Loops and Donuts
Yesterday I gave a presentation on self-checking math activities for the 2023 NCTM Virtual Conference. I find presenting at conferences valuable because it makes me reflect on my work and think deeply about why I'm selecting or creating certain activities for the classroom. I thought I was finished with my presentation when I saw a variation of a loop card activity that I decided to give a try.
Loop cards make for a great self-checking math activity. In a "loop card" activity, the rectangular cards each have two values or expression on them, much like a domino. Students evaluate the expression on the right of side of the card and find a different card whole value on the left side is equivalent. These cards are then joined together (for example, x+3=7 on the right side of one card would join with x = 4 on the left side of another card). Students continue this way until a loop is created with the cards. For a nice example of loop cards, check out Don Steward's post at https://donsteward.blogspot.com/2015/02/loop-cards.html.
The variation on this activity that I saw was NRICH's Doughnut Percents activity. In this activity, four students work as a team. Each student is given four cards and tries to make a small loop (for a total of 16 cards). They will typically not have the correct cards and need to trade cards with other members of the group. When complete, there will be four, four-cards loops created, one for each student. The twist with this activity is that the students have to do this quietly. They can just grab cards from other members of the group, but wait for the other student to give them a card that they need. It is very difficult for students to work together silently.
I was inspired by this activity to give it a try with other classes. I created a "Derivative Donuts" activity and gave it a try with a Calculus class to review power, product, quotient and chain rules for derivatives. The google slides file can be found here: https://docs.google.com/presentation/d/1lMrJhNWY3jPCqlZrui8EBPe5YeCUdLxt3-akhkGwPxU/edit?usp=sharing. The cards are organized by loop in this file... with a larger group, I would mix the cards up and then have students cut them out... many hands make light work. I used the percent activity above to teach students the rules for the activity using content that they were confident with. Then I gave them the derivative activity which they found challenging.
I thought that this activity was a nice change from a typical card sort or question stack type of activity. I like that students worked together as a team to try to create the loops. Because students weren't talking, one student wasn't able to dominate the team and take over the process. Each students had to participate for the group to be successful. The next time I try this activity, I'm going to try having teams of three students with five questions each. I think groups of three might be easier to form small groups (although I'll need more sets of cards). I also recently saw NRICH's Simplifying Doughnut activity which I'm looking forward to try with students.
Completing the Square
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.
Cracking the Code
I was a fan of Mastermind as a kid, envisioning myself like the mysterious gentleman on the box lid, using my superior intellect and logic to solve the puzzle and defeat the codemaker. In this two-player code-breaking board game, the codemaker sets a sequence of four colored pegs using any of six available colors. The codebreaker makes guesses and after each guess, the codemaker responds with key pegs. A black key peg indicates a correct guess in the correct place and a white key peg indicated a correct guess in the wrong place. A quick game with simple rules.
Kent Haines, in his blog Games for Young Minds gives some excellent suggestions for questions to ask children when playing a game of Mastermind. He states, "The best question to ask, in my opinion, is 'What did you learn from that guess?' The question is open-ended enough that you aren't guiding your child to a particular conclusion, but it keeps your child focused on the fact that she can learn something from every guess, even one that resulted in no pegs."
There are lots of similar guessing games where each incorrect guess gets you closer to the answer. Wordle has become an incredibly popular online word guessing game that has brought new attention to this style of game. There are dozens of versions of similar online guessing games. Dan Meyer wrote about the characteristics of this game in a post titled Why Wordle Works, According to Desmos Lesson Developers. One of the key characteristics is effective feedback. Like a good self-checking task in the mathematics classroom, effective feedback attaches meaning to thinking. Each piece of feedback supports the learner in getting closer to their goal.
Pico, Fermi, Bagel
Pico, Fermi, Bagel is a numerical guessing game with a long history. I first learned about this game from the book Math for Smarty Pants by Marilyn Burns (1982). In this book she called the game "Bagels." In this game the codemaker selects a three digit number. After each guess, the codemaker responds with clues. "Fermi" means that there is a correct digit in the correct place. "Pico" means that there is a correct digit in the wrong place. "Bagel" means that the guess has no correct digits. For example, if the code is 967 and the guess is 617, the clue would be "Fermi, pico." Fermi for the 7 (correct digit in the correct place) and pico for the 6 (a correct digit but in the wrong place). A guess of 167 would result in the a clue of "Fermi, fermi".
Let me tell you why I like this game...
Pico, Fermi, Bagel has been played in classrooms for decades. There is a reason that it has such lasting appeal.
A Pico, Fermi, Bagel Desmos activity
Here are some thinking questions you might ask students to consider.
Secret Message Activity
I'm a fan of self-checking activities. These activities give students immediate feedback on how they're doing. A well designed self-checking activity puts wrong answers to work. Like a game of Mastermind, each incorrect answer should give students some feedback on their attempt and prompt them to keep working towards the correct solution.
One type of self-checking activity is a "secret message" activity. You've probably seen this type of activity where students complete questions to assign letters to their answers and then these letters can be used to decipher a message. Often these messages are groan-worthy puns or jokes. My first year of teaching, I was bequeathed a well worn copy of Algebra with Pizzazz!, an entire book of just this type of secret message activity.
My issue with these activities is that students immediately try to decode the message without completing the math questions. The goal for these students was to determine the message instead of practicing the math (although they did improve their cryptogram skills). The point of the message is for student to use this to check their work. If the message is garbled, they know they've made a mistake and need to go fix it.
There is a relatively simple solution to this... just cut the message off the page. Give students the top part of the page with the question on it first. Once they've completed the question, hand them the bottom part of the page with the message on it to check their work. Additionally, instead of just a joke, I like the message to be some sort of instruction for students to do. On the activity below, the instruction is for the student to draw a silly picture in a box at the bottom of the page (in this case, "draw a picture of a crab at a birthday party"). This gives students a creative task while other students are finishing the work. Here is a link to the activity shown below.
Once I build the template, this activity didn't take long to create. I used ChatGPT to create the questions. Just stated, "Create the first 10 terms of twenty-six different arithmetic sequences." I also asked next, "Now write the equation for each of these sequences." Then I cut and paste the first four terms into the first page and the tenth term into my answer key... easy peasy.
Do you have a favorite secret message activity? How do you facilitate these activities in your classroom to get the most out of them?
Family Math Games
My "Mathy" Family Game Recommendations
Looking for some more recommendations? You can't go wrong with checking out some games at the Games for Young Minds blog. Do you have a favourite family game? Let me know about it.
I created the "Which One Doesn't Belong" discussion prompt for a professional development session with Mathematics 10 teachers. It has quickly become one of my favorite versions of this prompt.
Participants were quick to notice that there was a triangle that was impossible. I could watch the room and see their puzzlement slowly turn to a realization that there were several impossible triangles in this collection. This generated a lot of valuable discussions about a variety of different characteristics of triangles.
The conversation typically started with participants noticing that triangle C was the only triangle that "worked." In fact, with each group of teachers that I worked with, triangle C was the most commonly picked as the one that didn't belong. Then we went around the room and talked about what made each of the other triangles impossible.
What makes it impossible?
Triangle A - The angles of triangle A say "I'm an equilateral triangle" but the sides of triangle A say, "I'm an isosceles triangle." This disagreement is what makes the triangle impossible.
Triangle B - The 90 degree angle says that this triangle is a right triangle but the sides are not a Pythagorean Triple. Most participants, remembering the 3-4-5 Pythagorean triple, said that the side that said 12 would have to be 10 to make this a right triangle. That would make this an x2 enlargement (similar triangle) to the 3-4-5 triangle.
Triangle D - Participants noticed quickly that the angles in this triangle did not add up to 180. Less people noticed that this triangle also fails the Triangle Inequality Theorem which states that the sum of the lengths of two sides of the triangle is always greater than the third side. In this triangle 4 + 7 is not greater than 11 so this triangle would actually be a line. (Note that some versions of the Triangle Inequality Theorem allow the degenerate case where the sum of two sides is equal to the third side).
What else makes it not belong?
We can also talk about other reasons which each triangle doesn't belong. Triangle A is the only non-right triangle. Triangle B is the only triangle that doesn't have all three angles given. Triangle C is an vertex on the bottom instead of a horizontal side. Triangle D is the only triangle with odd side lengths and without a side of length 12.
I've changed how I facilitate discussions using the "Which One Doesn't Belong" prompt. As described in Geoff Krall's book Necessary Conditions, how a teacher facilitates a task is just as important as the quality of the task. The region that I work in has the benefit of having a Diversity Team with several Culturally Relevant Pedagogy Specialists. Working with them has shown me that we often don't need a different lesson to be culturally responsive. We instead need be intentional about how we might facilitate a task through the stance of being a culturally responsive practitioner.
I started this discussion by asking each participant to individually take a sticky note and decide which one they think doesn't belong and their justification. This let each individual have some time to consider their own reasoning. After recording their thoughts, I asked them to discuss their choice and their justification with others at their table. Did they choose the same triangle? If so, did they have the same justification or a different one? This allowed each participant to communicate their mathematical ideas and reasoning. Often, participants were sharing something that other people hadn't noticed. This allowed them to be positioned as someone with worthwhile mathematical ideas. After sharing, I asked each participant to walk to the screen to place their sticky note on the triangle that they had chosen. This gave everyone a chance to move, even if just a little bit.
This gave me, as the facilitator, insight into what ideas the participants had come up with. A sign of a good WODB prompt is that at least a couple of people select each different triangle.
Creating your own WODB prompt
Creating a WODB prompt can take some time. If you're looking for some ideas to help you get started, there are lots of examples at https://wodb.ca/. This website was created and is maintained by Mary Bourassa (@MaryBourassa).
Christopher Danielson, who wrote a children's book featuring WODB images, titled Which One Doesn't Belong? A Better Shapes Book, gave this advice in a recent "My Favorite Theorem" podcast: "If you ever try to design a “which one doesn't belong” set, what you want to do is think about whatever your domain is, so say it's shapes, you want to think about four properties of shapes, and then cover up the first one, and design one that has these three, but doesn't have the first one. And then cover up the next one, design one that has those three, but doesn't have this one. And by the time you're done, you'll either realize that your set of four properties is more intertwined than you had originally thought, and now you’ve got to go back and revise, or you'll have a set where you know for sure that there's at least one reason for each not to belong. But then extra, an important key to this is that you have to be open to the possibility that some kid will see a reason for a shape to not belong that wasn't the reason you'd intended."
Right Triangle Ratios and Art
There are lots of ways to introduce the primary trigonometric ratios to students. While brainstorming last week with a few teachers, we decided that this year we'd like to try to introduce it using an art project. This project comes from day 34 of Annie Perkin's #MathArtChallenge series. This "Similar Right Triangle Patterns" project asks students to make a design with similar right triangles that have been cut out of a sheet of paper. Here are some instructions:
Step 1 - Cut a sheet of paper in half along its diagonal. Set aside one triangle.
Step 2 - On your right triangle, fold or draw an altitude to the hypotenuse.
Step 3 - Set aside the smaller right triangle and repeat step 2 with the larger.
Ask students to continue this process a few times. They can then trade every other sized triangle with a partner that used a different color. Next, ask them what they notice and wonder about their set of triangles. Some students might notice that all the triangles are similar. In other words, they are all different sizes but have the same three angles. Students can prove this to themselves by aligning their triangles at each corresponding vertex.
Next, ask students to create an artistic design by arranging their triangles in pattern. There are lots of different interesting ways to do this.
For some cool pattern ideas, check out these from Becky Warren. You could also try arranging triangles using this Mathigon Polypad shared by Mark Kaercher.
Ask students to take their four largest triangles and measure the length of the two legs (adjacent to the right angle) to the nearest millimetre. Note that each time students fold and cut a triangle, small inaccuracies are introduced so using the largest triangles will give the best results. Ask students to take the smaller side length and divide by the larger side length (this is the same as asking students to find the slope, or "rise over run", of each paper "ramp"). Students will notice that these values are quite close. Ask students to average their four values. They can then compare their average with those from other students. Since all the triangles in the room are similar (assuming we all started with an 8.5"x11" sheet of paper), we should all have similar ratios.
Ask students to measure the angle at the bottom vertex of this triangle. They should get an angle of about 38° (given the diagonal of an 8.5"x11" piece of paper). Ask student if they think this is a special property of 38° right triangles or if all right triangles might have a common ration. Split the class up into groups. Have one group draw right triangles with a 15° angle. Have another group draw right triangles with a 30° angle and a final group draw right triangles with a 45° angle. What do they notice when they compare the ratio of the two legs (smaller side divided by larger side)?
At this point you might be ready to introduce the term of tangent as a function that lets you know the ratio of these two sides given any angle. Perhaps they could look up these ratios in a tangent table. There are lots of great suggestions for how to introduce trigonometric functions: Check out this idea from Jo Morgan or this idea about introducing trig through slope from Jon Orr.
Where would be your next step in teaching students about the primary trigonometric ratios? Additional explorations? Perhaps a more traditional worksheet to consolidate and reinforce their ideas?
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 10 M01 - Students will be expected to develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles.
The Rosenthal Summer Institute
I had the great fortune to participate in the first Rosenthal Summer Institute in July 2022 at the Museum of Mathematics (MoMath) in New York City. I was one of 14 invited educators from across the United States and Canada. The institute was an opportunity to work with these educators to discuss innovative lesson design and to learn more about the Rosenthal Prize and the library of winning lessons from the past 10 years of the competition.
During our time together, we had a number of discussions about designing lessons for students. One theme that emerged for me was the similarities between the design philosophy behind both the exhibits at the Museum of Mathematics and the Rosenthal Prize lessons. Below are a few of the shared characteristics that I noticed:
We had the opportunity to tour the MoMath exhibits with Tim Nissen, MoMath’s resident designer. Tim discussed the evolution of the design of several exhibits and how they had to anticipate how visitors, especially children, might interact with them. The safety of museum visitors is a critical factor. They had to anticipate how children might interact with the exhibits in unexpected or accidental ways to ensure that each exhibit was safe. Similarly, the Rosenthal lessons anticipated common misconceptions that students might have and how the teacher might respond to them in order to clarify or correct these misunderstandings.
Many of the museum's exhibits had descriptions of the mathematics principles involved displayed on a computer screen. The screens allowed the visitor to select one of three different levels of complexity for each description. This allowed each visitor to engage with the exhibits at the level to which they felt the most comfortable. Similarly, the Rosenthal lessons often contained directions for how the lesson could be adapted or extended for different students depending on their mathematical background. For example, in the Dice Auction lesson, there is a set of 15 "lots" to auction off. This is also an extended list of additional lots that can be used to supplement or adjust the number and complexity of the "lots." There are several additional "extensions and adaptations" listed for the activity that can be included as needed for the specific class you are teaching.
A museum is all about public engagement and exhibits have to be entertaining and enticing. Exhibits should draw visitors in and captivate their attention. Exhibits also should be intuitive and easy to understand. An great example of this is MoMath's Square-Wheeled Trike. Visitors can ride a tricycle with square wheels around a circular track made of inverted catenaries. Visitors easily understand the purpose of this exhibit. While they ride a host engages them in a conversation about the mathematics they are experiencing. The Rosenthal Lessons are also often designed to be engaging and interesting to students. They can draw students in and let them experience mathematics in a new or surprising way. The Random Walk lesson is a great example of this. Students can participate in a lesson that lets them physically see how random change can lead to surprising patterns.
Rosenthal Prize Lessons
In order to make the Rosenthal Prize lessons easier to access for Nova Scotian/Canadian teachers, I have organized below where several of these lessons might best be placed within the Nova Scotia/WNCP curriculum.
Can't Stop is a board game that was designed by Sid Sackson and released in 1980. It is a "press your luck" dice game in which players roll for dice and arranges them into two pairs. The sums of these pairs allow the player to progress their markers up the columns labeled with the values from 2 to 12. The goal of the game is to be the first to reach to top of three of these 11 columns. The height of the column is related to the probability that the sum will be rolled. The more likely the roll, the longer the column is.
I was excited to find a copy of this game at a second hand shop recently. I've played Can't Stop online numerous times (at Board Game Arena) but this is a game where playing in-person seems better than virtual. The physical process of rolling and arranging the dice as well as interacting with other players makes for a better experience. I think it plays best with 2 or 3 players. They game is made for 2-4 players but I've seen people use the square plastic pieces from the game Advance to Boardwalk to add additional players in different colors.
The Great Races
A nice user-created version of this paper and pencil game can be found at Board Game Geek. I like this version of the game because it is a quick game, with the "push-your-luck" aspect removed. Each player rolls just once, records their roll and then passes the dice to the next player. For a shorter game, you could play until 8 of the races are finished instead of completing all 11.
Game at a Glance
Because all the information about the game is contained on the game board, this is an excellent game for "Game at a Glance." Chad Williams has a collection of Game at a Glance images on his website, Beyond the Algorithm. Students are shown an image of a game in progress and could be asked a variety of questions.
I recently went camping and took this game along with me. I taught the game to several people (including kids) and because of the simple rules, they were all able to learn it in just minutes. Because all the pieces are plastic, they didn't blow away in the wind and if it got dirty, it could be easily rinsed off and cleaned.
I think that this game would make a great addition to a mathematics classroom collection. What are your favourite games for the math classroom? Let me know!
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 12 LR01 - Analyze puzzles and games that involve numerical and logical reasoning, using problem-solving strategies.
Mathematics at Work 12 P01 - Students will be expected to analyze and interpret problems that involve probability.
Mathematics at Work 11 N01 - Students will be expected to analyze puzzles and games that involve numerical reasoning, using problem-solving strategies.
Mathematics Essentials 10 G4 - Compare predicted and experimental results for familiar situations involving chance, using technology to extend the number of experimental trials.
Mathematics 8 SP02 - Students will be expected to solve problems involving the probability of independent events.
Mathematics 7 SP06 - Students will be expected to conduct a probability experiment to compare the theoretical probability (determined using a tree diagram, table, or other graphic organizer) and experimental probability of two independent events.
Hands On Related Rates
In previous years, I've taught related rates in calculus class by the book. As in, we worked through the example problems from the textbook together and then students individually worked on the practice problems. After working through some examples, I would give students some more challenging problems as an assignment.
Recently, a teacher reached out looking for some new ideas for teaching related rates. We brainstormed some ideas of how we could make related rates more engaging and hands-on. We decided to create some stations where students could experience and take measurements of related rates in action.
It turns out, this is probably the reason that related rates problems are in our textbook. While doing some research to prepare this activity, I found an article titled, "The Lengthening Shadow: The Story of Related Rates" by Bill Austin, Don Barry and David Berman. This article is from Mathematics Magazine (Feb 2000, Vol. 73, No. 1, pp. 3-12). In this article, the authors state that related rates problems originated in the 19th century as part of a reform movement to make calculus more accessible. By observing changing rates, students would be able to measure concrete examples and discover their relationships. Joshua Bowman discusses how he grounded his teaching of related rates using observations in his blog post Using calculus to understand the world.
Related Rates Stations
For this activity, we decided on four stations:
Station 1 - Blowing Up a Balloon. Blow 5 big breaths into a balloon. After each breath, measure the circumference of the balloon and calculate radius and volume. How are radius and volume related?
Station 2 - The Sliding Ladder. A metre stick is sliding down the wall. The bottom of the metre stick is moving away from the base of the wall at a constant rate. How fast is the top of the stick sliding down the wall? How are they related?
Station 3 - Building Fences. Build several “fences” (rectangles made with multi-link cubes) such that the length is twice the width. Put them in order of size. Measure each rectangle’s length and width. What is the rate of change of the area? How are perimeter and area related?
Station 4 - Driving Cars. Two toy cars are traveling at different rates in perpendicular directions. How fast are each of the cars travelling? How fast is their distance apart changing? How are they related?
After taking measurements at each of these stations, students drew a picture and created an equation to relate the quantities to each other. We used implicit differentiation to determine the relationship of the rates and then we tested our equations with our collected data.
Reflection and Resources
The activity went a bit long for one class period. Next time I would either split the activity up over two class periods or reduce the number of stations to three. This will allow more time to consolidate the learning at the end of the lesson. As Tracy Zager says, "never skip the close."
If you're interested in giving this activity a try, below are the files I used:
Related Rates Stations Google Slides
Related Rates Recording Sheet
Update - 13 April
After doing this activity a few times in classrooms, I decided to reduce the number of stations from 4 to 3. In a 75 minute period, students were able to complete the three stations with a few minutes at the end to consolidate the lesson. I also changed the set up for the driving cars question to make it a bit more interesting.
Here are the updated files that I've been using for 3 stations and a different problem for the driving cars problem.
Related Rates Stations Google Slides
Related Rates Recording Sheet
Nova Scotia Mathematics Curriculum Outcomes
Calculus 12 A3 - Demonstrate an understanding of implicit differentiation and identify situations that require implicit differentiation
Calculus 12 B14 - Solve and interpret related rate problems