Been trying to use spare time at work to make some more worksheets and activities – Been on a Completion Table Hype Train recently so that is the bulk of what I am making at the moment – In general just trying to make any worksheet that I need has been a fun challenge over the past few weeks!
Equations to Ratios
Our students were struggling with the two questions above – Both coming in their Mock from Edexcel Nov 2020 3H.
I decided to make a worksheet firstly just converting linear equations to ratios and then one where you had to factorise a trickier trinomial (Which I actually didn’t realise you could use splitting the middle term for so that was a nice surprise!)
Trying to find ways of scaffolding Compound Measures questions where you either have multiple journeys for Speed/Distance/Time or multiple objects for density calculation.
Been doing solving equations with a class and wanted some scaffolded practise of solving equations that contain brackets so made the two sheets below – One with unknowns on one side and one with unknowns (and brackets) on both sides
This question above caused some issues in a recent mock so tried to make an activity that could introduce the idea and give students a bit more practise – see below!
Saw this great question on the 2016 UKMT Senior Maths Challenge and wondered whether it would be possible to make some of my own. I had just been teaching Surds and wanted some nice enrichment that linked surds to indices.
I realised that you can nicely create your own by working backwards from the answer (See Below)
I have said before on this blog that I absolutely adore the UK Maths Challenge as a way of massively improving your problem solving skills and using preexisting tools in totally new ways. It’s also an absolute goldmine for interesting purposeful practice. One of my favourite way of creating resources is by choosing a UKMT (UK Maths Trust) question that I particularly enjoy and seeing in what glorious directions it can lead.
From Question to Exercise: Square Root Puzzle
Lets look at the question above – what seems like a relatively simple question has a wonderfully elegant solution. I will include the answer at the bottom of the post – although as usual I would recommend trying the question first!
The start of the worksheet
So to start with students are encouraged to just try the question themselves – they have the whole page to write down their thoughts and try different things and if they want some support they can turn to the next page.
An example and follow up question to help with the UKMT question
They can work through the simpler example I have given them and then go on to doing very similar questions to the original so they can practice the steps involved. These slight variations mean that students can use what they have used in previous examples to help when the questions get slightly weirded!
An interesting alternative question
The questions progress, using an differing increases before finally looking at a more general example!
The generalised example
But why?
Although the question feels pretty niche there is lots of maths to be practised here. The link between a percentage increase and the corresponding numerical multiplier, solving simple quadratic equations and some nice thoughts on fractions lead to an exercise that practises some key skills in a nice creative way! I plan on using it as some problem solving practice after having taught part of those topics.
Please feel free to try it by downloading here – The answers are available through the resources under the subheading “Problem Solving”
From Question to Exercise: Factors and Square Numbers
Another UKMT question with depth galore
How would you attempt the above question? You could just work out the multiplication and then square root that product to find out the missing value. But square rooting in notoriously a tricky function without a calculator and also where is the fun in that! It also will become an increasingly impractical method the larger the numbers get – for example in the question below…
Surely there is a nicer way of solving this!
Simplifying the question
The beginning of the worksheet
As with lots of the worksheets I make I like to start with simple questions for students to work through and throwing in an example to help but the aim is for it to be a relatively self-explanatory exercise that slowly builds to the intended questions that we want to solve!
These questions also came from a wonderful UKMT question that I saw and needs some knowledge about the structure of square numbers to answer it. In particular that the prime factors of square numbers have to come in even pairs. For example 16 can be represented as 4 x 4 but also as two pairs of 2 x 2. The number 18 on the other hand is built from 3 x 3 x 2 so you have a pair of 3s but then one factor left over so it is not a square number. How could we make 18 a square number just by looking at its factors? We need to find a buddy for that extra factor of 2 so by x 2 we end up with 3 x 3 x 2 x 2 = 36 which is a square number! You can even rearrange it slightly and write it as (3 x 2) x (3 x 2) to make it even clearer that it is a square number as it is something times itself.
Showing that this product gives us a square number using factors! Find a, b and c!
So for the question above you can rewrite 64 as 8 x 8 and 4 as 2 x 2 and by rearranging you then have (8 x 2) x (8 x 2) which means that 64 x 4 has to be 16 squared! A wonderfully simple way of solving these questions and also a great excuse to practise factorising (prime and not prime!)
Try these yourself!
You can try the worksheet yourself here and the answers can be found on our resources page. Its such a delightful little puzzle with a lovely solution – Even coming up with your own questions is fun so I implore you to try that too!
Post Credits
The UKMT is an absolute goldmine of maths and puzzles. Even more excitingly the UKMT has extended solutions to their maths challenges that include additional questions and investigations for each question so I would definitely suggest checking that out here
Below is the solution to the original problem:
As always thank you so much for reading and if you are interested in getting involved with us let us know here and you can also subscribe so you don’t miss out on any content by entering your email below!
Problem from the brilliant mindyourdecisions youtube channel
Firstly I would urge you to try the question in the photo yourself. It requires some knowledge of powers and solving quadratics and is wonderfully satisfying to find all SIX solutions.
Looking at easier examples
Questions like this are a great example of finding creative ways to practice fundamental skills. In this case one of the key skills being practised here is the ability to solve quadratic equations and rearranging them into a form that you can solve easily. So how to introduce this to a class?
A possible starting place
Lets look at the example in the picture – How can we can get to a solution of 1 on the LHS of our equation? The two cases that make this happen: Case 1 by setting (x-1) = 1 and case 2 by setting (x-1) = -1
Both these methods when squared will give us our answer of 1 and so that is where we will get our final values of x to be 2 and 0.
Taking it a step further
So taking it a step further we now an expression as our power which gives us a third case: Setting x+2 to zero. When the power is zero whatever is inside the bracket is irrelevant as any real numbers to the power of zero equal to 1.
Okay 1 to the power of anything is one so x-1 = 1 gives us our first solution: x=2
In case two we need -1 to an even power to get to one so we have x-1 = -1 which gives us a solution of x=0 which when put into the power gives us an even power so it must be a solution
Finally in case 3 we set x+2 = 0 which gives us our third solution and final solution which is x=-2 and it is solved! Now can we use this simpler example to work our way up to the six solutions of the first problem?
Creative Practice
Small worksheet building up to answering stacked quadratic questions
I used the sheet above as a starter in my first lesson with my new year 11s – as a little bit of a fun way to gauge both their ability and their desire to apply maths they definitely know in ways they might not have seen before. The questions don’t lead onto each other as nicely as I might like but once we had discussed the first few the whole class were able to get their teeth into the rest of them.
By working through this small exercise they were able to practise factorising and solving quadratics in a way that felt fresh and interesting whilst also giving me an opportunity to circulate and learn more about my class.
Post Credits
I will not explain the full explanation for the question as it seems only fair to direct you to the original video here for a lovely detailed work through of the solution. Once you have watched the video try the rest of the worksheet!
I encourage you to try each of the questions looking at each of the three cases. Then try and come up with some of your own that fit this format – I always think one of the best ways to improve is to play around with writing your own questions and these are a fun way to start!
If you want to download the worksheet as a word document click here
Spot the difference! Which wording are you used to and why?
Asking “What is Factorising?”
The majority of the time, when I hear this question asked in class I have heard the word “Brackets” come up in the response. As a teacher when I hear this answer I generally take it as correct – The student has clearly remembered part of the method behind factorising and is stating a key symbol they use during that process. But there would usually be silence when I asked how factorising related to its root – Factors. What are the factors of 3x+3 – why does it feel strange to describe (x+1) as a factor?
Using a Factor Tree to Practise Factorising
When students are taught to write numbers as the product of their prime factors we tend to use a “tree” to split the numbers (As shown below)
Example of Prime Factor Tree with Prime Numbers circled
I decided to see if I could use this same method to practise students ability to factor expressions as well and then create a link between what you are doing in both case – Splitting something into a product of their factors.
So I made a worksheet and gave it to my students to see what they would make of it. As the sheet contains a lot of structure I let them initially work through it in pairs and without giving them too much introduction so that they could do all the thinking themselves. Before this lesson we had previously practised how to factorise different simple linear expressions (Using the worksheet here) so we had already seen and discussed some of the maths behind expanding and factorising expressions.
The Worksheet
(The worksheet is attached here – I encourage you to try it yourself before continuing!)
The purpose of the worksheet is for students to recognise that when they are factorising the parts they have inside and outside the bracket are the factors of the expression. Intuition that will help with many similar problems later in their mathematical careers.
Worksheet starts with an example and builds from there – fill in the blanks!
The sheet uses the idea of a “Slow Build” to get students to start from something they are comfortable with and slowly build to deeper questions
Moving onto using the same method for splitting expressions into factors
After quickly discussing questions 4 – 6 with the class they were then left to attempt the rest of the worksheet while I circulated to check on progress. The sheet contains various questions and hints that students are required to write an answer for which lead to some fantastic discussion throughout.
Example of hints and questions for encouraging deeper thought. In particular looking at the different ways you can start to split equivalent expressions.
The rest of the worksheet contains more practise including looking at harder examples eventually moving to expressions with multiple variables that they hadn’t seen before (Although in the context of this worksheet were answered quite successfully!)
Is x+3 a prime factor? Does it matter?
This Question on the left gets the students to ask if there is a more efficient way of answer the same question leading to a factor of 6 on the outside of the bracket later on. As a class we discussed the differences between prime factorisation and splitting things into their factors and also using brackets when writing something in its factorised form. This question on the left encapsulates lots of these thoughts at once.
Factor(is)ing
I am undecided as whether I like using the word Factorising when looking at products of factors or whether I prefer the more american Factoring. The (is) in Factorising feel slightly unnecessary and Factoring seems to convey just as simply that we are going to be looking into factors. Let me know which one you use!
Post Credits
Overall this worksheet created some very interesting discussion with my students and also led to increased fluency when dealing with factoring questions in later lessons. The idea that factors can also be algebraic expressions and not just integers was also a great source of conversation in the classroom during the lesson.
I have included an image of the whole worksheet below in case you are unable to view the download. Answers are available through the Resources page.
Question taken from Brilliant.org – The most glorious of websites (Ans: a=4 b=3 can you show why)
As with most of the Maths that I think about I got pretty obsessed with Factorising by Grouping after doing a question on Brilliant.org. (I will not spend too much time here talking about how much I love Brilliant.org but I absolutely absolutely love Brilliant.org and think that anyone that is interested in Maths and Problem Solving would love scrolling through its hallowed courses)
I was also in the process of discussing the factor theorem with my year 11 students and was repeatedly asked “Do I just use trial and error to find a factor”. As well as this technique works it doesn’t feel all that satisfying and certainly there is more fun maths to be found here if we look around.
The Fun Maths
The start of the Slow Build worksheet – start with something they have seen before
When putting together exercises to try and practise this skill I started with using factorising by grouping to factorise quadratics – something that the students were confident in and had seen before.
As the worksheet progressed the questions slowly increase in difficulty, with the scaffolding being taken away in steps.
Cubics joining the party
Eventually moving on to some all together more tricky cubics that require a splitting of the middle terms. This is a technique that is often used to factorise quadratics where the coefficient of the x squared term >1 but I had never seen it used to factor cubics.
Its these questions that I think were particularly fun to play around with as it is not immediately obvious how to split our terms nicely. I would encourage you to try to complete these questions as well as the rest of the questions on the sheet.
Try these questions yourself!
Post Credits
As we continued to try this out with different cubics it felt like we were once again using trial and improvement to work out how to split these trickier cubics up. It didn’t feel like a fruitless exercise as working out why your choices weren’t working and trying to choose better options feels like it has great benefit in become more fluent in your algebraic manipulation and also was just a fun way to practise lots of smaller expanding and factoring skills. It is always fun to practise these skills using deeper problems!
I spent a long time trying to come up with a clear method for factoring cubics by grouping when then grouping isn’t immediately obvious and I am still struggling! If you come up with anything please let me know at chelekmaths@gmail.com – I am excited to keep learning!
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.
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!
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!
Bonus extension question sent to me by Woody L – show that p and q are both 6Another bonus question that can be solved using equating (Show that c is 3 and d is 13)
Chelekmaths 
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