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:

Mini-Math Lesson-Solving Equations/Inequalities and Normal Distribution w/fx-115ESPlus2 Scientific Calculator

I am always amazed when I remember how much power a scientific calculator has to do high-school and upper level mathematics content. I think, like many, it’s assumed once you get into those higher mathematics courses, you need a graphing calculator. But – you really can do so much with just a scientific calculator, except graphing (though the fx-991EX does that too!).  Today I am exploring just a tiny bit of the functionality that the Casio fx-115ESPlus2 scientific calculator has – specifically, I am going to demonstrate how to use it to verify solutions and to determine probability of a normal cumulative distribution. There is so much more that it can do, so consider this a teaser.

The two lessons I am sharing are ones that utilize the two features I am highlighting. What I love about these activities is it isn’t all ‘using the calculator’. In fact, the calculator is just a tool to verify work or help in the analysis of a situation, which is what we should be doing with calculators. They are not ‘answer givers’, but if used properly and appropriately, they help verify students work, they help them analyze mathematics and make decisions and these lessons do a nice job of making the calculator work with the student rather than do the work for the student. The first activity focuses on solving linear equations and inequalities in one variable and quadratics. In this activity, the calculator in this is a tool to verify student solutions to quadratic equations. The second activity is about representing data in various ways (dot plots, histograms, and box plots), and determining probability. The calculator here is used to here to help students find the area under the normal curve.

The video how-to that sort of goes along with these lessons is using the fx-115ESplus emulator software, which, if you check out the link below for the Casio Cares, is available as a free download with an extend free trial through August 31. Great tool supporting students in this remote learning environment.

Here are the links to the two activities that are overviewed, as well as the video which walks through the two main function needs in the activities.

  1. Backtracking
  2. Describe and Infer

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

Here are quick links:

Mini-Math Lessons – Systems of Equations

(The flowers have nothing to do with the math…just thought we could use a little beauty!!)

Today I am in a sense ‘cheating’ again, as I am using a previously made video and activity to share today. It seemed to fit in well not only with the advanced Algebra focus of the week, but also in light of the current pandemic crisis we are in and importance of our medical experts and workers who are giving so much every day. The activity shared today is an activity called “Not Enough Doctors”, and uses real-world data to explore our future supply-and-demand of doctors. What I like about this activity is that you will experience several different functionalities of ClassPad.net – i.e. data/statistics, graphing, text, equations, tables. It is a nice multiple-representation activity and allows students to see different ways to look at data and display data and answer questions about that data. A really important skill.

Here is the link to the activity and accompanying video:


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 – Confidence Intervals, Sample Size,

Ending the week with a couple more lessons focused on statistical analysis. These are at a high-school level, as they explore t-tests, z-tests, and confidence intervals, among other things. In both activities, students will use some of the interval and test templates that are in the ClassPad.net statistical menu. The videos that accompany each activity will discuss how to access and use these templates, though within the activities themselves, the templates are already set up. Additionally, in the second activity, students will be exploring changes to sample size and confidence intervals and what impact these changes have on the width of the intervals, as well as p-values. They will collect data and make scatter plots to help in their analysis, so they will be getting a lot of experience with many of the statistical tools and visualizations available in ClassPad.net.

Here are the links to the activities and the video overviews:

  1. Statistcs – One Proportion zTest and zInterval
  2. Statistics – Exploring Confidence Intervals

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 – Normal Cumulative Distribution

In keeping with this weeks theme, today’s activities focus more on statistics, and specifically the Normal Cumulative Distribution. The activities that I am sharing today are really one ‘tool’ – i.e. a simulation where you can adjust the mean, standard deviation, upper and lower bounds, to find given percentile values, and then one activity that uses the statistical functionality of finding the Normal Cumulative Distribution. One uses a simulation to change things dynamically, the other uses the functionality of ClassPad.net to find the percentile for you, as well as the upper and lower z-scores and is a specific real-world problem situation, where questions are asked in context to the results. The video overview shows you how to utilize both.

Here are the links to the two activities and the link to the video overview:

  1. Statistics – Normal Cumulative Distribution Tool for Explorations
  2. Statistics – Normal Cumulative Distribution 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 – Statistics and Regressions

Staying focused on statistics this week, today’s mini-math lessons are about analyzing data relationships and determining if there is a relationship that exists between two variables, and if so, what mathematical model best represents that data. ClassPad.net has robust statistical functionality (AP Stats teachers take note!), and today’s the activities center around regression. We are using real-world data and graphical representation to analyze whether relationships exist within the data. All three lessons shared today focus on regression, with the first activity, looking at a relationship between age and height. In the first activity, we also use a Linear Regression t-Test to determine if the linear model is appropriate for the relationship.  The second activity explores exponential regression, using data about Facebook and it’s users. And the third activity uses several different regression models to help students think about which models are the best fit for a set of data or range of data. In the third activity you will also see (if you watch the accompanying video) one of the coolest features of Classpad.net, the ability to separate and merge graphs – it’s a great way to help students compare and contrast representations.

Here are the links to the activities and the accompanying video overviews for each activity:

  1. Statistics – Linear Regressions and LinReg t-Test Activity
  2. Statistics – Exponential Regression Activity
  3. Statistics – What Regression Model Fits My Data? 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 – Corona Explorations – Exponential & Logistics

There are a lot of statistics out there and a lot of graphs, some accurate and some not, on the CoVid19 virus. Which can be confusing and scary. Today’s mini-lesson is just one activity and video from Ish Zamora (@seemathrun) that tries to provide you with an understanding of how the data impacts predictions. Using a logistical curve, you will see how changes in the numbers impact the steepness of the curve and the predicted end date. You will collect your own data and answer questions about the impact of those data points.

Be sure to watch the accompanying video first to get a sense of how to collect the data and observe the changes in the curve.

The links are below.

  1. Corona Logistic Video Overview
  2. Corona Logistics Activity

We will be back tomorrow with some more mini-math lessons focused on simple statistical plots and what they can tell us about data from the elementary perspective.

Remember – stay inside, wash your hands, and social distance and we can beat this thing!!


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: