size that is common to both.
1/6+1/8
1/6=2/12=3/18=4/24
1/8=2/16=3/24=4/32
This gives 1/6+1/8 = 4/24+3/24 = (4+3)/24 = 7/24
I introduce LCM after I ask them to add
1/2+1/200 at which point most children start taking a short cut into
100/200+1/200.
Of course nothing works for every kid, but I was able to address 90% of the kids this way.
As I took supplementary classes for the 7th grade kids, it was obvious that they were comfortable operating fractions when the denominators were the same (5th grade). Of course this just meant that they had a system where they added/subtracted numerators without worrying themselves about what fractions are/were and this was a gotcha in 6th. I am also working with the 6th graders to address the issue here itself.
Numbers are abstractions, but fractions are more so given that they are parts of a whole and a fraction can take different avatars depending on what the whole is. E.g. 1/2 of 1 kg is 1/2 kg, but 1/2 of 1/2 kg is 1/4 kg. A nice abstraction of a whole is something circular. It makes it very obvious when pieces cut diagonally are extra or are missing.
Pizza Party
Most children around Auroville actually know what a pizza is (not all like it) and I spent quite some time with a teaching aid called pizza party (Creatives). The kids that don't like pizzas assume that its a dosa. The game is fairly cheap (Rs.165) and is generally well done (though the suggested games need work).What it has:
Base cards of fractions 1/2, 1/3, 1/4, 1/6, 1/8. Pieces of the same proportions. A die that has these fractions (5) and creative written on it.
Modified/invented games:
1) Getting familiar with the pieces using base cards. The idea was to roll the die and pick up the corresponding base card. Then in turns roll the die till the fractional piece on your base card comes to complete the pizza.
Well the kid getting 1/2 needs 2 pieces and the one with 1/8 needs 8 so the game if far from fair.
I tried to even out the odds by allowing them to check if the piece fits into the pie and taking as many pieces as it fits, so a kid with 1/8 base card can take 4 pieces of 1/8 when he/she gets 1/2. Now, it seems 1/8 has the advantage. But, we added a no overflow rule i.e. if you get a 1/2 and a 1/8, then you get another 1/2 i.e. you can't use it. The only disadvantage is for 1/3 base card who really do need to wait for 3 such cards to complete the pizza.
2) Complete the rest of the pizza: Roll the die make the rest of the pizza with the pieces you have.
3) Selecting pieces to make a pizza in turns, but you pick the piece for the next person to play. The one who completes a pizza gets a point. You can keep the full pizza as part of the game to see when it gets used and who gives it to who :). Nice game to get into the kids psyche.
4) Pizza delivery game: Group game, the next piece of the pizza is determined by the die roll and the team tries to build 5 pizzas for delivery.
Initially, the children who were getting it wanted to play the game with less luck, but given the mix of kids they ended up playing many different ones. Some kids also wanted new games and we introduced ones with subtraction of two fractions and finding pieces that match the difference, or a piece that is just larger or just smaller than the difference.
One often wonders if doing these fraction games is really worth the time and I started a conversation with the kids of what we learnt (not what we did) from the activity. With feedback starting from 4 pieces of 1/4 makes a whole. This helps reiterate 1/4 means you cut the pizza in 4 pieces and take one. Similar ideas continue for 2 pieces of 1/2 and 8 pieces of 1/8. Then we move on to expressing one set of pieces in terms of another, 1/4+1/4=1/2, 3x1/6=1/3 and my personal favorite 1/2+1/3+1/6 is a whole pizza. Wow, made my day.
A note of caution for teachers using mixed fruit to teach fractions. We ask children to treat everything as 'fruit', adding apples to oranges to make up a whole. The whole is not obvious as a fruit can be added or removed and it would still be a collection of fruits. We then ask them to remember the 'fruits' individuality by asking what fraction of the fruits are bananas. The children get comfortable adding grapes to watermelons, but they will also add 1/2 a pizza slice with 1/8 pizza slice to give 2 pizza slices.Equivalent fractions
An often skipped section to work quickly towards factorization and LCM is the idea that a fraction can be expressed as equivalent fractions.
By now, most kids can tell stories about fractions.
What is the story of 1/4? You take a pizza and cut it into 4 pieces and take one piece.
The idea can be extended into the relm of 1 out of every 4 pieces. This helps build equivalent fractions. What if you had 8 pieces in the pizza then 1/4 would cover 2 pieces (from the pizza game). So
1/4=2/8=3/12=4/16=...
At this point you can reintroduce the idea of adding fractions with the same denominator say
1/8+3/8 = (1+3)/8 = 4/8
and remind them that the denominator indicates the number of pieces you cut the pizza into. You can add the number of pieces as they are the same size.
1/2+1/4 the pieces are not the same size and can't be added directly. This can easily be seen from the pizza game. With equivalent fractions we can talk about what 1/2 will be if the pizza is cut into 4 pieces. One out of every two gives 2/4 pieces. Now adding:
2/4+1/4 = (2+1)/4 = 3/4
Most smaller fractions can be added by writing them in equivalent fractions and looking for a size that is common to both.
1/6+1/8
1/6=2/12=3/18=4/24
1/8=2/16=3/24=4/32
This gives 1/6+1/8 = 4/24+3/24 = (4+3)/24 = 7/24
I introduce LCM after I ask them to add
1/2+1/200 at which point most children start taking a short cut into
100/200+1/200.
Of course nothing works for every kid, but I was able to address 90% of the kids this way.
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