Distributive Property and Percents

It was a great class. We were examining how to calculate the taxes and discounts on goods. Pretty simple stuff that all of the students were familiar with. After all who does not go the store to buy things and find out they are a little short on cash due to some taxes they did not figure into the final price. We started with a problem that went something like this … “Jim goes to the store to buy an iPod. If the iPod is $150.00 and tax is 12% what is the final cost?”

Most students recognized the need to calculate the tax and add it to the cost to determine total cost.

Total Cost = Original Price + Tax

However there were some that recognized that you could simply multiply by 112% (1.12) to find the total cost.

Total Cost – Original Price (1.12)

I asked the students why this worked and everyone was slow to answer. Then suddenly from the back of the room one of the students said “Distributive Property.”

I asked if he knew why it worked and he was pretty honest, “No Mr. DeMerchant, it was just the thing I remembered.”

This has me thinking of the application of the distributive property [e.g. a(x+y) = ax+ay ] in elementary and middle school mathematics. Early multiplication models and the development of number facts start the journey of the distributive property through the math curriculum. Repeated addition is in itself an application of the distributive property (e.g. 7×4 = 7(3+1)=7(2+1+1)=7(1+1+1+1)=28) although it is not often talked about in these terms. The applications of this extend into geometry, algebra and other mathematical topics.

In Grade 4 of the Western and Northern Canadian Protocol students use arrays for multiplication of numbers:

5 x 12 = 5×10 + 5×2 = 50 + 10 = 60 or

4 x 341 = 4×300 + 4×40 + 4×1 = 1200 + 160 + 4 = 1364

Mental mathematics comes much easier when the distributive property is used. I was witness to this when the class was working out the iPod problem.

Calculating $150 x 1.12 can be difficult to do in your head.

However, if you change the problem using the distributive property it becomes much easier

$150(1+0.1+0.01+0.01)=$150+$15+$1.50+$1.50=$168.00

As I tell my students the rules of mathematics do not change and in a later post I hope to expand on other applications of the distributive property which occur in the curriculum. Analyzing mathematical problems and applying the distributive property can tie parts of mathematics curriculum together.

Teaching Tech

Over the last few months I have been trying to incorporate technology into the classroom experience for students. This has occurred at many levels including projects which require the use of:

• PowerPoint
• Excel
• Scratch
• Dynamic Geometry Software
• Google SketchUp
• Gimp

Overall the use of these tools has been well received by students. Unlike some (dare I say many) people of my generation students are not worried about “breaking” these tools. They use them with ease and are not afraid to try new things.

I do find that there is one aspect of their learning that is missing. An understanding of some of the basics of using technology. These include (and are not limited to):

• Proper naming of files (untitled 1, untitled 2, etc. is common)
• Filing systems (“Where did I save my file?” is a common question)
• Keeping different version (“Sorry Mr. D. I sent you the old copy”)
• Application versus cloud computing
• Transferring files (“I didn’t know that you couldn’t email a 20 MB file.”)
• Reducing file sizes
• Working cross-platform (OS X, Win 7, IOS, Android)
• Internet safety
• Creating a positive digital footprint

I do find that integration of these skills in the classroom is possible  and I try to do this in the mathematics classes I teach.  At the same time I think there are better ways. I see real benefits to developing a specific course that focused on teaching of those skills which are cross curricular, cross-platform and universal in terms of application. This would allow for more time to focus on the specific tasks within a subject that assist in the learning of the subject’s content.

What are your thoughts? Which “skills” are universal to all platforms and courses?

Best Quiz Given

I had a great experience giving a quiz today. I like to do little quizzes to check for understanding as part of formative assessment. I have tried various approaches such as:

• working with elbow partners
• one or two questions written on the board with a response written on scrap paper
• oral questions
• creating a physical object (geometry) or representation
• drawing a diagram or
• creating a mind map

Today I tried something different. We were completing our unit on adding fractions and I wanted to make sure that students could use models to represent the addition of fractions and add them algebraically. I created a quiz which I thought I would take the better part of the period.

To start with I placed a line on the board for each student to write in their names. I let them choose the place to write their name. I then rolled dice to determine a number to assign each student making sure that two students had a 1, two students had a 2, etc. Since there was an off number I had three students with the same number. This formed my groups and was the point I told the students what they were doing.

I provide the students with one quiz between each group. Students were told the following “rule”:

• only one quiz per group was to be submitted
• the quiz was open book
• they could use calculators
• each student would be assigned the same mark per group if they were all on task
• students who were off task would have a mark deducted from their score and awarded to the other student
• the quiz must be handed in by the end of class
• all work must be ordered logically with work shown

Overall this went very well. While the students were completing the quiz I was able to observe them work and make anecdotal notes on each student. I was also able to observe the following:

• students learned from each other
• all questions were answered
• students did not need to look at their notes
• students did not use a calculator
• students who have graphomotor problems were able to have the other student scribe
• all quizzes were neat and easy to follow student thinking
• student who were not good at writing out their solutions and showing their work learned from their partners

For those who care about number the average mark on the quiz was around 90%. I guess this would not be a surprise however, all of the questions given on the quiz were much more difficult than what I would have given on a normal quiz.

Perhaps you have tried a similar strategy if so I would love to hear what has worked for you.

Math Challengers – Reflection

After making a couple of posts on Math Challengers I thought it would be a good idea to do a bit of reflection.

First of all it was great to see so many mathematics students come in to the lecture theater at the University of British Columbia to compete in Math Challengers. There was some genuine excitement in the room as participants filed in. The nervous energy was coupled with the excitement of a competition and the fear of the unknown, especially for those of us who had not been there. I felt really blessed to have a chance to go with someone who had taken a team there several times. Mr Williams was an awesome mentor.

While the regional meetings do prepare students for what is to come there is nothing like the provincials. Sitting in an actual lecture theater was something new for most students. The idea that over 200 students may be in a class was a bit of a shock and a far cry from the reality of even the largest K-12 classroom. As one student noted, “I can hardly see the board.”  This aspect of the competition was enough to make the trip worthwhile for students.

The competition itself is divided into four rounds. The first two rounds, Blitz and  Bullseye,  give students a chance to answer questions individually. The third round is one of my favorites when students have a chance to work as a team to solve mathematics problems. In many ways I think this round is closer to the nature of how “real” mathematicians work as people can bounce ideas off each other, propose solutions and come up with the best possible response. The final round, Face Off, is by far the most exciting as two students compete in a Jeopardy style contest with the fastest response winning the round.

The questions of the contest range from easy to “how did you do that” difficult. I was amazed how quickly students were able to solve questions in the buzzer round. In many cases I had not read the question before the response was given. It was truly inspiring.

I was asked by one person at the event what makes the students who go to these events “different?” How do they do so well? Reflecting on this I think there are two main factors.

1. The students have a vast repertoire of general mathematics knowledge. The know their “basic facts”, square roots, prime numbers and basic geometry. While some people may know this well these students know it like a language and can call upon it at will.
2. They can think beyond the lower level of Blooms Taxonomy. The students are able to take this general mathematics knowledge and synthesize it into new information. They can apply it to solve unique problems. In some cases the problems are similar to ones they have seen before but in many cases the problems may be unique and this calls for far more than just basic recall.

One may ask what I learned from the competition. While I would not say that I have improved mathematically I would say that my appreciation for mathematics has improved. I also think that these types of competitions are just as important for youth as soccer, basketball or hockey competitions. The best way to get better is to compete with the best people you can find and there were lots of great young mathematicians at this event.

If anyone else has attended a math competition I would love to hear your thoughts/advice. We will be back next year.

PS. A special thank you to all of the organizations who help to make these events possible.

Math Challengers

As I write this post I am in a hotel room in Vancouver with two other teachers and seven bright young students solving math problems for the Math Challengers competition tomorrow. It is great to see such a surge of excitement about mathematics in the room. Students solving problems, laughing, and having a good time learning from each other. It is truly a great thing to see as a teacher. It is the “stuff” that makes teaching a great job.

I am looking forward to the contest tomorrow when students from across British Columbia will come together to compete. I hope other math teachers have a chance to go to similar competitions as it is truly worth it.

Fraction Apps

I will be the first to admit that I am a bit of a techno-junkie. I can get lured in by the hype and new technology that is cool. However, this same technology is good for engaging people in learning and help to solidify concepts. One area in particular that I think opportunity knocks is in examining mathematics concepts. What area could benefit more from this than developing an understanding of fractions.

In my previous job I had lots of opportunity to discuss mathematics concepts with university professors, parents, teachers and students. In many of these conversation fractions was brought up as an issue for students. It also seems to me that fractions have been an issue for as long as I can remember. I know many people I attended school with found fractions difficult. I even met one person, around the year 2004) who cited a study which showed that student performance on fraction operations (addition, subtraction, multiplication, division and simplification) had not improved or gotten worse since 1971. I have since lost this study but if anyone who reads this knows of it I would love to find it again.

I think many of the issues about fractions are actually related to an understanding of proportional reasoning. The relationship between part of something (or a set of things) and the whole can be hard for students to understand.Developing benchmarks is essential when estimating and checking the reasonableness of calculations with fractions.

I also think that there are some apps which help develop this understanding. A couple of my favourites are listed below:

Motion Math- One of my favourites this app is great for developing benchmarks. It can also be addictive and make students forget they are playing a math game.  “Developed at the Stanford School of Education, Motion Math HD follows a star that has fallen from space, and must bound back up, up, up to its home in the stars. Moving fractions to their correct place on the number line is the only way to return. By playing Motion Math, learners improve their ability to perceive and estimate fractions in multiple forms.”

Number Line – This app is great for developing benchmarks for fractions, decimals and percents. The app relates these quantities using a number line. “Number Line is an educational game app to help students learn about fractions, decimals, and percents by ordering equivalent fractions, decimals, and percents on a number line. The app features multiple levels where the player must drag circles with either a percent, decimal or fraction onto a number line in the correct sequence. A score is earned based on the time it takes to put all the circles in the correct order (faster is better), plus points for each correct placement, minus points for each error. “
Fraction Factory – “Enter the Fraction Factory and use your math skills to place fractions into their correct positions on a number line. When the game begins, fraction gears will move across the screen on a conveyor belt. Use your finger and drag the fraction gear to where you think they should go on the number line at the top of the screen. “

I am sure that as I continue my search I will find more. In the meantime I hope you enjoy the ones listed below and … I am always open to recommendations.

Teachers Sharing – BCAMT New Teachers Conference

This afternoon I presented a session titled, “Middle School Mathematics That Worked” at the British Columbia Association of Mathematics Teachers (BCAMT) New Teachers Conference. I felt this was a fitting conference as it was the first conference I have spoken at since I left my previous position with the Ministry of Education. As a “new teacher” this seems really fitting. After all 12 years working outside of the classroom is a long time.

The new teachers conference has always been a favourite of mine even when I worked in the government. It is a great place to speak with student teaches and newer teachers who are keen to learn and have lots to share.  At my session I shared several examples of projects, teaching strategies and approaches to mathematics instruction and learning which have worked for me.  The majority of these projects came from working with other teachers in our school as we have developed a team approach to creating the best possible program for middle school mathematics. Some of the ideas include:

• Rubrics for students teaching each other
• Using Google SketchUp for teaching about 3-D objects
• Creating a Vacation Travel Holiday to teach about decimals, fractions and percents
• Using Scratch to develop logical thinking skills and incorporation of mathematics
• Scavenger hunt to demonstrate applications of the Cartesian Plane
• Developing a game to solidify concepts involving integers

While it was great to chat about the things that I have been doing in my class the good thing for me was the questions the student teachers asked. Their questions made me look at the activities in a different light than I had before. Typical questions included:

• How much time do you allocate to projects?
• How do you plan a lesson?
• How do you manage with a large class?
• How many in each group?
• What if you don’t have the technology?
• Are the students engaged?
• Can all of the students complete the project?

As a result of this experience today I would like to encourage all teachers out there to take a chance and share what you are doing. Speak at a conference. Send a tweet. Create a blog. Post some of your ideas on-line to get feedback from others. It can only make things better if we all share. These actions not only show teachers taking leadership but also show a willingness to learn.  It creates a  community and makes those joining the profession feel welcome. This is what we expect of our students and it is great to model it.

One place to do this is the new resources site on the BCAMT website. You can find information about the site here. I will be posting the projects that I have developed on this site soon and would really appreciate any feedback. If you are like me you have more questions than answers.