Working with Probability

December 5th - In our lead up to the winter break, we have been working with the concept of probability during our numeracy time. Our main objective was to help the students understand the difference between the following terms:

impossible ------- unlikely ------- possible ------- likely ------- certain

We introduced this concept by using spinners labelled with students' names. The spinners were divided into equal parts, and the students had to record the results of 100 spins. What we realized was that the results were not equal for all students. For instance, a group of 4 students would predict that each student's name would be landed on 25 times in 100 spins. What we found though was that some students' names were landed on as little as 17 times and as many as 36 times. This meant that it was more probable that the spinner would land on one student's name than that of their partners.

We explored the possible reasons for this. The students suggested that each person might be spinning it differently, or that the starting point was not the same on all spins. Others even went so far as to suggest that there might be a slight defect on the spinner itself that was causing friction in a way that affected it's ability to spin freely. These were all interesting  suggestions.

The main thing we learned was that, though it was possible for all students to have their name landed on with any given spin, it was (for some reason) more likely that one name in particular would be landed on. This allowed for a small amount of predictability for the students, which is really the basis of any study on probability. We plan to repeat this experiment using something other than spinners to see if we get similar results.

Exploring Factors

Today in math, we looked at factors. To do this, we used coloured tiles to make rectangles. I showed the students how a rectangle that was two blocks wide and ten blocks long could represent the equation 2 X 10 = 20 in the same way an array would. I then rotated the rectangle to show that the same rectangle also represented 10 X 2 = 20.

My intention was to allow the students to work with partners to create their own rectangles using the tiles, and to make the connection to multiplication facts. We set a few simple parameters first, both to keep the task from becoming too simple and to prevent us from running out of tiles. We decided that the smallest any side could be was two tiles, and that the largest any side could be was twelve tiles. We also decided that it was important to right each multiplication fact two ways so that we didn't end up making the same rectangles over again by mistake.

The students set right to work making their rectangles, and enjoyed the idea of using large quantities of tiles. I noticed right away that many groups would make a rectangle, draw it, write the facts, and then promptly destroy it and start again. I encouraged them to think of a way that they could change their existing rectangles by making the sides larger or smaller in order to change the facts represented. This both saved time and allowed students to gain a deeper appreciation of the relationship between these numbers. For example, without doing this, a student might not realize that the rectangle representing 12 X 3 is not that different from the one representing 4 X 10.

Our next step will be to explore the factors of the products that we created. None of the groups today created rectangles with the same products, so it will be fun to experiment with this. I plan to have each group look at the rectangles they created and try to find different way to make the same product. For example, if a group built a 6 X 4 rectangle, they will need to build 1 X 24, 2 X 12, and 3 X 8 rectangles as well. This will also be a good activity to look back on while studying surface area later this year!

Spiral! - Our new favourite multiplication game!

We have been working on multiplication strategies in our class lately, and have placed a big priority on learning our basic facts. One fun way to practice these facts is a new game that we have started playing, called Spiral!

The game is simple: Set up the cards in a spiral arrangement and start in the middle. When it is their turn, the students roll a die and move their game piece the correct number of spaces (or cards). When they get there, they have to multiply the number on the die by the number on the card. If they get the answer correct, they can stay there until their next turn. If they get the answer wrong, however, they have to return to the card they came from. The first player to get to the end of the spiral is the winner.

There are a few ways to alter this game. We like to use a 9-sided die so that we can practice with all facts from 1-9. A 6-sided die is fine too if that is all you have. We like to take out the face cards, but you could easily leave them in and assign values to each (Jacks = 11, Queens = 12, Kings = 13). We are not ready for these facts yet, but may add them in at some point.

Have fun practicing those multiplication facts!

Power of 10

During our Explore+4, Mrs. Casper has been running a station for small groups using a program called Power of Ten. It has been fabulous so far!

The basic premise of Power of Ten is to familiarize students with various ways of representing ten.  This might seem like a bit of a primary exercise for middle years students to participate in, but when we look at where there are holes in our students' numeracy learning, it is often with their number sense that these have originated.  If we are to fill the gap at all, we need to meet our students where they are at, especially during station-based activities.

So far, my students have been using ten frames as the basis for the games they play. I am encouraged by the growth that we have seen already. For example, a student that looks at two ten frames representing ten and four is more likely to quickly shout out "FOURTEEN!" today, whereas a few short weeks ago that same student may very well have counted each of the fourteen parts individually.

Think about the number sense related to that task! First, the student has to understand that a full ten frame will always represent ten. Then, they need to quickly recognize that four dots represents four, no matter where they are on the ten frame. Finally, they need to quickly process that ten plus four is fourteen, and feel confident about this.

Zero > 1, 2, 3...?

Today, Ms. Casper shared a picture book with us at the end of math class. It was called Zero, and was written by Kathryn Otoshi.

The message to the story was that we are wrong to assume that zero has no value, and that without zero our entire counting system would be completely different. I don't know about you, but I'm quite glad that we don't use a base-9 system for counting!

I remember being a young teacher and having the chance to meet the legendary constructivist math guru, John Van De Walle. He insisted, to the point of being offensive to some teachers in attendance, that we had no concept of the importance of zero to our number system. Throughout the course of an afternoon, he proved that he was right, and it was an important lesson to learn.

Building Bridges

Over the past week our class has been building bridges made out of bristle board. There was a lot of steps to this. First we had to choose what style of truss to use, there was a lot of people who chose he double warren truss.

The next step was to make a plan of what strips of bristle board we needed to make our trusses out of. After everyone was done making a plan, we all had to construct our trusses. This was the hardest step because one, we needed to hole punch the pieces, fasten them together, and make our roads.

Finally after we were all done our bridges we had to see which bridge could hold the most Lego pieces. The bridge that could hold the most pieces was able to hold 180 pieces.

Geometric "Guess Who"

For our Explore today, we played a fun variation of a very familiar game. Everyone knows how to play the game Guess Who?, and the skills involved were a perfect fit for the math lesson we were doing.

The students looked at a picture featuring many geometric shapes and chose one in their mind. Their partners then had to take turns asking "yes" or "no" questions of their partner until they correctly guessed the shape they were thinking of. Instead of asking about hair colour and whether they were wearing a hat or not, they had to base their questions on the attributes of the shapes. For instance, they had to ask questions about how many sides or vertices there were, parrallel or perpendicular lines, whether the angles were right, acute, or obtuse, or whether it had a line of symmetry.

Shape H, for example, is a six-sided shape (Hexagon) with 4 obtuse angles and 2 acute angles, it has 3 pairs of parrallel lines, no perpendicular lines, and one line of symmetry.

The students had a great time trying to stump their partners, proving once again that games can make any task educational and engaging!