Real-life Examples of Solids of Revolution and Cross-Sections

I get so much out of creating lessons. When you deeply understand a concept, you can talk about real-life situations and the math behind it. I’ve never thought about it until now, but a revolving door is a perfect example of a solid of revolution. Taking a rectangle and revolving it around a pole, creates a cylinder. What a beautiful example!

 

The last time I was at an AP Calculus seminar, I learned that another awesome example of a solid of revolution is a honeycomb decoration used at parties. These are perfect examples because you can see the 2-D version before it is rotated. This is exactly what we want students to know. What does the 2-D version become with you rotate it? If they can visualize that, then they get it!

 

Another concept that goes hand-in-hand with 3-D figures is the idea of cross-sections. Everytime we slice an orange, apple, or a loaf of bread, we have created a cross-section. In Algebra 2 and Pre-Calculus, we discuss cross-sections of cones. Depending on how you slice the cone, you can get a circle, an ellipse, a parabola or a hyperbola. 

 


 

 

 

 

A great hands-on activity for cross sections is to have the students create a shape out of play-doh. Take a piece of dental floss and slice the object horizontally, vertically or even at an angle. Be sure and have them make predictions before they perform the experiment! Students with phones can take a before and after picture so that other students can see. 

 

 

 

 

 

I enjoyed creating my Intro to 3-D Figures resource. It’s amazing how after teaching math for many years, that I can still pick up valuable insights and ideas. Math is infinite. There is no end to what you can learn! 

 Intro to 3-D Figures

Intro to 3-D Figures

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5 Ways to Gain Control of Your Classroom!

At certain times of the year, you may feel like you’ve lost control of your classes. Toward the end of the year, you’ve become lax and the kids are getting too comfortable. Students are talking while you’re teaching. You have a hard time getting their attention. Phones are visible along with earbuds inserted. You are miserable and little worried about how you are going to make it through the rest of the year. I have 5 easy things to try to get you and your students back on track. You will realize how easy these things are to do and wonder why you haven’t done them all along. Sometimes even teachers lose focus and go into survival mode. Survival mode is a hard place to stay. Try these 5 things and I promise you will be happy and you can finish the school year with a confident attitude.

I. A seating chart is a must!

It doesn’t matter what grade you teach. Seniors? Juniors? 7th Grade? Doesn’t Matter! You need a seating chart if you’ve lost control. Look at the seating arrangement of each class. Have they started sitting where they want to sit? Are they too comfortable with the students around them? Do you have them in groups? Have you ever had them in a seating chart?

My suggestion is to put your classes in rows for a while and decide where you want each student to sit. Who needs to be in the front? Who can sit in back? Who does not need to be by each other? Strategize! The kids will moan and groan, but ignore it because you’re going to see good results from this seating chart. You’ll probably have a few kids come up to you and say that they can’t see from their new seat or they can’t sit by a certain person. You will hear some excuses, no doubt. For right now, just tell them to sit where you put them. (You are in charge!) If you see a problem with where some kids are sitting, YOU DECIDE when to move them!

II. Getting Their Attention!

Now that they are in their new seats, it’s time for a discussion. You are going to explain that they will not have their phones out, unless they have permission. You do not want to see the phones or see the earbuds or you will take them up and either turn them into the office or not give them back until the end of class. Now, if you say this, you have to follow through. The first person that messes up needs to be made an example. This will get the attention of everyone. Don’t budge on this!

The other thing you are going to tell them is that since you’ve had problems getting their attention lately, you will put your hand up when you need their attention. When they see your hand up, they put their hands up too until everyone has their hand up and mouth’s shut! (Seniors can do this! Adults can do this! Kids can do this!) Tell them that you will not talk over them and until they are quiet, you will stand and wait. If they choose not to listen, then you will hold the class after the bell. (Yes you can do this. You don’t have to hold them long. 30 seconds to a minute will get your point across.)

III. Walk the Class!

From this day forward, you need to walk the class. Don’t stand in front of the room and teach. Be everywhere. Talk to the class, then walk around and see if they are doing what you asked them to do. You can teach from any part of your classroom. Let students go up to the board. You don’t have to always do it. If you’ve never walked your class, you will be amazed at how this will change your classroom. Try to talk to each student at some point during the class period. Don’t hover over any student for long though. Keep moving and act like you are really checking out what they are doing. If you really want to put on a good show, get a clipboard and act like you are writing stuff down that you see. You can even walk around and put stars, stickers or check marks on things that you see that are good! You will have fun and the kids will too. All kids like stickers, smiley faces and check marks!

IV. Calling Cards!

Another awesome trick to keep your students on their toes is to randomly call on people to answer questions. There are random selectors on the internet, but I like to use index cards. It’s very easy to do. Put each student’s name on an index card. Shuffle the cards and randomly select students to answer questions. When I’m having kids work an example problem, I’ll warn them and say, “I’m going to randomly call on someone in a minute.” If you call on a kid that sits and looks at you because they don’t know the answer, still make them give you something. You can give hints, or help them out, but don’t let them get away without answering the question. All the students are watching to see how you handle the kid that says, “I don’t know.” This will get your students engaged, because they don’t want to be the kid that doesn’t have an answer. If students give me wrong answers, I still praise them. I don’t want anyone to be afraid to answer a question because they are afraid of being wrong!

V. When it’s Time to Go!

Be in charge…even when the bell rings. Did you know that the bell doesn’t release the kids? YOU DO! I tell the kids that they are not allowed to leave the room, until they hear me say, “Have a nice day!” I keep them in their seats until I inspect the room. Is there trash on the floor? Did they put up any materials they used? One trick, I’ve learned to do and yes it works with any age, five minutes before the bell I say, “I’m inspecting the room and the cleanest, quietest, straightest row will get to leave first.” This works great on that rowdy class that always leaves your room in a mess. I usually walk around and point out trash on the floor and other things that need to be corrected. When the bell rings, I’ll pick one row to go. I don’t release another row until the first one is gone. When I’m down to 2 or 3 rows left, I usually say, “Ok, everyone can go now.” If you do this a few times, they respond well. If there’s one kid talking and keeping their row from leaving, I’ll say that everyone can leave, but “Johnny.” I’ll let everyone go and Johnny stays behind. It’s entertaining to say the least, but I promise it makes a difference.

Don’t be miserable, take charge. It’s never too late. Trust me, I’ve made every mistake possible. I’ve learned that being FIRM is the key. I left many days, almost crying because my day has gone terrible, but it was my fault. Take control. Kids will respond as long as they see you are firm and fair. I hope the rest of your year goes well. If you would like a complete discipline plan for secondary students, see the link below.

Area for Secondary Students

Area is taught in early grades. The concept of adding up all the square units that COVER a figure is actually pretty simple. Students need to understand WHAT area is before really delving into using the formulas and then taking them a step forward.

In secondary grades, we begin to use the formulas and study how areas of composite figures can be found as well as finding shaded areas. These types of problems bring in the “real-life” and “problem solving” component. I tell my students all the time, “You must know how to apply the problem and you must know how to think about it forwards and backwards.” (If you are given the sides, find the area. If you are given the area and one side, find the missing side.)

To take area a notch further, we throw in algebra. Instead of a side being just 4, we might make it 4x. We will talk about how area is quadratic, volume is cubic and perimeter is linear.

Before students get to pre-calculus, they also need to how dimensional changes affect the area. If all dimensions change, the area changes by that amount squared! So if all sides are doubled, then the new area will be quadrupled (or doubled squared). If one dimension changes, then the area is only affected by that amount!

I’ve been hard at work creating my Geometry Curriculum. My most recent resource covers area. The resource practices finding area of rectangles, parallelograms, squares, rhombi, triangles, and trapezoids. These figures are combined into composite figures and problems are worked forward and backwards. There is a special emphasis on regular polygons. Regular polygons are a great way to practice special right triangle rules and trig! There are tons of notes, practice and quizzes in this resource. If you need something like this, then go to my TpT store and check it out.
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Area of Polygons

Patterns in Nature

 A pattern that occurs over and over in nature is the Fibonacci Sequence. The pattern is this: 1,1,2,3,5,8,13,21,34,55… Do you know the next number in the pattern?

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When you make squares with those widths, you get this cool spiral:Fibonacci Spiral

Look at these beautiful images that follow this pattern:

Messier 83, a spiral galaxy located 15 million light-years away from Earth.

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A spider form formed using a spiral shape.

Image result for fibonacci in nature flowers

Image result for fibonacci in nature flowers

The pattern does not always appear in a spiral.

The Golden Ratio and Fibonacci Sequence go hand in hand…pardon the pun.

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Divide 8 by 5, Divide 21 by 8, Divide 55 by 34… choose any two numbers in the sequence that are next to each other and divide larger by smaller.

Pi Day Activity

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This coloring activity is a fun way to practice area and circumference problems. There are 16 problems in all. The students are asked to leave some answers in terms of pi. There are some problems that area is given and the student has to find circumference and vice-versa. The back of the activity has adding, subtracting, multiplying and dividing with pi. The variety of problems makes this resource fun but challenging!

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Buy This

Exponential Functions Activity

Slide1Exponential Functions Activity

This was my very first resource that I put on TpT. I love this activity. Students have fun making a poster and using multiple representations. They are given several exponential scenarios to choose from such as a basketball bracket, a cockroach infestation, a zombie takeover, friendship bread, chain letters and more. Here’s a sample poster of a football stadium after a game has ended:

go-tigers

After students look at their choices, they may come up with their own situation. This is what differentiation is all about…students making decisions based on interests!

When I was getting my masters, we were encouraged to study exponentials in depth. This opened my eyes to all types of situations that are out there. I was inspired to create Exponential Functions: 6 Stations. Look at the titles of the stations below. Have you ever played Tower of Hanoi? There are free phone apps and online math sites that have this interactive game. Click here to see what I’m talking about! The number of discs and the least number of moves to get all the discs to tower 3 is an exponential function! How cool is that! Kids love games, so when you introduce this they will want to play for hours. Some of my buyers have turned these stations into other activities. After you see them all, you can decide for yourself how it works best for your room. These are not quick stations, they are more like math centers!

6-stations

I still was not finished creating exponential resources. I couldn’t get enough! I wanted a unit! I wanted that unit to be about multiple reps and I wanted it to include transformations, compound interest and compound interest where the compounding is continuous which would bring in the constant e. I wanted another data collecting activity because there needs to be practice writing exponential functions. At the end of the unit, I wanted an assessment. I believe in differentiation, so I created 5 assessments that cover the same thing but with different difficulty levels. The assessments are a part of my unit in my TpT store, but I also sell them separately. The unit also contains warm ups and cool downs.

I make the unit a booklet and let the students work through it at their own pace. There are a few places in the unit, when the teacher needs to stop the class and have some discussions, but for the most part, the students can work through this on their own. Once they’ve made it through the booklet, we trade and grade. Students will know exponential functions forwards and backwards by the time they are finished with this unit.

Here are the links to my Exponential Functions Unit and my Exponential Functions Bundle (which saves you money and has all the resources in it plus 20 task cards),  but I’d like to leave you with a freebie. This is an exponential activity that I have done over the years. It’s fun, interactive and tasty! Enjoy!