Alex is understanding linear equations well, as shown in his increasing proficiency to calculate the gradient, intercept and equation of lines. He is also comfortable with applying these equations to real-world scenarios which is good because that is a recurring theme in graphical questions in general.

We didn't get the chance to go over the formula for completely finding the equation of a line from two points, a product of the formula m=(y_2-y_1)/(x_2-x_1) and y = mx+b, so I'll quickly cover that next lesson. It also appears as though Alex had never done any questions where the answer was a range of values (as opposed to just a single value, or a discrete number of values) so it would be good to practice more questions like that in the future. Next week will be a good chance to assess the rate at which the school teaches at and I will try to match that speed in our sessions.

Today, Luca was able to get the hang of finding the percentage of a part of an amount as well as finding the part of an amount when given the percentage - which was great. And it's early days but he seems to be getting the gist of percentage changes to amounts thus far.

We spent a while on the topic today so it's a matter of keeping up with school for now. Next lesson will be good to assess where what school is covering and how quickly they do so. The way the changes to percentages are taught into the textbook involves the relationship between percentages as a partial amount and that of part of 1, which Luca is still somewhat hazy on so it might be good to consolidate that before continuing. That said, it's not my preferred method so it may be possible to continue with the content without doing so. Very dependent on what the school teaches and when.

Alex was good at the worded problems and those involving perimeter and area, completely them with little difficulty. He was quick to pick up on my teaching of different ways to approach questions centering around algebraic manipulation of terms with multiple constants, implementing them shortly after.

The key to algebraic manipulation is just familiarity with the tricks and approaches needed to rearrange and solve equations. Attempting more questions will allow Alex to instinctually know how to approach these problems. It would also help to consolidate his understanding of general mathematical skills, such as finding a common denominator and when the use of brackets are necessary.

Luca was good at the calculation side of the topic, he could do pretty much all the conversions between percentages, fractions and decimals with little difficulty (besides the occasional slip up when dealing with very small numbers or 3+ significant figures).

He's good on the calculation side, it's just the theory that needs improvement and this is important because most percentage questions require understanding of how to use these calculations in real-world applications. It's about understanding what these percentages actually tell you about the amount we're dealing with, and how percentages can be acquired from samples. It's difficult to navigate between the worlds of percentages just as a number (i.e. the number between 0 and 1 that it usually is) and that of a proportion of a total amount, so, going forward, I think the best way for Luca to look at it would be as a representation of a proportion.