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:

image

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:

image

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:

image

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:

image

Then the gradient of PQ:

image

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:

image

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.

Advertisements

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:

image

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:

image

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:

image

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:

image

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:

image

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

A puzzle with possibilities

Brilliant’s Facebook page is a fantastic source of brain teasers, they post a nice stream of questions that can provide a mental work out and that I feel can be utilised well to build problem solving amongst our students.

Today’s puzzle was this:

image

It’s a nice little question. But when I use it in class I will only use the graphic, as I feel the description gives away too much of the answer. Without the description students will need to deduce that the green area is a quarter of a circle radius 80 (so area 1600pi) with the blue semicircle radius 40 (so area 800pi)  removed, leaving a green area of 800pi.

I find the fact that the area of the blue semi circle is equal to the green area is quite nice, and in feel that with a slight rephrasing the question could really make use of this relationship. Perhaps the other blue section could be removed or coloured differently and the question instead of finding the area could be find the ratio of blue area to green area.

Another option, one I may try with my further maths class on Friday,  could be to remove the other blue section and remove the side length and ask them to prove that the areas are always equal, this would provide a great bit of practice at algebraic proof.

Can you think of any further questions that could arise from this? I’d love to hear them!

This post was cross-posted to the blog Cavmaths here.

“What does the word ‘percentage’ mean?”

I love asking low-ability students or students whose first language is not English questions like this. Unpicking the etymology of words is something that can benefit all students but for low-ability or English as an additional language (EAL) groups this is of the utmost importance as they had anywhere near the same level of exposure to the subtleties, nuance and conventions of the English language. It is the job of educators to increase the level of exposure beyond what they would otherwise encounter.

IMG_7112

For young students especially I really ham it up when I reveal that ‘percentage’ means ‘of 100’ and let them know that they are part of a small secretive club that will refuse to use the word ‘percentage’ willy-nilly but will stick to its strict definition.

I usually then go on to tell students that they would now be able to have an educated guess as to the meaning of any word with the word ‘cent’ in and I ask for a number of suggestions. Again for younger students a touch of the theatrics can be useful here (think Sherlock Holmes references).

When I asked this question last week it was with a Year 7 group, most of whom hadn’t encountered percentages yet but who were beginning to gain confidence with fractions of amounts and equivalent fractions. They quickly cottoned onto the idea that 25%=25/100=¼ and were then able to find percentages of amounts.
Focusing on the fact that percentages mean ‘out of 100’ and treating percentages as a special instance of the fractions work they had already encountered was something of a long way round. However, initially and in subsequent lessons students seemed to ‘get it’ and were far better able to explain some of the intuition behind percentages. Additionally, it made subsequent work on converting between fractions, decimals and percentages far easier.

Also posted @NWMaths