Two questions from the ACT (American College Testing)
The benefits of doing weird maths
This is going to be a short post with some niche maths. Upon discovering that a good friend did the ACT exams instead of the GCSEs that I have done and taught I was immediately intrigued to see what the exam would look like. A bit of research led me to this document which is an absolute goldmine of maths. The questions above are two examples of questions from the ACT which you would just never see in the GCSE in england. They don’t require any new or radical Maths but are just phrased in such fun ways! I think giving students examples of what different Maths exams look like – Be that different exam boards or different continents exams – give students a chance to experience something weird that can set them up for the one inevitable weird question that might come up in their exams.
Another joyful ACT maths question
Doing maths makes you better at maths
I know that I am guilty of saying this in far too many of my classes but in general I really do believe that. In particular doing fun strange maths that gets you thinking and being playful with the maths you know will do wonders.
Here are a few more weird and fun versions of questions you might be used to seeing – Enjoy!
Like a simple factorial question or a complicated simplifying questionA triangle!
Post Credits
The attached document contains about 20 or so problems taken from the larger pdf linked above – Just a mix of some of the questions I particularly enjoyed.
All taken from the same 2017 UKMT Senior Maths Challenge paper – try them yourself!
These three seemingly very different questions all showed up in the same SMC paper in 2017. Try each one yourself and see if you can work out what one trick links them all together!
DOTS: A Party Trick
A joyful example of the efficiency of DOTS
I’ve always been thrilled by the ability of using the “Difference of two squares” to simplify problems. Whether its answering numeracy problems quickly and easily like the one above or using it to crack open harder problems like the three SMC problems above.
Difference of two squares (DOTS) formula
At the bottom of this post I will include some worksheets (not made by me) that practice using this skill in simple and fun ways as well a lovely geometric proof of the above formula. There will also be some extension questions as well as seeing how these techniques show up in the A level curriculum.
The Answers:
Puzzle 1:
Question 4 from the 2017 SMC
This question is a great example of just starting off trying to simplify your answers and seeing what pops out. Looking at the multiple choice options available here its not immediately clear how any of those could possibly be the correct answers.
The most interesting part of working through this is when you get to the following step Recognising that by definition 1 is a perfect square means that whenever you have a square number mius one you can use this formula. This give you the final piece of the puzzle which simplifies to answer E as shown below. Such a elegant question!
Solution to Q4
Puzzle 2
Qu 11 from SMC 2017
This one has a longer but no lesson elegant solution. There are plenty of ways to solve this one but the words “The difference between the squares of their ages” should immediately get the DOTS alarm bells ringing. Using DOTS simplifies this problem down in a lovely way and makes the algebra much more manageable as the answer below shows.
Answers for Qu. 11 SMC 2017
There is also some extension to this question which are a great way to practise the skills yourself – try them yourself and contact us if you have questions/answers!
Qu 11 Extensions
Puzzle 3: The finale
Qu 21. from SMC 2017
It is not immediately clear how DOTS could help you with this question. The first thing to say is that the main reason I thought to use this trick is simply that I have answered lots of problems before. There is no substitute for practise and increasing your mathematical toolkit but playing around with lots of different maths. Needing almost any excuse to use DOTS and additionally y squared in the question meant that I went down that route first of all.
I’m not going to go over every step of the question but the step where you set each bracket to different factor pairs of 15 is not an easy step to see – once again there is no replacement for practise and seeing this question will help with similar questions in the future – I will also include an A Level question that uses this exact technique later on.
Answer to Q21 SMC 2017
Honestly an absolutely glorious question. It is no surprise that it is one of the last questions in the Senior Maths Challenge (For school years 10+) when it requires lots of quite difficult techniques (also under a timed exam pressure! Very tricky!)
Extension questions for Question 21 – 21.5 is particularly juicy!
Bonus A level question:
This is a classic type of A Level proof by contradiction question that uses the same techniques as the question above
This A level question is a lovely way of seeing how theses skill show up in the school curriculum and are a great way to prove some seemingly not straightforward mathematical statements. Try it yourself! I have attached a link below to a document with a group of these questions and importantly the answers to them.
Difference of two squares is a wonderful trick that has a multitude of problem solving uses. The more problems you do and the more you can recognise this the easier (and more fun!) these questions will become.
I have included below a link to worksheets created by the legend Don Steward that can help you practise DOTS in numerical settings.
Lovely visual proof of Difference of Two Squares formulaTry doing these in your head! You’ll be surprised how easy they become (Click the link to download)
I was recently given this problem by a colleague and it immediately took my interest. As soon as I saw it my first thoughts were not “what is the answer?” but “is there an answer? Is there more than one answer? Can I prove this?”.
I spent that lunchtime in my classroom looking at the problem. I started by looking at the digits in the units column and noticed that since the value of the units in the answer must be z then x + y = multiple of 10. This gave me different possibilities to consider:
x + y = 0. Since x or y can not be negative (they are digits) then the only other possibility is if x = y = 0 and using quick inspection this clearly does not work.
x + y = 10. This is feasible since there are multiple ways 2 single digits can sum to 10.
x + y = 10n, where n is any integer greater than 1. This is not possible since x and y are single digit numbers so can not sum to any number greater than 18.
This was the first breakthrough in the problem as I now had an equation that places a constraint on the possible values of x and y.
I then adopted a similar approach with the tens column (x + y = 10 so don’t forget to “carry” the 10 into the tens column which I initially did!). Looking at the digits we can see that 10x + 10z + 10 = 100x which we can simplify to get z + 1 = 9x. Now since x, y and z are all single digit numbers I could then use the two constraints to deduce their values:
x + y = 10 (A)
z + 1 = 9x (B)
Looking at equation (B) x = 1 and z = 8 since x, y and z are single digit numbers
Substituting x = 1 into equation (A) we can see that y = 9
So I had obtained “the answer” and could see that it works. It’s always satisfying to find the answer to a maths problem but it’s even more satisfying to say with confidence that this is the only answer and prove it with reason. I presented this problem to a high ability year 8 class with no instructions and was intrigued to see how their thinking was to trial numbers and simply search for “the answer”. As expected, many of them were able to find the correct values for x, y and z but weren’t able to tell me with any conviction if these are the only possible answers.
I am sure there are other proofs to this relatively straightforward problem and I would be interested to see people’s suggestions!
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.
During this Lockdown period the wonderful youtuber 3blue1brown has created a youtube series called “Lockdown Maths” that have been a glorious set of lectures. These are two worksheets I created that directly use puzzles that are mentioned in this lecture: https://www.youtube.com/watch?v=QvuQH4_05LI&t=1501s
They are beautiful geometric proofs of some often (and less often) used Trigonometric Identities. The techniques used in the first sheet will help greatly in the second sheet and lead me to a fantastic aha moment.
For example in sheet 1 you end up proving pythagoras’ theorem from a trig identity – something that feels like the wrong way round!
images of worksheets (You can download them via the links below)
Recognising that by definition 1 is a perfect square means that whenever you have a square number mius one you can use this formula. This give you the final piece of the puzzle which simplifies to answer E as shown below. Such a elegant question!
Chelekmaths 
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