One of my favorite recent technology tools has been the ability to incorporate Mathigon Polypad screens into a Desmos classroom activity. Last week I had the opportunity to work with a high school class learning to change from standard form to vertex form of a quadratic using completing the square. I thought this would be a great opportunity use virtual algebra tiles in a Desmos activity. Students at this school have had lots of experience using physical algebra tiles in previous grades so these students were already familiar with the manipulative. The activity starts out with a review of perfect square numbers and then shows how these relate to perfect square polynomials. Then we moved to using virtual algebra tiles in Polypad to model perfect square polynomials. From there we progressed slowly towards completing the square. Once we built up to generalizing the process of completing the square, students had an opportunity to practice this skill. Instead of giving them feedback on the correctness of their answers, I was able to give them pictures of the algebra tile representation to compare their answers to. I think this helped students reflect on their answers instead of just looking for a green checkmark. I used the pacing tool to make sure most students had finished the question before moving to check their work on the feedback slide. Desmos had an amazing guide to help teachers reflect on the activities they've created. It is called "The Desmos Guide to Building Great (Digital) Math Activities v2.0". I encourage everyone to check it out. Students seemed to enjoy this Desmos activity and the dashboard allowed me to provide targeted support to the few students that appeared to be struggling. After this we did some additional practice and consolidation using a completing the square fill in the blanks sheet from Dr. Austin Maths (a great site for a variety of practice resources). Nova Scotia Mathematics Curriculum Outcomes Precalculus 11 RF04 - Students will be expected to analyze quadratic functions of the form y = ax² + bx + c to identify characteristics of the corresponding graph, including vertex, domain and range, direction of opening, axis of symmetry, x-intercept and y-intercept, and to solve problems.
EL
About a year ago, I signed up for Samira Mian's Udemy course on Islamic Geometry. I also purchased a copy of Eric Broug's book Islamic Geometric Patterns. I wanted to learn the basics so that I could determine if this might be a good way to satisfy the grade 7 mathematics geometric constructions outcome. I designed a short unit that I described last year. Recently, I decided to try replicating some of these patterns using the online Desmos calculator and geometry tool. I think having some experience drawing these patterns with a compass and straight edge was helpful. If you're looking for some Islamic geometric patterns to try, YouTube is a great place to get some ideas. There are some great instructional videos from Samira Mian and Nora Youssef, among others. The first pattern that I tried was a Star and Hexagon pattern that I learned from Samira's Udemy course. I learned that sticking with exact values are worth the effort. Rounding intersection points and slopes of lines to the nearest tenths or hundredths place work well at first but the errors compound and things start to get messy down the road. Interlacing the pattern gave me lots of practice with domain and range restrictions. 8 Fold RosetteNora Youssef has a nice video tutorial on for drawing an 8-Fold Rosette pattern. I did this pattern twice. The first time I constructed the basic pattern and the second time I added interlacing. I used the polygon function to add colour and figured out how to use trigonometry to rotate the polygons around the origin. This made it really efficient. I created a table with the vertices of the polygon and then just duplicated and rotated that polygon around the rosette. I duplicated the polygons multiple times to make the colours bold. Links:
Mathy MomentsYou can see from my notebook below that some of the math took me a few tries (this goes on for several pages). To make the weave for the 8 fold rosette, I made lines parallel to the original with a distance of 0.5 above and below. Each ribbon was then 1 unit wide. I was working with the equations in point-slope form. I'm pretty sure that there are more efficient ways to do these calculations but I haven't discovered them yet. I really like how these messy bits encourage me look for more efficient and elegant methods. Desmos Geometry ToolAfter working with the Desmos calculator for a while, I wanted to give the geometry tool a try. I decided to try a pattern that I saw on the Pattern In Islamic Art website. This site has some great resources. The pattern that I tried was from David Wade's book Pattern in Islamic Art. The geometry tool requires much less algebraic manipulation, but I find hiding the underlying grid is much more tedious than in the calculator. Everything has to be hidden individually instead of turning a whole folder on or off in the calculator. I've drawn this pattern in the past by hand and it would have been much more difficult if I didn't have that previous experience. Future ProjectsI've tried tiling some designs to cover the plane but I haven't come up with any good methods for this yet. I've also tried using sliders to dynamically adjust some of the relationships between the sizes of the pieces in these designs. These are great challenges and are helping me learn new features of Desmos. Dan Meyer wrote "If Math Is The Aspirin, Then How Do You Create The Headache?" I hesitate to call these graphing projects "headaches" because I enjoy the challenge. Regardless, this is a case where my need for mathematical solutions guide my learning and give me reasons to explore new graphing methods. EL
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