Tuesday, May 15, 2018

Open Middle Fraction Problems

I have often shared the Open Middle website as a source of good thinking problems with the teachers that I work with. This week I created a couple of my own and used them with a grade 6 class who have been working on equivalence in fractions, decimals and percentages. The first problem (given using verbal instructions) I gave was this: 

Use any of the digits 0 to 6 once only to satisfy the statement below.
We used visibly random groups to get the students into threes and gave each group seven tiles to work with. As we began to walk around the room, a few students asked 'What do I have to do?' Taking the lead from Peter Liljedahl's work, I smiled and told them to find out from a friend.
We could see some students getting stuck or making mistakes but resisted the urge to jump in and show them how to get a solution. 

We let them think their way out of it.

Sure enough, we soon heard some 'A-ha's around the room as students realised that the first space had to be a zero, and as they figured out the decimal equivalent of a fraction:
Once they had found one solution, I asked them to find me another. Some used two-fourths as the fraction, others used one-fifth or two-fifths:

One group bent the rules a little bit:

Another group tried something similar but put 2.5 instead of 0.25. I simply pointed to this and asked 'Is this more than one or less than one?' and walked away. When I came back, they had corrected it.
Another group tried this:
When I asked about this, they knew that two-sixths was one-third but they also thought that its decimal equivalent was exactly 0.3. As they knew how to use a calculator to change a fraction into a decimal, I asked them to do this for one-third. This made it easier for me to convince them that 0.3 is different to 0.333333333... 

As we had fifteen minutes left, I gave a variation of a question that I tweeted last week that proved popular:

Any thoughts we might have had about this question being tricky for them were soon dispelled:

As we circulated, we could hear the students justify their solutions and, if we were unsure, simply asked a question like "Can you convince me that two-sixths is less than five-eighths?"
One student asked if it was OK to use a fraction whose numerator was larger than its denominator. This led into a nice discussion about improper fractions and how these are all larger than one:
We could see a couple of groups who found solutions quite quickly so we gave them an added challenge by removing the '1' tile. They relished the added level of difficulty:

All in all, it was a really pleasing lesson that allowed the students to show us their problem solving skills as well as allow us as teachers to assess their understanding of fractional equivalence.

Tuesday, March 20, 2018

Creating Thinking Classrooms (3)

Having done some work on creating a thinking classroom with junior and intermediate students (see my last two posts), I was keen to see how high school students would react. Again, I wanted to gauge the impact of three of the optimal practices for creating a thinking classroom as outlined by Peter Liljedahl:

  • start with good questions
  • use vertical non-permanent surfaces
  • use visible random groups of three

The first problem I tried this with was actually one Peter Liljedahl had shared with us at the Ontario Mathematics Coordinators Association's conference.
Using the numbers 1 to 10, and the operations +, –, ✕, ÷ plus another one of these operations, create five number sentences that have the following solutions:
17     2     21     3      2
(For what it is worth, this question is a lot easier to understand when the instructions are given orally!)
The students got stuck into this problem immediately. Being free to move around the room I was able to listen to some impressive number sense and logic. I'd go so far as to say that the students were more accomplished at this than adults who I have given the task to.

When they solved it, I gave another set of answers:
2    2    2    2    9
and later
10   14   1   20   16
 Next, I gave one of the Problems of the Week from our @DCDSBMath twitter feed:
As students quickly solved this, I then asked them to consider a 10-by-10 array with the top square missing and, from there, to consider the general case:

 Many groups looked at of squares of different sizes and noticed that these were one less than a perfect square. It was a nice opportunity for me to introduce sigma notation:
One group gave me math bumps though as they saw the number of squares of different sizes in a different way which connected beautifully to the difference of squares:

Perhaps the strongest feeling I take away from this is the amount of math that the students are doing. They are not sitting passively copying a note, they are actually being mathematicians.
It is something I have seen before in a Grade 9 class (this post) and I also saw recently in a Grade 12 calculus class. Students were asked to sketch a graph of any function they wanted and to then use what they know of the derivative to sketch the graph of the derivative (without resorting to deriving by first principles). Students were first sorted visibly into random groups by being given a card and then having to find the corresponding representations:
They then got stuck into the task:

 and as they did so, they began to challenge themselves more:
Remember, the students had to choose their own graphs for this activity. The discussions that took place were a joy to behold. Independently, they began to hypothesize that the derivative of the graph would of a degree one less than the original graph. They carefully considered the key points of the original graph (e.g. turning points) and used these clues to plot where the corresponding values would be on the graph of the derivative. As they worked through this, it was clear to me that they were developing a solid understanding of the connections between the graph and its derivative: I am not sure this understanding would have the same clarity if they were to jump into deriving through first principles. The next day they consolidated their learning and their teacher, Leanne Oliver, sent me these photos:

The upshot of all of this is that these three components of creating a thinking classroom are having a real impact on students' learning: 

  • they are doing more math
  • they are taking more risks
  • they are developing collaborative skills
  • and they are enjoying it!
Our challenge now is to make sure that all students experience the power of a thinking classroom.

Thursday, March 1, 2018

Creating Thinking Classrooms (2)

Following on from my previous post, I want to gauge the impact of three of the optimal practices highlighted by Peter Liljedahl's research into creating thinking classrooms:

  • start with good questions
  • use vertical non-permanent surfaces
  • use visible random groups of three
I went into three classes (a Grade 4, a Grade 5, and a Grade 6) and gave the students a variant of the Precious Pentominoes activity. 
The Grade 4s were asked to use two pentominoes to create a symmetrical shape with the largest perimeter:

The Grade 5s were asked to use two pentominoes to create a symmetrical shape and then calculate its cost by working out perimeter multiplied by number of sides. They then had to find the most expensive design:

The Grade 6s were asked to do the standard Precious Pentominoes task and find the most expensive design:

Look carefully and you will notice three different methods that the students have chosen for multiplying!
In terms of the three practices outlined at the start, here is what I noticed:
1) The question (which I gave orally) engaged the students from the get go. Allowing the students to use pentominoes meant that the students had multiple entry points into the problem. And the problem itself allowed the students to use many (if not all) of the Mathematical Processes. In other words, it allowed them to think mathematically.
2) The VNPSs made it much easier for me to see what each group of students was thinking. Occasionally, I noticed that some students were not measuring the perimeter carefully (showing misconceptions highlighted in this post). I was able to quickly address these misconceptions by getting the students to focus on the line segments and not the squares. 
The VNPSs also meant that students felt that students felt more comfortable showing their work in the knowledge that if they made a mistake, then they could erase it. And having the students thinking on their feet (literally!) resulted in great discussion and problem solving: more so than I have seen when students are sat down.
3) The students had no trouble at all working in the random groups. The fact that they were in groups of three meant that I had a manageable number of groups to monitor and also allowed for a good exchange of ideas between the trio. Even students who teachers identified as having difficulties with Math rose to the challenge of the problem. From what I could see, every student made some contribution.

Each of these three practices certainly had an impact in creating a thinking classroom in each of these junior grades (like it did with the intermediate class in my last post). I left each of these classes amazed by the wonderful mathematicians I had just worked with.
Now, how would it look with high school students? 

Monday, February 12, 2018

Creating Thinking Classrooms (1)

I have been reflecting a lot on Peter Liljedahl's work in the past few months and have been more intentional about implementing his ideas in any of the classrooms I go into. In particular, I am trying to gauge the impact of using three of his optimal practices:

  • start with good questions;
  • use vertical non-permanent surfaces (VNPSs);
  • and use visible random groups.

This week, I went into a grade 8 class who had been working on measurement. I began the lesson by showing them a game called 'Prism or No Prism!' which involves me holding up a shape and the class deciding if it is a prism or not. For the most part, they were correct but about half the class said that a cube was not a prism. When I asked why, they said because it is a cube! As they weren't too clear about what a prism is, I shared with them my 'loaf of bread' analogy:
If a shape can be sliced like a loaf of bread from front to back and give exactly the same size and shape slice, then it is a prism.
"So that would make a cube a square-based prism then!" said one student.
Next I wanted to ascertain that they knew how to get the volume of a prism. The 'loaf of bread' analogy works well here to as we can connect it to layers (or slices) that can be made thinner which leads us to develop the idea that the volume of a prism is the area of one 'slice' multiplied by the 'number of slices' into the more generalised formula, V=Axh
In all of these discussions, we did not look at cylinders.
I then showed them the opening act of Dan Meyer's Popcorn Picker:
I asked "What do you notice? What do you wonder?" Some wondered if one cylinder would give more popcorn than the other cylinder. Others reckoned that the cylinders would give equal amounts of popcorn. So the task was set:
Decide which way you want to make your cylinder. It will then be filled with popcorn!
I used playing cards to create visibly random groups of three students each and then gave each group one marker each and had them work at VNPSs.
The students got stuck into the task immediately, even though they have never been shown the formula for the volume of a cylinder. Whilst there was the occasional dead end (one group got stuck on using V=lxwxh before realising that this wouldn't work with a cylinder!), the students soon reasoned that since the cylinder is a prism, they could work out the area of the base circle and multiply this by the height for each cylinder. Getting the area of the circle requires knowing the radius and some did this by direct measurement whilst others measured (more easily I'd suggest) the circumference of the circle (that is, of course, one of the sides of the rectangle and then divided this by 2π to get the radius.

One of the great things about VNPSs is that as a teacher, it makes it easier for me to see what students are thinking and any errors that they might make.
When we were satisfied that the students had reached a conclusion, we noticed that seven groups opted for the shorter, wider cylinder and one group opted for the taller, narrower one. I filled one of each of these cylinders and, by then emptying the popcorn on the table, we could see that visually most groups had got it correct. It turned out that the group that didn't had the right idea but made a calculation error.
With the students merrily munching on popcorn, I was able to summarise the lesson by using their work on the VNPSs around the room and got them to tell me the formula of a cylinder:
The use of good questions, VNPSs, and visible random groups certainly proved effective in getting these grade 8s thinking. I wondered how it would be for younger students.

Thursday, January 11, 2018

Precious Pentominoes

Here is a task that I made up that I have recently tried which provoked a lot of thinking with students and adults alike.

From a set of 12 pentominoes, choose two tiles to make a closed shape that has symmetry. For example these are NOT allowed:
Once you have a design, then work out the perimeter:
the area:
and the number of sides:
Now calculate a 'cost' for your design as follows:
What is the most 'expensive' design you can make?
The students got stuck into this immediately using the pentominoes to create a variety of designs:
They recorded their thinking as they went:


After a while, we recorded the costs that students had found on the board. Now the students were really keen on finding a more expensive design. There were shouts and screams of delight as groups found new, more expensive designs. After a while, all groups had reached a consensus of what the more expensive design was (I won't spoil it for you by revealing the answer!)

As there was still time left, we extended this by allowing three pentominoes and removing the symmetry constraint. 

As we wrapped up, we asked them what strategies they used. They noticed that the area was always the same but that they had to take care when calculating the perimeter and the number of sides to avoid 'double' counting:

They also noticed that it was important to maximise the number of sides as this was the multiplying factor. So long, 'straight' pentominoes were not as good as the pentomino shaped as a cross.
But perhaps the best comment was as I was leaving:
"Please come again soon and please bring some more questions that will give me a mathematical headache!"

We also tried this with our educators at a Capacity Building session today. Again, the level of engagement was high. We tried also bringing some of the ideas from Peter Liljedahl's research about using vertical non-permanent surfaces, giving verbal instructions, creating visible random groups, and only answering 'keep thinking questions'.

This is when one knows that it is a good task: that adults and students alike are captivated by it.