The concept of fractions as some number of equal parts begins in 1st grade per the Common Core (image below). There are students who struggle with the idea of equal parts and this could undermine student work in subsequent topics. The activity cited in this post is designed to develop the concept of equal parts.
Jamboard with Sharing Slides
The following images are from a Jamboard used as an introduction to equal parts activity (see photo at the end for access). The activity is chunked to incrementally present more of the ideas underlying equal parts. The use of the Jamboard can be viewed in a FB Reel and on YouTube.
First, the idea of equal is addressed by presenting a situation in which two students are sharing candy. Partitioning out pieces alludes to the set notation of fractions.
The idea of sharing equal amounts transition to sharing a single candy that can be broken into parts. The candy bar image is actually two images of parts. The a non equal sharing is used to unpack equal parts. This is continued for a circular shape and a triangular-ish shape.
Jamboard with Mathy Slides
There are additional slides to do more “mathy” work with equal parts. First, the students are asked to choose the shape that was cut into equal part (rectangle, circle, triangle). Then the students partition the shapes but with a dotted line as scaffolding.
Each shape can be connected to the food images from above. For example, the student may intuitively understand that a pizza is cut from the crust to the tip. I use pizza fractions to unpack the need for common denominators, which reinforces the significance of the concept of equal equal parts cited previously.
Here is an image of an accompanying worksheet. It draws upon the images from the Jamboard and follows the same sequence.
Accessing the Jamboard
The image below shows how to make a copy of the Jamboard in order to use it.
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.
To help students learn how to measure with a ruler, I focus on minimizing the number of tic marks on the ruler at first. The image below shows an excerpt from a WORD document with a halves ruler that I use and an instructional strategy. It also contains a quarters and an eighths ruler that students can slide around the WORD document as shown above and in a video explaining this artifact and how I created it.
This is useful for distance learning as well as in class. Here is a link to the WORD document with the rulers shown in the video.
It is easy to get caught up in the steps and rote memorization when working with fractions. The brain processes information more effectively when the information is meaningful. ADHD makes paying attention to rote memorization of steps even more challenging.
Below is an excerpt of work I completed with a middle school student who has ADHD. This was completed extemporaneously as intervention (you see his initial attempt was incorrect) but can be used as Universal Design in whole class instruction.
Here is a break down of how I helped the student after seeing his mistake in his initial attempt. First, I modeled the first mixed number as pizza pies.
Then I presented the problem in pizza terms. “You have 3 pies and 1 slice and you are going to give me 1 pie and 2 slices. Do you have enough slices?” <wait for response> “You don’t, so what can we do?” <wait for response> “We cut up one of the pies.” I have the student cut the pie into fourths.
I then make the connection with the mixed number and guide the student to taking away 1 pie and writing 4/4. This provides more concrete meaning for writing 1 as 4/4.
In turn, this provides meaning for the new mixed number and meaning for the subtraction of the whole numbers (pies) and the fractions (slices).
The photo below is courtesy of Robert Yu, Head of Lego Education China, as shared by Jonathan Rochelle, Director for Project Management at Google.
The use of Legos shown here is a classic (and wicked clever) example of manipulatives.
Before writing the actual fractions students can use drawings as shown below. The sequence of manipulatives, drawings then the actual “mathy” stuff constitutes a Concrete-Representational-Abstract (CRA) model. Concrete = manipulative, Representational = picture, abstract = symbolic or the “mathy” stuff.
We explain steps in great detail to students but often omit the underlying concept. The topic of adding or subtracting fractions with unlike denominators is an example of this.
The example above right is a short cut for what is shown above left. These short cuts, which math teachers love to use, add to the student’s confusion because these rules require the student to use rote memorization which does is not readily retained in the brain.
I suggest using what I call a meaning making approach. I present the student 2 slices of pizza (images courtesy of Pizza Fractions Game) and explain the following setting. “You and I both paid for pizza and this (below) is what we have left. You can have the pizza slice on the left and I will have the pizza slice on the right. Is that OK?” The student intuitively understands that it is not because the slices are different sizes. I then explain that when we add fractions we are adding pizza slices so the slices need to be the same.
I then cut the half slice into fourths and explain that all the slices are the same size so we can now add them. Then the multiplying the top and bottom by 2 makes more sense.
I have posted on how to effectively provide support for current math topics. Here is an example (below) of how support can focus on both the current topic and prerequisite skills.
For example, on the 22nd in this calendar the current topic is solving equations. The steps for solving will include simplifying expressions and may involve integers. The support class can address the concept of equations, simplifying and integers which are all prerequisite skills from prior work in math.
This approach allows for alignment between support and the current curriculum and avoids a situation in which the support class presents as an entirely different math class. For example, I recently encountered a situation in which the support class covered fractions but the work in the general ed classroom involved equations. Yes, equations can have fractions but often they do not and the concepts and skills associated with the steps for solving do not inherently involve fractions.
Fractions is one of the most challenging math topics. Many high school and college students struggle to some degree with fractions. The Common Core of State Standards (CCSS), despite all the criticism, includes components to address the conceptual understanding of fractions. Below is a photo showing a 4th grade Common Core standard regarding fractions along with an objective for a class lesson I taught at an elementary school in my district. I subsequently presented on this at the national CEC conference in 2014. Notice the bold font at the bottom, ¨justify…using a visual fraction model.¨ The photo above shows an example of a model I used in class.
The photo below shows a handout I used in the lesson. The first activity involved having students create a Lego representation of given fractions. These would eventually lead to the photo at the top with students comparing fractions using Legos. The students were to create the Lego model, draw a picture version of the model then show my co-teacher or I so we could sign off to indicate the student had created the Lego model.
The Lego model is the concrete representation in CRA. In this lesson I subsequently had students use fractions trips (on a handout) and then number lines – see photos below.
Fractions is one of the most challenging topics in math. Here’s an approach to help introduce fractions.
I show the photo above, explain to a student that he and I both paid for the pizza. We are going to finish eating the pizza and I get the slice on the left. I ask “is this fair?” This leads into a discussion about the size of the slices and what 1/2 and 1/4 mean. The pizza on the left was originally cut into 2 slices so the SIZE of the slices is halves. The SIZE of the slices in the one on the right is fourths. I have 1 slice left and it is a half so my pizza is 1 half or 1/2. He has 1 slice left and it is a fourth so his pizza is 1/4. The bottom number is the size and the top number is the # of slices.
We cannot count the number of slices because they are not the same size. So we need to change my pizza. So I slice my pizza and now I have 2 slices and they are cut into fourths. So now I have 2/4. Note: I don’t show the actual multiplication to show how I got the 2 and 4. I am sticking with the visual approach to develop meaning before showing the “mathy” approach.
Now that I have slices that are all the same size, I can now count the # of slices. “1, 2, 3…3 slices and they are cut in fourths.”