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.

Scaling Up

One of the biggest hurdles to mathematical understanding is moving out of additive thinking into multiplicative thinking. There are many reasons why students get stuck in an additive phase so what can we as teachers do to move them into a multiplicative phase?
I worked with a Grade 9 Applied teacher recently who noticed from her diagnostic tasks that many of her students could not think multiplicatively. As they were about to begin some work on ratios this was going to be a problem. We decided to adopt a concrete-diagrammatic-symbolic approach to move students on from additive thinking. 
We began with a simple problem:

The weights of two dogs as puppies and fully grown are shown:

Which dog grew more?

Without exception, the students said that they grew by the same amount (i.e. 6 kg). They were looking at how much weight had been ADDED.
So we then asked them, is there another way of thinking about this. After a bit, one student noticed that the first dog had DOUBLED in weight whilst the second dog had not increased by the same rate.

This was the platform we needed to build on.

I was clear with them: we need to learn how to compare things not just by addition but also by multiplication. I told the that we were going to do some activities that would help them how to see things in terms of multiplication and not just addition, and that this would make them better mathematicians. 
I also told them that we were going to do this in three steps: concretely, then diagrammatically, then symbolically. 

Each student was then given a set of cuisenaire rods.
I told them to find two orange ones and put them end-to-end. "If one of these is 10, how much will two be?" "20!" came the instant reply.
I then told them to put a yellow rod directly below the two orange ones (and showed this using the mathies.ca Relational Rods tool). I then asked them to estimate how many rods would be need to match the two orange rods. 

After they made some suggestions, I asked them to find out and then tell me how much a yellow rod was worth: they were able to tell me that it was 5.

I then asked them to write a number sentence for what they had just done. Over half wrote 5+5+5+5=20 so I then asked them to write a number sentence without using an addition sign. This nudged them toward multiplication and they wrote 5x4=20.

Again, I was clear with them: this is the goal of today's lesson...to think multiplicatively.
Next, I asked the students to do this again but this time with the purple rod. Seeing the students carefully lining up the rods to make sure they were equal to the two orange rods (and the four yellow rods) made me realise that maybe this is the experience that they had missed out on: the actual concrete act of creating equality using equal groups.

They wrote 4x5=20 without any prompting. One student then noticed something: "I can write it another way without using addition. If you split the rods up again you are dividing the 20 so you can write them using divisions!"

This led to related facts:

4×5=20

5×4=20

20÷5=4

20÷4=5

Next, I told them that as they were grasping this so well, it was time to scale up: now we need to use larger numbers and that these would be better modelled with diagrams. So I asked them to write a set of related facts for this diagram:

From this alone, they were able to write:

20×6=120

6×20=120

120÷6=20

120÷20=6

No-one wrote 20+100=120. We were seeing the students shift away from additive thinking.
Curious I wrote down my favourite math fact on the board:

37×3=111

and told them that we were about to scale up again. I asked them to complete the set of related facts which they were able to do even though they had not learned the 37-times table!

We then split them into visibly random groups and gave them a problem to try:

Two people do some decorating. Ann worked for 2 hours, Bill worked for one hour. Together they were paid $30. How much should each person get?

As the groups worked on this, it was clear that they realised that it would be unfair for the people to be paid the same amount. Most groups got the sense that Ann should get paid twice as much as Bill and used different ways to come up with an answer. We summarised their their thinking by using a bar model approach:

This allowed them to see the 'three-ness' of this problem and allowed them to see that each hour block is equivalent to $30÷3 or $10. We then challenged them with the following set of problems and encouraged them to use bar models to show their thinking.

It was pleasing to see many of them successfully use the bar models to solve the problems (though I wish I took more pictures of their work).

When I asked how they felt about this concrete-diagrammatic-symbolic approach at the end of the lesson, the students told me that it really helped them. 

Sometimes it takes just a well-timed nudge to move students on.

Polygon Angle Sums: Develop, Don't Give.

A common way to get students to see that the sum of three angles is 180° is to get them to tear the three angles and rearrange them to create a straight line. Rather than giving the angle sum formula for any polygon, I wondered if I could use this approach would work with other polygons so as to develop the formula instead. I tried this with a couple of grade 9 Applied classes.

I visibly random grouped the students in to threes and gave each group a different paper quadrilateral. It didn't take long for the to rearrange the angle to form a complete turn and for them to tell me that quadrilaterals have an angle sum of 360°.

Moving on to pentagons, I wanted to make sure that the angle would rearrange clearly into one and half complete turns. I figured that the best way to do this was to give pentagons that had two right angles like this one:

This allowed the students to rearrange like so:

...and then tell me that pentagons have an angle sum of 540°

In a similar way, I then gave each group of students a different, hexagon, heptagon,  or octagon whose angles could be torn off and easily rearranged into full and half turns such as these below:

This nudged the students into quickly rearranging the angles:

As the students were finding the angle sums, I recorded their results using Desmos so that we could all see what was happening:

I could now ask the class "What do you notice? What do you wonder?"

They quickly noticed that the angle sum was increasing by 180° each time the number of sides increased by one. Some students wondered if this was something to do with the angle sum of a triangle.

So, I then sent them in their groups to the vertical whiteboards and asked them (one polygon at a time) do choose a single vertex, and from there, draw as many diagonals as possible to any other vertex. 

As this decomposes the polygon into smaller triangles (each of which has an angle sum of 180°) they could then confirm their earlier results.

I now challenged them to predict the angle sum of a dodecagon. As they were now recording their data in a table, it made it easier for them to spot and extend this pattern:

I followed this by asking them to tell me the angle sum of a 102-sided shape and then to generalise for any shape. They were able to see that each polygon with n sides could be split into (n-2) triangles and so the angle sum is (n-2)×180°. 

This was pleasing as we had not only used the concrete-diagrammatic-symbolic continuum but also the next-near-far-any continuum for patterns. The students seemed pretty chuffed that they had 'discovered' this rule themselves. I finished the lesson by giving them these questions to consolidate what they had just learned.