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!

Happy Pi Day!

Archimedes would have been proud of us today! Our class celebrated Pi Day in full force, and it impacted many curricular areas. After a brief lesson about pi and the relationship between a circle's circumference and its diameter, it was time for Arts Ed. We made pony bead bracelets and, when finished, we wrote as many digits of pi as would fit around our wrists.

Next it was time for math, so we made "pi plates". We took regular paper plates and measured the circumference of them using yarn. This was trickier than we thought it would be. We marked the starting point for measurement and, in teenie, tinysections, inched our way around the plate. Then we took the length of yarn and measured it to the nearest millimetre using metre sticks.

Finding the diameter was considerably easier. We folded the plates as carefully as possible and measured across the center of the plates to the nearest millimetre. The students then took their circumference and divided it by their diameter (yes - we used calculators for this!) with the hopes of getting close to pi. Most students were able to fall within the range of 2.9... and 3.5... To our amazement, two students (pictured below) were able to calculate their quotient remarkably close to pi! Their responses were 3.141... and 3.147... Amazing!

In Daily Five, we worked on writing by creating characters from the pi symbol. Being a huge baseball nerd, I created Hall of Fame Mets catcher, Mike Pi-azza. The students came up with ridiculous ideas of their own! We had Broncos quarterback Pi-ton Manning, Pi-lots, Pi-rates, Sir Pi-saac Newton, and even a Pi-rannosaurus Rex! It was too much fun!

Then it was time to listen to reading, so I did a short read aloud of Cindy Neuschwander's amazing picture book, Sir Cumference and the Dragon of Pi. The students got all of the weird math jokes and we had lots of fun with the voices of the characters!

After such a great day of pi-related activities, it was time to have a little disconnected fun. Thanks to our awesome group of generous parents, we had lots of different pies to go around, complete with ice cream and whipped cream. The students literally gorged themselves on all of the delicious pies and we crushed 4L of vanilla ice cream! Apologies to all of the parents who had to hear their children say, "I'm not hungry. Math ruined my appetite!" ;)

Before:

 

After:

 

Been there!

I am absolutely addicted to the Kid Snippet videos, but this one perfectly captures that feeling of trying to explain a math concept so many different ways that, in the end, everyone's confused!

Garbage Can Basketball

5 March 2013 - Today we took a short break from the textbook and practiced some hands-on work with fractions and decimals! We had one student away, which left us with a nice, round 25 students - an easy number to convert into 100.

Each student ripped four pages from a magazine and crumpled it up, giving us exactly 100 basketballs. I then placed a garbage can on a table at the from of the class and we were good to go. We each took 4 shots while seated at our desks, and 11 of the balls went in the bucket.

First, we wrote it as a fraction over 100. (11/100) The students then had to convert it to a decimal (0.11) and a fraction using a denominator of 1000 (110/1000) and a decimal value to the thousandths place (0.110).

Then, we established a foul line in the center of the class and the students took turns shooting four shots again. This time, 18 shots went in, so we wrote the same four values.

 

18/100 = 0.18 = 180/1000 = 0.180

Then we made it a little more challenging, each taking four shots from the back of the room. This time, only 2 went in.

2/100 = 0.02 = 20/1000 = 0.020

To finish up, Mr. C. (Grade 8 teacher) and I faced off, taking ten shots each. He hit 2, while I hit <cough> zero. The students calculated his results as:

 

2/10 = 20/100 = 0.2 = 200/1000 = 0.200

What a fun way to practice fractions and decimals!

Jellybean Fractions

February 25th, 2013 - We have been working a lot with fractions in our current math unit. in particular, we have been working with writing equivalent fractions. Today, we decided to make things a little more fun by ditching the base 10 blocks and using jelly beans instead.

Each group was given a cup of jelly beans (no, I didn't count them out first). Their first task was to count the beans to determine how many there were in total, as this would be their denominator. They then had to sort the jelly beans by colour, using each value as their numerator in a new fraction. For instance, if a group received 72 jelly beans in total, and 15 were pink, they could write this as 15/72. They did this for each of the five colours of jelly beans.

Once the beans were sorted and the fractions were written, the students had to write two equivalent fractions for each one. Whenever possible, I encouraged the students to use division to reduce their fractions. In the example above, both 72 and 15 are divisible by 3, so they could write a new fraction like 5/24. This was not always possible, however, and some groups had a denominator which was a prime number, so they had to turn to multiplication instead. Some of our denominators became quite large as a result! This wasn't a bad thing, however, as it was further support for the concept that the overall ratio of any colour does not change no matter how many pieces you cut each one into.

The students had fun working with an unexpected manipulative, and did a little pigging out at the end of class!