The Big Ideas of Trigonometry (2)

In order to consolidate what they had previously explored on sinusoidal graphs, we gave the students the Marbleslides Periodics activity. Half of our students were with us and the other half joined us via Zoom (their teacher provided them with the class code). 

Engagement was not a problem: all students got stuck into the tasks straight away and we could use the teacher dashboard to see which students needed prompts and which ones did not. If a student at home needed the prompt, the teacher sometimes did this quickly through Zoom.

We had set the activity up so that we were both co-teachers which meant either one of us could use the feedback button to give feedback to the students working at home.

In addition to the conversational and observational evidence that we were getting we could also use the students' responses to some of the 'Predict' questions (shown below) to provide feedback.

By using the 'Snapshots' tool, we could take some responses for a particular question and look at these with the whole class (those in the room and those working at home). We could then give feedback as to how we could improve these answers by using more precise mathematical terminology.

It was during the Challenge Slides that we really could see that the students had a solid understanding of sinusoidal graphs: conversations were littered with suggestions such as 'Change the amplitude', 'Shift it up', 'Change the b value and it will stretch the graph out'.

Challenge 4 was a sticking point though: 

Students had no problem getting a graph that went through the stars but with the marbles being dropped vertically from the point (0,12), they just settled in a dip without collecting any stars.

At this point, I was a bit stumped too so we stopped the class to try and think our way out of this. I made the point of telling the students that I was stuck and that this often happens when you do good problems. I wanted to model what I do when I get stuck so that they could develop these strategies too.

Firstly, we confirmed that we had pretty much exhausted all the potentially useful transformations.

Then we mused: why don't these work? We agreed that it was the position that the marbles were being dropped from. This of course begged the question: 'if we could change something, what would we change?'

Well, in this case we would like to change the position that the marbles are dropped from. I do confess that I did try to drag the launch point but to no avail!

So if we can't change the position of the drop, what else could we do?

Then the penny dropped (or rather, the marbles): We could divert the marbles.

How could we divert the marbles? With a second sinusoidal and by using domain restrictions!

I then noticed the instructions on the page:

In the rows below, type as many equations of periodic functions as you need to collect all the stars.

Well played, Desmos. Well played.

After a quick exchange of ideas as to what this second graph could look like, the students came up with some great solutions:

It is one thing to hear the cheers of the students in the class as they are successful at each challenge, but it is another thing when one of the students working at home sent this message via Zoom:

Of course, this is all very well but if the students are just being entertained by marbles sliding down ramps and collecting stars, does that actually mean they have learned anything?

After the lesson, the teacher stated quite simply this:

'They get it. They totally get it.'

'It' being the overall expectation we were focussed on:

Demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numerical, graphical, and algebraic representations of sinusoidal functions.

'I can give them all level fours get it and not worry about giving them a test.'

This was important: with face-to-face time at a premium, any time spent on tests takes away from this. There will be situations where a product (such as a quiz or a test) might be a better way to assess or evaluate our students, but my sense is that if we look more closely at our overall expectations, we will see that many of them lend themselves just as nicely to activities such as this where we can use our observations and conversations to assess and evaluate.

More importantly, the first line in Ontario's Growing Success document should be guide our assessment and evaluation practices:

'The primary purpose of assessment and evaluation is to improve student learning.'

The Big Ideas of Trigonometry (1)

Teachers around the world are facing many challenges as we get to grips with new schedules involving online learning, cohorts, quadmesters and even octomesters. For many, the reduction of face-to-face time with their students (the most valuable relationship) has teachers feeling increased pressures in effectively delivering any particular curriculum. I wonder if, now more than ever, we need to focus on the big mathematical ideas of each course.

Last week, I worked with a teacher of a Grade 11 class who were about to begin trigonometry. There are many specific expectations in the Ontario curriculum which (if you only see your students face-to-face for two or three mornings every two weeks) can be overwhelming: where to start?

Instead, we looked at one overall expectation:

Demonstrate an understanding of periodic relationships and sinusoidal functions, and make connections between the numerical, graphical, and algebraic representations of sinusoidal functions.

What activities could we give our students so that they could demonstrate an understanding of all of these especially given the scheduling constraints (a 2.5 hour lesson on Wednesday and Friday for one cohort and a 2.5 hour lesson on Thursday for the other cohort)?

For me it boiled down to this:

• Mathematicians need to describe how things move in circular paths or behave in regular cycles.
• The Unit Circle is key to understanding trigonometric ratios and can be connected to what they know about right-angled trigonometry.
• The Unit Circle gives us the sine and cosine graphs, and we can see how these are connected.
• We can transform these graphs in a number of ways similar to what we have learned with other functions.

These are the activities that we did and (spoiler alert) allowed the students to demonstrate their understanding.

1) Graphing Stories

As a minds on we gave them two videos from the excellent Graphing Stories site.

Plot the height of the waist above the ground:

The distance of the person on a roundabout from the camera: 

Both videos are by Adam Poetzel (Twitter: @adampoetzel).

I made sure that students had the chance to watch each video four times so as to allow them to improve their sketch graphs each time. Once we had done these, I could point out that straight line motion results in graphs with straight line segments. Circular motion, on the other hand, results in a wavy graph that repeats itself.

I could now tell the students that we were going to learn about the second type of graph.

2) Redefining Sine and Cosine

As an interlude, I asked them to write down a definition of what they understand 'sine' means. This puzzled them for a bit, so I rephrased it and said, 'Draw a right-angled triangle and use this to describe what we mean by the sine of an angle and the cosine of an angle'. 

This worked much better: they were confident in telling me that:

 sin θ=opp/hyp and  cos θ = adj/hyp.

This begged the question: what is the largest value that θ can have?

After a bit of debate, they agreed that θ had to be less than 90˚.

I told them that if this is the case, then our current definitions of sine and cosine would not help us with this triangle:

We could use the cosine law, of course, to find the missing side but this means that we have to find the cosine of 140˚. To do this we have to redefine the sine and cosine functions.

3) The Unit Circle

I used a Desmos graph to show the students what a unit circle and by seeing the right-angled triangle formed in the first quadrant, we can now define the cosine of an angle to be the x-coordinate associated with that point and that the sine of the angle is the y-coordinate of that point.

From here, we can see straight away the values for the sine and cosine of 0˚, 90˚, 180˚, 270˚ and 360˚.

The unit circle is a powerful mnemonic device so I recommended that they get used to sketching it!

 

4) Connecting the Unit Circle to the Graphs of Sine and Cosine

I asked the students to sketch a graph and plot the known values for sin θ between 0˚ and 720˚. They got something like this:

Now, it was a case of asking them to 'fill in the gaps' and sketch the complete graph. They all did this successfully and I reckon that the graphing stories they did earlier helped them see this. I did the same for the known values of cos θ between 0˚ and 720˚:

Again, they were successful. Now I could show them this lovely Desmos graph to confirm what they told me:

With these two graphs established, the students could tell me:

• that the maximum and minimum values for sin θ and cos θ are 1 and ¯1.
• that the cosine graph is just a sine graph shifted to the right (and vice versa)
• that these neverending waves (which we can now call sinusoidal) are made up of repeating periods, and that the length of one of these periods is 360˚

5) Transforming Sinusoidal Graphs

Next, I grouped the students into threes and had them stand (socially distanced, of course) by a whiteboard, with each person having their own marker. They had downloaded Desmos on their own devices and I quickly showed them how to scale the axis so as to allow us to work in degrees.

Their task was to find out the effects of each of the following transformations:

• y=sin x +c
• y=a sin x
• y=sin (x – d)
• y=sin b x

Since they had already explored transformations of other functions, they were able to tell me what each of these effects were. With a bit of further provocation, they could tell me that the 'a' value gives us the amplitude of a sinusoidal graph (a measure of how 'tall' the waves are) whilst the 'b' value can be used to calculate the 'length' of one period of any wave.

I finished the lesson by asking them to make sure that they summarised what they had learned in a short note. I told them that this note would be useful in the next lesson when they would combine two or more of these transformations.

Afterwards, I chatted with the teacher and we both agreed that we were pretty convinced (through conversations and observations) that the students had a solid understanding of the overall expectation. The Desmos activity that I had planned for the next session would confirm for us how solid this understanding was.

Elapsed Time and Empty Number Lines

I cannot think of a better tool to help students understand elapsed time than empty number lines. I was in a Grade 4 class and began by asking them to show me various times on their individual analogue clocks. I have blogged before about the importance of making sure (as teachers and parents) that children are familiar with analogue clocks so I was happy to see how adept all these students were with showing me a variety of times. As it was 8:30, I set them a challenge:

What will you be doing in two-and-a-half hours?

As I moved around the class listening to and watching the students thinking, I noticed that some were using the clocks to help them find the new time whilst others seemed to be using their fingers to keep track of their thinking. There were two common answers: 11:00 and 10:00.  

I then told them how I 'see' this problem using an empty number line.

First, I mark the start time on my line:

Then I move forward two hours from this:

And finally add the extra 30 minutes:

One student then said that this was like how they have learned to add and subtract numbers: a good sign!

When we agreed that the time would be 11:00, they were able to tell me that they would just be beginning lunch.

I then gave them this challenge:

You are going to see a film that starts at 3:15 p.m. You know that it lasts three-and-a-half hours. What time will it finish?

As I moved around the room, I could see students making great progress by using the number line:

Some students used an analogue clock to help their thinking:

All were able to give me the correct time that the film finishes.
My next challenge was this:
Your train leaves Oshawa at 9:15 a.m. and takes 6 hours and 40 minutes to get to its destination. What time will it arrive. Again, the students used the empty number line effectively:

What I like about number lines, is that students use numbers that are friendly them and that these eventually will lead to more efficient strategies such that students (in my experience) begin to 'see' the number line in their head.
I ended the lesson by giving this challenge:

It is now 9:25 a.m. If school ends at 2:50 p.m., how many minutes are there before you go home

Unfortunately, I forgot to take pictures of their solutions, but they were able to get the correct solution despite not being shown how to do problems like this: they just used the empty number line, marked on the start time and end time then figured out the elapsed time in ways like this:

Developing Perimeter

I recently was in a grade 3 class and tried this activity with them to help them understand that perimeter is the sum of all the sides of a shape. To help with this, I cut five lengths from stir sticks with these measurements:

3cm, 4cm, 4cm, 5cm, 8cm

I used stir sticks as I wanted something thin so the students focus on the length of the segment and not its width (or even height). I also wanted to see how well these students were meeting the overall expectation: 'estimate, measure and record length, perimeter... using standard units'.

I started by asking the students to order the sticks from shortest to longest:

I then asked them to pick up the longest stick and asked them to estimate how many centimetres it was. I would say that about half the class had a reasonable estimate. To help them refine their answer, I gave them two benchmarks: firstly, that the width of their finger is about a centimetre; and secondly I held up a ruler and showed them what 10cm looked like. This helped all students refine their estimates and was worthwhile doing.
Then, each student was given a ruler and asked to measure and record the lengths of the five sticks. What particularly impressed me with this class was that every student used the ruler correctly i.e. by lining up the zero on the ruler with one end of the stick. It was clear that there teacher has done some fantastic work on this already.
I asked the students to round their  measurements to the closest centimetre (which helped cover the fact that sometimes my cutting was not as accurate as it should have been!)

I then set the students a challenge: 

Create as many triangles as you can using only three sticks and find the perimeter of each triangle.

Before they did this, I demonstrated (using longer sticks) how I wanted them to carefully place the sticks end-to-end.
For the next fifteen minutes, they worked really well on finding as many triangle as possible. I was really impressed as to how much care they took in putting the sticks end-to-end and also how well they recorded their results.

I had carefully chosen the lengths of the sticks to limit the number of possible triangles as well as to create situations where a triangle was impossible. When I realised that they have pretty much found all the possibilities, we recorded our results as a group. As I wrote these down, some students were able to explain why one triangle had the largest perimeter and another had the least perimeter. Some also made the point that some sticks didn't form a triangle so I gave them some time to think about why this was:

This is getting at a big math concept which I think often is not mentioned: the triangle inequality i.e. that two sides of a triangle always sum to more than the third side. Having the sticks in front of them made it easier for me to show them why this is true.
Now some students had earlier tried using four sticks to create a triangle so I gave them one final challenge: 

Create a rectangle with all five sticks and find its perimeter.

Most were able to do this but in the future, I would make it more accessible by asking them to create any shape using all five sticks and to work out its perimeter.

I was really pleased how well the students took to this. It is so important that they have this CONCRETE understanding of what perimeter is before they move on to more diagrammatic and abstract questions and, hopefully, will help them avoid misconceptions as outlined in this earlier post.

Unpacking Some EQAO Measurement Questions: Junior

With the recent release of some of the questions used in this year's EQAO Math tests, I thought it might be useful to share some insights as to how students performed on individual questions and what we might learn from these. Here are some measurement questions from the Grade 6 test which raise some concerns (for me at least) regarding how we teach measurement and how we test it.
Concern 1: Counting versus Measuring

As a province, 81% of the students got this correct. It is a classified as a Knowledge question. Initially, we might be encouraged with this, but I wonder if this is really a test of a student's understanding of measurement: maybe they are getting the correct answer simply by counting. As I have written about previously, there is a misconception that some students hold onto when they believe that a measure is discrete and not continuous. This particular question doesn't actually tell me if students really understand volume.

Concern 2: Teaching Conversions
About 70% of the province's students got this correct.

I worry about questions like this as I do not see this as a practical conversion. Who really needs to know how many millimetres a classroom is? Still, it is a better question than this one from a few year's ago:

I have yet to meet any firefighter that knows the length of their truck's ladder in decametres. Now, part of the problem with these questions stem from the specific expectation in the curriculum:

• select and justify the appropriate metric unit (i.e., millimetre, centimetre, decimetre, metre, decametre, kilometre) to measure length or distance in a given real-life situation
  

I have a problem with decimetres and decametres being included in the curriculum as there are precious few real-life situations that we actually use these. And fire trucks are not one of those. Nor is using millimetres to measure a classroom. A much better question would be to ask a student to measure a line and write the answer in centimetres and millimetres.

So why are decimetres and decametres included? Maybe it is to 'fill in the gaps' in the metric system but then we fall into the trap of using these mnemonics:

I always cry a little bit when I see something like this: try using it to convert 5 square metres into square centimetres and you'll see why. If we teach conversions this way, we end up with worksheets like this:

Asking students to change from cubic hectometres to cubic kilometres will not improve students' measurement sense. It is much better to get students to physically measure objects using tape measures, scales, measuring containers etc. that have the different units on them (e.g. metres, centimetres, and millimetres marked on a metre stick). We should never teach conversions without giving students plenty of practical experience measuring objects. But it would also be better to ask more practical questions like these from earlier EQAO papers:

 

Concern 3: Developing Formulas

I have written before (here, here, and here) on the importance of developing formulas with our students. Simply giving a student a formula to memorise is not good enough: Students must be given the opportunity to see why a particular formula works. This will necessarily involve a concrete-diagrammatic-symbolic approach (as outlined in this post).

So when half our students answer this correctly: 

I wonder how many of them actually learned about surface area by using cardboard cereal boxes and physically running their hands over the 6 rectangles present and seeing that there were in fact three pairs of rectangles: front and back; left and right; top and bottom.

Also, how many of our students also had to work with open-topped (and even open-bottomed!) boxes? For surface area, I wouldn't even give the students the formula SA=2(lw+lh+wh): it is much better to get them to see that the surface area is the sum of all the required faces.

Of all the released questions, this was the one that our students were the least successful in: only 37% got it right. 

Why is this? From the responses, we know that the students who chose the second option multiplied the 7 and the 4. Those who chose the triangle simply did 6 times 5. So did these students develop the formula with their teachers or were they just given it? It is too hard to say, but my sense is that those who have developed the formula are more likely to remember it and use it it correctly. But what about those who have developed the formula but in the pressure of the test, forgot it? Grade 6 students are not allowed to have the formulas displayed in the classroom for the test. Grade 9 students, on the other hand, are each provided with an individual formula sheet. I have yet to make sense of EQAO's rationale for this but I wonder how much more successful our students will have been if they developed the formula AND had access to the formula for the test. Also, I wonder if the question is a true reflection of the specific expectation: 

develop the formulas for the area of the parallelogram and the area of the triangle, using the area relationships among rectangles, parallelograms and triangles.

For example, the triangle above is not shown related to its generating 6cm by 5cm rectangle.

To end with, it is not just our students who need to see how formulas are developed. At a Math Café for parents earlier this year, I developed the area of a circle with them:

Afterwards, parents told me that they wished that they had been taught this way when they were at school as opposed to being told simply to memorise a set of formulas.

Unpacking Some EQAO Measurement Questions: Primary

With the recent release of some of the questions used in this year's EQAO Math tests, I thought it might be useful to share some insights as to how students performed on individual questions and what we might learn from these. I will start by looking at some measurement questions from the Grade 3 test:

I include both the English and French Immersion versions as there was a difference in the results. In English, 82% of the students were right whilst in French, only 68% were right. The most common wrong answer for the French Immersion students was 'mètre'. When I have shown this to some adults who know a bit of French but not a lot, they have also chosen 'mètre' as they did not know what 'clé' meant but knew that 'maison' meant house so figured that metres would be the best answer. Students are not allowed to use English-French dictionaries in the EQAO tests but I wonder if these French Immersion students would have performed better even if a picture of a house key was included?

Here is another interesting question:

Recent tweets from Steven Strogatz revealed his concern that students do not know about analogue clocks. In this question, 68% of the students were right with French Immersion students slightly higher than their English counterparts (76% to 67%). I wonder if this difference a result of the clock being more frequently used as a tool for developing the language in French Immersion classes. I also wonder if telling the time is best taught not as a separate unit but as an ongoing life skill throughout the day, throughout the year. A simple act of taking 30 seconds to stop the class and get them to look at the clock to tell the time, done five or six times throughout the day could have a massive impact on student learning. I've shared some ideas on why analogue clocks are fantastic tools for developing Mathematical thinking previously in this post. 

This question also reveals a common misconception:

Only 57% of students got this correct. Over a quarter of the students chose 240 minutes. This does not surprise me as I often see a common counting misconception where students are asked to count up (or down) by ones from a start number (bolded) and write:

157, 158, 159, 200

or

300, 259, 258, 257

I call this microwave math as I think it develops from situations such as looking at the decimal clack on a microwave, putting in your food, pressing '3 0 0' then start and, WOW! The next number is 259!

So how to address this misconception? Well, firstly we need to acknowledge it exists and not assume that all students (even if they can tell the time) know that there are 60 minutes in an hour. I have seen some teachers write the number of minutes alongside each hour number on their classroom clock, and then ask questions such as, 'It is two hours to lunch. How many minutes is that?' I have also seen some teachers use a double number line with hours on top and the corresponding minutes below. Such strategies done at frequent intervals throughout the year could go a long way to fixing this common misconception.

A final question which is revealing, is this one:

Here, just less than half the students (49%) got this correct. The most common wrong answer was the top one (one-quarter litre, one-half litre, 1 L, 2 L). I wonder how many of these students misread (or are used to seeing) it as a smallest to greatest question? If so, then at least they were kind of correct! What is more problematic are the students who chose one of the middle two answers. For the second one (chosen by 12% of students), I would assume that they are thinking greatest to smallest as this list has 2 L followed by 1 L. Do these students also think that one-quarter litre and one-half litre are therefore bigger than two litres? For the third option, again assuming that the students are thinking greatest to smallest (on account of 2 L being followed by 1 L), do these students think that one-quarter litre is greater than one-half litre? If so this reminds me of the McDonald's-A&Ws math misconception:

So how do we address this misconception? For students to really understand measurement, they need to spend a lot of time practically measuring things. They MUST experience measuring with rulers, tape measures, scales, or, in this case, by pouring water into containers to see the difference between one litre, one-half litre and one-quarter litre. It is actually a great way to develop fractional understanding too (How many quarter cups of water do you need to fill a full cup?)

One final thought: I wonder if any students were confused by the notation used i.e. 2 litres written as 2 L? I only mention this as I am not sure how often (if ever) I use an upper case L to stand for litres. I either write litres out in full or use a lower case l. 

More Than Adding Zeroes

 A common misconception I often see is that when you multiply by 10, you just stick a zero on the end of the number, and that multiplying by 100 means you just stick two zeroes on the end of the number etc. A problem with this rule given as is, is that it doesn't endure: 6.7×10≠6.70. Also, students get confused when a zero is already on the end of the number and will write something like 320×100=3200. Further to this, students will not see the connections between multiplying by powers of ten and dividing by powers of ten and so will get flummoxed by a question like 1234÷100.

This was the case when I visited a Grade 5 class recently and started with this warm-up question:

5.5×10=?

The students gave me four possible answers:

5.50

50.5

5.05

55

After agreeing that all four answers can't be correct, I used this app based on the old National Numeracy Strategy ITPs from Mathsframe.

As I did so, students were able to see that multiplying by 10 is equivalent to each digit moving one place value column to its left and dividing by ten is equivalent to moving one place value column to its right. As such the digits themselves 'stick together' when they are being multiplied or divided by powers of 10 so we cannot just stick random zeroes in the middle of the digits!

Going back to original question, all the students now agreed that 5.5×10=55. 
Doing it this way also allowed me to show that when we multiply by powers of ten it really is the digits that move and not the decimal point. That being said, I used a couple of examples with the moving digits to show how you how some people might see it as equivalent to the decimal point moving. The important question then, is to ask: "Is my answer going to be bigger or smaller?" If we are multiplying by 10, 100, 1000 etc. then the answer must be bigger and so we need to move the digits (or, if you see it that way, the decimal point) the appropriate number of places. The opposite is true for dividing by 10, 100, 1000 etc.
As the students then worked their way through some practice questions, I noticed that a couple of students were still having difficulties. Sensing that they would benefit from concretely moving the digits, we did a few questions with playing cards. After the lesson, I realised that this would have been even more powerful if I created the place value columns on the desk as shown

To multiply by 100, we just shifted the digits 2 columns to the right as we knew that the number had to get bigger. We then filled the empty columns with zeroes:

Going back to the original question, to divide by 100, we just shifted the digits two columns to the left as we knew the number had to get smaller:

This really helped the students who later were able to do these type of questions without the playing cards.

*         *         *

The next day, I went back into the class to try a new game that I recently created. It is called Triple Jump and the rules are here:

We got the students in pairs, gave them a deck of cards each and a piece of paper and pencil to jot their answers and away they went:

Within minutes there was an amazing buzz in the room: there were cheers and groans as rounds were either won or lost. Students were getting excited about multiplying and dividing by powers of 10.

Let me repeat that:

Students were getting excited about multiplying and dividing by powers of 10!

As the students worked on this, it made me realise that often one of the best ways to practise a concept is in a game-like situation and this certainly was the case here. As a follow-up, we got some students to try another version of this game called Decimal Triple Jump:

These games are part of the Math@Home kits that we created in our board. They are a series of different games and activities designed to improve students' number sense and are being used in class and also at home.