ProfSmudge

Amazon-speak for
Package was left on doorstep

Thank you - similar to my initial method, but using ÷6, x6 to give 11 x 10.

Thank you. When I wrote this I had in mind multiplying the mixed number to turn it into a whole number, like your last method, except I multiplied by 6, giving 11x10. Then other methods came to mind, like some of yours (I didn't see 2x66 – 1/3 of 66 !).
[Also 1&1/3 of 6 =10, so 10+100.]

A simpler version, perhaps.

Think of ways to solve this

In #MathsToday
Varying an expression and thinking about what happens to its value;
and when that didn't work, varying the expression to be varied....

Interesting idea

Why the music?!

Would you choose the same diagram or different diagrams for the two tasks?

But note also, that we usually draw a double number line with each line representing a particular measure space, in this case people along one line and pizzas along the other. That's the left hand DNL in my example.

Yes, the numbers form a ratio table. Note also that the 'vertical' or 'between the lines' multiplier is the same for all pairs of numbers, whereas the 'horizontal' or 'along the lines' multiplier is not. If we had 2 pizzas instead of 3, we would still divide by 5 but would multiply by 2/5 not 3/5

Actually, this makes A and B more equivalent

Which diagram do you prefer here?

Yes, something like that!
If for example one thinks of the scalar relations (pizzas to pizzas, and people to people), in A these happen 'horizontally' (along the lines), in B they happen 'vertically' (between the lines).

Thanks Matt. That's mind-boggling! It seems to 'work' even though the interpretation of the numbers doesn't fit their meaning in the story!

YouTube video by scryl "Ami, go home!" - German Anti-American Song

Can't help thinking of this. Our new song?!
[Might need to tweak the words a bit.]
www.youtube.com/watch?v=fJbx...

Opinions welcome!

And we could use Heron's shortest path problem:

- and for me it didn't click that it formed an ellipse!

Yes, it's not 100% obvious!
If one moves the top vertex horizontally and further and further away (so obtuse-angled triangle) the perimeter clearly gets bigger and bigger, but that doesn't immediately rule out that the right-angled triangle is a minimum.
What if the isos triangle is made of string?

If we slide the top vertex of the triangle on the right leftwards to form the triangle on the left, then the length of the red side increases by more than the length of the green side decreases....
[Because of the angles in the two small triangles sharing the 1 unit horizontal base....]

Nice task.
Looked at statically it is hard to tell: on the left, the vertical line is shorter than each of those on the right but the slanting line is longer.
Looked at dynamically, it is pretty clear (!) that the closer the top vertex is to the 'middle', the smaller the sum of the sides.

Ah, I see - quite nice!

? What's the nice conclusion ?

Yes, indeed! But it still shows that the small shapes each cover 1/5²

- and other possible arrangements of the smaller shape:

Or blobs....

Wow, thanks. Here's its cousin

It's nice that NCETM is using our ICCAMS task to clarify their thinking on oracy. But how far should one push the use of formal maths language? Is there not a danger that pushing formal language will work against inclusion?