Analyzing Systems

Grade 8 Science No Comments »

In my grade 8 science classes we have been learning about systems. The Ontario curriculum unit, Systems in Action, has the students go beyond learning about physical systems and has them explore social systems and the evolution of systems.

I began this unit by having my students analyze a social system which included identifying subsystems, inputs, outputs and side effects. Each student chose a different industry and analyzed the various parts of that industry and their effect on society, economics, and the environment.

The students are currently finishing presentations for their second assignment, the evolution of technological systems and their impact on society. Together we brainstormed a list of ideas to research, and then students wrote their top five choices. They were then assigned to groups based on their topic preferences. Amongst the favourite topics, students chose to research the evolution of the telephone, the computer, the camera, and mp3 players. Each student group had to create a visual timeline of their chosen system using digital or non-digital tools. As they presented the timeline, they had to explain the reasoning for each technological change and how the evolution of the system has impacted society.

Below is a copy of my assignment sheet. As my school is an IB World School, the rubrics are IB MYP style.

Systems Evolution

Once presentations are complete, the students will move on to the second phase of this assignment. They will consider the various presentations and determine which technological system they believe to have made the biggest impact on society. To do this, we will have discussions to determine which criteria would be used to make this type of judgment. Based on the criteria that we determine, the students will compose their thoughts.

After analyzing the evolution of systems, we move on to experimentation with work and mechanical advantage.
Have a great week.

Pascal’s Triangle and Magic Squares

Math No Comments »

We have been working on patterning in Grade 7 math.  We spent a lot of time looking for patterns in Pascal’s triangle and seeing how the numbers in the triangle work together. I asked my students to each try to find a different pattern in Pascal’s triangle, and they rose to the challenge. They came back to class excited to share what they found, and each student was hoping that no one else had found his/her pattern. At the end of the first day of presentations, most students had claimed a pattern, but there were a few students whose patterns were claimed by others and needed to explore further. The next day I decided to help them out, and gave a short lesson about figurate numbers and asked the students to find tetrahedral and hexagonal numbers in Pascal’s triangle. We then looked into fractals and how the Sierpinski triangle can be created in Pascal’s triangle by blacking out all of the odd numbers. I left them with another challenge – to see what happens when you block out even numbers, and numbers that are multiples of 3 and 4.  I also showed them some of the Pascal patterns discussed in The Number Devil, a book I mentioned in a previous post.

Here are some of the links that I used for this series of lessons:

Pascal’s Triangle and its Patterns

Pascal’s Triangle from Math is Fun

Patterns in Pascal’s Triangle from Cut the Knot

Pascal’s Triangle from Math Forum

Wolfram MathWorld Fractal Page

Wikipedia Fractal Page (Scroll down to see the changing fractal beside the history section.)

Sierpinski’s Triangle from Math Forum

As we were having so much fun with numbers, we went on to look at the Magic Square in Albrecht Durer’s paintings. In his magic square, the sum of all rows and columns is 34. We used the Powerpoint below (source unknown) that was sent to me by a friend. To start, I only showed the first five slides, and then I left it to the students to determine where else they could find the sum of 34 in the square. They made me proud and found all of the sums mentioned in the Powerpoint, as well an additional sum found through a zig zag pattern.

Albrecht Drer’s Magic Square

Hope you have as much fun exploring numbers as we did.
Have a great week.

Division of fractions…beyond the algorithm

Math No Comments »

I am currently working with multiplication and division of fractions with my grade 8 students. I have never been one to have my students just learn a set of rules, and so we always have discussions about the concepts and why the algorithms work. Year after year, the same issues surface. They have no problem conceptualizing multiplication of fractions, but division of fractions is always troublesome. They can follow the algorithm easily enough, but there are always those that have difficulty understanding why it works.

Here are some of my favourite resources for helping my students grasp division of fractions. If you have found others that work, I would love to hear about them.

First up is a neat little widget from Math Design in the Juniverse. I found this several years ago, and have kept it bookmarked ever since. I started using this before I had a Smartboard, and now my students can interact with it, as well.

Next is the division page on the Visual Fractions website. The first page gives one example, but when you click on “Investigate Division” you are taken to a PDF with several pages of examples to use. Although not interactive like the previous site, we can still put this up on my Smartboard and outline parts of the circles in various colours so that students see how many times I can take the pieces of the divisor out of the dividend.

The last resource is a lesson plan from the Ohio Department of Education that I only found recently. It gives several examples to do with students, along with prepared paper manipulatives for use along with the lesson. When I have more time, I intend to check out their vast database for other math lessons and activities.

Happy dividing.
Have a great week.

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