STEM – Floating Through Air (Mini-Math Lesson – Algebra 2)

This weeks final STEM activity is also a relatively easy one for students to do on their own. And it is a lot more exciting and realistic to collect your own data and analyze the results than to just use given data. Makes the math more ‘real’. However – as with all the activities this week, there is provided sample data.

Today’s activity has to do with falling objects and how gravity and area have an impact on the resistance and acceleration of that object as it falls. Materials needed are pretty basic – a book of some sort and a piece of paper, measuring tape on a vertical wall, and either a stop watch or a camera (video) to record the fall of the object when dropped from a 6 foot height. Video probably works best because you can slow it down and look at the height frame by frame.  At least two people would work best.

The underlying understandings for this activity have to do with gravity and an object falling towards the surface of earth. If gravity is the only influence acting, then acceleration is always downward and has the same magnitude for all objects. An object falling toward the surface of Earth will fall 32.18 fee per second faster every second (32.18 ft/s^2). Students will explore how area of an object increases the drag, which than impacts the terminal velocity (so parachutes have a lower terminal velocity than say a bullet).

Students will first drop a book and record it’s height as it falls. Then, they will drop a piece of paper from the same height and record it’s height as it falls. They will then fold the paper, and repeat the drop for each fold, which is decreasing the area of the paper and thus should decrease the terminal velocity. They will compare the data by making scatter plots and consider when the falling object might have a constant terminal velocity (speed). They will look at different parts of the graph to see where the data is in a straight line, which indicates when the force of gravity is equal to the air drag force.  It’s a fun little experiment and relevant as well.

This activity is adapted to ClassPad.net from the Fostering STEM In Education with Casio Technology, Casio 2013. The links below include the ClassPad.net version of the activity, the PDF version of the original activity, that includes more description and background information as well as calculator tips and strategies, and then a video overview of the ClassPad.net activity, showing how to create the scatter plots, zoom in and out with the plots to look for when the data is becoming linear.

  1. STEM – Floating Through Air (Scatter Plot, Drag, and Gravity)
  2. STEM Floating Through Air (PDF)
  3. Video Overview – STEM: Floating Through Air (Scatter Plot, Tables, Drag, Gravity)


The tool being used in these mini-math lessons is the FREE web-based math software, ClassPad.net.

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

STEM – Pendulum Exploration – Mini-Math Lesson (PreCalculus)

Today’s STEM lesson is at a PreCalculus level, where we are exploring the graph of a a pendulum swing. The image to the right is a short GIF of a pendulum swinging. It’s hard to tell because this is a short GIF, but the pendulum will swing because of gravity, over the center point and, if there were no air resistance friction, this swing would continue indefinitely. The motion of a pendulum has been traced before by others, and forms a regular periodic curve, and because of the damped motion (slowed by friction), the motion of the damped pendulum can be modeled by a sine function with a decreasing amplitude over time.

The activity today explores the proportion that the amplitude is decreasing, given by an exponential function. Students will graph the function and explore the position of the pendulum at given times. They will explore the swing positions of the pendulum using the graph and ratio of the changes in the peaks.

We will explore a bit more with pendulum and force in tomorrow’s STEM lesson as well, with a focus more on the force of a pendulum pushing on an object as it hits it.

Here is the link to the activity and also the video overview:


The tool being used in these mini-math lessons is the FREE web-based math software, ClassPad.net.

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below:

Mini-Math Lesson – Statistical Calculations and Fractions with the fx-260Solar

I am finishing out this week of scientific calculator explorations with a look at the fx-260 Solar. On the surface, this calculator looks really simplistic – almost four-function if you look at size and cost. But it can do so much so its looks are deceiving. The old ‘don’t judge a book by it’s cover” comes to mind!

Here are just a few of the capabilities of this little ‘engine that could’:

  • Statistical Calculations (mean, sum of values, sum of the square values, sample standard deviation, population standard deviation)
  • Fraction calculations (including mixed-number) and conversions
  • Constant Calculations (i.e. setting a constant and then using it in several different calculations)
  • Trigonometric and Inverse Functions
  • Exponents and Logarithms

There are more, but what I wanted to focus on today were two of these: Statistical Calculations and Fraction Calculations. I have made two quick videos. Having to use my document camera to record since this calculator does not have emulator software, so apologize if the lighting isn’t all that great. It is hard to record and see what you are doing with a document camera – I am learning a lot!!!

  1. Entering Statistical Data in the fx-260 Solar and Statistical Calculations

     2. Working with Fractions with the fx-260 Solar Scientific Calculator


Be sure to visit Casio Cares: https://www.casioeducation.com/remote-learning

Here are quick links:

Mini-Math Lesson – Time Calculations and Conversions with fx-300MS Plus2

Trying something new in this lesson – working with a document camera! So I apologize now for the awkwardness of it!!  I am trying to share the different scientific calculators, and one that really is a great tool and doesn’t require anything but the sun/light for power, is the fx_300MS Plus and Plus 2 This is one of the two scientific models that doesn’t have emulator software, so I am going to use the document camera to show you the steps, something I know many teachers have in their classrooms to show student work or do their own work for students to see.

The activity I am sharing is called Travel Plans, and it is all about time conversions and calculations. How do you determine how much time you need to do several different tasks (so adding time)? How do you determine the time you need to wake up based on having to do several errands and also leave at a specific time (so subtraction of time). Students often struggle with time conversions and getting decimal answers – what do the decimal values represent in time? If you divide 85 minutes by 60, you get 1.41666….  What does this mean? The common misconception is that you have 1 hour and 42 minutes (rounding), just assuming the decimal component represents minutes, instead of the actual understanding that the decimal represents a fraction of the hour, or a fraction of 60 minutes. 42 hundredths is not quite half an hour, so an estimate is a little less than half of an hour or 30 minutes, so maybe 25 minutes (which it actually is). Using the calculators ‘sexagesimal’ button (my new word of the day!), you can actually show the conversion of the 1.41666… to 1 hour and 25 minutes and 0 seconds. Back to the new word (for me) – it means “of, or related to, or based on the number sixty, which makes complete sense since time is based on 60 – i.e. 60 seconds in a minute, 60 minutes in an hour. So using this button that looks like the degree symbol and three commas, makes sense for doing time conversions.

Here is the link to the activity and the short video on using the sexagesimal button:


Be sure to visit Casio Cares: https://www.casioeducation.com/remote-learning

Here are quick links:

Mini-Math Lesson – HS Exploring Rational Exponents fx-115ESPlus2 Scientific Calculator

Today I am continuing my look at the functionality of scientific calculators by diving into more activities around expressions. I am using the fx-115ESPlus2 scientific calculator. This scientific calculator is really a powerful one for high-school level mathematics courses because of some of it’s more advanced capabilities such as being able to do numeric integration/differentiation and solve polynomials, just to name a couple. We are just going to look at two different aspects – the ability to use the calculator to evaluate expressions (in this case, exponential) relatively quickly, and the ability to make a table of values and compare two functions.

I am only exploring one activity today because there are a lot of important connections being made to student prior knowledge and also their ability to discern patterns. The activity is focused on applying prior knowledge and understanding of inverse operations and inverse numbers (reciprocals/additive inverses) and integer exponent properties, and how these all work to help make sense of rational exponents. These are important connections, and the activity provides students the chance to collect data, compare and look for patterns, and develop their own understandings about rational exponents and how they relate to radicals and integer exponent properties. The goal is for students to make the connections themselves and then be able to apply their understanding to expressions involving rational exponents.

Here is the link to the activity and a video that shows the two main features of the fx-115ESPlus2 that are utilized in this discovery lesson.

  1. Let’s Be Rational and Get To The Root
  2. Video Overview – fx-115ESPlus2 – Table of Values and Evaluating Expressions


Be sure to visit Casio Cares: https://www.casioeducation.com/remote-learning

Here are quick links:

Mini-Math Lessons – Middle School Equivalent Expressions and Exponential Expressions

If we are thinking about equity and getting technology in the hands of students, scientific calculators are powerful tools that can support math discovery and exploration (except graph, though the fx-991EX even does that!) and are really inexpensive (between $10 – $20). This week I am going to be showing some of the capabilities of scientific calculators that I don’t think many people realize they can do – i.e. equations, expressions, inequalities, and statistics.

Today I am going to share two  activities that utilize the fx-300ESPlus2, though you could use these with any scientific calculator from Casio (or graphing for that matter). Both these activities focus on expressions with a goal of understanding what expressions are, equivalent expressions, the role of variables. The first activity, It’s All the Same, is about using properties of operations (associative, commutative, distributive) to explore equivalent expressions. The calculator is used as a verification tool, where students use stored values for given variables that then allow them to check that different expressions yield the same result. The second activity, The Variable Game, focuses on exponential expressions, and helping students apply properties and write exponential expressions with given parameters. They use the calculator to store variable values and then use these as either the base or the exponent to try to create expressions that fit specific criteria. This allows them to get a better understanding about exponential rules and working with whole number exponents and bases. Both activities, as you will see, include standards, warm-up exploration questions, suggestions for student discussion and ways to extend. Each activity has an application and then some practice problems.

Here are the links to the two activities and a video overview that goes through some of the functionality of the fx-300ESPlus2 that is needed in the activities.

  1. It’s All the Same
  2. The Variable Game


Be sure to visit Casio Cares: https://www.casioeducation.com/remote-learning

Here are quick links:

Mini-Math Lessons – Proportional Reasoning: Comparing Rates and Looking At Scaling

What size frame do I need? Why is that candle burning faster than the other one? These are questions we are going to explore today. And it all has to do with rates, and proportions, along with other factors such as type of wax for the candles. Proportional reasoning comes into play in seemingly mundane things, like determine the size frame needed for a picture that you might be enlarging (or shrinking). Like yesterday, it’s about comparing and using mathematics to help understand and model real-world situations. What I love about these types of problems is that they can be approached several different ways, and each way can provide a different perspective and answer because you get more and/or different information. This is what modeling with mathematics is really all about.

Both activities today, as with all the activities this week, are adapted from Fostering Mathematical Thinking in the Middle Grades with Casio Technology, Casio 2011. I have made ClassPad.net version of them, but if you have handheld calculators, these same activities are available in the free Math Activities under graphing calculators for middle school at the Casio Education Website. The first activity has to do with two candles, the same height to start, but burning at different rates due to different types of wax. Students will explore fractions by looking at the fraction of each candle that is burned. They will compare using tables and graphs and use proportional reasoning to determine things such as when is one exactly half of the other. The second activity has to do with wanting to frame an image, and depending on the room it is to go in, the image will be sized-up or sized-down, so how much framing is needed and how much glass is needed?  This is a perimeter and area ratio problem and there is some nice simulations that students use to collect data on side length, perimeter, and area as side length increases. From experience, I know students struggle with the understanding that if you double the dimensions (length/width), that perimeter also doubles but area quadruples (exponential). The data collection and looking at the tables and graphs

Here are the links to the two activities and the video overview that explores the activities and some of the ClassPad.net skills/features:

  1. Proportional Reasoning – Burning Bright
  2. Proportional Reasoning – Stretch That Picture

 


The tool being used in these mini-math lessons is the FREE web-based math software, ClassPad.net.

Remember – if you want to save and/or modify any of these activities, create a free account.  Some useful links below: