ProfSmudge
@profsmudge.bsky.social
School maths should be more than tables and algorithms. I try to write materials that show that. Dietmar Küchemann
Would you choose the same diagram or different diagrams for the two tasks?
December 2, 2025 at 1:33 PM
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.
December 2, 2025 at 10:35 AM
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
December 2, 2025 at 10:33 AM
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
December 1, 2025 at 10:32 AM
Actually, this makes A and B more equivalent
Which diagram do you prefer here?
December 1, 2025 at 10:19 AM
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).
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).
December 1, 2025 at 9:39 AM
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).
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!
November 29, 2025 at 5:25 PM
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!
And we could use Heron's shortest path problem:
November 28, 2025 at 4:56 PM
And we could use Heron's shortest path problem:
- and for me it didn't click that it formed an ellipse!
November 28, 2025 at 4:10 PM
- 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 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?
November 28, 2025 at 1:28 PM
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 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....]
[Because of the angles in the two small triangles sharing the 1 unit horizontal base....]
November 28, 2025 at 11:36 AM
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....]
[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.
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.
November 28, 2025 at 9:23 AM
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.
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!
November 26, 2025 at 8:49 PM
Ah, I see - quite nice!
? What's the nice conclusion ?
November 26, 2025 at 8:26 PM
? What's the nice conclusion ?
Yes, indeed! But it still shows that the small shapes each cover 1/5²
November 24, 2025 at 3:29 PM
Yes, indeed! But it still shows that the small shapes each cover 1/5²
- and other possible arrangements of the smaller shape:
November 24, 2025 at 11:19 AM
- and other possible arrangements of the smaller shape:
Wow, thanks. Here's its cousin
November 23, 2025 at 1:25 PM
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?
November 21, 2025 at 1:06 PM
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?