Math Matters

December is my favorite time of year. Most people are hopeful about the new year, still focusing what they are most thankful for in life, and merry with the spirit of the holidays approaching. As I reflect this time of year my motto would be "All things in proper proportion." This applies not only to mathematical ideas, but just as a way of living. Like many I struggle with balance and keeping the perspective in sync between the way things are now and the way I would like them to be. While living in the moment is important, so is renewal and small improvements that make the moments even better. In light of this idea, this month's post is about this motto, All Things in Proper Proportion, in the mathematical sense.

First off - what is a Proportion? The common definition is that a proportion is a mathematical statement about two equal ratios. Therefore, the key to understanding a proportion is to have a firm understanding of equivalent ratios or equivalent fractions. The introduction of equivalency with fractions typically begins in grade 4 and by grade 6 students are expected to use this idea in problem solving with ratios such as in scaling, similar figures, or other simple ratio situations. Modeling equivalent ratios can often be done using area or grid models. For example, using area models of the same size whole and various numbers of rows and columns, illustrate a simple fraction such as 1/2 in as many ways as possible. Examples from students might be a square or rectangular model showing 1/2 of the number of unit squares shaded. This could be 1 out of 2 parts, 2 out of 4 parts, 3 out of 6 parts, or even 50 out of 100 parts. This simple idea is often the visual students need to really comprehend that a ratio is the number of shaded parts out of the number of total parts in the whole, and that a proportion is a situation where to ratios, or in this case two models, show the same ratio in simplest form of number of parts shaded to the number of parts in the whole.

Have you ever really thought about how many math topics tie back to the concept of proportions? Let's see - equivalent fractions, percents, probability, geometric transformations (dilations), scale factor, similarity in geometry, direct variation, slope as a rate of change, linear functions of the form y = ax, just to name a few. I'll bet something you are teaching right now can be connected to proportions in some way. Given that the understanding of setting up and/or solving proportions, or just using proportional thinking, is so pervasive and thus so valuable as a math tool, be sure to take the time up front to ensure that students have a solid foundation to build on. Sometimes the easy way found in tricks and shortcuts isn't always the best way when students need to use the concept behind the shortcut at a later point in mathematics.

Using this modeling idea quickly can lead students to understanding decimal fractions, which supports decimal-fraction conversion; then comes percents as a proportion using 100 as the whole, and then the possible applications are endless for success as a math student!

The word portion is a derived term from the word proportion (its a portion of proportion lol). As a noun, the word portion references an allocated amount while a proportion is a quantity that can be counted. A portion is a part of something and a proportion is the relation between the dimensions of something. Perhaps the simplest way is to think about a portion as a part of a whole and the proportion as a fractional ratio of the part (portion) to the whole set.

For a quick activity, use the concept of a plate of food. The portion is the amount of one item or category on the plate, such as the portion of vegetables. The proportional example would be to examine the relationship of the amount of vegetables compared to the total amount of food on the plate.

Formative assessment comes in many forms but basically involves small tasks that help discover students' current understanding and diagnose any misconceptions during the course of learning. These tools allow us as teachers to make course adjustments so that by the end of the learning process, all students GOT IT!

One way to see how students are progressing with proportions is to provide them with problems and examples of how another student set up the problem. Ask students to examine how the problem was set up and to determine if the proportion is set up correctly in order to solve the problem. If the problem is not set up correctly, then have students "fix" it. This task allows the teacher to analyze the student's thinking and if they understand that the unit labels are important for setting up the problem.

Whenever I've worked with students on proportion, whether its using grid paper to illustrate fraction models, using the squares on graph paper to help students keep numbers in problems organized and in line, or to use graphs to draw dilations, proportional linear equation examples, or even to model scale drawings, I always needed different sizes of graph paper. Rather than buying paper and photocopying, or even buying a special book with all the graph paper sizes to photocopy, I love using Dynamic Paper from NCTM Illuminations. And the best thing about this tool is that it does so much more than just graph paper! Check out this easy to use online resource from NCTM.

Math Cut Ups are designed as a hands on way to engage students to do more math. No one wants to work problems on a boring worksheet. Math Cut Ups take the problems off the worksheet and make them interactive and more fun for students.

This month I'm featuring a a title keeping to our theme of proportions, Proportion Warm Up + Line Up. The set actually includes 2 different activities made for use with your whole class. First all students each get one of the 4 different cards. Students with the same problem card group together to talk about the problem and work out how to explain it to others. Then this group separates and students form a group of 4 with students who all have the different 4 cards. They take turns explaining to each other the problem on their card and its solution method. This task helps "warm up" students by reviewing how to solve a proportion problem.

The second part of the activity set is the Line Up activity. For this activity each student gets one problem cards. Each of the 30 cards presents an application problem for which a proportion should be set up and solved to determine the answer. Students work independently on their problem and then as a whole class, order their solutions numerically from least to greatest. When they do, small letters on the corner of each card will provide a self0checking phrase for the class to know they lined up correctly.

Each package of Proportion Warm Up + Line Up contains the 2 activities and enough problem cards for 30 students. All materials are pre-printed in color on cardstock and for just $10 comes ready to laminate, cut, and use!

Visit my square market page for purchasing or visit my full website for more titles and ideas. I also have a Facebook page, Twitter account, and a Pinterest page if you are interested in following me there.

Modeling is a critical part of the teaching process. Using visual models helps students get a clearer picture of the concept behind the topic being covered and makes learning deeper, easier to connect to other knowledge, and more interesting for students. Throughout the instructional modules of Edusmart, visual modeling also plays a key role and is prominent as part of the teaching process. While using the instructional module video, teachers can show how a concept is modeled, then pause and work additional examples as needed using our built in white board. You can also incorporate the available virtual manipuatives into instruction to provide even more clarity and model with students how the process might work in a hands-on way.

Click here for a sample of how Edusmart models the concept of proportions with scale drawings.

If you are interested in learning more check out EduSmart Math today. And if you have questions or thoughts, please email me anytime!

K-12 Mathematics Specialist

Mathematics Enthusiast

Math Cut Ups creator

Edusmart Mathematics Director

Email: integralmathematics@gmail.com
Website: www.mathcutups.com
Phone: 214-471-5760
Facebook: facebook.com/Math-Cut-Ups-155205867883398/timeline/?ref=hl
Twitter: @mathcutups