53-4=49
49/3=16
16x2x3 =96
96+8=104
My daughter answer

Every two triangles can be paired to form one rectangle.

Two ways to arrange the rectangles :

① Put the 3cm side aligned with the 12cm side i.e breadthwise

12cm ÷ 3cm = 4 (so you can put 4 sets of rectangles)
53cm ÷ 4cm = 13 R 1cm (so 13 sets can be put lengthwise)

Total number of triangles = 13 × 4 × 2 = 104

② Put the 4cm side aligned with the 12cm side i.e breadthwise

12cm ÷ 4cm = 3 (so you can put 3 sets of rectangles breadthwise)
53cm ÷ 3cm = 17 R 2cm (so 17 sets can be put lengthwise)

Total number of triangles = 17 × 3 × 2 = 102


So, option ① is better since more triangles can be put and less remaining space is wasted.

But the sch teacher give 106
Any idea why? Thanks

Your daughter's way is acceptable also just that her arrangement of the triangles involves both the horizontal and vertical orientation.



12 ÷ 3 = 4
4 × 2 = 8


53 - 4 = 49

Take note of the presentation for the following :

49 ÷ 3 = 16 R 1 and not 16

106 is incorrect.

That's obtained by finding the area of the whole cardboard.

53cm × 12cm = 636cm²

Then, area of each triangle = ½ × 4cm × 3cm
= 6cm²

636cm² ÷ 6cm² = 106

But this is not acceptable as there are leftover pieces of the cardboard that cannot be joined together to form new triangles.

Be cautious, sometime the last few rectangle can change direction to fix more but not for this case.
Example:
At length 53cm can fix
16 x 3cm = 48cm and 1 x 4cm = 4cm
Used up 52cm.

At length 12 cm can fix
3 x 4cm or 4 x 3cm

16 × 3 = 48
1 × 4 = 4
Total = 52 then x2 = 104.
After change orientation, still same for this case.
Need to check 3x4 and 4x3 in order not to miss any additional rectangle can be fixed.

Can't be 106.
Is either
53÷4 =13 R1 then x4 (side 12÷3) = 52 (104)
Or
53÷3 = 17 R2 then ×3(side 12÷4)= 51 (102)

The most is 104.

Thank u. Can u key something on the qns to close this qn? As i will not be able to post another qn with u solving it. Appreciate. If not i have to delete the post

Yup.

What's interesting is, 13 × 4 also gives 52.
16 × 3 + 4 = 48 + 4 = 52 also.


Note that :

13 × 4 = 12 × 4 + 1 × 4 = 48 + 4


The first 48 cm by 12cm can actually be made out of either 4cm by 3cm or 3cm by 4cm orientation.


The result is 4 '12cm by 12cm big squares'


The LCM of 4 and 3 is 12 (to be learnt at higher levels)

So no matter horizontal or vertical, this 48cm by 12cm is obtained.

Done as requested.

Remember to take note of presentation :

49 ÷ 3 = 16 is mathematically incorrect as it should be 16.333333..... or 16⅓ (exact answer)


If there is a remainder and it needs to be expressed/shown ,

Then you have to write 49 ÷ 3 = 16 R 1

This shows that there are only 16 complete sets possible.


Likewise,

53 ÷ 4 = 14 R 1

53 ÷ 3 = 17 R 2