Ratio problem

[Erijkhan]

Without shorthand

you know:
1 cake + 2 biscuits = 450 calories
2 cake + 3 biscuits = 800 calories

therefore, from the first fact
2 cakes + 4 biscuits = 900 calories (by doubling)

therefore the diff is 1 biscuit so 1 biscuit = 100 calories

not an undefined variable in sight

Thanks!

[Guest55]

It's still wrong as it's the calories that is the variable.

[Yes]

I was explaining this type of problems to my son using pictures.
Draw 1 cake and 2 biscuits and write 450 near it.
Below draw 2 cakes and 3 biscuits and write 800.
And finally 2 cakes and 4 biscuits

discuss this drawing

[anotherdad]

I was explaining this type of problems to my son using pictures.
Draw 1 cake and 2 biscuits and write 450 near it.
Below draw 2 cakes and 3 biscuits and write 800.
And finally 2 cakes and 4 biscuits

discuss this drawing

Food prices post-Brexit?

[Guest55]

I was explaining this type of problems to my son using pictures.
Draw 1 cake and 2 biscuits and write 450 near it.
Below draw 2 cakes and 3 biscuits and write 800.
And finally 2 cakes and 4 biscuits

discuss this drawing

I like this approach ...

[solimum]

Pictures are much safer than algebra-style shorthand at this stage - I hadn't realised the potential problems with "fruit salad" algebra before taking some OU Maths Education modules in the last year, I suppose because as an adult who "can do maths" I know what I mean by the various symbols, but the potential for confusion in learners is immense. The example that brought it home to me was:

"in a school there are 30 pupils for every teacher. How would you write this as an equation with symbols p for the number of pupils, t for the number of teachers"

Almost universally learners will write 30p = t (I even tried this on a GCSE pupil aiming for a level 5)

Stop and think what's wrong there and maybe it's clear why Guest55 is getting so agitated!

[RedPanda]

I was explaining this type of problems to my son using pictures.
Draw 1 cake and 2 biscuits and write 450 near it.
Below draw 2 cakes and 3 biscuits and write 800.
And finally 2 cakes and 4 biscuits

discuss this drawing

Yep, like that. I used to find (and still do with my kids), that setting the question out in a way that makes the problem easier to understand is often the first step.

The problem with diving straight into algebra is that the problem isn't always understood and we just end up with 'robotic' maths.

I'm a big fan of bar models.

[Surferfish]

Pictures are much safer than algebra-style shorthand at this stage - I hadn't realised the potential problems with "fruit salad" algebra before taking some OU Maths Education modules in the last year, I suppose because as an adult who "can do maths" I know what I mean by the various symbols, but the potential for confusion in learners is immense. The example that brought it home to me was:

"in a school there are 30 pupils for every teacher. How would you write this as an equation with symbols p for the number of pupils, t for the number of teachers"

Almost universally learners will write 30p = t (I even tried this on a GCSE pupil aiming for a level 5)

Stop and think what's wrong there and maybe it's clear why Guest55 is getting so agitated!

Surely it should just be p = 30t ?

(The way I'd think of it was there are going to be many more pupils than teachers so you'd need to multiply the number of teachers by 30 to make it equal to the number of pupils. I don't quite understand though why many people would think it was the other way round? )

Regarding the original question suppose for my answer I wrote:

A. One cake and two biscuits provide four hundred and fifty calories.
B. Two cakes and three biscuits provide eight hundred calories.

Doubling everything in sentence A gives the following sentence C:

C. Two cakes and four biscuits provide nine hundred calories.

Subtracting everything in sentence B from everything in sentence C then leaves:

One biscuit provides one hundred calories.

Would that be considered an acceptable non-algebraic solution (even though it is basically the same as the standard algebraic one except that words are used in place of letters, numbers and mathematical symbols)?

[Yes]

It is t/p=30

Visualization helped us a lot. Algebra is too abstract for this age.
My son was struggling when he met such type of questions, I tried to explain him - it didn't work. But when I draw - he got it very quick.
Same approach I was using explaining him equations. He couldn't get the concept, but when I draw balancing scales and put apples and numbers on it, he got the idea.

[MSD]

This question was for 11+ and I would think the idea is to get the right answer in the minimal time available. Some children will like the picture approach more than algebraic and others may find algebra concept easier; and they don't necessarily need to know the specific approach being used is referred to as 'Algebra'.

I would assume OP welcomes all ideas here and glad to use the one that best suits their child and discard the rest.