Chelekmaths

As it says in title – alternating and funk sequences

Firstly try the questions!

These series are a lovely in many reasons. Looking quite bonkers at first glance they can both be lovingly rearranged or broken apart to make them more accessible. All the series in this post can be split into two smaller more manageable series – for example in the first question by taking alternating terms you have a sequence which contains the sum of all odd numbers (1+3+5+7…) and an arithmetic sequence (-2-4-6…). Find the sum of 25 terms in each sequence, add them together and then the series is complete!

I used the worksheet with my year 13 class as a way of practising finding sums of arithmetic and geometric series but in a slightly more interesting context. Alternating sequences have also been known to come up on A level papers so its good to get exposure to them in lesson.

Additionally the extension of finding the sum of the first 51 terms adds in the challenge of working out what that 51st term would be (or just summing 26 instead of 25 terms in one of the series!) which is also some added fun.

You can click on the image of the worksheet below to download and then try it! (There are formulas for sums of geometric and arithmetic series that would definitely be helpful and can be found anywhere online)

Also the last question using Logs is pretty tricky – There was a summation of logarithms question in the 2019 A level exam and it threw both my students and I! Actually a really lovely question that just feels scary if you hadn’t seen series with logs before.

Show that the LHS = RHS

Beautiful visual proof that the sum of n odd numbers is just n squared. (Image comes from youtube channel EpicIQ https://www.youtube.com/watch?v=MM7fb1HaRYQ)

NJK

vector-geometry-scaffolded Download

Part 1: The Question

This rather innocuous looking question came at the end of one of our GCSE mock exams and at first glance didn’t seem like it would cause to many problems. Part a is simple enough but part b was basically unlike any question I had seen at GCSE – definitely not a question I had adequately prepared my students for. The marksheme doesn’t even reveal the true glory of this question

Markscheme for part b – the P1 for complete process to equate coefficients is the most brutal 1 mark I have ever seen

Every attempt to explain this question to students led to whiteboards full of confusion. So with the help of some wonderful colleagues we created some resources to help actually teach the skills required to achieve these few marks.

Part 2: Equating Coefficients

Start of the equating worksheet given to students (Can be downloaded from link at bottom of page)

We couldn’t find any resources for teaching equating so we decided to create one ourselves. We created a worksheet using the principles of the “Slow Build” where students slowly work through examples starting from examples they have seen before and getting progressively more difficult. Also always a shout out to Craig Barton and VariationTheory who I have big time fan girled over ever since I sat next to him in a session at BCME 2018. It felt fun to use the idea of collecting like terms – something that the students were very comfortable with – to explore a much deeper method.

Students have slowly built up to these more difficult questions where they need to use simultaneous equations to solve them. The second question in the photo is actually the exact equating problem they would have needed to solve in the earlier Vector Question

Working in pairs the students were able to carefully work through all the questions with minimal input from me – drawing a few students to our glorious whiteboards when they were in need of a nudge.

Example of using equating to factorize quadratics

The use of examples they recognized helped with the transition to more tricky and interesting questions on cubics later on an fed in nicely to our work with cubics (details in a future blog post)

Once I felt confident that the students were comfortable with the ideas behind equating coefficients we moved on to the main event (in a different lesson)

Part 3: Getting to the problem

The question that started the next lesson

Working out that this problem was relevant to our initial question was such a wonderful aha moment for me. We were just playing around on some whiteboards wondering what the simplest form of the vectors question might look like – a classic problem solving technique that always brings me joy – when we settled on this. Although you can solve this with similar shapes as well as straight line graphs (I will leave those as an fun extension) the vector proof feels not only very elegant but also leads directly into its more harder variations. Below is how it was presented to the students after some discussion and attempts from them.

This is the beginning of the Slow Build Vectors worksheet that the students were given – made using the wonderful equation editor on Word

Once the students had attempted this question we worked through it together as a class making sure that there was a consensus of understanding they were then encouraged to work through the rest of the worksheet in pairs using the whiteboards around the room to play around with the questions.

Generalizing the previous problem to rectangles of any size

Eventually they had tackled a few simpler questions and they were faced with the same (albeit more structured) vectors question they had seen in their mock exam.

Adding in a twist before leading on to the main event

Part 4: The Main Event

The most brutal P1 mark I have ever seen

Seriously this was two marks or something. The students could have completely left this out and still done phenomenally well (as they did!) but the students have every right to want to understand everything that could possibly come up on their GCSEs and also maths is so cool and the JOURNEY. The JOURNEY. Such an absolutely joy. Big shout outs to the one student in the whole year group who got the marks as well as to some awesome colleagues for dealing with me pestering them about the question repeatedly over the course of a week.

Part 5: AOB

Whether a question like this will every actually come up again seems very doubtful, and whether my students would have thought to use these skills if it had come up is also doubtful but I think there is such a joy in deeply exploring one question and seeing all the other Maths that falls out.

Below are the links to the full worksheets – feel free to click, download and try them yourself!

Find all the missing coefficients using Equating

vector-geometry-scaffolded

If you are interested in seeing more Slow Build worksheets I am currently in the process of adding them to the Slow Build page on this website!