I recently had the opportunity to work with a student to investigate parabolas and quadratic functions. We used one activity to investigate two different quadratic relationships. First we observed the shape of the stream of water coming out the side of a water bottle and then we observed the rate the water drains.
The Water Fountain
I set up a cylindrical bottle of water on a crate. The bottle had a whole in it covered with a piece of tape. I asked students for some predictions. What will the shape of the water coming out of the side of the water bottle look like. What will happens to the stream of water as the water level goes down?
I noticed that the student drew the water stream coming out of the bottle like it comes out of a water fountain (where we had just filled the bottle). We took the tape off the hole and then watched the water come out while making some observations and taking some photos. We selected a good photo (the black bulletin board in the background really helped) and loaded into Desmos. Then we used a table to record some points along the steam of water. After that we did a linear and then a quadratic regression on the point to see that the parabola was a much better fit than a line. We then had a chat about parabolas and projectile motion.
Draining the Tank
We set up the water bottle again but this time instead of looking at the shape of the stream of water, we focused on how fast the water level fell. I asked the student to predict what this might look like. You might ask students to predict what a graph of the water level might look like over time for the two situations below. How would the graph look when filling the tank compared to emptying the tank?
The water flowing into a tank should rise at a linear rate. Students should expect that when the water drains from an open tank, the flow will be greatest at first and then gradually decrease as the water level decreases. (This is an application of Torricelli's Law).
Next we taped a measuring tape to the side of the bottle and collected some data as the water flowed out of the bottle (A similar experiment is described in Canavan-McGrath, Foundations of Mathematics 12, 429). We used the stopwatch on my cell phone to record the time at each centimeter of height. This wasn't as accurate as I had hoped due to some distractions in the room. We set up the experiment again and the second time I recorded the water falling using a video (I used the CoachMyVideo app). We were able to get much more accurate values this way.
We entered the data in a table on Desmos and then did a quadratic regression to fit a curve to our points. I was a bit surprised at how well the data from the video analysis on our second attempt fit to a quadratic curve (R^2 = 0.9999.
I really liked how we could use the exact same setup to investigate two different quadratic relationships.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 11 RF02 - Demonstrate an understanding of the characteristics of quadratic functions, including: vertex, intercepts, domain and range and the axis of symmetry.
Pre-calculus 11 RF04 - Students will be expected to analyze quadratic functions of the form y = ax^2 + 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.
I've been thinking a lot lately about the role of practice in the mathematics classroom. Reading Mark Chubb's blog post made me reflect on my teaching. Practice provides students an opportunity to enhance and refine newly acquired mathematical concepts and skills. I have lots of questions about practice:
I think that it is important to be reflective about what I devote time to in class. You don't want to invest time in something that is not going to pay dividends in student understanding. Jon Orr has an insightful Ignite talk where he talks about "being picky" with the technology tools he uses with his students. I think that teachers should think critically about practice routines as well. Below are a few criteria that I consider when making decisions about student practice.
Characteristics of Effective Practice:
It is not desirable for students to spend time practicing and have no idea if they are producing accurate and correct mathematics. Self checking activities allow students to know immediately if they are on the right track or if they need to ask for clarification. One of my favourite activities that are self-checking are row games. I first learned about row games from Kate Nowak's blog. In a row game, students work with a partner. Each of them completes a different question, but the answer to both questions are the same. If they don't get the same answer, they collaborate to find out where the mistake is. About a year ago I wrote a blog post describing a number of other self checking activities including row games, add-em up, tarsia puzzles, and question stacks.
Purposeful practice is practice with a goal to achieve. An example of a question with purpose is an Open Middle question. The Two Fractions Challenge from Michael Fenton is a great example of this type of question. In this problem students create an expression using 4 digits and one operation. The goals is to make the value of this expression the largest, smallest, or closets to zero. Students will evaluate a great many fraction expressions as they hunt for an optimum solution. Another practice activity with a clear goal is a Tarsia puzzle. I explored a few math practice routines with purpose in a previous blog post.
Just about every student loves a good game or puzzle. The challenge is to find a game that is easy to learn and targeted to the math skill you want to practice. An example is playing the card game war to practice adding integers. Remove the face cards and jokers from a deck of cards. Shuffle and deal the cards to two students. Red cards are negative numbers and black cards are positive numbers. Each student lays down two cards and adds them together. The with the largest value wins the cards. A couple of puzzles that I've seen used a number of times in class are KenKen and Shikaku (aka Rectangles) puzzles. Both of these puzzles come in a variety of difficulty levels and require lots of number sense and logical reasoning.
You might also consider adding to movement to an activity in order to boost engagement. For example, have questions posted around the room (i.e. a math scavenger hunt) instead of printed on a handout. Another way to add a bit of movement is with stations set up around the classroom that students move between.
When practice includes one or more of the criteria above, I believe it will be more effective. Once you've though about how you're going to practice math, the next step is to thing about what you're going to practice. Often it is the topic you're exploring in class but sometimes you might include some cumulative review as well.
I've been exploring retrieval practice lately and looking for strategies to incorporate it into classroom practice. The goal of retrieval practice is to cement understanding in long term memory.
"Retrieval practice is a strategy in which bringing information to mind enhances and boosts learning. Deliberately recalling information forces us to pull our knowledge “out” and examine what we know." - https://www.retrievalpractice.org/
One strategy is to start the class with four quick questions for students to do in 5 minutes. These questions relate to a mixture of outcomes from previous units of study. Students have to reach back into memory in order to determine the methods and strategies to solve the questions. After students have had a chance to work on them spend the next 5 minutes reviewing solutions.
I've been taking Mandarin Chinese lessons with my son for the past year or so. I know that if we don't practice in between weekly lessons then we quickly forget what we've learned. Instead of working through workbooks and study sheets, I try to include some conversational practice throughout our day, while eating breakfast or in the car on the way home from school. Our favourite ways to practice are playing games and singing songs. We've made up a couple of our own games to practice together. The practice helps keep our skills fresh and helps solidify our learning. My next challenge is to learn to knit. My son is learning at school and is trying to teach me. He makes it look easy... I've got a lot of practice to do!