When our 3rd child was born, we decided to buy a used Honda Odyssey as 3 young kids were not fitting into a sedan. Being the stats geek I am (master’s in statistics at the University of South Carolina – total geek) I collected mileage and price data for all the used Odysseys for sale on dealer sites throughout South Carolina. I then created a the scatterplot shown below. I went to a dealer, showed an agent my graph, and he immediately exclaimed “Where did you get that? We create graphs like that every week!”

It was this experience that led me to the idea of using used car data to introduce linear functions. Shopping for a used car has proven to be a relevant, real life activity the students enjoy.

Here is a link to a comprehensive activity that walks students through various components I use for introducing students to linear function topics.

Used car shopping to collect data on 10 used cars of a single make and model.

Creating a scatterplot for price vs mileage of the used car of choice.

Creating a line of best fit (regression line) to model the data.

Creating a linear bi-variate equation (regression equation) to model the data.

The activity is presented on a WORD document (feel free to revise). It shows screenshots to walk student through the Carmax website (subject to Carmax revising their website). The screenshots make it easy for the student to navigate, which increases independence. (NOTE: there is an ample number of Youtube videos on using Google Sheets for this activity.)

The end product looks like this. Note the importance of using 1000s of miles as the slope is more meaningful, -$140.64 per thousand miles, as opposed to 14 cents per mile. I would start with the scatterplot alone to unpack the variables, the relationship between the variables, and the ordered pairs. Then the line and equation can be introduced to show a meaningful use of the line and the equation. The y-intercept has meaning with “0 miles” equating to a new car (I do not explain that new cars have miles already accumulated until we unpack the math).

Here are excerpts from two handouts I use to help students understand how to write multiplication and rate word problems as math expressions. The image, below at top, shows a problem from the first handout I present. The students draw a single group represented by the rate expression (for elementary school word problems the term rate is not used). The image at the bottom is the same problem with scaffolding to write the multiplication problem. I find that students working on rates and slope in middle school, high school, and even in college struggle with this topic. I use this approach as part of a review of prerequisite skills before getting into rate and slope.

Several special ed teachers identified solving multi-step equations as the most challenging math topic to teach in middle school math. Here is my approach to teaching multi-step equations like 3m + 4m + 1 = 15. .

First, I use a task analysis approach to break down the math topic like we cut up a hotdog for a baby in a high chair. MOST of the steps involved are prior knowledge or prerequisites skills. I present these in a Do Now (warm up, bell ringer, initiation) – see image below. This allows me to fill in the gaps and to lay the foundation for the lesson. The prerequisite skills include simplifying expressions and solving 2 step equations. I also present meaning for the equation with a relevant real life problem that is modeled by this equation. By attempting the walkathon problem without the “mathy” approach, the students will more likely understand the equation and why they add 3m and 4m.

After reviewing the Do Now I use Graspable Math, which is a free online application that allows users to enter their own expressions and equations. These can be manually simplified and solved by moving parts around. Here is a tutorial on how to do this. This allows them to manually work with the simplifying and the equation before working on the handout, in a concrete-representational-abstract approach.

This is followed by a scaffolded handout with the use of color coding. I have student work on the first step in isolation as that is the new step (the other steps are prior knowledge and were addressed in the Do Now). This avoids all the work on the other steps that can result in sensory overload and allows me to address mistakes in the new content immediately.

This handout can have the equations removed and be used as a blank template to follow. In turn this would be followed with regular solving worksheets.

In education, math especially, there exist a learning situation I call the patting head and rubbing belly phenomena. In this phenomena students are presented a math problem that consists of several steps they know how to do and then maybe one or two additional steps that are new. Adding the additional step is like adding the task of patting your head while you rub you belly. The additional math step seems so simple, but attempting it simultaneously with an additional task can make the entire effort exceedingly challenging. A related scenario is generalization to different settings, but that is different. This is true for all types of math, whether it is the general curriculum or life skills/consumer math.

This phenomena plays out in life skills math or consumer math in a stealthy manner because the steps or tasks seem so simple. For example, many of us have worked with a child or student who was learning to count money. When learning about a nickel or a quarter, the coin name and value are easily identified. Once both are introduced, many students confuse the two and may even freeze while attempting the work with the coins.

There is an ABA based process for addressing this using a task analysis and chaining in which steps are worked on in isolation before connecting (chaining) the steps together (and not all of them at once until the end). One related strategy to help implement this approach is through scaffolded handouts in which the steps are enumerated and the structure of the handout isolates the tasks. I have used this approach for 1 to 1 correspondence up to AP Statistics (see below).

When working out a draft of an IEP, I suggest having the task analysis and chaining explicitly identified in the accommodations page and ask for an example of what this looks like (using an example math topic).

There are several layers to solving equations that can be unpacked using a task analysis approach. This includes written and mental steps (such as what we teachers mean when we tell a student to do the same thing to “both sides of the equation”). Here is how I develop the concept of equations and solving.

To unpack the layers for students, I have had a lot of success with the scaffolded handouts below (see last photo of example of what to write.) Here is a link to a Dropbox folder with all 4 handouts, in WORD format. Feel free to use and revise as desired.

A pseudo- concrete representation of a sales price problem is shown below. This is what I use as an entry point for teaching these problems.

The entire shape represents the total price of $80. This is 100%, which in student language is “the whole thing.”

The discount rate is 25%. Cut with scissors to lop off the 25% which also lops off $20, which is the actual discount. Explain to the student that this 25% is part of the “whole thing.”

What remains is 75% or $60. This is the “new price” which is called the sales price.

I introduce solving equations by building off of the visual presentation used to introduce equations. The two photos below show an example of handouts I use. Below these two photos I offer an explanation of how I use these handouts.

First I develop an understanding of a balanced equation vis-a-vis an unbalance equation using the seesaw representation.

I then explain that the same number of guys must be removed from both sides to keep the seesaw balanced.

I then apply the subtraction shown above to show how the box (containing an unknown number of guys) is isolated. I explain that the isolated box represents a solution and that getting the box by itself is called solving.

I use a scaffolded handout to flesh out the “mathy” steps. This would be followed by a regular worksheet.

I extend the solving method using division when there are multiple boxes. I introduce the division by explaining how dividing a Snickers bar results in 2 equal parts. When the boxes are divided I explain both boxes have the same number of guys.

The students are then provided a scaffolded handout followed by a regular worksheet.

The photo above shows a scaffolded handout to break down elapsed time for a student. The problem is divided into 3 parts: time from 10:50 to 11:00, time from 3:15, time from 11:00 to 3:00 (see photo below). This is based on how we may compute elapsed time by focusing on minutes then on hours. Notice the 3 clocks (in photo above) with no hands which allows the student to engage the clocks by having to determine and show how many minutes passed, e.g. 10:50 to 11:00.

The photo above shows a screen for a hundreds table was shared by one of my students in a Math for Children graduate course. She found it on Pinterest for use a class presentation. I love this idea and came up with some revisions I think can make it more effective. It seems to me that this screen may be too busy with 4 different numbers showing. Additionally, the view of the other numbers outside the screen could be distracting.

List all the steps for the objective. Use this table (above) as a pretest to identify gaps.

Provide instruction on the gaps. In the photo below I used color coding to show what to multiply and scaffolding to align the digits in ONES and TENS place. NOTE: I provide the problems with some steps already completed to focus on the steps forÂ which gaps were identified.

After providing instruction on the steps with gaps data is collected on mastery of these isolated steps. NOTE: The problems are identical in nature to the gaps and the problems used in instruction. (Link to the data sheets – WORD so you can revise.)