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! 😄

Mind Maps

While learning about statistics and the different ways to display the data, one of our assignments was to construct a mind map. I had honestly never even heard the term "mind mapping" before so I had to do some internet research to even figure out how to construct one. A mind map is basically a thought bubble with several different branches and subcategories relating to the original thought. Click on the link for some helpful mind map guidelines.

Here is the list the teacher gave us of mandatory terms we had to use in our mind map...

Mind Map Required Words: 

Bar Graph 

Pictograph 

Pie Chart / Circle Graph 

Histogram 

Stem and Leaf Plot 

Line Graph

Scatterplot 

Dot (or Line) Plot

Frequency Table 

Box and Whisker Plot 

Categorical Data

Numerical Data 

Discrete Data 

Continuous Data 

Mean 

Median 

Mode 

Range 

Mean Absolute Deviation 

Optional Words: 

Data, Outlier, Cluster, Gap 

Just looking at that list gave me anxiety because I knew that several of the words could relate to multiple different items so my mind map was going to be so clustery and I am not a clustery type of person. I like neatness and order so I was dreading having to do this assignment. 

The teacher said we could create one on paper if we liked but also gave us several different websites that have mind mapping software you can use such as popplet, mindomo, mindmeister, and bubbl.us. In the end I chose bubbl.us to use but when I tried to construct my mind map strictly on the website I found it very difficult to organize my thoughts so I ended up making a rough draft on paper first. Below is a picture of my rough draft. It may be hard for you to read it because I started to run out of room so I had to write pretty small for some of bubbles. I also had to circle each bubble in colors that matched each stem so that I could differentiate between the different categories since everything ended up so close together. All of the bubbles are colored in with pink because that was how I knew I recorded it when I was finally putting it into the final draft on bubbl.us. 

I'm not going to lie, this rough draft took me several hours to construct. I had to go back over the material, read the chapter in the text book, and look things up on the internet because I was still cloudy on exactly what type of information each data display used. I mean, I new what a bar graph was but I didn't know EXACTLY what the difference was between a bar graph and a histogram was until this assignment. 

Once I had my rough draft completed it was time to make my final draft on bubbl.us. This took me about 1 1/2 more hours to construct because not only had I never used this software before,  I also needed to clean it up a bit. This honestly gave me even more opportunity to really comprehend the information. Here is link to my completed mind map. I got full points for the assignment but my teacher did let me know that I had a line connecting variance and MAD and I shouldn't have since they are not related so just ignore that part. It honestly ended up being the best study guide for the mid-term. 

Even though this assignment took a long time to construct, I learned more about data displays in those couple of hours than I had in any math class ever so the assignment I had been dreading because it wasn't catered to my individual learning style ended up being one of the most helpful math assignments I had ever done. The lesson I learned as a future teacher was to offer activities and assignments that cater to all different learning styles because even the students that think they will get nothing out of the assignment will end up learning something new in the end. Plus, teaching students different ways to learn just opens up the door to creative thinking and reasoning in their future endeavors so don't be afraid to teach a learning style that may not be your forte because your students will ultimately benefit from it. Happy teaching! 😊

Graphing Activity

In my math class (teacher prep math class - second semester) we do LOTS of activities. These activities are designed to not only reintroduce the material to us students but to also demonstrate how to teach the material to our future students in ways that help them to understand the concepts better. Last semester when I realized that we would be doing activities nearly every class for two full semesters I wanted to just die because I hate group activities (I know, I know, teaching is one big group activity so no need to point that out to me haha.) The first few weeks were rough, I'm not going to lie. First of all, I purposely sat at a table of four that already had three out of the only four guys in the entire class sitting at it because I figured guys are usually good at math so they could help this old lady out with refreshing her math memory. So here I am, the oldest person on class (and I'm pretty sure you can include the teacher in that calculation as well) sitting at a table with guys that are younger than own my son. Luckily for me, they all were really cool and that semester was a blast! Seriously, I have never had so much fun in a class or with group activities in my life. Shout out to Uriel, Jacob, and Michael for breaking me out of my group activity phobia. You guys are awesome!

Probably one of the most popular group activities this semester was graphing M & M's. The teacher handed out an activity sheet and a Halloween size bag of M & M's to each student in class. Before opening the bags, we were asked to make conjectures as to what color we thought would occur most often, what color we thought wold occur least often,  and if we thought our bag would match the rest of the class data. After we made our conjectures we were instructed to open our bags and record the data - first by placing the M & M's on the graph provided in the activity sheet and then by coloring in the graph with crayons like this...

This type of graph is called a pictograph which is basically a picture graph. Once this was complete we were allowed to then eat the M & M's....yum! Once everyone had completed their pictographs it was time to collect all of the class data. We did this using another kind of graph called a dot plot. Dot plots are a super easy way to collect data from a group. We have 24 students in the room but if you had a really large group it probably would not be the way to go since you have to be consistent on the placement of the x's otherwise it would be kind of hard to read. Basically you make a line graph, add your data points and then call on the students one by one to give your their individual data. In the picture below, each student gave a number for how many of each color they had in their individual sample.

Be careful when calculating the totals though because it is really easy to want to just count each individual x but one x counts for the number it is representing. For example, one x above the number 5 actually counts for five and not one.

Now that we had the class data it was time to move on to a pictograph and a bar graph using the new data set.

Notice the difference in the pictograph from my individual data set and the class data set? Since the class data set was much larger, we had to have a key at the bottom showing that each circle represented 10 M & M's. We had 92 green M & M's so had we not done this our pictograph would have either been too small to read (in order to fit on one page) or would have taken up several pages which is impractical. However, because we had 92 green M & M's and not a even multiple of 10, we had to color in part of a circle to represent the extra two. This can make pictographs hard to read exact data values.

Our bar graph was horizontal but you could also make your bar graph vertical, which is what most people are used to seeing. Note that the bars on a bar graph do not touch. If the bars touched this would be a different kind of graph altogether called a histogram. Bar graphs measure categorical data like the data we have here. We have our categories (colors of M & M's) on the horizontal axis and their frequency occurring on the vertical axis. Histograms measure numerical data (numbers) and their frequency occurring. For example, let's say we were wanting to measure the weight of every student in class. In my class, we have 24 students so in theory we could end up with 24 different weights. That would be a pretty boring chart because we would have 24 bars on the horizontal axis (bottom) and each bar would only go up to 1 on the vertical axis (side). It would be better to have a weight ranges (like 100 - 125, 126-150, etc.) and then chart the frequency occurring for each range. Since both the vertical and horizontal axis are numerical in value, our bars would touch each other and therefore our graph would be called a histogram and not a bar graph.

I hope I was able to explain the activity and the differences in the types of graphs in a way you were able to understand. It really was a fun activity and you really do learn concepts better with activities such as this. I also came across this website Teachers Pay Teachers where teachers buy and sell original education materials. Here is a link to a creating pictographs and bar graphs page on the site. The site has tons of other resources that could be helpful to you in your classroom as well. Happy teaching! 😊