If you have taught algebra, you have likely experienced this error. We know many students will make this type of error and we can help many students avoid it by being proactive.
Below are excerpts from a Google Jamboard that can be used to unpack the underlying concept of the division or simplifying shown above. First, start with prior knowledge students can relate to, presented as manipulatives.
Then move, in a CRA fashion, a step towards more symbolic representation of the concept.
Finally, represent the situation in symbolic form. The focus here is to show the problem as two separate division problems to emphasize that both terms are divided. Then write out the simplified expression below.
The approach shown above is an entry point to simplifying rational expressions, with the same type of common errors we see there as well.
An effective instructional strategy is to make the new math topic meaningful. A fellow Facebook group member asked about teaching the topic constant of proportionality. My suggestion is to use hourly wages as an introduction.
I created a handout that starts with students finding a job with an hourly pay stated and then completing a time sheet.
This is followed by unpacking the relationship between hours and pay.
This establishes a context and a situation that many if not most students may find interesting and to connect to the math topic. This handout is intended as an introduction and not the formal unpacking of the term.
Some educators and parents of students with special needs are unclear about what is meant by the term inclusion. Some think it is having the student with a disability in the same location as “nondisabled peers.” Some think it involves doing the same exact tasks or academic work.
Sesame Street figured this question out years ago. The girl in the red shirt in the video below (video set to start with her) was experiencing inclusion, not because she was next to the other kids. She was not jumping rope but was most certainly included and appeared to love it! (Note: “inclusion” is not defined in IDEA, so formally this issue would be one of least restrictive environment.)
Below is a genius representation of inclusion (not my idea).
It appears that inclusion is sometimes viewed as a dichotomous choice. For example, I observed the student in a school who was the most severely impacted by a disability sitting in a grade level history class during a lesson communism. This was an effort to provide inclusion but was he was experiencing proximity.
Below is an example of inclusion for a student with autism in an algebra 1 class. Below left is a typical math problem. To the right is one I created for the student with autism. It was designed to help him understand the concept of matching inputs and outputs without using a lot of the math terminology. In his case, the focus in math was on concepts.
Plotting points is surprisingly challenging for some students. Here is an approach originated by one of my former math teacher candidates in a methods class I taught. This approach uses the analogy of setting up a ladder.
First, determine where to position the ladder, then climb the ladder. (brilliant and not my idea). Plot the point on the ladder, then pull the ladder away. The context includes green grass for the x and yellow for y because the y axis extends to the sun. This is shown on a Google Jamboard with moveable objects (you can make a copy to edit and use on your own).
Next, fade the ladder but keep the color – note the color of the numbers in the ordered pair. 3 is green so move along the grass to the 3. Then yellow 5 so move up 5, towards the sun.
Now, keep the the colored numbers and still refer to the green grass (faded) and sun (faded).
Finally, on a handout students can use highlighters as necessary to replicate the grass and sun numbers. The highlighters can be faded to result in a regular plotting a point problem.
One step in reading and analyzing scatterplots is simply identifying what the dots on the graph represent. If students do not understand the dots (including the position) how can they analyze. An approach I have used is start by having students create their own scatterplot for mileage and price of used cars they shop for on Carmax.com. This allows them experience the scatterplot from a data and context point of view.
Then I present the scatterplot of used Ford Mustangs on a Jamboard (image above) with ads for two used Mustangs along with a cutout of each car. The goal is to help the students understand the reasoning behind the position of each dot.
First, I take the cutout of the first car and “drive it” along the x-axis (top 3 photos in gallery below). This helps them understand the horizontal axis placement. Then I move the car up to the appropriate price (bottom row left). Finally, I replace the car cutout with the bigger blue dot that was placed by the ad with the car. We then discuss that a dot can be used to represent that car and the location on the scatterplot is based on the two values in the ordered pair (which can be typed into the ( , ) in the Jamboard next to each car.
The same steps are used for the other Mustang (see it “driving” along the x-axis below).
The next step would be to identify additional points on the scatterplot. I then revisit driving the cars and show that driving the car more miles results in a lower price and driving the car less miles results in a higher price. Finally, we discuss that this is a general trend but that it is not always true for each car. I highlight a couple points where one of the cars has more miles and a higher price (below). This leads into a discussion about additional factors influencing price.
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.
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).
The scatterplot above is an approach I use to introduce systems of equations. Here is the process I use. (Note: I have found that students like math associated with buying a car – relevant, real life application for them.)
In my class, students would have seen a scatterplot with mileage and price for a single car. I explain that we will now compare two cars.
To review, in a do now or initiation at the start of class I would have one group generate a scatterplot for the Toyota Camry data and the other groups, Mustang (Excel sheet for all of this note: this data is old). Then they would share with each other
We would revisit the relationship shown and revisit the idea of depreciation.
I show a Camry and Mustang and ask two questions: Which car do you think costs more brand new? Which do you think depreciates faster and why?
Then I show them the scatterplot above and ask which car has higher dots at the far left? Explain what this means (Mustangs start off with a higher price). Then I ask about the dots at the far right.
The students are then asked to estimate when the cars have approximately the same value.
Then I present scatterplot below, with lines of best fit (trend lines) and they are asked the same question. We estimate the specific mileage and price and write as an ordered pair.
Finally, I explain that this is known as a system of equations and the ordered pair is the THE solution. The entire unit will focus on finding an ordered pair as a solution.
IXL.com is a site that provides online practice for math (and other topics). It has a hidden feature that allows for very effective differentiation. This can be highly useful in a general ed math class and in settings for special education services. This includes special ed settings with students working on a wide ranges of math topics, for algebra students who missed a lot of class or enter the course with major gaps, and for the general algebra population to meet the range of needs. IXL can be used before the lesson or after, for intervention.
By way of example, assume you have a student or students working on graphing a linear function using an XY table (image below). Using a task analysis approach, this topic can be broken up into smaller parts: completing an XY table, plotting points and drawing the line, interpreting what all of this means. I will focus on the first two in this post.
IXL has math content for preschool up to precalculus. For the topic of graphing (shown above) many of the steps are covered in earlier grades. For example, plotting points is covered in 3rd grade (level E), 4th grade (level F), and 6th grade (Level H). To prepare students for the graphing linear functions, they can be provided the plotting points assignments below to review or fill in gaps.
The tables used to graph are covered starting in 2nd grade (level D) and up through 6th grade (level H). These can also be assigned to review and fill in gaps.
When it is time to teach the lesson on graphing a linear function, IXL scaffolds all of the steps. For example, the image below in the top left keeps the rule simple. The top right image below shows that the students now have an equation in lieu of a “rule.” The bottom image below shows no table. All 3 focus on only positive values for x and y before getting into negatives.
The default setting on IXL is to show the actual grade level for each problem. I did not want my high school students know they were working on 3rd grade math so I made use of a feature on IXL to hide the grade levels (below), which is why you see Level D as opposed to Grade 2.
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.
Top left is a scaffolding I use to help students learn to solve math problems using multiplication (3rd grade). The situations are typically rate problems (e.g., 5 pumpkins per plant or $2 per slice of pizza) although the term “rate” is not used yet. The same concept of rate plays out in high school with slope of a line, applied to real life situations (top right).
These types of problems start in 3rd grade (below, top left), play out in 6th grade (below, top right), into 8th grade (below, middle), and into high school algebra and statistics (below, bottom). I referenced this connection previously regarding word problems and dominoes. This highlights how crucial it is that strategically selected gaps in a student’s math education be addressed in context of long range planning.