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Week 5: Computer-based math

In this task, you search for a video (Part 2) and an interactive (Part 3) related to a math problem or exercise (Part 1). It may be easier to start from Part 3, because there are more, many more videos than there are interactives, and you can do Part 1 for pretty much any problem.

Part 1

Use Wolfram|Alpha to pose and solve some math problems or exercises. Then copy the url of one of your solution and share it here. For example, here is the solution to a system of two equations:

You may need to search the web for the syntax of Wolfram|Alpha, using multiple search words such as "Wolfram Alpha trigonometry" or "Wolfram Alpha limits." Or you can start by exploring excellent examples the site provides:

Part 2

Find a decent YouTube, TeacherTube or Vimeo video about the type of math problem or exercise you shared. Say a couple of words about what you like about the video, in a comment on the video if they are allowed: authors need to be encouraged. You may need to create an account to leave comments. Copy your comment here too.

Part 3

Find a math game, a virtual manipulative or some other interactive about your problem or exercise. Briefly explain what you like about it. Again, if comments are available, leave a comment for the author of the interactive to see.

Part 4

Pose a higher-order, deeper question about this same type of math problems or exercises. For example, here are somewhat deeper questions about systems of polynomial (just x's, y's and their powers) equations with two variables that I used in Part 1:

  • How many solutions can the system have?
  • Most linear systems (just x's and y's without powers) have one solution, but some have none. How can you make linear systems with no solutions?
  • How can you make systems that have "nice" (small whole number) solutions, like most textbook exercises have? Where and how do textbook folks get their examples?

Part 5

Briefly explain how you would help students explore your deeper question using technology. Would it be helpful for them to use solvers like Wolfram|Alpha, or programming environments like GeoGebra and Scratch, or videos, or interactives? When and how would you recommend your students to use these different types of tech?

Part 6

Conrad Wolfram, the more ed-oriented of the two Wolfram brothers, leads ongoing discussions about computer-based math, contrasting it with computer-delivered math. Use examples from Parts 1-5 to compare and contrast computer-based and computer-delivered math. For example, would you use one or the other to start a new topic? 

Task Discussion

  • Green Machine   Feb. 25, 2013, 10:34 a.m.

    Part 1


    Solution for a item discounted 25%.


    The explanation of their solution could have been a bit more detailed for my liking. I can see some scholars becoming confused if they aren’t familiar with the steps.


    Part 2


    This video provided a multitude of real world percent applications. It would be great for introducing the concept to scholars who can work independently on an advanced level or reviewing/reinforcing. I enjoyed how they explained to solve each exercise, along with written calculations and formulas.


    Part 3

    This game allows the scholar to choose a maximum of 5 items to purchase and they must calculate the final price after applying the displayed discount. I love how they go through each step for solving the final price.


    This game allows scholars to burst balloons with the answer to “percent of” exercises.


    The game allows scholars to calculate the tip of their bill from a meal. They have various degrees of difficulty, which the scholar may select before hand.


    Part 4

    What’s the interest paid on a $5,000 loan over 42 months with a rate of 5%?


    What is the number of degrees in the circle graph for the percentage of people who like almond milk which is 42%?


    How are proportions related to solving various percent exercises?


    Part 5

    I would have scholars experiment with Wolfram|Alpha once they beginning using calculators and once they develop a strong mathematical vocabulary foundation. I believe a website like is excellent for extending certain concepts or exposing scholars to various applications of math. GeoGebra would be great when entering the realm of geometry and trigonometry. I would also use GeoGebra for solving some perimeter and area exercises. There are an abundance of technological resources out there and I believe they are best used when integrated with the lesson. I don’t believe that scholars should depend solely on technology to complete all their exercises. It takes proper guidance and proper classroom management to receive the benefits of technological enrichment in the classroom. I would also encourage scholars and adults to purchase various programs and items to help foster the use of the technology out there.


    Part 6

    In my opinion, I believe that computer-based math falls under the umbrella of computer-delivered math. I think this because computer-based math is math that is delivered or completed on or at a computer. Some of the differences that I have noticed is that computer-based math is completed on the computer or pieces of technology and its interactive, while computer-delivered-math may be a video or article that is delivering or supplementing a certain concept. I lean more towards computer-delivered math to introduce a lesson and then computer-based math to enrich, review, or extend the concept.

  • Maria Droujkova   March 2, 2013, 7:38 a.m.
    In Reply To:   Green Machine   Feb. 25, 2013, 10:34 a.m.

    You are making a good point about the taxonomy here: computer-based math being a sub-category of computer-delivered. It makes sense and it explains why people have hard time finding aspects of CBM that would place it "outside" of computer-delivered realm. Thank you!

  • Katherine Hanisco   Feb. 17, 2013, 8:12 p.m.

    Part 1: The topic I chose was horizontal and vertical shifts of functions. I think it’s important to really take some time with this topic so that students get really comfortable with graphs of common functions and how algebraic transformations are related to graphical transformations. I used WolframAlpha to graph the familiar function f(x)=x^2 on the same graph as f(x)=x^2-2 and f(x)=(x-2)^2. The results can be found here.

    My initial graph showed x and y values that I thought were a little strange so I had to play around with the syntax so that it displayed a graph with a domain and range that give a better picture of the graphs. I thought this was a good example of how it is important to use technology to enhance understanding about graphs of functions and not as a replacement for knowing the shapes of common functions, learning how to find intercepts by hand, etc. If students used WolframAlpha to graph functions but did not have the basic understanding of what a quadratic (or cubic, square root, etc.) function should look like, they would not know if a particular set of x and y values on the graph was giving an inaccurate picture of the function.

    Part 2: I found this video that discusses vertical and horizontal transformations. My comment: I really like how you approached horizontal shifts. A lot of the videos I've watched on this topic just gloss over it as being "backwards" but you provided a nice explanation.

    Part 3: I found this interactive graph that would be useful for this topic.  Although it has the limitation of only being able to graph two functions at once, I thought it had some user-friendly features that made it easier to use than WolframAlpha. For example, you can easily shift the viewing window around by re-centering and zooming in and out, and you can get back to the basic window with the click of a single button. There are also some clearly written instructions directly underneath the applet that make it easy to use. Like WolframAlpha, you can save and copy a url with a specific graph. Here is a graph showing f(x)=x^2 and f(x)=(x+2)^2.

    Part 4: Here are two higher order questions about this topic:

    1. Why is it important to understand the basic shapes and features of functions when we have so many ways to graph using technology?
    2. Why does the graph of f(x)+a move the graph of f(x) in the positive direction for positive values of a, and the graph of f(x+a) move f(x) in the negative direction for positive values of a?

    Part 5: I talked about question 1 in part 1 above, and I think this is something that students could discover for themselves very easily with the help of technology. For example, I could give them functions to work with that give an incomplete picture of the function if graphed on a standard viewing window. Rather than me telling them why it’s important, they could discover for themselves that it’s important to know the graphical features of basic functions. Graphing tools such as a graphing calculator or the interactive tool I linked above would be good for this task. Although now that I’m thinking about it, WolframAlpha or graphing calculators might be a better choice for this task than the applet initially since they both require the user to input the graphing window, while the applet makes it easy for students to randomly click around and zoom in and out. I think it’s a great user-friendly feature to have, but it might make it too easy to see the whole graph without understanding the graphs conceptually!

    With question 2, I feel like this is one of those things that is often glossed over so students end up memorizing that one transformation shifts the graph the “logical” way and the other is “backwards”. As I was browsing videos, I saw several that just kind of handwaved it away as an exception or counterintuitive, and I think that does a disservice to the topic because I believe it does make sense if you approach it the right way. I came across this blog post that has tips on how to present this topic to students, and I really like how it shifts (ha!) the focus to the intercepts. I would approach the idea of horizontal shifts as shifting x-intercepts, x-intercepts occur when y=0, what happens to the equation of the function when y is 0, and how do we solve for the x values (and the same idea for vertical shifts as shifting y-intercepts). I think the result becomes more intuitive if students do the algebraic manipulation to find intercepts to reinforce how when x is 0, the y-intercept is the constant term, and when y is 0, the constant term has to be moved to the other side of the equation to solve for x, and compare that result to the graphical shifts. For this question, I think that graphing tools would be a great enhancement to the material, but I think this is one of those times when students need to manipulate the equations algebraically by hand to get to the point where the direction of horizontal shifts is intuitive rather than “backwards”.

    Part 6: I’ve been thinking about the difference between computer based and computer delivered math since our discussion on Wednesday and trying to get my head around the differences. Based on my understanding, I would say that the video in part 2 is computer delivered since the computer is only using technology to make the lesson more widely accessible. There is nothing in that lesson that can’t be taught with an old-fashioned chalkboard. My example in part 3 would be computer based, since the learning comes from using the mathematical capabilities of the computer: graphing functions, shifting viewing windows, etc.

    After thinking my way through the two questions I discussed in part 5, I feel like even though my suggestions for how to use technology is similar for both questions (graphing functions) the first one is computer based and the second one is computer delivered. For my suggestions for the first question, the computer is where the learning is happening because students are using technology to manipulate graphical models. For my suggestions for the second question, the learning comes from the algebraic manipulation, which is outside the computer, and the technology aspect comes in as a way to quickly verify results. So for the first, the computer is where the significant learning happens, but for the second, the computer is used as a convenient way to check answers.

  • Maria Droujkova   Feb. 20, 2013, 7:54 a.m.
    In Reply To:   Katherine Hanisco   Feb. 17, 2013, 8:12 p.m.

    Katherine, this is an important observation:

    "My initial graph showed x and y values that I thought were a little strange so I had to play around with the syntax so that it displayed a graph with a domain and range that give a better picture of the graphs."

    This "playing around" is an action students can be doing, after you model it for them. For example, you can purposefully zoom in so close to a parabola that it looks like a straight line. As you note, novices have no idea how far in or out they need to zoom for a representative picture. To some degree, they can use computers to find out. However, a graph can have some weirdness going on far away, for example, (x+1000)*x^2 - someone who looks near zero will only see a nice parabola. So the analysis of graphs (made by hand point-by-point, or by machines) has to go together with the analysis of formulas, as you describe.

    Here is a pedagogical question to follow up your deeper questions:

    How can we help students analyze formulas?

    For example, I know that multiplying x^2 by (x+1000) will introduce interesting twists around x=-1000. Or that replacing x with 1/x in sin(x) will cause wild oscillations near zero. How can I help my students learn such lore?

    You make a good point about explanations. You will find a lot of textbooks glossing over things, as well, which hurts math learning a lot. By the way, small world - I just recently reviewed a book by Jeremiah Dyke, the author of the blog post you liked!

  • Gina Mulranen   Feb. 17, 2013, 7:37 p.m.

    Part 1

    I wanted to use WolframAlpha to calculate the first derivative of x^2+2x+3. I first put in “derivative of y= x^2+2x+3” and the first derivative was calculated. Here is a link to the solution:^2%2B2x%2B3

    However, I wanted to see what would happen when I just put in the function y= x^2+2x+3 in. Wow! It showed the graph of the function, classified it as a parabola, showed alternate forms of the function, calculated the implicit derivatives, and gave the coordinate of the minimum value of the function. I also clicked on the More button next to the implicit derivatives and it gave me the second derivative of the function as well. Here is the link to the solution:^2%2B2x%2B3

    Now this is what I consider computer-based learning. The students can investigate more about a problem than just the solution to the question. So cool! I love this site!

    Part 2

    This is a video from, which is a fantastic tutorial website that I have given my students as a resource when reviewing or exploring a new concept. This video is an introductory video to what a derivative represents and how the definition is defined using the slope of a curve between two points.

    What I LOVE about this video is that he explains what a derivative represents and how the definition of a derivative relates to finding the slope of a curve between two arbitrary points on the curve. He first uses previous knowledge about the slope of a line and then uses the concept of slope to explain how to find the slope of a curve – showing what the derivative is. This is a really great introduction to what a derivative is before students start using the different methods of solving for the derivative of a function.

    The Calculus: Derivatives 2 video then guides students through the process of finding the derivative of a function. There is also a Practice This Concept button at the top right of the screen where students can complete practice problems to test their understanding of the concept they just learned. I really really love this website for all the different ways it teaches and re-emphasizes math concepts.

    Here is the comment I left on the video:

    What a great video! I think these introductory videos are SO important for students to see before they start getting their hands dirty with actually solving for the derivatives of a function. I think it also gives purpose to solving the problem. It’s not just “do this and get the answer.” It shows students what the derivative represents and how we can use it to find the slope of any point on the curve.  I think this video can also be used as a reinforcement to remind students about what they are solving when they are finding the derivative. Sometimes when you work with a concept for so long, you can lose sight of what you are actually doing.

    In terms of the structure of the video, I really like that you start off with the concept of finding the slope in order to tap into the students’ prerequisite knowledge and then expand off of that point and apply it to finding the slope on a curve. How you derived the formula for the derivative of a function was also very easy to follow. Thank you for all these great tutorial resources!

    Part 3

    This is a game that I found that practices with matching the graph of a function to its derivative.

    I like this game because it does not just practice finding the equation of the derivative; it practices with identifying the graph of a derivative. This is a great way to get students to think about the concept of a derivative, not just the algorithm to get the equation. It also has different levels of difficulty so students can challenge themselves on their own skill level.

    There was not a way to leave a comment about the game on the website.

    Part 4

    A higher-ordered thinking question about derivatives that I thought would spark some good conversation about different types of derivatives is: Can a function have the same graph as its derivative? What type of functions would they be? Students can they play around with some functions and their derivatives to figure out if any function would fit that description.

    Part 5

    I think WolframAlpha would be a great technology resource for students to research different functions and their derivates to see if any function has the same graph as its derivative. With the WolframAlpha software, students can see the derivative and the graph of the original function. I would have the students work in groups so the students can test different functions and then report back to their group members.

    Part 6

    I think the WolframAlpha site explored in Part 1 is primarily a computer-based math since the calculations are done by the computer. The computer-delivered math would be more like the KhanAcademy site that I used for my tutorial video of derivatives. The video that I linked in Part 2 would be fantastic to introduce the concept of derivatives. The tutorial website delivers a great explanation about how the definition of a derivative is derived by expanding on the definition of slope. I think I would lean more towards the computer-delivered math to introduce a new concept since the students are the ones in control of solving the problem and learning the information, not the computer. However, I think computer-based math is very important for solving a lot of real-life situations where the numbers can get very large or the accuracy is very tedious to calculate by hand. I think this type of math would come after the students have learned about the concept and they are ready to apply it to more challenging situations.

  • Maria Droujkova   Feb. 20, 2013, 8:16 a.m.
    In Reply To:   Gina Mulranen   Feb. 17, 2013, 7:37 p.m.

    This is such a good question to ask in math, in general, Gina:

    "However, I wanted to see what would happen..."

    Computers can lower the price of exploration. For example, kids often ask, "What would happen if I replace y=x^2 with y=x^3, y=x^4, etc.?" Just the other day we found the derivative of x^2 by hand (through limits) with my kid, and that was exactly the question. It took us some fifteen minutes to work through the algebra of that limit. So my kid's next question was, "Can we use Wolfram|Alpha and look at the patterns?" 

    Starting from what students already know is such a solid practice - and often neglected! Khan Academy has provided valuable innovation in the "framing" of math. The bite-size lectures, and the practice tool (with badges and other "gamified" task management) - these two aspects make a huge difference. I would like to caution that Khan Academy, like most other content, should be approached with a critical eye. There are better and worse quality videos and explanations there. We can even invite students to remix and improve videos they find; what is nice about Khan Academy is that the license is Creative Commons and allows remixing.

    Agreed on your Part 3 praise. Working with multiple representations of the same idea (formulas and graphs of functions, in this case) is powerful for the analysis of that idea. 

    Part 4 - oooh! Do you think students could be coached into re-discovering the number e, for example? In any case, students will look at greatly many functions in their quest for an answer, and undoubtedly notice a lot of patterns.

    I have a follow-up question to you about Part 6. Is it a good idea to use exploration - with the help of computer-based math - to introduce some concepts, and what sort? For example, I tried, with mixed results, to use a grapher tool to help students explore maximum and minimum points of polynomials. The task I assigned was simply, "Make a graph with maximum points at 2 and 5." But it was more like "Computer- plus teacher-based" than "Computer-based" or "Computer-delivered." 


  • Gina Mulranen   Feb. 20, 2013, 7:26 p.m.
    In Reply To:   Maria Droujkova   Feb. 20, 2013, 8:16 a.m.

    I absolutely agree that computer-based math can be so beneficial for explorations in math. I have used graphing calculators to explore the shapes of different functions and even Excel functions to explore a large list of data or sequences.

    I was not aware that Khan Academy has a license that allows remixing. And I also like the idea of students creating their own tutorial videos! So with Khan Academy, can the students download the videos and edit them? How does that work?

    To answer your follow-up question for Part 6, I think that computer-based math can be a great tool to help introduce concepts because it allows students to explore concepts without actually knowing how to create or solve the problem yet. By just focusing on the patterns or aspects of a topic instead of the algorithms to solve it, students will make important preliminary connections to formulate what the concept is. Then they can start working with and learning how to solve for different aspects of the problem or topic, knowing WHAT they are solving for because those connections were already made. I do think that these explorations need to have guidance from the teacher in order to make sure the students reach the correct conclusions or redirect them if their explorations are going off the right track.

  • Maria Droujkova   Feb. 21, 2013, 7:11 a.m.
    In Reply To:   Gina Mulranen   Feb. 20, 2013, 7:26 p.m.

    Gina, Khan Academy's content license says it's Creative Commons Non-Commercial Share Alike, on this page: license description reads:

    You are free:

    to Share — to copy, distribute and transmit the work
    to Remix — to adapt the work
    So yes, your students could download the video and use parts of it in theirs - say, add their own examples, or argue with Sal Khan, or overlay some music and visuals. They could share the results online, as long as they attributed the source to Khan. 
    The software Khan Academy uses for exercises is also open source, so a company or an individual could take it and use it for their own collection. 
    I like how you formulated the intro idea - looking at patterns without attempting to solve problems yet. Some people equate all of math with problem-solving. This pattern-finding is a good example of math that is something other than problem-solving.
  • Gina Mulranen   Feb. 21, 2013, 7:36 a.m.
    In Reply To:   Maria Droujkova   Feb. 21, 2013, 7:11 a.m.

    Thank you for the insights into the Khan Academy video sharing. I would love to offer the idea to students of remixing or creating their own video tutorials!

  • Lisa Ritt   Feb. 17, 2013, 3:58 p.m.

    Part 6:

    This is a tough one. As were talking about it on Weds in our live forum, I guess I wa thinking mostly about computer delivered math. So, I think my understanding of computer-based math is when you are getting your computer or whatever technology to essentially do the calculations for you. For ex, in a spreadsheet, when you put formulas in & then the math is computed for you.

    This is what I'd say is mostly going on with the Wolfram Alpha website. I actually was frustrated at how little step by step was being shown through each input or problem I asked it to solve. I get nervous when technology is doing the math for you when I think of students in elementary or middle school. But, I think as you move forward with higher thinking math concepts, it makes sense to use technology to jump you long as your knowledge foundation is there.

    I think I feel uncomfortable with this for higher math for myself right now because I'm not there yet as a mathmetician where I feel like I fully grasp the beginning steps so I don't want to have them calculated for me. I want to go through the steps so I'm assured of understandinh the end result.

    I see this with the middle school kids I'm working with now. They are very attached to using their calculators, yet they can't do simple multiplication equations. This is the question for me. WHEN or HOW much knowledge do you need to have before you let a computer do the calculation for you? And, how can you demonstrate that knowledge?

    So, as I dive in to this class, I continue to realize more and more how many foundational skills I've forgotten. I'm spending plenty of time practicing fundatmentals. When I think of solving a polynomial equation, I honestly don't remember graphing them. So, I don't want to have Wolfram Alphra do it for me until I know, I can do every step in between myself.

  • SueSullivan   Feb. 17, 2013, 6:29 p.m.
    In Reply To:   Lisa Ritt   Feb. 17, 2013, 3:58 p.m.

    Lisa, I too was disappointed with WolframAlpha; the answer was provided but no explanation given.  Great for verifying answers, not so great for explaining/teaching.

    I agree with you wanting to learn things for yourself so that you can learn how they work, rather than just being shown an answer to memorize.  To me, situations like that remove the teacher from the situation - the student isn't taught anything; the student is left to teach themselves, in spite of the fact that if the student already knew, they wouldn't be a student but a teacher.  Student exploration is essential to the learning process,but is ineffective if a student doesn't have access to someone who can answer their specific questions, in whatever terms the student uses.  I don't think that WolframAlpha can do this as efficiently as a teacher that knows their students.

  • Maria Droujkova   Feb. 21, 2013, 8:41 a.m.
    In Reply To:   Lisa Ritt   Feb. 17, 2013, 3:58 p.m.

    Lisa, you pose excellent questions about CBM (computer-based mathematics). You write:

    Computer-based math is when you are getting your computer or whatever technology to essentially do the calculations for you.

    WHEN or HOW much knowledge do you need to have before you let a computer do the calculation for you? And, how can you demonstrate that knowledge?


    Here is something to contemplate. What math do YOU do while the computer does calculations?

    For example, compare these two student tasks.

    Task 1. Solve (x+3)^2=0

    Task 2. How can you make a quadratic equation have two different solutions? A single solution? No solutions?

    You can use your computer to assist you with both Task 1 and Task 2, by doing calculations and graphing. But what YOU do with computer-generated answers will be very different in Task 1 and Task 2, right? What are the differences?

  • SueSullivan   Feb. 17, 2013, 3:38 p.m.


    PART 1:  Circumference might seem like an elementary topic, but I wanted to discuss something that's actually taught in middle school.  My WolframAlpha page explaining how to find circumference is very straightforward and simple:

    PART 2:  I didn't see an area on TeacherTube to leave comments, so I'll post them here for the video I found about the concept of circumference at  I feel this video is a useful teaching tool because the visuals use contrasting colors and things are very easy to see, students can see exactly how the teacher is using the software to create the drawings (a big plus if students will be using that exact same software themselves), circumference is found using a step-by-step process, pi is explained as being a Greek letter (rather than 'just a symbol'), and a non-scientific calculator is used for calculations using the approximation of pi=3.14 (students may not have access to scientific calculators and need to be familiar with the approximation).  I also liked the fact that it was presented by both male and female teachers, as unfortunately there is sometimes some bias regarding the capabilities of women when math and science are concerned. 

    PART 3:  I found an interactive at that shows the relationship between diameter and circumference.  This interactive deals only with circumference and is rather short.  There's also a game on the parent site that reviews additional basic circle facts (such as chords, central angles, etc.); students click on a balloon-type shape, and it pops open to ask a question about circles - it's at

    PART 4:  Higher-order questions that I might ask about the topic of circumference (and circles in general) include:  What is the difference between diameter and radius, how to find the diameter (or radius) when given a circumference, what type of triangle results when the endpoints of a diameter are connected to a third point on the circle, pi in general, and the angle that results when two radii are combined to form a diameter, just to name a few.

    PART 5:  I feel that software such as Geometer's Sketchpad or GeoGebra would provide students with both a visual and numerical explanation of the questions in part 4.  Interactives, virtual manipulatives, and software such as Geometer's Sketchpad or GeoGebra can provide a hands-on introduction to concepts such as circumference and basic circle geometry.  Programming environments can also be useful for more advanced activities (such as representing visuals mathematically) as it allows the user to actually see what type of graph their functions produce.  Solvers can be used to check one's work, or simply as a time-saving tool to perform complex computations quickly.

    PART 6:  WolframAlpha provides both computer-based and computer-delivered mathematics.  Students can use the site as as a tool for computing, but topics/problems are delivered.  The videos from TeacherTube and other sites generally deliver a lesson that might be the the basis for the student to perform computer-based math learning.  The circumference activity that I chose for Part 3 is primarily math-delivered as the lesson is delivered.  The student explores the concept by placing diameters on an unraveled circle, which is technically a math-based activity, though it is extremely brief.

    I feel that either computer-delivered or computer-based could be used to introduce a topic - students might view a video about circumference, or they might be asked to explore the concept via non-computerized exploration (tracing circumference of various circles with a string, measuring the strings, and comparing these lengths to the respective diameters to try and find a relationship/constant).  Which method a teacher would use depends on the learning environment (availability of tech would be an obvious factor).  I personally think that tech offers opportunities to make things more aesthetically appealing (i.e. visuals, sounds), which might be a more effective approach to get the attention of uninterested students.  The way I interpret it (and please correct me if I'm wrong!) is that computer-delivered math uses tech as a tool to make something available; computer-based math uses tech as a tool that allows the user to manipulate it.

  • Maria Droujkova   Feb. 21, 2013, 7:23 a.m.
    In Reply To:   SueSullivan   Feb. 17, 2013, 3:38 p.m.

    Sue, I would like to commend the choice of topic. I agree that circumference is very accessible to young kids ("elementary" is a good word) - but at the same time, it can go as deep as calculus and beyond. That's because Pi can be found through limits. Oh, you can find it experimentally by measuring (talk about math vs. science). But how do you PROVE that Pi is what it is? That's deep and interesting and fun.

    I think some of your questions are higher order than others. For example, "What type of triangle results when the endpoints of a diameter are connected to a third point on the circle?" is a factual question, with one-word answer, but it can lead to very beautiful high-order explorations, so I consider it to be "up there." On the other hand, "What is the difference between diameter and radius?" is more about terminology - I'd say it at the level of Understanding or so. However, as usual, you can start with this question and fly higher. For example, you can ask kids some WHY questions. Why do you think people felt the need to name both radius and diameter, if they are so closely related?

    I would like to share a recent cute "provocation" by Vi Hart, a math artist. She created a video manifesto for abolishing Pi. It has to do with the question of radius vs. diameter! Check it out:

    "User manipulating math" is a good phrase to describe computer-based math. You talked about modeling before, which is another relevant term. I think math ed people would understand modeling, but if you need to explain things to parents, "manipulating" may be an easier word for some.

  • Lisa Ritt   Feb. 17, 2013, 2:07 p.m.

    Part 5 Continued...

    This is a really hard concept for me...may seem silly, but it is. So, after much research & in trying to imagine breaking this binomial equation for a student in technology, I think I'd try to have them creat patterns similar to the necklace pattern at the bottom first in say a spread sheet format. 

    Then perhaps, look at moving that onto a graph, say on geogebra, or something like that.

  • Maria Droujkova   Feb. 21, 2013, 8:33 a.m.
    In Reply To:   Lisa Ritt   Feb. 17, 2013, 2:07 p.m.

    Just so others see what we are discussing, here is the picture of the binomial (10+3)^2 using Montessori beads:

    The idea of using a spreadsheet together with this visual seemed very interesting to me, so I gave it a try. I don't know if this is similar to what you had in mind, Lisa. Likewise, different students will probably implement it differently, even if they use the same tech (spreadsheets). So here is what I did:

    Verifying formulas with spreadsheets

    This is a somewhat low-level (probably Understanding-level) activity I occasionally do with students. That is, when they verify an identify, they program the two equivalent formulas into the two columns, and then try them with different numbers. They like to try large numbers or negatives or fractions, which is easy using a spreadsheet. So here I programmed two different formulas for the binomial. I am showing the extended formula.

    What did you have in mind for spreadsheets, Lisa? Something like this, or something else? I want to point out this Part 5 question is supposed to be hard, if you take it seriously. All design questions are hard!

    With GeoGebra, you could create "virtual algebra tiles" that work like Montessori beads. I found an example on their wiki:

  • Lisa Ritt   Feb. 17, 2013, 11:53 a.m.

    part 5:

    I'll tell you, I have a REALLY hard time with the concept of thinking of an equation in REAL LIFE. This link above shows a binomial cube. I think this is very helpful in thinking of what is actually happenning inside of an equation. In Math class, I don't remember any teachers helping me relate equations to actual physical things or images. This to me helps you understand how math moves into the world & its logical.

  • Lisa Ritt   Feb. 17, 2013, 9:41 a.m.


    PART 4:

    So my deeper question is: What does a polynomial look like? Is this possible to see? & What's the difference between a linear equation & other polynomial equation images?

  • Maria Droujkova   Feb. 21, 2013, 8:15 a.m.
    In Reply To:   Lisa Ritt   Feb. 17, 2013, 9:41 a.m.

    The first question can lead to a very nice class project, where different students visualize polynomials in different ways (graphs, algebra tiles, color-coding and other diagrams), and compare and contrast their methods. Some people even compose music based on polynomials, so you can ask - "What does a polynomial sound like?' cool

  • Lisa Ritt   Feb. 17, 2013, 9:27 a.m.

    PART 3:

    I really like this interactive because the author does a great job of explaining how to "factor a trinomial, but she also explains WHAT FACTORING IS. In many cases when a watch a video, there is just an explaination of how to solve something...but not actually WHAT it is that you are doing and how it relates to other math lessons or components...This is very important in learning!

    The auhtor also highlights whatever she is speaking about which I appreciate. First she gives you an example of how to factor a trinomial & then she tells you to try the next one on your own & hit continue when you think you're done. Then, when you continue, she goes through the problem so you have an opportunity to check your work. LOVE IT!

    Unfortunately, there was not a place to leave a comment online. There was another website somewhat linked to this website through "classbrain"..which was Here, I was able to sign up as a teacher & look over lessons, create assignments, etc. I LIKE IT!

  • Maria Droujkova   Feb. 21, 2013, 8:13 a.m.
    In Reply To:   Lisa Ritt   Feb. 17, 2013, 9:27 a.m.

    You named two strong principles of teaching, one general (WHAT the topic is) and one specific to math representations (highlighting of important parts).

    When you evaluate games, videos and interactives, you develop a list of such good practices in your mind. So when you look at a new object, you can run through your list of good practices to see if the object fits your criteria.

  • Lisa Ritt   Feb. 17, 2013, 7:46 a.m.

    PART 2:

    Here are the comments I left after setting up a YouTube account... first time for commenting. 

    "HI. Thank you so much for being willing to take your teaching to the MASSES :) Your method here is CLEAR & straight forward for a novice!! I wonder if there is a way to also put the "Rules" (like using the last + or - Rule) somewhere within the video. I find that these are the things that are the hardest for me to remember when I'm not factoring trinomials on a regular basis. OR possibly saying, "You may want to hit pause to write down the RULE so you remember it." Hope you don't mind the suggestions. Again, thank you so much!"

    Honestly, I felt a little out of my comfort zone in my comments making suggestions feeling like "who am I to suggest" ..but.. I wanted to be honest & what would have helped me as I learn this if I'm a student (& I truly AM right now the student), is to have some time to write down, think & remember the RULES you need to know. I did look at many videos & what I found most common is that if you're not someone who is pretty familiar with regular math terms, most videos move pretty fast.

    Most videos are built on some math foundations being assumed. This one though I felt was pretty thorough. I really feel like there are so many solving equation RULES that need to be remembered & thats the hard part as your math gets layered & layered!!

  • Maria Droujkova   Feb. 21, 2013, 8:10 a.m.
    In Reply To:   Lisa Ritt   Feb. 17, 2013, 7:46 a.m.

    "First time for commenting" - right on! Good for you. Teachers' voices need to be heard, both in local communities and in online networks.

    Your comment is polite and concrete. Most authors LOVE comments of that sort. Of course, occasionally you may run into an insecure person who doesn't. But I found that most of the time people are thankful and even become online colleagues or friends if you reach out and talk with them about their work. Of course, if you do like their work overall. Commenting on things you dislike is futile.

    You are right about "assumed foundations." We can't start from explaining what 1+1 is in every lesson - but what DO we assume? Something to contemplate...


  • Lisa Ritt   Feb. 17, 2013, 6:56 a.m.

    PART 1:

    Above here is my inputs & solutions.

    I'm having trouble having the exponent input working on my macbook. Any suggestions? It only seems to work the first time I try it in any equation. When I go to put it on a second number sequence, it doesnt work ! UGGGG

    My higher math understandings are limited at this point as I'm just starting back to school so I wanted to stick with something fairly basic like learning how to factor trinomials.

    Also- this website doesnt give you too much step by step....even when you ask for the "step by step." for us recently re-born to math 

  • Maria Droujkova   Feb. 17, 2013, 7:04 a.m.
    In Reply To:   Lisa Ritt   Feb. 17, 2013, 6:56 a.m.

    Lisa, the exponent sign looks just how you input it, ^

    But try using the multiplication sign * between coefficients and x's

    Like this:*x%5E3-45*x%5E2%2B10*x-22%3D6

  • Lisa Ritt   Feb. 17, 2013, 8:01 a.m.
    In Reply To:   Maria Droujkova   Feb. 17, 2013, 7:04 a.m.

    I figured it out! I stupidly didnt see the exponent sign just above the "6" on my keyboard. Clearly I need more coffee! Thanks Maria!

  • MgnLeas   Feb. 15, 2013, 9:51 p.m.

    PART 1

    Help me please!  I cannot figure out how to pose questions and give solutions. I have found many examples on the topics I wrote about for the rest of this assignment. However, I am lost as to how to solve one myself for this part of the assignment. If anyone can give me some guidance I would appreciate it!

    PART 2

    Here is the YouTube video I liked for solving systems of equations. The teacher is explains everything in simple terms. At one point he asks if everyone is confused and the students say yes. He replies it’s ok it’s only been 30 seconds, and they laugh. I can tell his class is at ease learning a new topic. He jokes with them while he writes on the board. I like this because they are having fun while learning math!


    PART 3

    I really like this site for help with solving systems of equations. There are many visuals and lots of examples. There is also an interactive section where students can move two lines and the equations change as well as the solution. On this page there is a link to games as well.

    The next sites are for inequalities. I am putting them in here as well because for part 4 I was thinking about how to expand on systems of equations.

    This game has students solve inequalities and the Genie tells you if you are right or wrong. I like the endless examples. The student also gets instant feedback and the opportunity to try again if incorrect. With traditional examples they have to wait to get the answers, most of the time the next day. Everyone likes to know instantly if they are right or wrong.

    This is more like a video game. You (the student) are a space ship and you shot the correct solutions to the inequality. I like this game because it is fun. The solutions come at you faster as you go up levels and you have to think quickly. I was laughing as I was playing. (Played longer than necessary!)

    PART 4

    So I got to thinking about higher order thinking skills and one thing I came up with was expanding on the topic. This led me to think about inequalities. Students would have to apply their knowledge of how to solve equations to the inequalities. The solutions are different so it takes some thought and understanding of the solutions. Some questions I would pose would include; What is the difference between no solution for equations and inequalities? Do systems of inequalities have many solutions or only one? When would a system of inequalities have only one solution?

    PART 5

    For the questions I posed, I would want my students to first try to solve them without technology. I think the concept of what solutions work and don’t is one that once they see it, the students will understand the real world applications better. For example if you run a pizza kitchen and have a budget for crust, sauce and cheese, a system of inequalities will help you see how much of each item you can purchase to maximize profits. If they find the answer by searching, I feel they might not have as good an understanding as they could. However, once they have an understanding or starting place for an answer, they could search online for more concrete examples. Any graphing software would be able to help them visualize the answers. Graphing inequalities can get messy, especially if the student uses pen or has a bad eraser. So to graph these systems with technology would be beneficial. I would have the students use technology to help them with this process.

    PART 6

    So I went on a search to make sure I understood the difference between computer based and computer delivered math. I came across this site There is a video of Conrad Wolfram talking of computer based math. After watching it, I feel like my answer to part 5 is somewhat an idea of the past. Conrad makes great arguments and gives good examples of why we should let the computers do the calculating part of math and have the students do the thinking and posing questions. Computers are a part of the everyday life of almost everyone.

    I feel like computer based math and computer delivered math are quite different. When I think of computer delivered, I think of a situation like our class. The content and assignments are delivered via technology. We do use technology obviously, but we are doing the leg work so to speak. I would use computer delivered math to start a new topic. In a classroom setting, I could have assignments or directions on the smart board when the students come in and they could just start without direction from me. I have heard of schools starting to utilize tablets and ipads more frequently, I could also send assignments to the students for them to work on. As far as computer based math, I feel this is more the computer doing the leg work. As far as the equations, the students could try to come up with very complex systems that might be impossible to solve by hand, but the computer can solve it. This would be a great way to get the students to do more thinking about what the equations represent instead of just doing the solving and graphing. They could spend more time pondering why there is only one solution, or tow or infinitely many and what the solutions really mean in real world situations.

  • Maria Droujkova   Feb. 16, 2013, 1:06 p.m.
    In Reply To:   MgnLeas   Feb. 15, 2013, 9:51 p.m.

    What question would you like to pose? An example of an inequality? Maybe you can write your question here in words, and we will work on the Wolfram|Alpha syntax together.

  • MgnLeas   Feb. 17, 2013, 10:57 p.m.
    In Reply To:   Maria Droujkova   Feb. 16, 2013, 1:06 p.m.

    Ok so here is the question I posed, 2x + 5 < 3y, 3x -2 > 3y. I wanted to see the graph which would show the soltions that make both inequalities true. Here is the solution What I like is that the graph only shows the area that make both inequalities true as apposed to shading all the solutions for both inequalities. This can confuse students because not all the solutions hold true for both inequalities, I have my students color the two areas in different colors, this seemed to help.

  • Maria Droujkova   Feb. 21, 2013, 10:35 a.m.
    In Reply To:   MgnLeas   Feb. 15, 2013, 9:51 p.m.

    Meagan, I interviewed Robert Ahdoot (the creator of the video you liked) in 2011: Your observations are spot on - he says videos help students to say at ease, but also to focus better. Also, Rober believes laughter helps with math, so he jokes a lot.

    Your Part 4 questions are strong overall for inviting investigations. What I like about those questions is that you can ask them again and again about different math objects. Then students can learn to ask these good questions about new topics, too.

    - What is the difference between solving ___ and ____? (Equations and inequalities; quadratic and cubic equations; problems with numbers and problems with variables; questions about formulas and questions about graphs of those formulas...)

    - When does ___ have more than one solutions? (Equation, intersection of graphs, optimal value of a variable...)


    I am not sure about your answer to Part 5 being outdated. It can work well. Quite a few prominent educators at Conrad Wolfram's first and second meetings in London offered similar designs! That is, they suggest you explore the physical world before using computers to analyze it more in-depth.

    We can strive for certain flexibility, though. What if we played with exponential vs. power functions on a grapher before exploring the trajectories of falling water by a fountain? What if we tweaked variables in a simulator of a chemical reaction before heading for the lab? 

    Again and again, the question we can ask as teachers: What are our students doing? So, the computer computes... but what math does the student do with the results of the computations? Does the student just turn the results in? Does the student use the results to address a deeper question, to notice patterns, to propose conjectures?

    The example you have in Part 4, "When do systems of inequalities have many solutions?" can be investigated by tweaking parameters in a computer solver. But it will be the student observing patterns and conjecturing...

  • MgnLeas   Feb. 23, 2013, 9:22 a.m.
    In Reply To:   Maria Droujkova   Feb. 21, 2013, 10:35 a.m.

    It is cool that you interviewed him!

    i wonder sometimes about, what are students doing? When it comes to using computers. However, I then think about the fact that they need to know what they are putting in and about what they should be getting out. If they imput a formula for a parabola and the graph they get is a circle, the student should know something went wrong and be able to figure out what. As I mentioned graphing can get messy, the neatness a computer provides will allow for the student to spend more time on different questions about the graph rather than 20 minutes getting the picture right! That would be a waste of great learning time. Also if they get frustrated in the graphing they might not want to move on to harder thinking questions. So in this case the computer would free up mind space for actual thinking!