## Introduction to Unit Cost (Unit Rate)

Unit rate (e.g., hamburger meat on sale for \$2.39 per pound or you make \$13 per hour) is an incredibly important topic in middle and high school. First, unit rates and unit costs are common in life. Second, in the Common Core State Standards math categories you can see that Ratios and Proportions (which includes unit rate) are a 6th and 7th grade topic and are then replaced by Functions in 8th grade. Below is a photo showing a graph of a function you can see that the slope in an application is a unit rate.

The unit rate is also conceptually challenging whether it is in a function or is a unit cost at the store. This is a major sticking point for many students in special ed who have fallen behind. To address this, I used the approach below.

First, I present a pack of items the student likes (4 pack of Muscle Milk for this student). Use a Jamboard to show a 4 pack and the price of the 4 pack (photo on left). Then I “pull out” the 4 individual bottles and divide the \$8 among the bottles to show \$2 for each bottle. Finally, I have the student shop for packs of items at a grocery store or Amazon and compute the price for 1 item using a mildly scaffolded handout.

I Follow the same steps for ounces or pounds but show how 4 oz is divided into single ounces (in lieu of a pack divided into single items). Then the student shops for items that can easily be divided to get a unit cost.

## Rate of Change in Real Life

61 cents per ounce is a rate of change. Graph the line modeled by this (y intercept is 0) and it becomes slope of the line. In referring to algebra we often hear, “when will I ever need this?” My response is “all the time!” Our job as teachers is to make this connection for students. ## Making Slope Less Complicated Slope is the rate of change associated with a line. This is a challenging topic especially when presented in the context of a real life application like the one shown in the photo. The graphed function has different sections each with a respective slope.

One aspect of slope problems that is challenging is the different contexts of the numbers:

• The yellow numbers represent time
• The orange numbers represent altitude
• The pink numbers represent the slopes of the lines (the one on the far right is missing a negative)

Before having students find or compute slope I present the problem as shown in the photo above and discuss the meaning of the different numbers. What I find is that students get the different numbers confused and teachers often overlook this challenge. This approach is part of a task analysis approach in which the math topic is broken into smaller, manageable parts for the student to consume. Once the different types of numbers are established for the students we can focus on actually computing and interpreting the slope.

This instructional strategy is useful for all grade levels and all math topics.

## Introduction to Slope 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. ## 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. ## Chunking a Lesson

Chunky peanut butter means you there are pieces or chunks of peanuts. The peanut butter is broken into distinct parts. Chunking in teaching means breaking a concept or lesson into distinct, smaller parts. This makes it easier for students to process the lesson and the concept. They focus on less content at one time which is important given that working memory has a limited capacity, especially for many students with disabilities.

Here is an example of how it can play out. On a lesson on slope (compute slope of a line given two points from the line) the steps are as follows for (1, 5) and (4, 11):

• write the formula y2-y1/x2-x1 (rise over run)
• plug in the ordered pairs (11-5)/(4-1)
• compute 6/3 = 2

But there is also the conceptual understanding that is often lost on the students. These steps do little to help with conceptual understanding.

In the photo below the teacher is presenting a conceptual piece as a chunk of her lesson before she gets to the steps listed above.  She has drawn a triangle to represent going downhill, another going uphill and a horizontal line segment representing no hill. The rise and run are listed for each (the horizontal line segment has no rise). The students wrote a rise over run ratio showing the slope for each then practiced this before the teacher moved onto to the steps listed above. Upon completion of this chunk the teacher can give a practice or pop quiz to help students fill in the gaps individual students may have before moving on to the next chunk.

The next chunk would not yet involve the formula. A slope triangle would be drawn under the line to visually represent the rise and run. Students would practice finding slope using this approach with the next step bringing in ordered pairs and the formula listed above. 