Division of fractions may be one of the most abstract concepts in middle school math. Here is an approach to address the concept using a Google Jamboard (you can make a copy which allows you to edit it), which would be a foundation for the ensuing steps. I will preface this approach by stating the obvious. Because this is very abstract and challenging for students, the approach is more complex – no royal road to dividing fractions.

To unpack this concept I start with the concept of division itself. One interpretation is distributing a collection of items into equal groups to determine how many items in each group. That lends itself well to dividing by a fraction. In the example below, I show 6 cookies divided into two groups to get 3 cookies per group. That is the goal, identify the per group amount.

Then we introduce a fraction. 6 divided by 1/2 can be stated in the group context as 6 cookies for half a plate or for half a group.

But we want a whole plate, a whole group. How do we get that? We need another half group which ends up revealing that we multiply by 2. (Keep in mind that the goal here is to unpack the concept and not so much the actual steps yet.)

Now we can turn our attention to the full dividing fractions situation. The approach is the same as the whole number divided by a fraction; we start with the fractional item in the fractional group. Then we build the whole plate (group) which results in building the whole cookies. At the end I take a stab at showing the mathy steps but I am unsure how I would unpack the steps at this point – again, focusing on the concept in this activity. I think I would not show the steps and have the students simply do hands on building a whole group, by manipulatives and subsequently by drawing.

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.

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).

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.

I have frequently encountered the presentation of absolute value as a positive value or opposite. This is part of the repertoire of memory devices we (certainly I have in the past) use as a short cut to learning how to do the steps for a problem. The meaning of the absolute value of a number is it’s distance from 0 (below).

Below is an image of a Do Now or Initiation handout I use to introduce absolute value. From the start I focus like a laser on the meaning of distance for absolute value. I start with a situation that may be prior knowledge for them. Then take a step towards the mathy part with the numbers and slowly make my way to the symbol.

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.

My first step in presenting a new topic is meaning making. For scientific notation, the underlying idea is NOTATION – “the act, process, method, or an instance of representing by a system or set of marks, signs, figures, or characters.” We can represent numbers in different ways, one of which is scientific notation. This is useful to represent very large or very small numbers (as happens in science). It is useful because in lieu of writing out a bunch digits, the power of 10 can be used as a shortcut. In the image above you see that 4.5 x 10^{4} has two parts, the decimal and the 10s.

Before I get into these big or small numbers, I address the concept of notation because that word is in the topic. To introduce a concept, I typically start with a related topic that is relevant for students. In this case it is money. To mirror the two parts of scientific notation, I list the bills and how many of each. In the left image below, I show both parts and pair combinations that are the same value (a single $10 bill and ten $1 bills). I then show how I can convert a single $10 bill by dividing by 10 and then multiplying the number of bills by 10 (middle image). This previews the steps used in scientific notation. Then (right image) I show the same approach for dollars and cents (which previews decimals). Note: to help flesh out the dollars and cents I would first use the linked Jamboard.

The image below left keeps the concept of money, but the images are faded. The students are still working with money and how many but now with numbers only. The middle image introduces decimals, but the same steps are used (divide by 10 and multiply by 10).

Finally, the matched pairs shown in the previous handout pages (images above) are presented with an explanation of the parts of scientific notation (below left). I explain the idea of scientific notation as a special way to write numbers, list the two parts, and then I show examples by circling the ones in each pair (bottom left) that fit the criteria. Then they identify numbers that are written in scientific notation (below right).

Following this introduction lesson, I would explain the applications (linked above) and go into more detail on how to rewrite the number in scientific notation.