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.

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.

A hidden treasure is the Common Core of State Standards Math Coherence Map. It is an interactive flow chart that shows connections between the various standards.

If you are teaching math, you can see the connections between what you are teaching, what was taught previously, and how you are preparing students for their future math education.

If you are a special education teacher, you can see the progression of prerequisite skills. If you write IEP objectives for grade level standards, you can address the prerequisite standards and progress made through these prerequisites can show progress towards mastery of the IEP objectives.

In this post, I show the progression from 1st grade standard on the = sign and 2nd grade standard of repeated addition all the way to interpreting slope in high school math.

After clicking “Get Started” you will narrow your focus to the grade, the cluster, and then the math domain.

The flow chart shows connections between a selected standard and others, including prerequisites. In this case 8.F.B.4 – 8th grade content that is a prerequisite for the high school math work. Click on the 8.F.B.4 standard and it pops up (below right).

In turn, the 8th grade standard is connected to a 7th grade prerequisite regarding ratios and proportions.

Notice that the 7th grade standard, 7.RP.A.2, appears to be a dead end (bottom left). I picked up the path by clicking on Grade seen at the top left of the screenshot and made my back to that standard and the connections to prerequisites appeared. (Same happened in 3rd grade shown further down in this post.)

The path continues from ratio and proportions in 7th grade to unit rate in 6th grade, multiplication word problems and multiplication in elementary school.

I want to emphasize that students are working on unit rate and slope problems in ELEMENTARY SCHOOL! 3.OA.A.1 below addresses groups of objects model for multiplication and 4.OA.A.2 addresses word problems involving multiplication.

I was recently working with a student entering middle school on multiplication word problems. To unpack the word problems and the concept of multiplication in context, I had her draw (photo below) groups and groups of objects to help her identify the unit rate (although we don’t use that term yet). This work will impact her math education through the high school math and even into college (slope has been a common gap for the college students in the math courses I taught at various colleges and universities).

This approach I used with the student could be used for high school students, especially those with special needs.

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.

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.

The orange circle on the right looks bigger, but in fact both are the same size. The deception is based on the additional sensory input.

Similarly, the prerequisites for taking algebra are often considered to be basic skills. This is largely an illusion. I routinely encounter students who are referred to me for help as they have been caught in an infinite loop of working on basic math such as number operations (adding, subtracting, multiplication, and division) before moving on to algebra, with limited progress. I am not suggesting basic math skills are not important but am focused on the context of prerequisites needed to engage algebra. Many of the students I have helped who were in this situation. We worked to quickly move them into algebra where they were successful.

One student worked on half a year of 4th grade math during her 7th grade year. During the spring of that 7th grade year and the subsequent summer, I worked with her on algebraic thinking and algebra topics. She successfully completed algebra 1 during her 8th grade year.

The Common Core of State Standards (CCSS) for Math maps out the prerequisites as seen in the CCSS math domains (below). Throughout elementary school, Operations and Algebraic Thinking topics are covered. The Algebraic Thinking standards establish for the students a foundation for algebra taught in middle and high school. A focus of algebra is to model or represent patterns or relationships in real life situations using equations, tables, and graphs. These include quantities modeled by variables.

Below is a break down of this foundation in elementary school. If you are supporting a student in middle or high school who is taking algebra and has major gaps in his or her math education, look to these standards for the essential prerequisite skills.

First Grade: represent situations in word problems by adding or subtracting, and introduce equations (and equal sign).

Second Grade: Represent, solve word problems, introduce multiplication as groups of objects.

Third Grade: represent, solve word problems, explain patterns

Fourth Grade: Solve word problems, generate and analyze patterns

Fifth Grade: Write expressions (equations are 2 expressions with an = in between), analyze patterns and relationships