# 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.

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 – 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.

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

# 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.

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

# 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:

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 – Modeling the Pythagorean Theorem with the fx-CG500 CAS Graphing Calculator I am going to end my week by looking at a very traditional model of the Pythagorean Theorem – i.e. a right triangle with three squares attached to each side of the triangle that visually and mathematically demonstrates why a^2 + b^2 = c^2. The Pythagorean Theorem comes into play in so many areas of mathematics. Since I’ve been focused on triangles the last couple of days, seemed to make sense to end on such an import theorem that is shows up consistently throughout a student’s journey with mathematics.

The tool I have used for all of this weeks activities has been the fx-CG500 CAS graphing calculator. A really powerful device (and yes, it’s allowed on SAT and AP assessments and states as well, though usually has to be put in Exam Mode so that CAS is not on). The emulator software is really great because you can put it in a resizable mode that basically gives a huge working screen, which makes it a nice tool for distance teaching. For students using the handheld, the screen is also large (compared to other handhelds). Having a full-on geometry software in your hands (well, really full-on all-over math software) is nice. As many have come to experience in this new age of remote learning for everyone, not all students have wifi, or computers, so handheld calculators are still an important and more affordable option than a laptop or tablet.

So – today is just one video, a little longer than the ones previously this week, about constructing a model of the Pythagorean Theorem. In this you will see how to construct perpendiculars, rotate objects, measure angles, sides, and area and change colors of objects to make a visual representation of the Pythagorean Theorem.

# Mini-Math Lessons – Points of Concurrency with the fx-CG500 CAS Graphing Calculator Keeping with my geometry focus this week, and my triangle focus in particular, I am going to explore how to construct the four points of concurrency using the geometry app in the fx-CG500 CAS graphing calculator. You can do the same constructs on Casio’s other graphing calculators as well, on ClassPad.net or other dynamic geometry software. I wanted to highlight the fx-CG500 since it is a handheld calculator, which means more accessible to many students, it has a really large view screen, with the stylus for doing the dynamic dragging, and, with the emulator software (free trial of this available through August 31, 2020), you can really enlarge the view screen, which is great for remote learning and teaching.

Points of concurrency are really interesting, and are great explorations for students. You can approach them in several ways – i.e. has practice for constructions, since they involve constructing angle bisectors, medians, perpendicular bisectors, and altitudes. But, they also have properties that can help in understanding other relationships within triangles and with other constructs, such as inscribed and circumscribed circles.

Here is a brief reminder of the properties: 1. Circumcenter
• The intersection of the perpendicular bisectors of the three sides of a triangle
• The circumcenter is equidistant to each of the vertices of the triangle
• The circumcenter is the center for a circle that circumscribes the triangle
2. Centroid
• The intersection of the medians of the vertices of a triangle to their opposite side
• The centroid is the center of the triangle where you can ‘balance’ the triangle (try it – balance a cut out triangle on a pencil at the centroid – fun to do with students)
• The distance from the centroid to each vertex is 2 times it’s distance of the centroid to the side of the triangle
3. Incenter
•  The intersection of the angle bisectors of the 3 angles of a triangle
• The incenter is equidistant to the sides of the triangle
• The incenter is the center of a circle that intersects the three sides of the triangle only once (inscribed circle)
4. Orthocenter
1. The intersection of the altitudes from each of the 3 vertices of a triangle

Obviously there is more to all of these – i.e. equations, Euler Line, etc., but I am not delving into these extensions – my focus is just on constructing and exploring these four points and doing a comparison. Helping students understand what each is, how to find it, and some of the basic properties related to location and distance to sides and vertices.

Each video below demonstrates how to construct, using construction tools of dynamic software, the four points of concurrency, using the fx-CG500.

Constructing the Incenter of a Triangle:

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

# Mini-Math Lessons – Basic Triangle Properties with the fx-CG500 Graphing Calculator Today I am taking a different approach to things, and not giving you a ready-to-use lesson. Instead, I am going to do to videos that show how to create basic triangles with the fx-CG500 graphing calculator. I am going to be exploring this tool a lot more because it is a really powerful handheld (CAS), and because of the stylus and screen size, you can really work with the Geometry App, which is a full-on dynamic math software. And, with the emulator, you can actually make the work space even bigger, which in this time of remote learning makes this tool another fantastic math software to use with students to demonstrate and help visualize. Right now, with Casio extending the free trial period for all it’s emulators until August 31, 2020, this is another amazing tool to add to your distance learning resources.

I decided on triangles because they are such an important geometric construct, not only in geometry itself, but as an underlying concept for distance, trigonometric functions, and many other algebraic concepts. The rest of this week, my ‘lesson’ are going to really be more how-to use the fx-CG500 geometry application to work with triangles in different ways. Today, we are going to focus on basic triangles – scalene, isosceles, equilateral, obtuse, right, and acute. The basic triangles students first learn about. From there we will go into right triangles and then points of concurrency. In the process, I will show you how to use the fx-CG500 calculator and Geometry App specifically, to construct perpendiculars, angle bisectors, bisectors, measure sides and angles and more. Each video will be a mini-lesson and how-to all in one.

Here are today’s videos:

Accessing the Geometry App on the fx-CG500:

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