President’s Day – Sample Lesson (Statistics and Measures of Center)

In honor of President’s Day this coming Monday, it seems appropriate for this months’ lesson feature to be center around information about presidents. There is a lot of data that you could explore with students about presidents – age, number years in office, age when they died, number of children, etc. But, I simply chose one statistic – i.e. age at inauguration, to make this really a focused lesson on using a set of data to explore data representations and measures of center.

The Lesson

The idea of this lesson is to help students look at different ways of representing data (table, dot plot, histogram and box-and-whisker plot) in order to make some conjectures and observations. And depending on the representation, really thinking about the information you are able to glean. Can you estimate measures of center from visual representations of data? Is one representation better than another for a given set of data.

ClassPad.net is a great tool for a lesson like this because it allows for students to see all representations in one spot,  continually add on representations, and also provides a place to write down observations and conclusions. You will see that the shared paper (the complete lesson) has explanations, questions, directions all in the one place, and students are doing the math and making their conjectures after each step. It’s an all-in-one, multiple representation activity. Here is the link to the shared paper, that you can use freely with students. If you want to create a duplicate copy to save, you must create a free Classpad.net account and ‘duplicate’ the paper there once you open it. This will create your own copy that you can modify and then share with your own students. Either way, the idea is that students can do multiple representations and measures of central tendency very quickly and make some mathematical connections and conclusions about the given presidential data.

ClassPad.net Lesson In Action

I’ve made a short video that explains the lesson a bit more and walks through the how-to’s of creating the different plots and shows the different aspects of the lesson (i.e. the text stickies, tables, plots, etc.). As usual, you will notice a lot of questions built into the activity, where students are asked to observe, notice, compare and write down their observations. My suggestion for this activity is to do each step and really pause after to allow students individual think time, but then also bring them back together (pair up first, then whole class) to make sure everyone is on the same page before moving to the next step.

Again – here is the shared paper link: ES/MS President’s Day Lesson

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Weather and Integers – The Importance of Real World Connections

A lot of my math teacher friends have been posted images from weather reports on FaceBook and Twitter, like this one to the left from @seemathrun, showing the real-world application of integers due to the extreme weather conditions that are happening across the country right now. It really is a perfect opportunity to show a true application of mathematics that students can definitely relate to, especially if they are in those freezing climates. Add in the wind chill, and you have some interesting data and comparisons and a chance to talk about the relevance of math and understanding numbers.  Here’s an image to the left showing wind chill, temperature, and frost bite times someone else shared that could help explain why so many schools are closed, even though there may not be any snow on the ground, (which is usually the reason behind winter closures). I know one of my colleagues and friends, @ClassPadnut, was sharing with me yesterday that with the wind chill, it was -60 where he lives.  Yikes!!!

There is obviously a lot of different math concepts you could explore with students, dependent on grade level and questions asked. I find the wind chill graph the most interesting. Looking at the wind chill chart, the drop in temperature is almost, but not quite, constant, like you would think – i.e. You will note that there is an equation for the calculation of wind chill at the bottom of the image. I was  curious about whether students could find that connection from the data alone -something to challenge students with. How would they graph this data? Could they? Thinking of statistical tables, what would they enter and what statistical plots would be appropriate? If students are in areas where schools actually closed, you could talk about how the data supports the decisions, and what is the ‘cut-off’ temperature/wind speed that might influence the decision? Lots of things to explore.

I found another image that showed the lowest temperatures reported in each state, so you could do a comparison across states. Even Hawaii is cold!!!  Crazy.  Below is the image, which I then used to enter the data in a table in ClassPad.net, and then make two different plots to represent the data – a histogram and a box-plot. You can see from the box plot five-number summary that the median temperature in the U.S. for this day in January is -40.  Wow!!! (And boy, don’t want to be in Alaska at -80!) Again – think of the interesting class discussions about integers, about how these temperatures will impact things such as the orange crops in Florida or the tourism in Hawaii or California. (Here’s a link to the Classpad.net paper that has the image, table, and graphs shown below: https://classpad.net/classpad/papers/share/b61b70a0-0eed-47da-947a-580e1d835f8d.

As you can see, using what is actually happening right now in our country, i.e. REAL world connections of weather (temperature, wind speed, wind chill), is an amazing opportunity to help students see the relevance of integers and statistics and how this data is being used to make important decisions, such as do we close schools? Who should not venture outside? How long before you get frostbite? The visuals help students ‘see’ mathematics in action, and particularly if we focus on the integer aspect, provide a clear connection to integer addition (and subtraction, depending on the questions asked), something many students struggle with.

Whenever possible, we should be trying to connect the math concepts students are learning and using to a real-world application. Here’s a perfect opportunity, no matter the grade level, to have some great class discussions about the impact of weather on our world, about the relevance of integers, and about how statistical information is important to decision making.

 

Math Hardware versus software – Similarities & Differences with Casio

Students using technology as part of learning math is important because of the extension of learning that is possible, the visual connections, and explorations that become possible as a result of technology. The most common technology students use these days are their phones, tablets, computers, and of course, hand-held devices such as calculators. It all depends where you live, what schools you attend, what’s allowed or not allowed, and also what resources are actually available and understood by both teachers and students. From my own research, some schools/teachers have a multitude of resources, but most schools have limited options. And – even if there are many technology tools available, teachers tend to utilize the tool (s) they are most comfortable with, and that the majority of students have access to. Basically, it comes down to choosing a technology that is going to support the learning and that students and teachers can use relatively efficiently, so that time is not lost to ‘tool logistics’. Often times, again, based on my own research (dissertation), teachers choose tools that may NOT be the best choice for learning because they know how to use it over a much better, more appropriate tool, that they are unfamiliar with or uncomfortable with, so many times better technology tools go unused because of the ‘learning curve’.

What I wanted to use this post for today was to show how Casio has really recognized the ‘learning curve’ issue and tried to keep functionality consistent across handheld models and even in their software, providing intuitive steps and menu options right within the graphing menu itself that alleviate some of that ‘learning new tool functionality’ concerns that teachers and students often face when using technology. Our graphing calculators basically use the same steps, buttons, layout, even from the very basic ones (fx9750) (fx9860), to the more advanced ones (CG50), so if you know one, you know them all. And, even the new software, ClassPad.net, is built along the same lines, though obviously with more features and capabilities.  But there is no ‘searching for menus’ – relatively intuitive no matter the tool. Obviously, as you get into the newer models and then into the software, the functionality and options increase – we go from black-and-white displays to color, we go from intersection points on the graphing calculators to union/intersections on the software. But knowing how to use one tool makes transitioning easy, and if you had students with several different models of the handhelds, you could still be talking about the same steps and keystrokes.

The best way to compare and demo is to show you how to do the same thing on the different models. I’ve chosen to show graphing two inequalities, so that you can see, even on the older models, that shading and intersections occur. But also to show that as you progress into the newer and more powerful tools (i.e. memory capacity, color, larger screens, resolution, etc), allowing for more options and learning extensions.

Here are the two inequalities that are being graphed in each of these short GIF’s:

Each GIF below graphs the two inequalities and finds intersection points of the two graphs. The software extends that to allow for finding the Union and the Intersection of all points.

 

fx9750

fx9860

CG50

ClassPad.net FREE Math Software

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Be sure to check out the free software that does calculating, graphing, statistics and geometry: ClassPad.net.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Systems of Equations – Sample Lessons and Resources

For this months lesson feature, I am going to focus on Systems of Equations. I chose this topic because I just did a workshop with Algebra 1 teachers in NJ, and this is where they were in their pacing guide, so I am making an assumption that many algebra teachers might also be focusing on this content as well this time of year. I am using a problem from Fostering Algebraic Thinking with Casio Technology in order to provide a real-world problem-solving experience (and I have the resource), but I have altered the problem so that I can utilize the all-in-one capabilities of Classpad.net (tables, graphs, equations, geometry, text).

The Problem

In 2010, there were approximately 950,000 doctors in the United States, and approximately 350,000 of them were primary care doctors. It was estimated that more than 45,000 new primary care doctors will be needed by 2020, but the number of medical school students entering family practice decreased by more than 25 percent from 2002 to 2007. With laws reforming health care, many more people will be insured in the United States. 

For many reasons, including a growing and aging population, the demand for doctors will likely increase in future years. The number of doctors available is also expected to increase. But, due to the high cost of insurance and the fear of malpractice lawsuits, many have predicted that the increase in the number of practicing doctors will not keep up with the increase in demand for doctors.

The table to the right provides data from a study conducted in the state of Michigan. These data approximate the number of doctors that were or will actually be licensed and practicing in Michigan, called the supply, and the number of doctors that were or will be needed by the people of Michigan, called demand.

The question is, will there be enough doctors to provide all the services? The shortage of doctors is a problem that challenges the entire country, not just Michigan.

The Lesson

A shared paper has been created in ClassPad.net called Systems of Equations Help! Not Enough Doctors, which you can access by clicking on the title. The idea behind this problem is to provide a real-world context where students can use tables, graphs, and equations (along with calculations) to create a system of equations. They can solve these using methods such as substitution, elimination, and graphing. Students will also be practicing how to model with mathematics, applying what they know about relationships and being able to create a system of equations that fits the context of the situation in order to find a reasonable solution.

In the activity, there is obviously some focus first on getting students to really understand the problem and what the numbers represent, and then the idea is to have them look for patterns and relationships as they look for a solution. First in the table, then by looking at a scatter plot of the data, where they again try to determine a solution based on a visual. Continuing to look for trends, they use prior knowledge to recognize linear relationships, create equations that model the data, and then graph those equations to find a more precise solution. Then, as a check, they solve their system of equations algebraically. It’s all about multiple representations and helping students see the connections between all the representations, and depending on whether you want a specific, precise answer or just a generalized answer, you might choose a different representation.

ClassPad.net – Lesson In Action

The video below shows the activity and does a brief walk through of some of the components and what it would be like doing the activity from a student perspective. I am a big believer in the think-pair-share approach, so I would suggest having students do the Notice and Wonder individually first, then pair up, then share so that you can make sure that any misunderstandings about the context, and clarification about the numbers is figured out before students start solving. Then I would suggest small groups for working on the problem itself.

Other System of Equation Activities and/or video links

• Algebra I Resource Guide (Systems on page 7)
• NCTM’s Problems of The Week (requires member log-in)  Type in ‘systems’ in the search and there are a ton of excellent problems.
• EngageNY Resources on Systems

 

Quadratic Functions – Sample Lessons and Resources

I am starting a monthly feature where I will be focusing on some specific math content areas and providing some resources, in the form of how-to videos (both calculator and Classpad.net) and some ready-to-use math lessons (either PDF or links, depending on the tool used). I know math teachers are always searching for resources that will help them provide more open-ended math activities, where students are collecting and using data, using multiple representations to analyze and solve problems, and where students have to make decisions and support their decisions with mathematics. And integrate technology as well! So, at least once a month I am going to be picking a math content to focus on and provide some technology options as well, sometimes both calculator and online, and sometimes one or the other, depending on content.

This week I would like to focus on quadratic functions and helping students use a real-world context to work with quadratics. I am going to utilize Classpad.net, which is FREE web-based dynamic math software where I can do statistics, graphing, and calculations in one place (geometry as well, but for this activity, our focus does not include geometry). I am using this technology for a few reasons:

1. It’s free, so all of you should be able to access the created activity, including your students, as long as you have a mobile device with internet access.
2. I am able to create a complete activity (i.e. directions, tables, graphs, and place for students to show work) in one place and then share it easily via URL.
3. Everyone who opens the activity can create their own copy of it (as long as you have a FREE account on Classpad.net) by duplicating into their account. Then you can modify, answer the questions, etc. and create it’s new URL to share with others (or for students to share with you). To learn more about duplicating activities, click here.

The Problem

You are fencing in a rectangular area of your yard to create a garden. You have 36 ft. of fencing, of which you plan to use all. You can cut the fencing into whatever lengths are needed, as long as you use all 36 feet. 

What dimensions should you use for your garden?

The Lesson

I have created a shared paper on Classpad.net called Quadratic Functions – Area of a Garden which you can access by clicking on the title. The idea behind this problem is that there are actually multiple solutions since the question is rather vague. I did NOT ask what is the largest garden, so students can work on collecting and analyzing the data and come to different conclusions depending on what they think is important. Some might choose largest area for the garden, some might choose largest perimeter, some might only want a rectangle some only a square, etc. By leaving the question a little more open, you are giving students a chance to explain their reasoning and come to multiple solutions based on this reasoning.

In looking at the activity (click the link above), you will note as part of the lesson, students use multiple representations. They first use their prior knowledge about dimensions of a rectangle, perimeter, area, and an understanding of feet and inches to record different dimensions for the garden. In the directions, students are asked to create at least 10 different rectangular gardens that use all 36 feet of fencing, where some of the width and length dimensions are fractional/decimal numbers and where width is sometimes larger than length. They record their dimensions in a table to start with, and then use those table values to calculate area (and perimeter if they choose to do the Extra Challenge), and use those table values to create statistical plots (scatter plots), and from the scatter plots and tables, create functions and graph those functions to fit their data. At different points along the way (after the table and scatter plots, and then after plotting their functions), students are asked to answer the question about what the dimensions they would choose for their garden and back up their reasoning using the information at that time. The idea here is to help them see that each representation provides insight into the dimensions, and some representations help you be a bit more precise or see the relationships between the quantities a little better. And also, depending on your goal for the garden, your reason for choosing certain dimensions may differ from others. There is also an extra challenge at the end (this is a way to support students who finish early, don’t need as much teacher guidance, and/or want to explore more), where students explore how the problem might differ if there was a fixed perimeter.

ClassPad.net – Lesson In Action

This is a video that shows using the activity and parts of doing the activity to get a feel for how this looks with students. I would recommend students working in pairs or small groups (3-4). All students can be recording on their mobile devices, or if you have one per group, choose a recorder.

Other Quadratic Activities and or video links. 

Here are a few more links that are focused on quadratic functions and also utilize ClassPad.net

• Activity: Quadratic Transformations
• Activity: Parabola Roots Investigation
• Activity: Is the cross section a Parabola?
• Video How-to: Focus on Parabolas

 

Complex Numbers – Support for Calculations

I received a question on one of my Youtube video posts on the Casio Fx991 scientific calculator asking if it was possible to do complex number calculations on this calculator. The answer is of course yes – which then prompted me to make a quick video today on exactly how to do that with the fx991. See the video below:

This of course then made me think of our other technologies and that perhaps I should show how to do complex numbers with these tools as well.

Here’s the steps on the graphing calculators (any of the Casio models, since they all basically work similarly – the beauty of Casio, the buttons are relatively consistent). This example uses the CG50, but see fx-9750, fx-9860, etc).

And finally, on ClassPad.net, the FREE online math software that does it all – statistics, geometry, graphing, and of course calculations. (You can sign up for a free account (ALWAYS free) – here’s a quick how-to).

The question of course arises, when are we even using complex numbers? Or why do we need them? As I never really taught math content that required students to utilize complex numbers, I don’t feel I am able to answer these questions with authority, so I did a bit of research. For one, if we just go from a ‘content/standards’ perspective, if you are in states that incorporate The Common Core Math Standards (or a version of, whether renamed or not), then it is actually part of the High School: Number and Quantity standards which state, “Students will…”:

• Perform arithmetic operations with complex numbers
• Represent complex numbers and their operations on the complex plane
• Use complex numbers in polynomial identities and equations

But, that of course doesn’t really get at why do we need them. So here are some things I found in my search for this answer. I admit I can’t explain these any more than just listing them, but it at least points to places where complex numbers are in fact important and needed.

• Complex numbers are used in electronics to describe the circuit elements (voltage across the current) with a single complex number z=V+iI
• Electromagnetic fields are best described by a single complex number
• People who use complex numbers in their daily work are electrical engineers, electronic circuit designers, and anyone who needs to solve differential equations.

Hopefully this is helpful to those of you who are in fact doing complex calculations for whatever reason!

The Power of Math Exploration

If I had a dollar for every time I hear “I would do more hands-on, inquiry, problem-solving, collaborative learning, in math class if I ________________________ (insert any one of the following):

• had more time
• didn’t have as many students
• didn’t have to get through the ‘curriculum’
• had students who would actually talk
• if I didn’t have to make sure they were ready for the test
• if I didn’t have to review all the things they didn’t learn from last year…..
• ….the list goes on…….

I would be a very wealthy woman. What is mind boggling to me is there is so much research out there that shows students do better when they learn for understanding and not for memorization, which means learning through context, through inquiry, through problem-solving, through struggle. Time is one of the biggest ‘road-blocks’ teachers throw out there, and granted, there definitely is a time crunch to get all the content in before those dreaded assessments. What I try so hard to get across to the teachers I work with, is that you can  save time by taking time – you actually can ‘cover’ more ground by teaching from a more contextual, experiential, problem-solving way. As students make connections and problem-solve, they are able to learn more efficiently and more than one concept at a time because they are working from a connected-math view point instead of the single-skill/concept at a time approach we traditionally provide.

An example from Geometry: (this is using Classpad.net, free math software) 

Concept – identifying polygons, and then what’s the difference between congruent-sided polygons versus regular polygons (identifying what a regular polygon is).

Activity: Using the drawing tool, have students draw examples of 3-side, 4-sided, 5-sided (and more….) polygons.  At least 2 of each kind that look ‘different’. Can be convex or concave

• Have students compare their shapes noting similarities and differences and coming up with definitions – attaching specific words to their definitions like convex, concave, closed, etc.
• Now have students use the arrow tool, and select one of their triangles, and the Adjustment menu to make all sides congruent. Then, choose a second triangle and Adjustment and make the shape a ‘regular’ polygon. What do they notice? Have them measure sides and angles and compare to others.
• Do the same for two different 4-sided figures (so Adjust congruent, then adjust regular), the 5-sided, etc.  Each time compare the two on their paper, and then compare to others, and try to come up with what the difference is between congruent-sided polygons and regular-polygons.
• Come to group consensus, and by the end of class students have manipulated, explored, collaborated and defined several things: polygons, convex polygons vs. concave, triangle, quadrilateral, pentagon,….regular polygon, congruent sides, etc.

An example from Algebra: (this is using CG50 Graphing Calculator (CG10 is similar):

Concept: Parent Function and Vertex From of a Parabola 

Activity: Students graph the parent function of a Parabola (y=x^2) and then graph another in standard form using variables for coefficients.

• Have students use the modify feature of the graphing calculator to animate the different coefficients (one at a time)
• Observe what changes in that coefficient does to the parabola by comparing the modified to the parent
• Make conjectures and compare with other students till consensus is reached.
• Do this with all the coefficients.
• Have students then test out their conjectures by providing them several equations of different parabolas and, based on their conjectures, determine the shape, direction and location of the parabola BEFORE they do anything, and then test their guesses by entering in the calculator.
• Time saver: Doing this activity with linear equations first will then give students a general understanding of transformations of functions which they then extend and solidify with quadratics, which then can be easily extended into other equations, like the absolute value function. Time saver!

Obviously I am using technology here, because technology allows for conjectures to be made and tested very quickly. But technology is just a tool that is appropriate in some instances, but there’s so much that can be done without technology as well. You can make math much more of an exploration just through your own questioning (i.e. why do you think? can you explain that more? Are there other ways to do this?) and by providing students a chance to puzzle things out on their own, ask questions, use tools (so objects, paper, pencil, etc).

One of my favorite things to do is to provide them with a situation that has lots of information, but no question (basically, find a rich math task, but don’t give students the question(s)). Students then write down all the things they notice, such as quantities, relationships, etc. and then come up with their own wondering’s and questions. Then you let them choose a path they want to explore (this works well with small groups or partners). Usually it ends up that there are several different questions and solutions generated and explored using the same information. When students then share their findings, you find that there is a lot of math going on, which leads to some really interesting class discussions – some you yourself might not have thought of. You can then maybe even give them the question that might have been given in the problem – by that time students may have already explored it and if not, by now they have a real sense of what information in the problem will help them and they are more willing to actually solve the problem.

The key here – students only become problem-solvers if they are given the opportunities to explore math, make their own connections, and collaborate with others to verify their thinking. The more you give them opportunities and provide tools and resources and challenging problems, the more efficient they become at using math, connecting math concepts, and viewing math as a connected whole instead of isolated skills and facts. Take the time….it’ll come back in the end.