About a week ago, I gave my students a handout to practice multiplying polynomials. It was just a quick check in to practice some of the skills we were learning in class. We were just starting this unit and some students already had some prior knowledge so I decided to give three different levels of challenge to choose from. It was a Friday afternoon last class so I didn't have high expectations for the level of engagement. To my surprise, nearly every student was working diligently on their chosen level of problem. Many students chose the most difficult level. About half chose the "Spicy" questions while the other half chose the "Mild" questions. A couple of students chose the "Medium" level or selected a few questions from several different levels. Letting the students choose their level gave them some agency in their learning. A few students who picked the "Spicy" questions worked hard to master the challenge. They asked lots of questions but felt really good about their accomplishment when they succeeded. I collected these sheets looked through them. I didn't mark them but instead used them to identify and share common misconceptions about factoring. I used examples from the students work (rewriting them in the style of "My Favorite No" from Leah Alcala) as the starting point for the next day's lesson. We discussed what we saw that was wrong but also identified some parts that were on the right track as well. I've been trying out different ways to offer more choice to students. It is important to offer a challenge for students that are ready for it but not to overwhelm students that are still learning. Students in my class come from a number of different feeder schools and have a wide variety of past experience. I created a factoring "scavenger hunt" activity last week for students with two levels of challenge side by side. Students could select the "mild" side if they wanted basic practive or the "spicy" side if they were ready for additional challenge. About 20% of students chose to do the more challenging side. Everyone in the class was very engaged and was able to complete the hunt with instant feedback (if they couldn't find their answer on another sheet, they knew they had made a mistake and worked to correct it). You might also notice letters on each question. I also added a ciphertext message for students to decode when they completed the loop. I've also been including choice in warmup questions and chapter review questions.
Let me know if you have a favourite way to offer students choice in their work and practice. 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 StationsFor 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 ResourcesThe 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 AprilAfter 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 EL
Some colleagues recently told me about an activity they had used in class called "Math Market". I'm not sure who originally created it. The teacher who shared it with me learned it at a math conference several years ago. I decided to give it a try with a Calculus class that was just finishing up a unit on integration. Here is how the activity is run. Students work in small groups (we had groups of three). Each group starts with $5 and selects a captain who can buy questions of different levels of difficulty from the market. Easier questions cost less and have a smaller profit. More difficult questions cost more and have a higher profit. The captain takes the purchased question back to their group to solve. Once they all agree on a correct solution, the captain returns to the market to sell the solution for a profit. The card is added back to the bottom of the market pile and some other group will have an opportunity to buy it. If their solution is correct, they buy a new question and continue working. If the solution is incorrect, they have to buy the question again to attempt a revised solution (or they can purchase a new question at a different level of difficulty). We decided to purchase the solution at a reduced price ($1 less) if they forgot to include the "+C" at the end for the constant of integration. The easiest questions were free so that if groups went bankrupt with an incorrect solution, they would still be able to "buy" another a problem. I printed the questions on coloured card stock and cut them out. Each question was marked with its level of difficulty. I also added a letter to the card so that it would be easy to find its solution to check the answers. Resources
How it WentI like that students got immediate feedback on their work. If it was wrong, they had to work with their group to correct their mistake. This was a test review for the class so there were lots of different types of problems mixed together and students had to determine what strategy would be best to solve each problem. It is a nice way to introduce some interleaved practice. This activity could be done with nearly any topic but it worked really well for integration as the questions were challenging and took them some time to solve. This made the market area less crowded.
I'm sure there are lots of variations of this activity. If you have some suggestions, I'd love to hear about them. EL
I am trying to be purposeful and organized in taking the "Calls to Action" from the recent NCTM conference in San Francisco. I am going to play with "Deleting the Texbook" over my next few blog posts (watch Meyer's most recent NCTM presentation here for elaboration). Since I attended my first conference about 7 years ago, I have been heavily influenced by the work of Dan Meyer. His Ted Talk "Math Class Needs a Makeover" really changed the way I thought about teaching math. I want to share something I did with first as a math coach with a class of grade 3 students and then as a presentation for teachers at the Nova Scotia MTA conference a few year back. I feel like I have been taking this particular "Call to Action" for many years now; I am excited about the evolution of this thinking and I wanted to show how I started with this work. The students in this class were a few weeks out from writing a provincial math assessment. The classroom teacher wanted to do some questions from the practice booklet. This was the original question: Instead of posing the question as is, I took out as much information as possible while keeping the math I wanted the students to learn in mind. The students needed to get curious before I hit them with the math. All staff at schools need to wear an ID badge and laynard (wait for it...it will make sense in a moment). Lots of people keep keys, change, notes, etc in the plastic badge holder. Instead of asking the above question, I announced to the class that I had some change in my ID holder (I put the correct amount in and shook it for dramatic affect!). Their job was to figure out how much money I had. They were a little confused at first, so I prompted them; I told them they could ask me questions about the coins. I gave them a few minutes to think, then to confer with a partner before I took questions from the class. I recorded and answered their questions. Here are the questions they came up with along with my answers in brackets:
Each time I took a question, I asked students "Why do you want to know that?" or "How will that information help you solve the problem?" Once they had all the information they felt they needed, they set off to solve the problem. They basically constructed the original question on their own. And because they had a part in building it, they really understood the context of the problem and had ideas of how they might solve it. When they were confident that they had the solution, I let them empty my ID badge and count the coins. That is much more satisfying than just giving the correct answer. Students got to prove that they were right, rather than the teacher being in charge of all the answers. I often hear from teachers that their students don't understand word problems and that it's not the math, it's reading comprehension. I see this strategy of building a question with the class as a way to help them get inside the problem before they start. Another reason I like this strategy so much is that I was able to create it by modifying a resource (practice booklet) that I already had. We can get bogged down trying to find appropriate resources for students at appropriate levels. By using a resource I already had, taking out parts, and layering them back in as needed I was able to differentiate on the fly. No tiered lesson plan, no separate handouts. It was one problem that we worked together as a class to first build, then solve. The Most Interesting Question I Almost Overlooked"What kind of coffee do you drink?
I didn't understand where the student was going with the question but I asked the usual questions of him: "How will that help you solve the problem?" and "Why do you want to know?" The student thought I probably kept my coffee money in my ID badge. If he knew the coffee I drank, he would have the problem solved.** Based on his question, we created a series of extension questions to keep the problem solving going. **All Canadians know the price of a medium double double. :P KZ |
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