## Place Value Representation

**Tagged**base ten blocks, concept, concrete, concrete representational abstract, CRA, hands-on, mathy

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Here’s a common word problem used for linear functions and equations (y=mx+b):

There are 6 inches of snow on the ground. Snow is falling at a rate of 2 inches per hour. Write a linear equation showing total snow as a function of time (in hours). The equation would be y=2x + 6.

Often the word problems like this are presented on a sheet of paper in isolation as an attempt to make the math relevant and to develop conceptual understanding. For students who have trouble with conceptual understanding, words on paper are likely too abstract or symbolic to allow applications like the one above to be meaningful.

The real life application is useful if presented more effectively. Here’s an approach to use the same scenario but in a more relevant and meaningful presentation. The photo above shows the current amount of snow – call it 6 inches. Students can be shown the photo to allow for a discussion about accumulation and for their estimates of the amount of snow shown. The photo below shows an excerpt from a storm warning. Showing this warning and a snow fall video can allow for a discussion about rate of snow fall and the purpose for storm warnings. Combined, this approach can lead into the above word problem.

Once the application is presented students can be asked to compute snow levels after 1 hour, 2 hours etc. Then they can be asked to determine how long it would take for the accumulation to reach 18 inches (the prediction for the day this post was composed). After computing the answers WITHOUT the equation the students can be shown how to use the equation – the “mathy way.”

Slope is one of the the most important topics in algebra and is often understood by students at a superficial level. I suggest introducing slope first by drawing upon prior knowledge and making the concept relevant (see photo above). This includes presenting the topic using multiple representations: the original real life situation, rates (see photo above) and tables, visuals, and hands on cutouts (see photos below).

A key aspect of slope is that it represents a relationship between 2 variables. Color coding (red for hours, green for pay) can be used to highlight the 2 variables and how they interact – see photo above and below.

The photo below can be used either in initial instruction, especially for co-taught classes, or as an intervention for students who needs a more concrete representation of a rate (CRA). The clocks (representing hours) and bills can be left in the table for or cut out.

Factors are “things being multiplied”, e.g. 2 and 3 are factors of 6.

Factoring polynomials like 5×2 + 15 above is one of the most challenging algebra topics, especially for students with special needs. To make these problems more accessible a more concrete approach is possible. The photo above shows how finding a greatest common factor (GCF) is possible using a hands on, visual approach. Click on this link to a folder with a document explaining this approach and with a document that is a master for these card cutouts.

Educators teaching math typically start with the “mathy” stuff first. For example, for finding the sales price teachers may start with showing students the steps to calculate (photo below).

I start with the concept, either with a pictorial representation or actual objects to represent the underlying concept. In the photo above, I show an object (related to the student’s interest – this student is into weight training) on sale. The $50 circled in yellow represent the original price. I explain the concept of being on sale and discount and show that 20% is $10 to take away (marked out). This leaves $40 (in green) which is the sales price. This allows for conceptual understanding before showing him the “mathy” way of doing the problem.

Math is an esoteric subject for most people. Good instruction makes information meaningful. One method for making information meaningful is to connect new information to prior experience.

In this situation the new information involves determining whether shapes are similar (see photo below). One example of student prior experience with this topic would be shrinking people down. In the photo above I use Mini Me and Dr. Evil and their respective (and fabricated) weights and shoe sizes as measures that will eventually give way to measures of sides of a polygon (below). When working on the problem below the students can be prompted by recalling the analogy of Mini Me and Dr. Evil.

This is a photo of a handout from a special ed conference shared by Dr. Shaunita Strozier (sstrozier@valdosta.edu) of Valdosta State University. It shows the use of algebra tiles to provide a concrete level of the concept of an equation and solving the equation. The photos on the left show the R or representational level of the concept. Her approach is called SUMLOWS which is an acronym explained in this handout.

Students have trouble with irregular shapes largely because they cannot visualize or determine the measures the dimensions of the individual parts. The photos below show a hands on activity to help students with these challenges. The students are cut out the individual shapes and write in the dimensions. This method allows them to see the individual parts and the respective dimensions. (The second has calculation errors.) This activity is followed by a handout in which students can shade in the different parts which is a step towards more abstraction – CRA.

This is a photo of matching cards for the topic of percent discount and percent tax.

Students are given a card with an item to buy, a percent discount and our state tax rate of 6% (not listed). We’ve identified 4 steps for to find the total amount to pay the cashier: compute discount, compute new price, compute tax, compute the total cost. There are two cards for each step, one that shows the step completed correctly and one with an incorrect step. Students are to identify the correct steps and arrange them in order.

This activity is useful for a variety of reasons.

- By having the steps listed there are no calculations. The students can focus on the steps and the concept.
- To figure out the correct order students engage in higher level thinking. They analyze and justify their choices.
- Because the problems are worked out and students simply focus on choosing cards they can do much more on their own. This is engaging.
- While they are engaged the teacher can circulate and ask leading questions.

It can be very surprising how poorly understood discount and tax can be for students yet most have bought something and paid tax, even items on sale. As such this is an important topic.

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