Teaching Tricky Trapezoids: Inclusive vs. Exclusive Definitions

Why Your Students Might Not Be
Classifying Trapezoids Correctly 

Do you teach quadrilateral classification? If so, did you know there are THREE ways to define a trapezoid?

Americans use either the inclusive or the exclusive definition depending on their curriculum. To complicate matters even more, teachers who live outside the United States define trapezoids in a completely different way! Believe it or not, the British English definition is the exact opposite of the two American definitions!

Which definition are you supposed to be teaching? If you're not sure, it's entirely possible that you're teaching the wrong definition! But don't feel bad if you discover this to be true because you are not alone. In fact, until recently, I didn't even know which definition was used by the Common Core State Standards!

Before we dig into this topic, you need to know which definition you're currently teaching. To find out, answer the trapezoid question below before you read the rest of this post. Then read the information under the 3 polygons that explains what your answer means.

What Your Answer Reveals

Because there are three ways to define a trapezoid, there are three correct answers to the question. Your response will reveal the definition you use to classify trapezoids.

• If you only chose polygon 3, you use the exclusive definition which states that a trapezoid has EXACTLY one pair of parallel sides. This is the definition that I learned, and it's the one I thought the Common Core used (but I was wrong).  
• If you chose polygons 1 AND 3, you use the inclusive definition which states a trapezoid has AT LEAST one pair of parallel sides. Many educators favor this definition because the other quadrilateral definitions are inclusive. For example, a parallelogram is a 4-sided figure with both pairs of opposite sides parallel, which means that squares and rectangles are also parallelograms. 
• If you only chose polygon 2, you're using the British English classification system which states that a trapezoid is a quadrilateral with NO parallel sides. You teach your students that a quadrilateral with one pair of parallel sides is a trapezium, not a trapezoid. 

Which definition SHOULD you be teaching?

Now you know which definition you use to classify trapezoids, but is that the definition you're supposed to be teaching? If you aren't 100% sure, make a note to check on it. Until recently, I thought the Common Core used the exclusive definition, but I discovered that the CCSS actually uses the inclusive definition! I posted a question on my Facebook page to find out which trapezoid definition most teachers were using, and over 180 people responded. I was surprised to learn that most teachers who follow the CCSS teach the inclusive definition.

How to Teach Kids to Classify Tricky Trapezoids

If this is the first you've heard that there are three ways to define a trapezoid, you might be wondering how much to share with  your students. I mean, quadrilateral classification is challenging enough to teach without having to explain that there are three different correct ways to define a trapezoid!

I recommend that you find out which trapezoid definition you are expected to teach, and only teach that ONE definition. You could tell your students that they might learn a slightly different definition at some point in the future, but if you go into too much detail, your students will end up more confused than ever.

After you know which definition you're supposed to be teaching, how do you introduce it to your students and help them learn to classify trapezoids correctly?

I've found that the best way to help your kids teaching those tricky trapezoids is with a simple sorting activity. There are two versions of this activity, and it's best to use both of them if possible. The first is a printable, hands-on activity for math partners which is great for guided practice. The other is a Google Slides activity you can assign in Google Classroom for additional practice or assessment. Both activities are included in the Sorting Tricky Trapezoids (or Trapeziums) freebie below. The directions in this post explain how to conduct the teacher-guided partner activity; directions for using the Google Slides version are included in the freebie.

Trapezoid (or Trapezium) Sorting Directions for Partners:

1. Begin the activity by introducing the characteristics of a trapezoid (or trapezium) according to the definition you are expected to teach. 
2. Next, pair each student with a partner and give each pair one copy of the Quadrilaterals to Sort printable. Ask them to work together to cut out the polygons and stack them in a pile. 
3. Explain that they will take turns sorting the quadrilaterals into one of two categories using the T-chart. Give each pair a copy of the T-chart or have one person in each pair draw the T-chart on a dry erase board. 
4. Before guiding them through the sorting activity, assign the roles of Partner A and Partner B in each pair. Then ask Partner A to select the first quadrilateral and place it in the correct column on the T-chart. Partner A then explains the quadrilateral's placement to Partner B who gives a thumbs up if he or she agrees. If Partner B does not agree, the two students should discuss the proper placement of the quadrilateral and move it to the other column if needed. 
5. Partner B then chooses one of the remaining quadrilaterals, places it on the chart, and explains its placement to Partner A. Partner A must approve the placement, or the two students discuss the definition and placement before continuing. 
6. Students continue to switch roles throughout the activity. If they aren't able to agree on the placement of one of the quadrilaterals, they should set it aside for the time being. 
7. As students are working, walk around and observe them to see if they are classifying the trapezoids correctly. Stop to help students who are confused or who can't agree on the placement of one or more quadrilaterals. 

Hands-on Activities for Classifying Quadrilaterals

This simple sorting activity is actually one of the most effective ways to teach kids to classify any type of quadrilateral. In fact, it's so effective that I developed a complete lesson for classifying quadrilaterals based on this strategy. Classify It! Exploring Quadrilaterals includes several introductory activities as well as a challenging game and two assessments.

One reason I wanted to bring the tricky trapezoid situation to your attention is that I've recently updated Classify It! Exploring Quadrilaterals to include all three definitions. There are now THREE versions of the lesson materials within the product file.

No matter which definition you're supposed to be teaching, Classify It! Exploring Quadrilaterals has you covered. You'll find lessons, printables, task cards, answer keys, and assessments that are aligned with the quadrilateral classification system used by your curriculum. Not only are these activities engaging and fun for kids, the lessons will help them nail those quadrilateral classifications every time! If you don't believe me, head over to see this product on TpT where you can read feedback from 400 teachers who have used Classify It! Exploring Quadrilaterals with their students.

By the way if you already own Classify It, you can download the updated version for free by clicking over to the Classify It! Exploring Quadrilaterals page on TpT. If you're logged in, you'll see a link at the top that says "Download Now! You own it!"

If you teach quadrilaterals and haven't purchased it yet, take a few minutes to preview it on TpT. If you use it with your students, I think you'll agree that Classify It is the most effective and FUN way to foster a deep understanding of quadrilateral classification!

How to Teach Addition of Fractions Using LEGO Bricks

Guest blog post by Dr. Shirley Disseler

We know that current math standards require students to learn through modeling using manipulatives. I have been using LEGO bricks for many years to teach students math concepts throughout the elementary and middle school curriculum. It’s a perfect math manipulative, and students love using the bricks, since many students are very familiar with them. I’ve developed specific strategies for teaching math using LEGO bricks for modeling and have been thrilled over the years to watch students’ test scores improve after they learn math using these strategies.

In recent years, I’ve taught many graduate students at High Point University how to teach with these methods, and they also report great success for their students when they use the techniques as new math teachers. I’ve recently published a series of books that show how to utilize LEGO bricks to teach all the major math topics in elementary school: Counting and Cardinality, Addition, Subtraction, Multiplication, Division, and Fractions.

Free LEGO Fractions Book
I’d like to share an example of how to teach using LEGO bricks. This is a strategy for teaching how to add fractions that have like denominators. It's one of the lessons in my book, Teaching Fractions Using LEGO® Bricks, which is a part of my Brick Math Series. If you'd like to see more fraction lessons, you can download the entire PDF of this book as a sample of the series! Click here to request your free copy.

Adding Fractions with Like Denominators
Teaching students to add fractions can be a challenge. Students must first understand that a fraction shows part of a whole. This method of modeling fractions with bricks helps students see clearly what the parts of the fractions mean, and how only the numerators are added, since the two fractions are part of the same whole.

Let’s add the fractions 1/6 and 2/6 together to show how the process works.

1. First, build models of the two fractions on a baseplate using LEGO bricks. The baseplate is an important component of Brick Math, because it keeps all the bricks in place. 
2. Start to model the two fractions, denominators first. Use a 2x3 brick (6 studs) to model the denominator of 6. Use two 2x3 bricks that are the same color, to help students understand that the denominators are the same. Leave a little space between the two 2x3 bricks.
3. Model the numerator of the fraction 1/6 by placing a 1x1 brick above the first 2x3 brick. Model the numerator of the fraction 2/6 by placing a 1x2 brick above the second 2x3 brick. Using different color bricks for the numerators helps to show they are not the same.





4. Now it’s time to model the action of adding the two fractions. Take another 2x3 brick and place it at the bottom of a baseplate. Place the 1x1 brick above this 2x3 brick. Then place the 1x2 brick above the 1x1 brick. Your model now shows 3 studs over 6 studs. Take three 1x1 bricks and stack them on each stud of the combined numerator bricks. Have students touch each stud to count 3 as the numerator of the solution fraction of 3/6 .

5. If your students are ready for it, you can demonstrate how 3/6  = 1/2 . Place a 1x3 brick on top of the three 1x1 bricks in the model and show students that the 1x3 brick (modeling the numerator) is 1/2 the 2x3 brick (modeling the denominator).





6. The final step in the process is to have students draw their brick models on baseplate paper. Drawing the models they have built helps students reinforce the visual depiction of the mathematical concepts. Baseplate paper is included in my book, Teaching Fractions Using LEGO® Bricks, which is a free sample of my Brick Math Series books.

When you take students through the modeling process, you give them a powerful way to visualize the action of the math. For both visual and tactile learners, this method helps student understand how to add fractions that are part of the same whole.

See Two Fraction Lessons in Action on YouTube
Watch the YouTube video below to see two fraction lessons demonstrated step by step.


Learn More 
If you want to learn more about how to teach using LEGO bricks, check the Brick Math program website. The books in the series are available as both printed books and as PDFs, and can be purchased on the website, on Amazon and Kindle, and on TpT. Brick sets that have been designed for the program are available from that site as well. You can also purchase individual LEGO bricks from LEGO Pick a Brick, or from online resellers of LEGO bricks such as www.bricklink.com or www.brickowl.com.

Dr. Shirley Disseler is an associate professor at High Point University and chair of the Department of Elementary and Middle Grades Education, and the STEM coordinator for the BA to MEd program. She is a LEGO® Education Academy Trainer and has been instrumental in developing and testing several LEGO® Education products. Disseler serves on the LEGO® Education Ambassadors Panel and is the trainer for the High Point University Teacher Academy for LEGO® Education. She has over 25 years of educational experience from elementary school teaching through higher education, including gifted education and exceptional children. She has recently started a new business called BrickEd on the Move that offers camps, field trips, and events based on learning with LEGO bricks.