# 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 – Proportional Reasoning: Juggling Peppers (Scale, Distance, Application) We’ve been focused on proportional reasoning all week. Hopefully you have noticed that rarely were any of the activities the ‘process’ of solving proportions – i.e. ‘cross-multiply then divide’. While that may have come up at some point to help answer a question, the activities really were about using ratios and proportional comparisons in real situations to make decisions or model a situation. Today’s lesson is more of the same, and I would say fits in the category of an application. It’s not straight forward – instead, students have to use their understanding of proportion and scaling to come up with their own scale for a situation.

The image in this activity is a fun one – a chef juggling peppers. For those of you who think this is unrealistic, I will tell you that my husband (an amateur chef for sure) juggles whatever produce he can find, be it eggs, vegetables or utensils (thankfully not knives). The activity starts with an image on a coordinate grid and the first thing students are asked to do is come up with reference points that would help them make an estimate of realistic distance from chef’s shoulders to his head. This involves so much more than just looking – i.e. where on the head? Do you include the hat? Where on the shoulders? What is an average distance? Would I need to measure people around me to get a more realistic measurement? Once they determine this realistic distance, then they look at the y-axis and determine what an appropriate scale would be (i.e. how many inches does each mark on the y-axis represent?). The cool thing about this is different students will have different interpretations, so you are not going to get the same answer, which leads to amazing discussions about reasonableness and approximations and proportion.

After students determine there scale, then they actually look at the peppers and determine the height each pepper is above the chef’s hands. This again is more involved than it looks. Where do they place the points on each pepper so that they are consistent? What distance are we measuring? Where on his hands are we measuring from? How does the scale enter in to our calculations?

It’s a fun problem and there is no one ‘right’ answer, which to me is the beauty of this because students are forced to justify their choices and work based on their own understandings and interpretations of the situation.

Here is the link to the activity and also the video overview of the activity.

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 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 – Proportional Reasoning: Comparing “Deals” and Making Decisions I am only going to focus on one proportional reasoning lesson today. Mainly because this particular lesson is something I think is really relevant for students in their every day lives. Not the specific topic – i.e. comparing the cost of online video games (though many would argue that is definitely important in many students lives). But more because this type of problem, where you are having to compare rates/costs of similar offerings and make a decision about what is the best deal, is such a prevalent part of our lives in so many areas. Think about buying a car – you may find the car you love, but how much you put down, how much you pay a month, may be completely different depending on different dealerships, and then should you lease or buy – that’s a whole other set of options. Basically, whenever you you are shopping around for the best deals, you are faced with this type of proportional reasoning situation, where you have to weigh initial costs with monthly costs, compare different company offerings, etc. It’s a skill that is important in life decisions, so this particular lesson today, while simple on the face of it, is really an important building block for future decisions.

In this situation, there is a choice to be made between three different online gaming companies that offer different deals for the online game you want to subscribe to. You have to determine which one is the best deal, which really factors in so many things – are you going for cheapest at the outset (short on cash) or are you going for cheapest overall or is length a consideration versus cost? In the activity itself, we help students look at unit rates and use different representations to make sense of the three deals and mathematically sound reasons for their choice. The nice thing is, there really isn’t a right or wrong answer – it depends on what considerations you are making and what is the goal. Definitely a life skill!

Here is the link to the activity (which is also available in the handheld graphing calculator free activities under middle school activities) and a video that walks through how to do some of the mathematical steps in ClassPad.net.

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 Lessons – Proportional Reasoning (Ratio and Scale Mapping) Another week, and more math ahead of us! Switching the focus this week to proportional reasoning, which is such an important mathematical skill, as it appears in the real world in so many different ways. I am going to be using ClassPad.net in this weeks lessons, though all the activities are modified from Fostering Mathematical Thinking in the Middle Grades with Casio Technology, (Casio, 2011), and therefore are also available in their hand-held calculator form as well at the Casio Cares site under math activities.  I have simply made ClassPad.net versions of them for my purposes here.

Proportional reasoning is often avoided by many teachers or skimmed through quickly because of the dreaded rational numbers that are involved – i.e. fractions! But, proportional reasoning is really such a crucial area in mathematics, and students experience it in things like map reading (though I guess with GPS, perhaps not as often?), measuring, baking, determining rates, etc. Too often, we approach the teaching of proportional reasoning as a process vs. true understanding – i.e. teaching them the ‘trick’ of cross-multiply then divide rather than focusing on the relationships, what these ratios represent, what assumptions are being made, what model best represents the situation, etc. This week I am going to share several activities that utilize proportional reasoning, and have students really using the context to answer questions and make decisions.

To start off, we are going to first explore an activity that focuses on ratio relations and making sense of what these relationships really mean, how to express them mathematically, and how to use these ratios to answer questions. The second activity then extends this into looking at a practical application of ratio relationships, looking at maps distances and actual distances. The two activities are below and then I made a short video overview of each to show how to work with ratios, tables, and graphs, in ClassPad.net.

Here are the links to the activities and 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: