Word problem Singapore way
Laura had 24 clips more than Holly. After she gave 5 clips to Holly, Laura had twice as many clips as Holly. How many clips did Laura have left?
This is from Singapore Challenging Word Problems book 3 (which they are now discontinuing, I heard).
I found two ways to solve this using the bar diagrams.
Solution 1. Notice that this is showing how I solved it initially, and at one point I had to adjust the length of the bar.
Solution 2.
In either case, once you get that x = 9 (the amount of clips Holly had in the beginning), it's easy to solve that Laura had 28 clips left after giving 5 to Holly.
Algebraically:
Initially Laura has L, Holly has L - 24. (Obviously you could also choose to use H and let Laura have H + 24.)
Then Laura gives 5 to Holly, so now Laura has L - 5 and Holly has L - 24 + 5 which is L - 19.
At this point Laura has double as many clips as Holly: L - 5 = 2(L - 19) and we can solve.
L - 5 = 2L - 38
L = 33.
Laura had 33 - 5 = 28 after she gave 5 to Holly.
There are probably other ways to solve this as well.
This is from Singapore Challenging Word Problems book 3 (which they are now discontinuing, I heard).
I found two ways to solve this using the bar diagrams.
Solution 1. Notice that this is showing how I solved it initially, and at one point I had to adjust the length of the bar.
Solution 2.
In either case, once you get that x = 9 (the amount of clips Holly had in the beginning), it's easy to solve that Laura had 28 clips left after giving 5 to Holly.
Algebraically:
Initially Laura has L, Holly has L - 24. (Obviously you could also choose to use H and let Laura have H + 24.)
Then Laura gives 5 to Holly, so now Laura has L - 5 and Holly has L - 24 + 5 which is L - 19.
At this point Laura has double as many clips as Holly: L - 5 = 2(L - 19) and we can solve.
L - 5 = 2L - 38
L = 33.
Laura had 33 - 5 = 28 after she gave 5 to Holly.
There are probably other ways to solve this as well.
Comments
Have you tried doing this with Cuisenaire Rods?
Just for comparison, I wish that I could show you how to think of this in the CSMP curriculum I use with my kids, but I can't draw pictures in a text comment.
But, the point that they are solving for needs to have a name or a star or something by it, so kids don't lose track of what the original question was asking.
I used the following equation:
x + 19 = 2(x+5)
x + 19 = 2x + 10
2x - x = 19 - 10
x = 9
Therefore, Laura has left x + 19 = 9 + 19 = 28
(x representing Holly's initial possession)
This is how I did the above:
Since Laura gives away 5 chips to Holly, then Laura now has x + 24 - 5 = x + 19 and Holly now has x + 5
And, since we know Laura has twice as many chips as Holly after giving away 5 to Holly, then x + 19
should be 2(x + 5)
I do not know how I can help children learn this with understanding, but to me it is definitely an application of the concepts of addition, substraction, multiplication and equations, combining to solve a problem. The use of bars was definitely helpful to me.
In solution 2, why not just compute the difference between H and L? 19 - 5 = 14. this is half of L, then L = 28
Laura has L.
Holly has L-24
then: L-5=2(L-24+5)
L-5=2(L-19)
L-5=2L-38
L-5+38=2L
33=L
The question was how many left to Laura after she gave 5 to Holly which is L-5 so 33-5=28.
you have Holly+(Holly +24)
2h+ 24
After the trade you have
(Holly+ 5) + 2(Holly +5)or
3h + 15.
There are the same number before and after, so
3h+15 = 2h+24
-2h -2h
h+ 15= 24
-15 -15
h = 9
Laura has 2h=10 left after the trade or 2(9)+10= 28.
Your diagram was very informative in solving this problem, and some of the text comments has illustrated more information and simplified this diagram even more.
Jeff