Polyhedra, Geometric Solids, and Euler's Formula

One of the great things about my math class is that we use a lot of food in our math activities. Seriously, you should see the excitement on college students faces when they find out that they get to eat the math "tools" when they are done with the activity. It definitely gets everyone motivated to start the activity. If it works that well for a bunch of college students, just imagine how well it would work for elementary school students. 

In our latest "eat your work" activity, we started out making gumdrop polyhedras. A polyhedron is a three dimensional figure with flat polygonal (at least three straight sides and angles) faces, straight edges, and verticies (point where two or more curves, lines, or edges meet.) Here is the activity sheet we were given in the first station, along with a bag of gumdrops and a box of toothpicks.

As you can see, the gumdrops are supposed to represent the verticies and the toothpicks are supposed to represent the edges. We had five people in our group so we each made one shape. Here is a picture of our completed polyhedras.

We were also provided these cards to fill out for each shape to record the information such as the number of verticies, faces, and edges.

And when we were all done, we got to toss the toothpicks and eat the gumdrops....yum!

At the next station, we were given nets (a two-dimensional figure that can be folded into a three-dimensional object) to cut which, when folded properly, formed polyhedras. You have to put some thought into the shapes on the paper because if you fold it the wrong way it won't form the shape you are intending to construct. It is a good way to tap into your students critical thinking abilities. Here are some pictures of our completed nets.

The next station played off of the puzzle shape station but included the use of technology. The station had ipads that were set to this activity on The National Council of Teachers of Mathematics website. Here you are given several different nets and you have to decide if each one will fold into a cube or not. After you pick yes or no for each shape, the interactive will show you if it will or if it won't. There is also a timer and a final score, so the more competitive students can race each other to see who gets the best score in the fastest time. Try it out for yourself, it's kind of addicting!

Another station built off of the first station where you look at a polyhedron and figure out the number of faces, verticies,  and edges. On the table were physical examples of each (sorry, forgot to take a picture of that) and this activity sheet.

Included on this sheet was the section "Where in the World?" This is good for students because it connects the math they are doing to real life examples. Based off the activity sheet, can you find the relationship between the number of verticies, faces, and edges? Yes? Well good for you! No? Well don't feel bad. Only one person in my class did. The relationship is something referred to as Euler's Formula. It states that Verticies - Edges + Faces = 2. If you look at my sheet, this holds true for all (except the sphere.) My math teacher taught it to us this way... V + F - E = 2. It works out the same way mathematically and I kind of like that way better because you tend to not end up with a negative number in the beginning. The classmate that figured it out came up with V + F - 2 = E, which also works out the same way. 

So there you have it - four activities you can have your students do to help learn about polyhedras and Euler's Formula. We were able to complete all stations in one day, but our class is an hour and forty minutes long so you would probably have to split this up into several days. Have fun with it and happy teaching! 😊  

Interior Angles of a Triangle

The one thing all triangles have in common is that all interior angles of a triangle add up to 180°, no matter what type or size the triangle is. We also know that a straight line is 180°. A really great way to help your students understand this correlation is with this simple math activity. First, give each student a piece of paper about half the size of a piece of copy paper. I suggest you use colored copy paper since it makes the activity more fun (seriously, a table full of college students was fighting over the "best" color of paper to grab!) Next, give each student a ruler and tell them make and then cut out a triangle. Tell them they can make any type of triangle they want. Once each student has their triangles cut out, have them use their angle finders (explained in a previous blog post) to identify what type of triangle it is (acute - all angles less than 90°, right - one angle that is exactly 90°, or obtuse - one angle greater than 90° but less than 180°.) You can also have them identify their triangles by their sides (equilateral - all three sides are the same length, isosceles - at least two sides the same length, or scalene (all three sides are different lengths.)  

Now that the students have all constructed and identified their triangles, you are going to have them rip them up. Yes, you read that right. Tell them to rip each corner off of their triangles. This will be really hard for some to do (it was for me) but they need the angles to be separated for the activity to work. They HAVE to rip them off too, not cut them off, because you need the jagged edges to know exactly where the angles of their triangles are. If you cut them off you just have three mini triangles with three different angles and this activity will be a lot harder. Here is an example of my triangle with the corners ripped off.

After the students have ripped the corners off, have them arrange the pieces tip to tip like this

Each and every student, regardless of the type or size of their triangles, will all have made a straight angle (180°) when they arrange their corners (interior angles) tip to tip. This activity helps to drive home the fact that ALL triangles have the same sum of all of their interior angles no matter what each angle's individual measure is. Once the students know this, they can figure out the measure of angle x when given the measure of the two other angles. 

Once the students know how to figure out the measurements of the interior angles of a triangle, this activity also helps them when figuring out the measurements of the exterior angles as well. Because this activity also taught them that a straight line is 180°, students can extend any one side of the triangle to form a straight line and subtract the measurement of the supplementary interior angle from 180° to get the measurement of the exterior angle. This you tube video explains that concept in better detail. 

I hope you find this activity helpful. Try it out with your students and see if it works for you. Happy teaching! 😊

Angle Finders 

There are four different types of angles

• Straight angle - 180° (basically a straight line)
• Acute - Less than 90°
• Right - Exactly 90° (basically a straight corner)
• Obtuse - Greater than 90° but less than 180°

Examples of angles

A really fun, and frankly super cheap, activity to do with your students when learning about angles is to make angle finders with them. The only materials needed are index cards (we used 4 x 6 ones) and some colored markers (because some students will want to make them colorful.) Here is a picture of the angle finder I made in class. Mine is kind-of boring, but one of my table mates made a super cute one. 

That's it! Super easy but it works. Confused on just HOW it works? Well, so was I at first. Here is how it works - you line a corner up with the vertex (point) of your angle and the edge of one of your rays (lines). If one the other ray lines up with the opposite edge of your angle finder then the angle is a right angle. If the other ray can not be seen anymore because it is now covered by your angle finder than it is an acute angle. If the other ray is sticking out past your angle finder then it is an obtuse angle. Here are some pictures of what I mean. Here is a picture of two angles - ∠ABC and ∠ABD.

When you put your angle finder on the vertex of ∠ABC, you see that ray AB is no longer visible. Therefore, ∠ABC is an acute angle.

However, when you put your angle finder on the vertex of ∠ABD, you see that ray AB is still visible. Therefore, ∠ABD is an obtuse angle.

Like I said, super easy, inexpensive, and fun to make. Have your students take their newfound math tool around the room and measure angles they come across. They can measure the angles of chair legs, computer screens, doorways, anything really. It's a great activity to not only tap into their creativity (by decorating their angle finders) but also gets them up and moving around, which always helps with learning comprehension. Happy teaching! 😊

Quadrilaterals

A quadrilateral is a four sided polygon. Probably the shape most people think of when thinking about a quadrilateral is a square but there are other shapes that fit into this category as well such as a rectangle, a rhombus, a trapezoid, and a parallelogram. Can a quadrilateral have two (or more) classifications? Let's say your friend Sally McKnowitall told you that a square is a rectangle. Would you believe her? In order to figure that out we must first look at the characteristics of each type of quadrilateral. Below is an example of a sheet you can pass out to your students to have them fill out.

To see if Sally is right or not let's compare a square's characteristics with a rectangle's characteristics.

Square

• 4 equal sides
• Opposite sides are parallel
• 4 right angles

Rectangle

• 4 sides
• 4 right angles
• Pairs of opposite sides are equal
• Opposite sides are parallel

As you can see, a square has 4 equal sides. A rectangle has four sides and opposite sides are equal. A square has opposite sides that are equal, they just happen to be the exact same size as the other pair so technically that characteristic of a square does fit into the rectangle classification as well. Both the square and the rectangle have to have right angles and have to have opposite sides that are parallel to each other so a check goes in both those boxes as well. So, your friend Sally McKnowitall was correct - a square IS a rectangle as well. However, the only rectangle that is also a square would be a rectangle that has four equal sides since if the sides are not all the same length it does not classify as a square.

Now let's look at the characteristics for a rhombus.

Rhombus

• 4 sides
• 4 equal sides
• opposite sides are parallel

A rectangle can not also be classified as a rhombus unless the rectangle has 4 equal sides. Most rectangles would not fall into this category since most rectangles have parallel sides of different lengths. A square could however be classified as a rhombus as well. A square has four equal sides and opposite sides are parallel. A square just happens to have 4 right angles as well so therefore, the only rhombus that could be classified as a square also is a rhombus with 4 right angles as well. 

Now let's look at the characteristics for a parallelogram.

Parallelogram

• 4 sides
• Pairs of opposite sides are parallel
• Opposite sides are equal

A rectangle, a square, and a rhombus all have opposite sides that are parallel. They all also have opposite sides that are equal so all three can also be classified as a parallelogram as well. 

Lastly, let's look at the characteristics for a trapezoid.

Trapeziod

• 4 sides
• one pair of unequal parallel sides

Since a square, a rectangle, a rhombus, and a parallelogram all need to have opposite sides that are equal, none of them can also be classified as a trapezoid. The only thing that can be a trapezoid is a trapezoid and a trapezoid can not be classified as anything other than a trapezoid. 

I hope this helped you learn quadrilaterals. Having your students fill out the sheet really helps them to understand the material better because having a list like that all in one spot makes it easy to compare characteristics. You could also have your students fill out a quadrilateral family tree like this to help them understand the relationships better. Happy teaching! 😄