Post 22/29 in the Staffrm #29daysofwriting challenge: An interesting way to investigate HCF and LCM
For my remaining full week, I'm going to try and post one rich resource and discussion. I found a nice prompt yesterday in a pack of really old investigations, and thought I'd look into it in a little more detail today. I'd got this filed under HCF and LCM, and have no idea of the origin, but it looked pretty simple so I sat down with pen and paper to try and figure it out.
It turns out it's not quite as simple as I thought; there's definitely plenty of scope for practice using factors, multiples and systematic working with this one.
Essentially, it comes down to:
Essentially, it comes down to:
Number of cuts = (Length + Width) - HCF
In cases where the length and width are co-prime (have a highest common factor of 1), it's quite easy to spot.
It's interesting to explore the links between the 3x2 grid and the 6x4 grid. Multiplying the length and width by 2 has the effect of multiplying both the HCF and the LCM by 2. If pupils connect that grid size = HCF x LCM, there's an interesting discussion to be had here around length and area scale factors.
I also really like the connection to prime factors. It's easiest to see on a bigger grid - we can split the cut on a 12x9 into 3 lots of a 4x3 cut, creating a building block for other sizes - so you could ask pupils how many crossings in a 16x12, or 20x15, or just get them to find other grid sizes that fit with this "family".
If you examine the 4x3 block further, you can see that it's made of two 2x3 blocks, one a 180 degree rotation of the other. We can view the 2x3 as a base building block - we put two together to give the 4x3 block, then fit 3 of these along the length of the rectangle to complete a side length of 12 (see the diagram, it's much easier on there!). This gives a lovely visualisation of the product of primes for the side length of each shape.
It's interesting to explore the links between the 3x2 grid and the 6x4 grid. Multiplying the length and width by 2 has the effect of multiplying both the HCF and the LCM by 2. If pupils connect that grid size = HCF x LCM, there's an interesting discussion to be had here around length and area scale factors.
I also really like the connection to prime factors. It's easiest to see on a bigger grid - we can split the cut on a 12x9 into 3 lots of a 4x3 cut, creating a building block for other sizes - so you could ask pupils how many crossings in a 16x12, or 20x15, or just get them to find other grid sizes that fit with this "family".
If you examine the 4x3 block further, you can see that it's made of two 2x3 blocks, one a 180 degree rotation of the other. We can view the 2x3 as a base building block - we put two together to give the 4x3 block, then fit 3 of these along the length of the rectangle to complete a side length of 12 (see the diagram, it's much easier on there!). This gives a lovely visualisation of the product of primes for the side length of each shape.
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