Interpreting Slope Intercept

Slope is one of the most important topics covered in high school algebra yet it is one of the least understood concepts. I have two observations about this. First, slope is often introduced with the formula and not as a rate of change. Second, students intuitively understand slope as rate of change conceptually when presented in a relevant, real life context. The challenge is compounded when slope is presented with the y-intercept.

In the photo I present slope and y-intercept in a context students can understand (money is their most intuitive prior knowledge). The highlighting makes it easier for them to see the context, specifically the variables. I have the students work on this handout and I circulate and ask questions.

Here’s a typical exchange – working through problems 11, 12:

• Me: “Look at the table, what’s changing?”
• Student: “the cost”
• Me: “How much is it changing?”
• Student: “20″
• Me: “20 what?”
• Student: “20 cost”
• Me: “What are you counting when you talk about cost?”
• Student: “money…dollars”
• Me: “So the price is going up 20 what?”
• Student: “Dollars”
• Me: Show me this on the yellow” (student knows from before  to write +$20)
• Me: “What else is changing?”
• Student: “People”
• Me: “By how much”
• Student: “1 people…person”
• Me: “write that on the green”
• Me: “Now do this same thing on the graph. Where do you start?” (they put their pencil on the y-intercept
• Me: “What do you do next?” (they typically know to move over and up)
• Me: “Use green to highlight the over” (they highlight)
• Me: “How much did you go over?”
• Student: “1…1 person”
• Me: “Now what?” (Student goes up.)
• Me: “Highlight that in yellow.” (They highlight.)
• Me: “How much did it go up?”
• Student: “2…20…20 dollars”
• Me: “What is a rate?” (I make them look at their notes until they say something about divide or fraction or point to a rate)
• Me: “So what is the rate of change?”
• Student: “$20 and 1 person”
• Me: “Look at the problem at the top. What is the 20?”
• Student: “$20 per person.”

I point out that you can find this rate or slope in the equation, the table and in the graph.

Intro to Slope as Rate of Change

Slope may be the most challenging concept to teach in algebra yet it is one of the most important concepts. I use the following sequence to introduce slope: rate of change, rise over run, rise over run as a rate of change. The first photo is a map of Manhattan with directions on counting city blocks. This builds on prior knowledge to introduce rise and run.

The photo below builds on the map and transitions students into coordinate planes. They are introduced to rise over run and positive and negative as indicators of the direction of a line.

The photo below combines rise and run with rate of change. The hourly wage is prior knowledge they can much more easily comprehend. A major issue is getting students to include units and to understand what units are. This would have been addressed in the previous unit on rates and proportions.

This is the handout used with a train activity in which I use battery operated trains and time them as they travel 200″. I project a stop watch on the screen as the train moves. The kids pick up right away that Percy is “slow.” As Percy is traveling I ask them how they know it is slow and get answers like “it takes a long time.” This is a concrete representation which they can draw upon as they work with the graph and calculations.

The photo below shows a scaffolded version of a Smarter Balance (Common Core assessment) test question. The original question simply shows the graph and asks for average rate of change from 0 to 20 years. Even with the scaffolding many problem areas appear: units vs variable (student wrote “value” as opposed to $), including $ with the 1000, finding unit rate, and even identifying the part of the graph at 0 years.

Intro to Measurement

The photos below are used to introduce length and area as part of a CRA approach. First a student is asked to build a Lego garage. He first builds the bottom row of a wall and the teacher asks for length in terms of how many Legos are lined up. After building a wall the teacher asks for area in terms of how many Legos are used in the wall. Then the student is given a handout with the following photos. Following this handout the student finds length and area of tiled floor and walls made up of cinder blocks, if available. Eventually a ruler is introduced and multiplying to find area is presented at the end.

For the photo above the student is asked to count Legos to compute length. In the photo below the student is asked which is longer and to explain.

In the photo above the student is first asked to determine the area of the red wall in terms of number of Lego squares. Then the student is asked which wall has more area. This is followed by the photo below. This allows a different perspective of area. 

CRA for Solving Equations

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.

Standards Based IEPs

At the turn of the century education has seen a standards based reform movement, e.g NCLB. IDEA 2004 reflects this with a change in the focus of an IEP towards curriculum standards. What does this mean? The focus of academic based IEP goals and objectives should be strictly aligned with the curriculum standards. This helps to make the general curriculum accessible for all students – to the extent possible.

For example, in the past a math IEP objective may be written based on weaknesses found in psychological tests regardless if this was in the curriculum covered in the life of the new IEP. Maybe a student had trouble with calculations with fractions on the testing so an objective would focus on improving these types of calculations even if the class was not going to cover calculations with fractions.

These new types of IEPs are called Standards Based IEPs. Here are a couple of links.

• “A Seven-Step Process to Creating Standards-based IEPs”
  
• Standards-based Individualized Education Program Examples

Linear Functions with Comic Book Context

This is a follow-up to a previous post about using comics to engage a student with autism. In the photo above the comics are on sale for $2 each which is a situation modelled by a linear function. The student finished this assignment independently, quickly and accurately (aside from the bars on the graph). (Note that he did not need the table at the top that shows the different number of comics.) In another previous post I explained the different levels of representation (CRA) with the equation being the most abstract. In this handout the equation was the last item addressed and was computed using the graphing calculator (Linreg function).

The following photo is from a warm up given prior to the assignment above. It addressed prerequisite skills for the intro to linear functions assignment. This student confused the x and y axes (you can see some of the points from his initial effort) and I used color coding to clear this up.The following photo is the student’s effort on a follow-up problem, which he completed correctly (and again did not need the table at the top).

Real Life Relevant Context for Students with Autism

 

In  a lesson on functions I provided a student with autism the input and output mapping on the left as opposed to the problems like the one on the right. This student, as I’ve written before, loves comics and superheroes and villains. He was tasked with matching (mapping) people with super powers with the organization to which they belong. He completed the assignment eagerly and quickly.

Self-Help Skills Video Presentation

See the video (link below) to find out about this photo.I believe one of the greatest problems in education is the challenge students have in taking an active role in the learning process.

This is a video of a presentation on issues related to academic self-help skills and how to develop them. This is especially important for students with special needs.

The sound quality is not what I want it to be but the slides may help make up for this. I intend to re-record this presentation.

Cutouts for Area of Irregular Shapes

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.

Number Lines to Represent Percents

I am following the blog Adventures at the Pond (author has some great stuff!) and the following photo was posted on this blog. 

This is an excellent alternate representation of a concept. I do not know how the author would present this in class but here’s what I would do. I would give the students the number line with the initial information and ask them to estimate the total minutes – no calculation.

I would then explain the factor in the percent part (multiply by 5 to get to 100%) and lead the students to carry this factor over into the minutes:This would lead into proportions nicely. I would present the number lines and have the students work out problems using this approach first (Universal Design for Learning) then redo the problems using proportions.