Making Ends Meet

I have recently finished a budgeting activity with my grade 8 math class titled, “Making Ends Meet”. (The document is attached below.)

Each student was given a “job” with an entry level salary. The first step was for them to determine their after-tax monthly income. They then needed to determine how they were going to allocate their income to the following categories:

• Food
• Housing
• Utilities
• Transportation
• Medical Expenses
• Entertainment
• Sports/Fitness
• Clothing
• Miscellaneous
• Savings

Students came into class with a report that outlined the distribution of income in their budget. For the summative task they were then presented with a series of challenges and unexpected problems to consider. These were not shared with the students beforehand.

It was a time consuming task, but well worth the learning experience. My students now have a sense of the value of the dollar, the importance of getting a good job, and the reality that life is more costly then they realized.

Have a great week.

Making Ends Meet 2013

App Review – Math Doodles and Symmetry Shuffle

There are two apps by Carstens Studios that I have loaded onto our school iPads.

The first app is called Math Doodles and it sells for \$2.99. The user is given three challenges (a fourth is in development) that revolve around addition, logic, and algebraic thinking. In the first challenge, Sums Stacker, the user needs to manipulate values within three piles in order to reach a target sum. In the second challenge, Connect Sums, the user must select values that reach a target sum. In the third challenge, Unknown Square, the user must find the missing value in a 3-by-3 array of numbers. One of the things I love about this app (in addition to the awesome graphics) is the ability to play in a variety of number systems. The user can choose to play with values represented as dice, fingers, holes, ten frames, tally marks, binary system, Braille, number prefixes, polygons, US coins and dollars, a variety of fraction types, Roman numerals, numbers shown in  either Chinese, Arabic, Gurmukhi, Hindi, Hebrew, or Spanish, or a mixture of all of the above. There are different levels of difficulty, as well. All of these options allow the app to be used across a number of grade levels.

The second app is called Symmetry Shuffle and it sells for \$1.99. The user must either rotate (turn), reflect (flip) or translate (slide) the image so that all targets have been matched. The user can select from 12 possible images to “shuffle”, and can also change the size of the “shuffle” grid. Its features are not as diverse as on the first app, but I still find it a great addition to our math apps on the iPads.

Both apps allow the user to track the number of moves they have used so that they can attempt to solve the puzzle in the fewest possible moves, which is another great feature for differentiation.

Have fun playing.

A Day for Social Action

On Friday I took a group of students to We Day. It is a day organized by Free the Children, a charity which inspires youth to take action and be agents of positive change in the world. Founded by Craig Kielburger, the main missions of this charity are to assist the impoverished with education, clean water and sanitation, health, alternative income, agriculture and food security. Full day events will occur this year in Toronto, Vancouver, Alberta, Manitoba, Waterloo, Montreal and Saskatchewan. The day is filled with musicians and motivational speakers who want to inspire youth to get involved in social action. In Toronto, we listened to Jennifer Hudson, The Tenors, Al Gore, Justin Trudeau, Hedley, Martin Sheen, Nelly Furtado, Spencer West, General Romeo Dallaire, K’naan, Justice Sinclair, and The Honourable David C. Onley. Our students left feeling inspired by the stories they heard and energized to rally for others less fortunate than themselves.  In the spirit of the day, I would like to highlight a few math resources that focus on social action.

I discovered a new resource called “Real World Math: Engaging Students through Global Issues” from Facing the Future. I tried one of their sample activities in the spring, which linked sustainability to surface area and volume. I decided to order a copy for this year, and I am excited to try more of their tasks. The resource focuses on issues such as waste and recycling, poverty, population growth, youth conflict, global health and carbon emissions. There are a variety of other resources to explore on their website, including web-based and print resources.

For many years I have been a fan of an organization called The Southern Poverty Law Center, whose mandate is to fight hate and intolerance. Their Teaching Tolerance program assists educators in preparing youth to live in a diverse world.

The Global Education website is based in Australia, and contains resources for a variety of global issues such as clean water, cultural diversity, human rights, sustainability, poverty, international aid, food security and the environment.

It is also worth checking out a few other resources that I have previously mentioned. “Math that Matters” from the Canadian Centre for Policy Alternatives focuses on connecting math and social justice so that students can make connections between what they learn in the classroom and the world around them. Radical Math is a website resource for integrating economic and social justice issues into the math classroom.

Have a great week.

Playing with Probability

I had to plan for the last 6 teaching days with my grade 8 math classes, and after that we are into exam review and end-of-year trips. We had not yet covered probability, so I thought that I could design some mini-activities to carry out over these six days.

Here is my plan for the six days:

Day 1:
-Introduce terminology (probability, theoretical probability, experimental probability)
-Each student is given an activity to carry out with either dice or spinners (see attachment below)
-Discussion of theoretical and experimental probability as related to the dice and spinner activities

Spinner and Dice Activities

Days 2/3:
-Introduce game assignment (see attachment below)
-Allow time for students to decide if they are working alone or in small groups
-Planning time for students to organize the activity

Game Assignment

Days 4/5/6:
-Class discussions of how each activity went and how other factors might have come into play. Classmates suggest ideas for improvement.

We just finished the first day of activities in one of the classes, and already students are learning how to modify their activities based on how the first ones went.

We will play some more tomorrow.

Pythagorean Theorem…Take 2.

This week I begin Pythagorean Theorem with my grade 8 students. I intend to use many of the same applets as last year (see Fun with Applets), with a few new additions.

Illuminations Proof without Words – This is similar to Puzzle 1 from the National Library of Virtual Manipulatives. The difference here is that this applet runs for you and asks you to figure out the proof from what you see. In the NLVM applet, you manipulate the pieces yourself. I still prefer the NLVM applet, but this is a nice alternative.

IES Applet – This is similar to their applet that I shared last year. In this applet, one of the squares gets transferred whole, while the other one is broken into pieces. The whole square and the pieces must fit into square “c”.

Learning Math – This site from learner.org has some features that I like. In Part A, students are led through some inquiries and then the theorem is explained. Part B then leads students through a few different proofs. Part C and the Homework section have some interesting questions to solve.

Wolfram Math World – This site has some of the proofs already mentioned on other applets and sites, they are all just put together in the same place.

I plan on showing my students a few of the proofs, and then providing them with the websites so that they can explore. They will need to choose one that makes sense to them, and then find a way to display it with reference to a real-world problem of their choosing. In the past, students have used foam board or bristle board and made pieces that they could move around and fit with Velcro. Other students created their own digital demonstrations of one of the proofs. Some simply created diagrams. Again, I will leave it up to them to choose a method they can work with.

I can’t wait for the fun to begin.
Have a great week.

Playing with Platonic Solids

This week I am starting to explore platonic solids with my grade 8 students. The key question that I want them to answer is “Why are there only five platonic solids?” (For a brief explanation, see the MathsIsFun website.  For a more detailed explanation, read this entry from The University of Utah.)

I want this to be a true exploration activity, and as such, I will give my students limited information. I will not volunteer this information, but I will give it only after they determine the right questions to ask.

First, the students will be given nets of the platonic solids so that they can build them and use them in their exploration. I will be giving them the copy from the learner.org interactives.

They will also get scissors and a handout with the regular polygons. They may cut out the polygons and use them as manipulatives. There are eight copies of each polygon, from three-sided to eight-sided figures.

Regular Polygons Handout

I have also created a Notebook file for the Smartboard. This will be open for the students to come and explore with, as well. It is not fancy. On one side of the page are the platonic solids for the students to see. On the other side of the page are the regular polygons, set up as infinite clones. In the middle of the page is a play area. Students can thus pull out copies of the polygons, turn them around, and see how they fit together. (The polygons were created from the tools in the program, and the platonic solid images were taken from Wikipedia. If you click on each image on the second page of the file, you will be taken to the home site for that image. )

Platonic Solids Notebook File (Unfortunately, this is what the Notebook file looks like as a PDF. WordPress will not allow me to upload the Notebook file. Help anyone?)

Should students get frustrated, I will begin to lead them through the following thought process:

• Consider the regular polygons. Starting with the triangle, what is the measure of each interior angle? Continue for the rest of the polygons.
• What do you notice about the sum of the interior angles of the polygons, as you go from three-sided figures up to eight-sided figures?
• Which of these polygons are able to tessellate? Why are they able to tessellate?
• Which of these would be able to be constructed into a polyhedron? Why wouldn’t all of the regular polygons be able to be constructed into a polyhedron?

They can then go play on the Learner.org website.

The final task will be for them to submit an explanation as to why there are only five platonic solids. I will accept written work or digital work – students can choose which method suits them best.

Have a great week.

The Escalator

I came across a neat resource from the University of Toronto. As an educator who lives in Toronto and a University of Toronto alumnus, I am surprised that I had never heard of it before. The resource is called The Escalator. It is an outreach site from the University of Toronto, with an emphasis on math and science.

Under the Math tab at the top there are two options: Mathematics and Fields. Click on the Mathematics link and you are taken to the University’s Department of Mathematics page. Here you can find links to math competitions, teacher resources, and other tidbits.

There are two links under the Physics tab. The Physics link takes you to information for the Physics Outreach program and the Physics Olympiad Preparation program for high school students, complete with practice problem sets. The Candac link takes you to the Canadian Network for the Detection of Atmospheric Change, which has a variety of links and information, as well as a teacher resource page. The Chemistry tab also takes you to an access page for the Canadian Chemistry Olympiad for high school students, again complete with practice problem sets. The Engineering tab takes you to a list of robotics competitions and a variety of summer programs for students in grade 5 and up.

Click on Universe under the Astronomy tab, and you are directed to University of Toronto’s public portal. Here you can video chat with astronomers and send them questions, or book a planetarium visit or speaker. There is also a link here to the Transit of Venus. On June 5, 2012 Venus will pass across the sun. This has not happened since 2004 and will not happen again until 2117.  (Alternately, you can read about the Transit of Venus here.)

The resources tab has a few areas to explore, including a link to the teachers’ resource page of the Canadian Mathematical Society, which has its own database of resources to search through. The curriculum link is still being developed, so check back to see its full potential. Currently you can find the link to the Science Rendezvous for Educators site. The Science Rendezvous is what first led me to The Escalator website. It is a one-day science festival, hosted on university campuses, research institutions and community sites across Canada on Saturday, May 12, 2012. The database on the educator page is not yet built, but again, I am curious to see what will be included there.

Have a great week.

Pascal’s Triangle and Magic Squares

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

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

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.