Isosceles triangles and deeper understanding

When marking paper 3 of the Edexcel foundation Sample Assessment Materials recently I came across this question that I found interesting:

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It’s a question my year tens struggled with, and I think it is a clear marker to show the difference between the current specification foundation teir and the new spec.

The current spec tends to test knowledge of isosceles triangles by giving a diagram showing one, giving an angle and asking students to calculate a missing angle. This question requires a bit of thinking.

To me, all three answers are obvious, but clearly not to my year 10s who do understand isosceles triangles. The majority of my class put 70, 70 and 40. Which shows they have understood what an isosceles is, even if they haven’t fully understood the question. They have clearly mentally constructed an isoceles triangle with 70 as one of the base angles and written all three angles out.

What they seem to have missed was that 70 could also be the single angle, which would, of course, lead to 55 being the other possible answer for B. One student did write 55 55 70, so showed a similar thought process to most but assumed a different position for the 70.

I already liked this question, and then I read part b:

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Now students are asked to explain why there can only be one other angle when A = 120. Thus they need to understand that this must be the biggest angle as you can’t have 2 angles both equal to 120 in a triangle (as 240 > 180), thus the others must be equal as it’s an isoceles triangle.

The whole question requires a higher level of thinking and understanding than the questions we currently see at foundation level.

In order to prepare our students for these new examinations, we need to be thinking about how we can increase their ability to think about problems like this. I think building in more thinking time to lessons, and more time for students to discuss their approaches and ideas when presented with questions like this. The new specification is going to require a deeper, relational, understanding rather than just a procedural surface understanding and we need to be building that from a young age. This is something I’ve already been trying to do, but it is now of paramount importance.

There is a challenge too for the exam boards, they need to be able to keep on presenting questions that require the relational understanding and require candidates to think. If they just repeat this question but with different numbers than it becomes instead a question testing recall ability – testing who remembers how they were told to solve it, and thus we return to the status quo of came playing and teaching for instrumental understanding, rather than teaching mathematics.

What do you think of these questions?  Have you thought about the effects on your teaching that the new specification may have? Have you any tried and tested methods,  or new ideas, as to how we can build this deeper understanding? I’d love to hear in the comments or social media if you do.

Further Reading:

Teaching to understand – for there thoughts in relational vs instrumental understanding

More thoughts on the Sample assessment materials available here and  here.

Cross-posted to Cavmaths here.

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A lovely angle puzzle

I’ve written before about the app “Brilliant“, which is well worth getting, and I also follow their Facebook page which provides me with a regular stream questions. Occasionally I have to think about how to tackle them, and they’re excellent. More often, a question comes up that I look at and think would be awesome to use in a lesson.

Earlier this week this question popped up:

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What a lovely question that combines algebra and angle reasoning! I can’t wait to teach this next time, and I am planning on using this as a starter with my y11 class after the break.

The initial question looks simple, it appears you sum the angles and set it equal to 360 degrees, this is what I expect my class to do. If you do this you get:

7x + 2y + 6z – 20 = 360

7x + 2y + 6z = 380 (1)

I anticipate some will try to give up at this point, but hopefully the resilience I’ve been trying to build will kick in and they’ll see they need more equations. If any need a hint I will tell them to consider vertically opposite angles. They should then get:

2x – 20 = 2y + 2z (2)

And

3x = 2x + 4z (3)

I’m hoping they will now see that 3 equations and 3 unknowns is enough to solve. There are obviously a number of ways to go from here. I would rearrange equation 3 to get:

x = 4z (4)

Subbing into 2 we get:

8z – 20 = 2y + 2z

6z = 2y + 20 (5)

Subbing into 1

28z + 2y + 6z = 380

34z = 380 – 2y (6)

Add equation  (5) to (6)

40z = 400

z = 10 (7)

Then equation 4 gives:

x = 40

And equation 2 gives:

60 = 2y + 20

40 = 2y

y = 20.

From here you can find the solution x + y + z = 40 + 20 + 10 = 70.

A lovely puzzle that combines a few areas and needs some resilience and perseverance to complete. I enjoyed working through it and I’m looking forward to testing it out on some students.

Cross-posted to Cavmaths here.

An Interesting Conics Question

My AS further maths class and I have finished the scheme for learning for the year, leaving oodles of time for review, recap and plenty of practice. Currently we are revisiting topics that they either didn’t score too well on the last mock in or that they have requested we look at again due to lacking confidence.

Today we were looking at conic sections,  by request of the students, and this past paper question caught my eye:

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We had recapped the topic quickly together then the students attempted so past paper questions. This one was one they rook to well, all competing the first 2 sections quickly and without trouble. Likewise, part c was mostly uneventful – bar one or two silly substitution errors and someone missing a difference of two squares factorisation.

Then they got to part d.

This caused them a little bit of worry so we worked together on it as a class, after we’d made sure everyone had a, b and c right.

Here’s what the board looked like when we’d done:

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I asked the class what we should do to start, one suggested drawing a diagram all this nagging about always drawing a diagram, especially if your stuck is paying off! We drew it, but it didn’t help much:

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Then one said, “if they’re parallel then one gradient is -1 over the other” – I refrained from scolding and calmly said “indeed, those gradients will be negative reciprocals of each other”. I think asked what the gradients were and quickly had “y1 – y2 over x1 minus x2” thrown at me.

So we worked out the gradient of the line join in n to the origin:

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Then the gradient of PQ:

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One then suggested we put the equal to each other,  but he was corrected by another student before I could react. So we set one equal to the negative reciprocal of the other and solved:

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A lovely question, with a lovely neat answer and a load of fun algebra on the way. I enjoyed watching the students tackle it and was glad I didn’t need to put too much input in myself.

We then discussed the answer and the class expressed surprise that 1 was the final answer and that a complicated algebraic journey could end so simply. It was a nice discussion and a nice way to start the day, and the week.

This post was cross posted to Cavmaths and One Good Thing.

Interesting questions

I’ve written before about the SAMs (Sample Assessment Materials) for the new GCSE, and currently we are swaying towards Edexcel. We have recently given year ten the SAMs to see how they got on with them and a couple of questions that stood out for me. First was this one:

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Students need to find change, fair enough, but them part b seems to be purely testing their understanding of the word “expensive”. This seems a really bizarre question in my opinion, and I’m not sure it fits well on a maths exam. It’s not even really a mathematical term.

Another that stood out was this:

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I think this one is a great question that approaches the assessment of fractions knowledge in a new way, it requires a deeper analysis but I do think there is a limitation to it if it isn’t thought out. If this type of question is regularly asked about fractions, then it becomes a “when they ask this you say this” sort of question. This could be combated by asking this type of question about different topics. It’s certainly a question I enjoyed seeing, and is much better to assess deeper learning than he current GCSE,  particularly the foundation tier.

The final question that caught my eye was this one:

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It’s similar to the type of question on forming and solving equations we see how,  but the interesting bit is the additional bit of reasoning students need to apply at the end, ie to work out if the amount of marbles that Dan and Becky have together is odd or even to work out if they can have the same amount.

Have you noticed any interesting questions cropping up? Have your students attempted the SAMs? If so, how did they get on? I’d love to hear.

This was cross-posted to Cavmaths here.

Upper and lower bounds with Trigonometry

This week I was planning to cover upper and lower bounds with year 11 as on the last mock a lot of them made mistakes so I felt it would be a good topic to revise. As part of the planning process I had a look through the higher textbooks our department has bought for the new specification GCSE (we bought the Pearson ones, the full suite at KS3 and 4. Some great questions in them and the online version, activeteach, is great to take questions and place into your lessons. I’d definitely recommend it, if used correctly, but I will admit to being disappointed to see a formula triangle being advised…) to see if there were any good questions I could pilfer, and I came across the section on using upper and lower bounds in trigonometry.

My first thought was, “that’s a nice topic”, and then the full spectrum of the topic began to unfold.

Initially, I had like the idea that students would be required to think about the fraction, and how minimising the denominator actually maximises it, but thin I remembered the nature of the cosine function! This example shows what excited:

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Not only would students be required to understand the nature of a fraction, they’d also need a deep understanding of the cosine function itself, to understand that the bigger cos x is, the smaller x is, and vice versa  (where x is between 0 and 90 of course). This could be a real deep understanding of the graph, or the unit circle, or just the geometry of a right angled triangle.

The example itself is very procedural based, which is a shame,  but it does give a teacher a good frame to start discussions. I wouldn’t use textbook example as teaching anyhow, just as an additional example to talk through one on one with students who were still struggling.

The textbook goes on to pose this awesome discussion question:

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A real nice prompt to get an in depth discussion around the trig ratios going. I often use similar prompts when looking at maximum values for sine and cosine “what’s the biggest opp/hyp can ever be?” for example. This often gives a nice discussion focus.

I think that this topic shows how different the new specification will be. Students are going to need a much deeper relational understanding if they are to achieve the top grades with questions like this being posed.

What do you think of bounds being questioned in relation trigonometry? Have you used prompts like this before? How have you found them?

This post was cross-posted to Cavmaths here.

What Questions Are You Asking Yourself?

This post is my contribution for Week 3 of the ExploreMTBoS Blogging Initiative.  Join in on the fun!

Last week, my honors course began graphing compound inequalities on number lines.  During the first day of our work with the concept, I used a series of everyday experiences to drive the lesson (speed limits and minimums on expressways, percentage ranges for letter grades, etc.).  From the success most students had with these tasks, I was comfortable turning them loose on a mess of problems I posted around the room.

As I circulated about the classroom to talk with students, I kept noticing errors with the directions of comparison symbols, choice of symbols, shading of points, and directions of arrows.  I was surprised at the mistakes students were making since they accurately solved context-based tasks earlier in class.  When I asked students questions to guide problem solving, I began to notice a common trend among students that was causing the errors.  I first asked students what questions they were asking themselves about the graphs or problems.  I got puzzled looks or responses like, “None,” to my inquiry.

Now I knew the root of the mistakes being made: self-questioning.

The next day of class, I told my class about my realization.  I shared how I kept asking students questions like, “What are we comparing in this problem? What quantities are being used in the comparison? Is the inequality true when x is more/less than this number?  Can x be between these numbers?  Is the inequality true when x is one of the points?”  I asked in a blind survey how many people asked themselves these questions.  Without surprise, few students raised hands.

I was humbled by this experience for many reasons.  First, I’ve been trying to instill this need to self-question during problem solving from the start of the year.  Second, students just got into the habit of asking themselves questions when writing equations for word problems.  Third, I felt like it made everything from the previous day worthless.  If students did not question themselves during the introductory activities, I questioned if my lesson (for how intuitive I thought it was with its grounding in familiar contexts) developed student thinking abilities alongside content knowledge.

After seeing the sparse amount of hands raised, I proceeded to write the questions I posed on the board for students to reference.  I directed students to ask themselves these questions for the practice activities that followed, but I stressed that even simpler questions like, “What are we finding?  What information do I need?  How is this related to that?” should become a normal part of their thinking for all problems.  Inequalities became pretty simple for students as the day progressed, since they were taking the time to ask themselves questions about the problems they faced.

My experience with questioning this week stirred within me a deeper thought I have about teaching.  If teachers are aiming to develop thinking abilities and reasoning strategies in students, how do we develop these qualities besides repetitive modeling?  In my experience this week, I ended up modeling self-questioning and directing students to self-question later in the day.  How can we create situations and experiences with mathematical content that naturally push students to pose questions about their thinking about given problems?  Bryan Anderson mentioned noticing and wondering as a means of developing student questioning abilities, but most of those questions are directed outward (formulating problems) rather than inward (the metacognitive task of asking yourself a question about a predetermined question or task).  I would appreciate any thoughts or comments on this subject.

[cross posted to trigotometry]

Implicit Procedures

My students are amazing, especially at solving big-idea-type problems, but they’ve been getting a little sloppy with some basics. Today is the first day of second semester, so (instead of worksheets) I figured they needed an activity to review significant figures and other simple skills. 

I came up with The Popcorn Lab. In small groups and the span of 55 minutes, they had to create some sort of procedure, data table, and results to answer the following questions: 

  1. What is the mass, volume, and density of a popcorn kernel before and after popping?
  2. What accounts for the differences between those values?

It was possibly one of the most beneficial things I’ve done recently in terms of getting them to ask their own questions and critique ideas (and as a side benefit, they’re using lab notebooks for, ya know, notes instead of filling in blanks). There was a lot of scratching out and revising, especially for the “after” kernels. A few kids looked up ways to measure volume of popped popcorn on their phones rather than using water displacement. 

Tomorrow, I plan to ask them about:

  1. Easy/straightforward parts
  2. Problematic parts
  3. Differences between groups’ procedures and would they get different answers/results
  4. Sources of error, and how they would be different for different groups

(Cross posting to my own blog.)


“What would a more difficult question on this topic look like?”

Cross-posted to @NWMaths

A classic ‘extension’ activity that Maths teachers often use is to ask students to create a question on a topic when they have finished their work. It’s an easy win for teachers; they keep students busy whilst supposedly ‘stretching and challenging’ them by encouraging them to work on the so-called higher order skills required to engage in the creative process.

Creating questions is usually a more difficult skill to master than answering them, particularly when you want a ‘nice’ answer to emerge. Think for instance about the knowledge and understanding required for creating a trigonometry question giving an integer answer compared with merely answering such a  question.

However, I prefer to ask certain questions and give particular prompts in order to refine this process and move it away from a ‘keep them busy’ or box-checking activity and move it towards a learning activity. For instance: “what would an easy question on this topic look like?”, “why is question a harder/easier than question b?”, “what would you expect to see in a more difficult question?”. Students can then use these prompts to create easier, medium and harder questions. They are forced to engage with the material and considering the different difficulty involved in each question really develops their metacognitive skills.

Here are some examples of the work that my year 10 class carried out on rearranging formulae:

I was especially pleased with the ‘hard’ example on the far right hand side- putting the intended subject as the denominator was a subtle but important difficulty this student grasped.

The question “can you create an easy, a medium and a hard question on this topic?” is a useful and powerful way of refining the process of students creating questions.

 

How are we questioning our students?

This month’s maths journal club is based in the article “Contrasts in mathematical challenges in A – level mathematics and further mathematics, and undergraduate mathematics examinations.” By Ellie Darlington

I found the article quite interesting overall. It looks at the differences in examination questions between. A level mathematics and undergraduate mathematics,  it starts off with the idea that. A level mathematics is tested in a manner that involves routine questions and that as such this doesn’t prepare students for undergraduate mathematics, which it presumes is tested in a higher level. I think this is one of the issues with A level mathematics and I hope that when the new curriculum appears this will have been addressed. The problem is even worst at the transition point between GCSE and A level though, but again, I have hope that the new specification will address this.

Interestingly, my own experience of undergraduate mathematics was that there were a lot of courses that were tested in a routine manner, and that learning the lecture notes by rote and practicing the past papers for a course could allow people to score well despite not understanding what was going on and not being able to apply their knowledge in other contexts. There were some of my peers who had no conceptual understanding of some of the modules yet still scored high enough to achieve firsts.

That said, I still feel that the procedural nature of the GCSE and A level papers is a massive problem. In recent years we have seem a change in the A level papers towards questions that are not answerable in a routine manner, but it needs to go even further.

There are many problems with these procedural questions. My main issue is they allow students to score well without understanding the mathematics behind the questions. This in turn can allow teachers to skip teaching for a relational understanding and just teach an instrumental or procedural understanding, which let’s down the learners,  especially if they are hoping to go into mathematics or a mathematical based subject at higher education.

So what can we do?

Well, rather than waiting for the changes we can be implementing these questions in our classrooms, ensuring that we are teaching for relational, or conceptual, understanding rather than teaching purely procedures. Take the time to ask the questions that require application in new contexts.  Take the time to teach the concepts, the why behind the what. Enrich the curriculum with tasks that involving thinking about the box and questions framed in a way that the correct method isn’t always immediately obvious, perhaps try some of these puzzles?

Other points in the article

There was a lot early on that I thought I already knew, but it was nice, and useful,  to see references and studies to back up some of the ideas.

The MATH taxonomy in this explicit form is new to me and I’m interested to look further into it and see how I can apply it myself.

I was a little purplexed to see that the article stated that questions can change their position on the MATH taxonomy with time, but then have no explanation of how these questions were classified in the research.

All in all a very interesting read that I will re read and digest in more detail later. I’d love to hear your thoughts on it also.

This post was cross posted to the blog Cavmaths here