Rhombus vs. Square

Is a Square a Rhombus?: The Great Debate & How to Approach it in Geometry Class

​Does a square qualify as a rhombus?  This big debate has been going on for as long as I can remember!  Depending on who you talk to, you will get different opinions; some will say yes, some will say no.  It all comes down to your preferred definition of a rhombus.

​Inclusive vs. Exclusive
 
In math, we have two separate definitions of a rhombus:

- Inclusive: a rhombus has four equal sides.
- Exclusive: a rhombus has four equal sides, BUT has no right angle. 
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Based on these definitions, the inclusive INCLUDES squares, because a rhombus can include right angles.  The exclusive definitions of rhombus EXCLUDES squares, because there cannot be a right angle.
 
Why You Should Consider Using the Inclusive Definition 

• Generally speaking, mathematicians tend to use inclusive definitions, and students will encounter more instances using inclusive definitions in math class in the future.
•  Inclusive definitions better match how we think visually or kinesthetically.
• Most geometry software, (think Geometer’s SketchPad) follows the inclusive definition; if you construct a rhombus and then drag the points, you are able to construct a square.

 
 
Teaching Opportunity
 
So, which definition do you use?  You may have a strong opinion based on your own belief or how you were taught; this is totally fine!  However, if you’re debating on which definition is correct, why not let your students explore this topic and hold their own debate?
 
It’s also a great opportunity to explicitly teach the above idea.  Let students in on the fact that there is indeed a debate, and explore the differences between inclusive and exclusive definitions.
 
Can your class come up with more examples of inclusive and exclusive definitions in math?  This is a great time to investigate a controversy in mathematics! 
 
You may even want to take it a step further and have teams of 2 students find sources on the web that prefer each definition.  Let them analyze the argument and decide which they feel is a better, stronger definition and share why.
 
Don’t miss this chance to integrate debate skills, research skills, and analyzing a source within math class.  These opportunities are rare!  Students learn best when they cross their curriculum-specific skills over into different subject areas.
 
Additional Ideas for Teaching Distinctions Between Quadrilaterals
 
If you are looking for other creative ideas to teach your students about quadrilaterals and their classifications, you have come to the right place!
 
1) Always, Sometimes, Never: Quadrilaterals
 
If you head on over to the Math Giraffe Teachers Pay Teacher’s store, you can find a super fun activity for teaching quadrilaterals!   This challenge is perfect for getting your students thinking about parallelograms, trapezoids, squares, etc.

The download comes with two versions: a worksheet and a sorting activity.  On the worksheet, they determine whether each statement is always, sometimes, or never true and color accordingly.  They end up with a design that you can check for accuracy.
 
They can work in pairs or small groups for the sorting activity.  Students sort the cards with statements into the correct category, (“Always true, sometimes true, or never true”).

 
2) Quadrilateral Fun!
 
Nichole from The Craft of Teaching provides a great inquiry activity for beginning to learn about quadrilaterals.  She cuts out various quadrilaterals, and has her students work in pairs to sort them in a way that makes sense to them on a pre-made graphic organizer.
 
I love how this activity is student-driven; they have the opportunity to discover the properties of various quadrilaterals themselves!  This makes it so much more meaningful for your students, which improves retention.
 
3) Quadrilaterals Bundle
 
This discounted bundle on the Math Giraffe Teachers Pay Teachers store has a variety of activities to teach a fun, interactive lesson on quadrilaterals for your High School Geometry Class!  It includes a blend of puzzles, activities, and proofs that add the perfect mix of fun and rigor.

​These supplemental activities can be spread throughout your quadrilaterals unit!
 
Here's what's included:
Quadrilaterals: Always, Sometimes, Never
Quadrilateral Proofs
Quadrilaterals Card Sort
Quadrilaterals (Algebra in Geometry) - GridWords

​Do you have a passionate stance on whether or not a square is a rhombus?  Or any great ideas for teaching Quadrilaterals?  Please take a minute to write a comment below!  We would love for you to weigh in.

To be extra clear, I sometimes like to write "(non-square)" along with "Rhombus" when I want students to classify by type.  This way, their answers are consistent and we can agree to use that as our category while acknowledging that there could be some controversy or confusion if we are not clear on this.  (see image below)

You may want to try a similar strategy in your sorting activities or definitions when you introduce the concept of a rhombus vs. a square.

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