So here are a couple of nice resources I've made, really getting underneath prime factors, HCF and LCM. If you just want to go straight in, the resources are here:
We start of with some really basic getting underneath the concept of what is a prime factor decomposition, using non examples and examples. And then slowly work through a series of tasks, getting kids to pay attention to the product part of the product of prime factors, getting kids to write the product, given the decomposition, fill in missing parts of the decomposition before actually doing it themselves. Then in the second PDF, we start to look at using prime factors for HCF and LCM.
But before that, we spend ages getting students familiar with the Venn diagrams themselves. So this involves just identifying the numbers from a Venn Diagram, reading the HCF and LCM from a Venn diagram, populating the Venn diagram etc before actually doing it all themselves. I'm certain many lower attaining students will find these scaffolds invaluable.
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For anyone who just wants to get straight to the resources, here is a huge booklet of fraction calculations. It starts by getting students to manipulate and simplify them before working up slowly to the four operations on mixed numbers.
For any one interested in a bit more detail, read on. This was one of my first experiments with a whole bunch of 'presteps' or a nascent version of atomisation. Historically I would have jumped into teaching fractions with some simplifying, but actually there's just a whole set of things we need students to think about around manipulating them. So I start the booklet with a bunch of activities like this: where we're really just getting students to pay attention to the fact that a fraction is a thing you can manipulate in this way. Then we do some "reverse simplifying" as it were, first with untiary fractions, so students don't have to think too much about what they're multiplying by, and then with non unitary fractions. If you look through the whole document you'll notice that sections 13 and 14 are about finding a denominator on the left hand side only, before working up to finding numerators and denominators on either side before working up to a more challenging equivalent fractions task. This is where some judgement is definitely needed. I certainly have taught classes who could jump straight into task 19 (which on the 6th page of this booklet!). However, my belief is that any student can learn any bit of maths. The only question is how much do they need it broken down and how much practice do they need. Lower attaining students also probably lack a lot of confidence, so teaching them one highly specific thing e.g. multiplying a unitary fraction to find an equivalent denominator only, isn't going to hurt. In fact, it's going to give them the confidence to tackle the harder questions we want them to. It's us explicitly talking them through every possible bit of meaning and aiming to secure 100% success. Then once students are familiar with manipulating equivalent fractions we do some more standard simplification: Then something slightly different again. Ultimately I also want students to be able to convert between mixed number and improper fractions, so it's possible you might jump straight to here: However, I want to be 100% sure that all student can do this so between section 20 and 27 there are a whole series of tasks where we get students just to think about what a whole number with a certain denominator is: and hopefully this will really build their confidence. Once that's done we really slowly work through the four operations, starting with addition and subtraction with the same denominator where the result can't be simplified: All the way up to mixed operations on mixed numbers: I'm certain this resource has something for everyone. It might be you just want to use the last page as some revision for a high attaining KS4 class, especially the mixed numbers questions. Alternatively, you might have a low attaining Year 7 class who you spend a few weeks working through all these activities with, supplementing with stuff from elsewhere to help them really understand! Currently I haven't done the answers, but I will...
My next big project was to try and write an exhaustive booklet about expanding single and then double brackets. This is all pretty standard stuff, it just breaks it down into a huge amount of detail and includes curveballs with negatives. It aims to really make a distinction and have students explicitly practice all the varieties of single bracket: e.g numeric coefficient, algebraic coefficient, numeric and algebraic coefficient, index laws required, with negatives, without negatives etc. And then slowly works through doing something very similar with double brackets. Building from: All the way up to: However, what I was finding was that students were having trouble just using the grid method itself (my preferred method for expanding). So I wrote a separate document where we just get students to practice using the grid. Starting as simple as getting students to just read off the expansion from a completed grid: Before having expand themselves with all positives. Then repeating that process with negatives: Before slowly building them up to blank grids: The next (incomplete) part of this project, will be getting the students to work backwards from the brackets: as a precursor to factorising. This insight around separating the procedures out more has led to some of my subsequent better in depth resources.
This was my first big AI project, to play around with timetables grids. At first I just wanted to make pretty standard timetables grids and really I was just learning LaTeX. But then I started experimenting with larger timetables. And then I started playing around with other possible applications of using the standard timetables facts. These are things for example, simple fractions, powers of ten, decimals etc. So all the questions use only what I would think of as the key multiplication facts (2x2 up through 12 x 12 and sometimes up to 16 x 16) but puts it into increasingly novel and complex settings so that even high attaining students can continue to benefit from just practicing their timetables. The file is attached as a pdf and the full code is as a word document.
