Percentages

  Its amazing how many of my students have trouble with the idea of percent being something per 100.  My father years ago told me that students do much better if you put it in terms of money rather than teach it as a decimal. 

So many times I remind them to think of 100 pennies and the percent is talking about how many pennies you have out of the 100.  I realize that its not precise but it helps my ELL students understand it better.

When I taught percentages at a community college, I learned to use the is/of = #/100 to solve those "What is 20% of 18" type problems.  It made it easier for students to solve that type of problem because they knew where to put the various numbers.

Better Lessons has a nice visual lesson on percents. This is a complete lesson plan that relies on a video, worksheets with percent model and a smartboard presentation. The video is more of a reflection but you do get enough information to present the lesson.

The one thing I love saving sales flyers from the Sunday paper for an assignment.  I have students "shop" for things and they have to calculate their savings from the shopping trip.  This makes it more realistic, especially if you are having them grocery shop.  Other times, I might have students plan to remodel a room and calculate the savings they received on the materials.

Another time, I had students decide to be a business.  They had to research the wholesale cost of the item and calculate the mark-up so they knew what they would have to sell it for so they could meet expenses. 

I just found a lovely article from George Mason University on using Art to help teach fractions, decimals and percentages. The idea is students use three colors of construction paper to create art on 10 by 10 grids much like the American artist Ellsworth Kelly from the 1950's.  Once they've created the art, they have to figure out the fraction, decimal and percentage of each color.   For more advanced students have them use an 8 by 8 grid. This makes it so they are not using a base of 100 for their percentage.

So now I have two new ideas for teaching percentages. The biggest issue my students have living in such a remote area is the lack of stores with sales.  The sales flyers I have to use are from the Bush Sales Flyer or the flyer from the supermarket in the hub two about 135 miles away.  The Bush Flyer is great because it includes sales info from Loews and Home Depot.  Its the best I can come up with so we make do with it.

Have fun trying the art activity.

Visual Math

  After reading the paper about visual math, I wanted to know more about it because it explains one reason I should not always get on students when they use their fingers to help them remember things.  It wasn't until I was an adult that someone showed me the trick with your fingers for nines.  I wish I'd known it when I was a kid in school.  It might have saved some frustrations on the 7 x 8 and the 9 x 6 calculations.


I discovered a website dedicated to visual math.  The site has lessons, exercises, and games for mostly Pre-algebra topics.  I checked out the integer chapter which has material spread out over several pages in small chunks. In addition, the pages are set up so a narrator reads each page.  This means someone with a print disability can still learn the material and allows for more independence.  Each page has lovely moving graphics that illustrate the concept presented on the page.  The exercises and games are geared more for elementary students but there are a couple that are worth using because they require the students to use higher order thinking skills. For instance, one game has students choosing the shortest route they can take from start to finish while crossing a square grid with assigned values for each segment.  That game takes a lot of thought.  I don't know if it requires flash because I used the site on my computer.

This site is dedicated to teaching fractions visually.   This site has a ton of material starting with lessons in pdf or power point form showing division, multiplication, addition, subtraction, comparing, renaming, etc of fractions.  The material is all visual and well done.  There are also games that use the help students practice using fractions and come in both a flash and non-flash version.  Each topic comes with worksheets to practice fractions and matching answer sheets. The worksheets use either circles, bars or both so students are able to see everything in more than one way. 

There is a short lesson plan for each topic that starts with an online pretest, the lesson, three online practice sessions, certain worksheets, and a post test so that the teacher does not need to do much planning.  There are additional worksheets available should a student need it to get ready for the post test.

This is a unit that could be used from elementary all the way up to high school age.  I wish I'd known about this at the beginning of the year because my Pre-Algebra class because all of them were weak in fractions and this would have allowed them to work at their own pace through the material. I'm going to keep this in mind for next year.  I'm also going to recommend this to other teachers in middle school and upper elementary as  scaffolding for their students.

Free This Week

Jo Boaler of You Cubed over at Standford University is offering the materials for the new visual math network without a login for the week.

The material consists of a 17 page paper in which she discusses how important visual representations are to people learning math.  This means all levels from very young to high school and beyond.

The paper discusses what the brain science research has to say about this particular topic.  One interesting thing I found in this part of the brain has to do with how the brain uses fingers, even past the normal expected use in school.  They talk about embodied cognition or how we use our body such as gesturing, pointing, etc to help convey mathematical ideas. If we don't have the words, we often will draw something in the air to convey it.  Apparently, it is much better for individual students to create their own gestures, rather than using ones supplied by the teacher because it helps them remember better.

In fact the paper gives two or three examples of visual math that were used in the classroom.  The paper concluded three important things.

1. Replace the idea with  strong mathematical learners memorize and calculate well with the idea that visual representations actually help students learn better.

2.  Successful mathematicians use finger representations in their minds. 

3. Mathematical instruction needs to include more visual representations.

Please down load the paper itself and read it so you can get the full picture. It was so interesting to see I'm doing some things right.  Today, I had my Algebra II class draw representations for completing the square.   A couple of my students after the example I put on the board had light bulbs go off and were off and running.  A few more could see it better and when we start doing the problems on Thursday, they feel they won't have that much trouble doing it.

The second item to download is a 26 page paper that has details on the visual math activities used in research for the paper.  This pdf has everything needed to complete the various activities.  Some activities are designed for young children to develop their finger sense and discrimination while others work for Algebra and other classes. 

The final item is a check list for the teacher to use to see if they use enough visual math now.  Its like a self reflection on your current teaching.

I plan to print off the paper, highlight and read it to see how I can improve my teaching.  I love trying to find visual representations for the topics I teach.  I admit, some topics such as dividing fractions can be very difficult to figure out but I've done it.  So go check it out, read it and enjoy it.

Arcs and Sector Areas

Currently in Geometry, the students are working on area, circumference, arcs, area of sectors, etc.  They are struggling to relate the topics and try to look at each topic as a separate topic.  I want to find ways that might help them learn to flow from one topic to the next.

I found this 39 page Pdf which contains a whole unit on arcs and area of sectors. It has everything from the pre-assessment to the lesson plan to the activities and the post assessment.  This is great in that it introduces radians so it would be perfect between circles and the trigonometric ratios.  I wish I'd found this about two weeks ago when I was starting this.  That being said, I could easily use parts of this to help my students learn the material better through the use of activities rather than relying on worksheets.  It is labeled as a concept development packet.

This site has a great pdf of cards with real life problems for finding circumference, area, radius, and other interesting things.  For instance one card has you calculate a sheet of gold to cut coins from but the sheet has to be of the smallest size so there is less waste.  I like the variety of problems since the problems are not all circumference, or area, or diameter.

This slide share shows where circles are found in real life from theater to car tires, to the earth, to manhole covers.  It has some good information and would be a great introduction to the card activity.

The Department of Education in Virginia has a great 5 page lesson plan and activities for arc length.  There are two activities included.  One finds the length between two ends which are actually arcs and requires use of the Pythagorean Theorem while the other is an activity that requires students to cut a 10 inch cake into 12 pieces.  The questions require thought so it will take them a while to complete.

This activity has students working to figure out the area of a large cookie that is at least 8 inches in diameter.  They end up calculating area and circumference of the cookie, the area of their slice and the length of the edge (arc) of the cookie.  When done they can eat it.

Finally, this power point presentation has some great practice problems for students that are done in such as way as to have a friendlier feel than most worksheets.  There are instructional slides scattered through out the problems.  I like the way its organized.

Have fun checking it out.

Mathmatical Modeling part 2

Yesterday, I wrote about a great pdf that would make a great introduction to the topic. Today, I"m looking at several places that offer lesson plans on mathematical modeling.

One of them is the University of Indiana in Bloomington.  The page offers 40 modeling lesson plans tested by teachers and staff for students in grades 7 to 12.  The lessons actually include a senario and ask that the students create a proposal that includes the mathematical model to show why their suggestion is the best one.  Others are either experiments that require them to come up with rules for certain things while others are performance tasks based on real life situations such as needing to protect art in a museum.

The teachers college at Columbia University has a wonderful 25 page sample of the  Mathematical Modeling Handbook with the introduction and the first project with everything needed to conduct the lesson.  The introduction gives some everyday situations and discusses how it modeling should be taught in schools. 

Annenberg learner has a great lesson on mathematical modeling using circular movement, and transmission ratios followed by one on mosquito population and exponential growth.   The lessons come with everything you need and utilize hands on activities to help students see connections.

Back to the Plus Maths site for a complete package on mathematical modeling. This package shows some of the ways mathematical modeling is used in medicine and nature, economics, politics, and human interaction, games, sports, and art.  They offer three levels - explicit has students using the math, middle ground that introduces mathematical modeling and gives a glimpse of equations, the bigger picture that looks beyond what can be done in the classroom, and the try it yourself are problems from NRICH to do in the classroom.

The University of Arizona has two semesters worth of posters showing student work from a class on mathematical modeling.  The posters are one page summaries of the work they did in the class and contains everything from the problem to results, references and potential applications of the results.  Each poster includes a listing of scientific challenges for the problem.  Some of these are beyond my ELL student's understanding but a few, such as the one which explores why the same store might charge different prices for an item is something they can relate to.

Finally from the University of Texas at Austin is a nice page of mathematical modeling projects with units on science and engineering, art, music, and entrepreneurship. Each unit comes with everything needed from teacher instructions to the worksheets to samples showing how the results can be put together.  Each topic has several subtopics such as under fine arts - geometry, there are two different lesson plans, one on symmetry of design while the other looks at kaleidoscopes.

I think I'm going to see where I can work this in my math classes so students get a chance to experience mathematical modeling.  I know I can use one on population growth anytime I teach exponential functions.  So I have a start.  Have a good day.

Mathematics Modeling

  Yesterday while searching for information on the math used by the CDC, I came across a great 72 page pdf on math modeling .  It's eight sections cover it all from the general introduction to putting everything together.  Every chapter has a summery and an activity at the end.  The activities are set up so by the end the student has completed their own mathematical model.

Each chapter focuses on a specific aspect of mathematical modeling.  Chapter two is solely on defining the problem statement including the types of problem statements and possible interpretations of those questions.   The authors use quite a few mind maps to illustrate the process used to help define the problem.

Chapter three looks at the process of making assumptions for the topic.  This chapter puts the material from the previous chapter into focus and helps create a clearer view of the topic.  It gives some great information and shows step by step how the assumptions fit together.

Chapter four focuses on what aspects are the variables in the problem, both independent and dependent.  The authors provide three different examples so students can see how independent and dependent variables are designated.  The authors explain that the problem determines which is the independent and dependent variables.  By this point, students have created their initial mathematical model.

Chapter five concentrates on approaches to using model to create initial data. The authors give a set of questions to help the student proceed with this step. Again, there are three examples and each example is shown with two or three different approaches so they can see how this works. The examples include any math and graphs to show results using these specific approaches.

Chapter six examines analyzing the data in two ways.  The first is does the answer make sense and the second looks at ways to look at a more detailed analysis of the data.  Again using examples the authors show how to analysis the data in both ways and provide a list of things to check for.  They even state that students might want to refine the model and run it again if there is time.

Chapter seven ties everything thing together to complete the process including how to write the results up.  It includes do's and don'ts for writing the results, and the recommended structure.  At the very end, there is a final write-up of a model done on recycling so students can see what the end results should be.

This is a well done guide put out by the Society of Industrial and Applied Math.  Check it out, download it, use it.  Tomorrow, I'll talk about sites with suggested lesson plans on mathematical modeling.

Center for Disease Control

Yesterday a lunch, we ended up discussing how the influenza of 1918 caused native populations in the area to decline rapidly. One of the guys brought up how a Japanese balloon during WW II landed and brought disease with it that killed quite a few locals. 

Every time there is a break out of something, the CDC or Center for Disease Control gets involved.  It doesn't even have to be a disease. Years ago when they put that additive M something in the fuel, Alaska had health issues because a byproduct was formaldehyde.  Due to the cold winters, the formaldehyde stayed at ground level and people developed problems.  The CDC came up an investigated.  So this asks the question - "What math does the CDC use?".

The Washington Post ran a great article that discusses the spread of Ebola.  The article includes some great infographics and discusses the math involved. It would be easy to create a worksheet or guide to accompany the article so students can write down information as they read.

This article discusses what can happen when a wrong value is used in calculating risk. A business sold flooring made in China that may not meet U.S. air quality standards.  In addition, this article explains more about the error.  Its quite good in that it shows how a small error can cause underestimation of the danger. This article showed the underestimation and how it changed when the error was corrected.

This pdf discusses the ways statistics is used within infectious disease epidemiology.  It is a 17 page document that discusses frequency distribution, ratios, proportions, and rates, morbidity frequency rates, point vs period prevalence and other topics.  There are lots of examples because this is used to train health officials.

From We Use Math comes an interview with an epidemiologist who explains the types of math they do and how she uses it in her job.  It is a short article that could be used as an introduction.

I found a great pdf for math modeling I am going to share tomorrow because its perfect for a entry on that topic.