In our IB Mathematics Interpretations and Applications class, students are not only learning new math concepts, but they are also using these concepts to make instruments and to solve problems they would not have been able to otherwise. The following is a project one of my students completed by constructing and using a clinometer to answer the given questions. A clinometer is an instrument for measuring angles above or below a horizontal line (angles of elevation or depression). A metal tube allows the user to sight the top (or the base) of an object, and a protractor-like device on the side measures the angle. It is used to take measurements of accessible distances and angles, and then use them to calculate inaccessible heights.

 

The clinometer

 

The road sign

The roof 

The road light

 

Measurement process

 

 

 

What factors do you think could have affected the accuracy of your measurements and final result?

The distance between me and the object must be as accurate as possible, because this distance is estimated by us, and the angle of elevation can be measured by protractor.

Suppose you are standing on the third floor of the building and want to find the distance to a building across the street. What information would you need and what measurements would you need to take in order to calculate the distance? Suppose you have the necessary information and have obtained the measurement. Draw a diagram and explain how you would find the distance. How would you calculate the angle of the depression if you know the side measurements?

I need the height of the bottom of the third floor where I am, the eye-level height, the distance between me and the building which I have to measure, and the angle of depression.

 

How are angles of elevation/depression calculated?

Look at the numbers on the protractor, and use laws of sin/cos/tan to calculate the length of the side.

Why are angles of elevation/depression useful?

You can directly put it into the formula to get the answer, which can omit complicated calculation steps.

Why some height measurements might be difficult to make?

If the object is too high to see the top of it, you have to stand farther, if you do that, that will be hard to measure the distance between you and the object.

 

AP Calculus BC 

One of the great things about teaching AP Calculus is helping students learn new methods of problem-solving. The first half of calculus is all about rate of change, with questions on related rates a typical problem calculus students will encounter on the AP test. I recently assigned a small project where students had to write an original related rates problem with its solution. Here is one of my student’s projects about an ice cream cone that is quickly disappearing!

 

Related Rates Project Problem

By Kurt Schimmel

Hello, I am Kurt Schimmel and I’m currently in 9^th grade taking AP Calculus. I was assigned to do a short project which involves making an original related rates problem that is creative, unique, and entertaining. The project that I made is about ice cream leaking out of the bottom of a cone. I got the inspiration for this when I was talking to my friends and we were chatting about what we had done in the summer. I thought of ice cream and thought I could use it for this project. From this project I gained a better understanding of related rates and how I could use them and I would recommend this to other students that are learning related rates.

Little Jimmy goes to the beach with his family on a particularly hot day. He plays in the sand for hours and then gets bored and dries off to go to the playground. He hears the sound of an ice cream truck in the distance and rushes to his mom to beg her to buy some for him. After about 15 minutes of whining and compromising, Little Jimmy finally got his strawberry ice cream. As he was walking back to the playground, his he noticed that his hands were getting sticky. He checked his ice cream cone and noticed that ice cream was dripping out of the bottom at a rate of 2 cubic inches per second. He panicked and ran back to his mom for help. He remembered that the ice cream lady told him that they only sold cones that were 8 inches high and that the top radius was 4 inches. By this time the ice cream level was halfway down the cone. Little Jimmy had no idea what this meant, but he was curious on how fast the ice cream level was dropping. Do Little Jimmy a favor and figure out the rate at which the ice cream level is falling.