Tag Archives: fractions

Lego Fractions

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.

Lego fraction model

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.

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Sausage Fractions – Real Life Example

I have 3 kids and was cooking sausage for them.

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There were 5 sausage links available (below). How do I give each the same amount? Fractions!

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Each child gets a full sausage link.

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I then cut  the remaining 2 sausage links into 3 parts, 1 for each child. 1/3 of a link.

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Each child gets 1/3 and another 1/3 or 2/3. So they get 1 full link and 2/3 of a link or 1 2/3. This is an entry point into mixed numbers (whole number and a fraction).

dividingupsausage4

 

 

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Common Denominator – Why?

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.

subtracting unlike denominators     adding unlike denominators

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.

fourth pizza slice         half pizza

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.

fourth pizza slice          2 fourths pizza slices

 

 

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Example of Using Support Class to Support Current Math Content

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.

examples of support class prerequisites

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.

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Fractions! Meaning Making for Comparing Fractions

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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.

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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.

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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.

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FRACTIONS! Meaning Making for Adding Fractions

original-adding-problem-unlike-denominators

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.

cut-pizza-to-get-like-denominators

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.”

add-pizza-slices

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Fraction Word Problems Scaffolded

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Above is a handout used to scaffold student work with fraction word problems (more on this below).

Here is a type of word problem I recently encountered in working with a student with special needs: There are 60 students. 3/5 of the students are girls. How many students are girls? The student struggled with this problem in regards to the concept of fractions and in determining a fractional amount of a total.

To address the concept of fractions I used the handout seen in the photo above. (The handout can be printed in color to show the actual colors of the birds – see this handout.) A pink highlighter (red is too dark) is used to help the student connect the actual red birds with the number of redbirds used in writing the fraction.

To work out the 60 students problem the following approach is used to develop conceptual understanding – see photo below. (Note: for students in upper grades who struggle teachers often turn first to showing how to solve by multiplying. Students who are working at below grade level typically need the conceptual piece to be addressed first.)

  1. Break the all the items into groups based on the denominator, in this case it is groups of 5.
  2. Mark the fractional amount in each group, in this case mark 3 of the 5 circles.
  3. Find the total number of circles marked in.

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To scaffold this approach I use a task analysis approach and break the strategy into steps.  First I use a handout focusing students on circling a fractional amount in each group – see photo below.

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The next step is to count the number of items circled (or marked).

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The next step is to use a situation where the items to circle are not colored (the circles in the student word problem are not colored but are hand drawn figures). In the photo below Students are tasked with circling and counting.

 

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Then students take the next step is to answer the question to find the total number.

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Finally, students are tasked with creating their own drawings before circling and counting.

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Fractions, Decimals and Percents Number Line

Good resource for students who struggle with the relationship between these three forms.

fractions decimals percents number line

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