Tag Archives: math

karengreenhaus 10:08 am on June 5, 2020
Tags: , , , Drag, Gravity, math, , , , Table,   

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:

karengreenhaus 10:00 am on June 4, 2020
Tags: , ball bounce experiment, , energy, , , math, , , , simulation, ,   

STEM – Conservation Is Not Just About Recycling (Mini-Math Lesson On Energy)(Bouncing Balls, Data & Regression)

Another STEM experiment today that students can easily do with just a few tools/materials:

  • Four different types of balls (so think tennis ball, basketball, ping-pong ball, racket ball, golf ball….)
  • Paper to cover an area of a wall so you can put measures on the wall (or use tape, or tape a measuring tape to a wall vertically).  We are going to be dropping the balls from given heights and record the height of the first bounce, so need to measure vertically.
  • Measuring tape to measure and mark the wall in 1-inch markings up to 6 feet

The idea behind the lesson today is to explore the difference between expected kinetic  energy and observed kinetic energy. Students will record the data of the balls dropped from different heights and their rebound and look at different scatter plots (Drop Height, Rebound) and (Rebound Height, Drop Height), find regression lines and analyze the meaning of the slope in the context of the situation.

There is also an extension activity, where they look at successive rebound heights of a balls bounces when dropped from a given height. This time they will see an exponential relationship (versus linear) and talk about what this means in terms of energy. The whole experiment is exploring the conservation of energy and momentum.

In both parts of the activities, students are encouraged to use their own materials and collect their own data – this obviously makes things a lot more fun and engaging. However, sample data is provided as well if they don’t have the materials. There is even a ball-bounce simulation provided for the second part (successive bounces), using the ability to insert images into ClassPad.net and sliders to control movement.

The activity used is adapted from an exploration in Fostering STEM Education with Casio Technology, Casio 2013. I have made a Classpad.net version here, link provided below, and also provided the PDF of the original activity which goes into more detail and provides some hand-held calculator tips and suggestions. The link to the PDF is also below along with a video overview of the activity in the ClassPad.net version.

  1. STEM-Conservation is NOT Just About Recycling (ClassPad.net Data & Regression)
  2. STEM Conservation Is Not Just About Recycling (PDF)
  3. Video Overview – STEM – Conservation is NOT Just About Recycling (Data Regression Simulation)

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:

karengreenhaus 7:30 pm on June 3, 2020
Tags: , , , F=ma, , , , math, momentum, Newton, pendulum, , ,   

STEM – Newton Knew Forces (F=ma) (Mini-Math Lesson Algebra/Algebra2)

Yesterday’s lesson was related to the motion of a pendulum and it’s graph as its swing was impacted by friction. Today we are looking at force, using a pendulum as the push (force) on an object (scooter). This is a fun little experiment that you could do with students, and in this time of home-schooling, it would be a relatively easy experiment for parents to set up with their kids. You just need some rope, a bucket full of water or even a jug of water to use as your pendulum, a door frame to swing the ‘pendulum’ from, and then a scooter or something with wheels that can move when pushed (so a wagon or a skateboard). You will also need a measuring tape, some weights to add to the scooter (three different weights), and something to mark start lines and height release for pendulum.

If it’s already sounding like too much work, don’t worry!! There is sample data in the activity, so you can still explore the mathematics if you don’t have the time to set up!!

The idea is to set up the pendulum so that it hangs from a door-frame/beam at the height of the scooter (so place scooters back end directly at where the pendulums lowest point so that the pendulum will hit the scooter and push it). Measure the weight of the pendulum bob (bucket/jug of water) before you begin. Measure the weight of the empty scooter. You then decide on a release height for the pendulum (will be the same height for each release). Mark the start position of the scooter (back end), position the scooter (empty) and release the pendulum from the designated height. It should hit the scooter and push the scooter. You then measure the distance the scooter traveled after it was hit (so from the start position) to where it ends (be sure to measure at the back of the scooter, to be consistent with the start position). The next step is to add some weight to the scooter (make sure it is secured on so it doesn’t fall off when pushed!). Repeat the experiment, record the distances and weights.  Do this for at least five different weights of the scooter – empty to heavier. (Don’t worry – I have included the PDF as well with all the detailed instructions).

Once data has been collected,then you will graph the data and look at how force and mass impacted acceleration. The goal of this experiment is to show that the rate of change of momentum of an object is proportional to the resultant force acting on the object and in the same direction.  Students will explore their table of data, make scatter plots, look at the relationships. The experiment has to do with Newtons’s 2nd Law, where Force=mass x acceleration. There are a couple things to keep in mind:

  • The pendulum weight represents the force (because you don’t change the drop height)
  • The scooter weight is proportional to the mass (because gravity is constant through the experiment)
  • The average distance the scooter moves is proportional to the acceleration
  • Weight is used in the experiment instead of mass, but for this experiment it is acceptable to deal with weight because the force of acceleration is constant throughout the activity. The weight in this activity is always proportional to the mass.

This activity comes from Fostering STEM Education with Casio Technology, Casio 2013. I have converted part of the activity (the pendulum component) to a ClassPad.net activity, which is shared in the link below and also overviewed in the video below. I have attached the complete activity, which includes a wagon-pushing activity. The PDF is the whole activity with a lot of description and calculator suggestions, as well as sample data.

  1. STEM – Newton Knew Forces (F=ma) ClassPad.net Activity
  2. STEM Newton Knew Forces (F=ma) (PDF)
  3. Video Overview – STEM Newton Knew Forces (F=ma) Mini-Math Lesson (Data & Regression)

 

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:

karengreenhaus 6:59 pm on June 1, 2020
Tags: , , , , , inertia, , math, , real-world STEM, , , ,   

STEM – Inertia, Force and Velocity – Newton Knew Inertia (Mini-Math Lesson)

I wanted to focus on some STEM lessons this week, using ClassPad.net, since it is so great for collecting data, showing statistical plots, and it’s ability to quickly change things to see the impact and more importantly, to do everything in the one activity (i.e. calculations, data collection, graphs, and explanations). So, I am going to share a different STEM focused lesson each day this week.

STEM is an acronym that stands for Science, Technology, Engineering, and Mathematics. It’s really about ensuring educational experiences that blend these four areas, so that learning is not in isolation, but rather a connected learning on real-world concepts, that require students to problem-solve, collect data and analyze results, and communicate their findings and use critical thinking. There are many definitions and reasoning behind the push for STEM in education – here is one article that I think gives a good overview if you feel you need more information.

For me, STEM means learning and problems that are real-world, that students can actually do or experiment or relate to, that require them to use science, math, technology and engineering in realistic ways or situations. It’s looking for and collecting evidence, and then modeling this information, and applying understandings of the subjects to make sense or make decisions. The activities I am going to focus on this week come from Fostering STEM Education with Casio Technology, Casio 2013. It is a resource with some really great real-world problems and explorations, and these can be done by students or there is also sample data provided if you don’t have the materials needed to do the experiments yourself. So you can make these activities as hands-on as you want, but if not possible, still have the great discovery and conversations and critical-thinking experiences needed for deep learning and application of STEM concepts.

Today’s activity is one that you could very easily do with students (and it even adheres to social distancing rules!).  In the image above, you will see it involves six students – five to stand at designated and one to push the object in a straight path.  This activity explores how the amount of push impacts inertia and acceleration. There is an object that starts from static position (so a disc (like a frisbee) or ball), and then a student pushes it in a same line with as much consistent push as possible, and as it passes student at the set up positions, they time when the object passes.  The force is changed for three trials and then students compare the data in several ways. So, a fun activity, but, if you are unable to do this with all the materials, or students, then you can use the sample data provided.

Here is the link to the activity, both the ClassPad.net version and the PDF that can be used and a 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:

karengreenhaus 10:00 am on May 29, 2020
Tags: , , , , math, , , mixed numbers, , ,   

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:

karengreenhaus 5:26 pm on May 28, 2020
Tags: , fx-300MS Plus2, math, , , sexagesimal, , time calculations, time conversions   

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:

karengreenhaus 5:57 pm on May 27, 2020
Tags: , , , Evaluating expressions, exponent rules, fx-115ESPlus2, , math, Rational exponents,   

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: