Students get scared of long-winded problems. For example, this problem doesn’t even have the questions yet, but they freak out!
I used the same activity about the Crop Duster as last year to start out this block class activity. The premise is to get students to think about the situation without initially knowing what the questions about the situations are. Once they organize the information into “What is known?” “What are the possible mathematical relationships?” and “Sketch the situation” students are then given the actual questions to solve.
In using the follow-up activity, I’ve found that the hardest thing for students to determine are the potential mathematical relationships. Once they have the question, they totally forget these relationships and almost blindly try to solve without thinking about them. This activity slows them down and gets them really thinking about the mathematical constructs in the problem. They also learn that all potential mathematical relationships may not be used, but recognizing them helped clarify how to approach the questions.
I hope to do this activity with different function types throughout the year so that I have some student work examples for a presentation I hope to give at the Texas Instruments International Conference in Orlando Florida next February. Here is my submission:
Using a Problem-Solving Activity To Develop Mathematical Habits of Mind
Have you ever wondered how to help your students to think thoughtfully about a non-routine problem situation before diving in to solve it? Or help them persevere during the needed productive struggle phase? Or encourage students to use meta-cognition during and after the problem-solving process? So did I!! This student-centered problem-based collaborative learning activity requires students to read the problem thoughtfully and then obliges them to work and think together to organize what they know (including a graphical representation) generate questions determine an answer and finally communicate the solution in a cohesive and understandable way. Come and enjoy the fun!
I’m a little nervous at this point, but I know as the year progresses I’ll have more student work to share and some suggestions for how to use in other classes besides precalculus.
What professional risks are you planning to take on this year?
Last year I used the crop duster problem in my precalculus class as an introduction to the eight math practices. I was very happy with the results, and found that my students were able to identify the practices readily throughout the rest of the year. After reflecting on the activity I decided I wanted to do some changes for this year.
One of the changes was to take away the coordinate axes in the picture , so the students had to think about how to impose a coordinate axes. We had a great discussion about various locations, discussing the pros and cons for each of those locations. Naturally the students decided to use the center of the circular field as the origin, but they were the ones that decided the location rather than being told.
A second change I wanted to do was to take away the questions. I didn’t want the students distracted by trying to solve the problem before the event understood what the situation entailed. Upon reflection, I wonder if I should also remove points P and Q. This would cause the students to realize at some point that they need those points labeled. Anyway, I gave them this information and picture:
After giving them the picture, we took some time to look at the information and determine what was known. After brainstorm eight as much as we could, we then look for possible usable relationships inherent in the problem. They initially balked at the idea saying, “We don’t know what the question is, so how can we possibly come up with relationships?” With a little prodding, they’ve really got into it. Here is a list of those brainstorms:
The third change I wanted to incorporate was to have students generate possible questions rather than being told the questions. By having the students do this, I was asking them to use math practice 2: Contextualize the situation.
Once we brainstormed the questions, we talked a little bit about how we would actually go about answering these questions. The really exciting part of this process is that once the students are given the questions they realize that they came up with most of them on their own AND had figured out how to go about solving the answer them. Awesome stuff!
What are some activities you have streamlined in order to get students to do more thinking and sense-making on their own without scaffolding?
Over the years in AP Statistics, I have had students create a poster using the ASA (American Statistical Association) Poster Competition as a guide. I have really enjoyed using this venue to get students thinking about data visualization. The structure of the competition made for an easy implementation into the course. As I looked back at the posters, they had lots of wonderful things going on! Here’s on looking at the influences on the ability to become a US president:
Another compared Personality and Activity Preferences. I asked my students to think of ways to incorporate their 5 supporting questions with their main question and to create intriguing and complex graphs that draw the observer in to analyze the information more closely.
A drawback of the whole process is that once the posters are sent off to the competition, we never see them again, not get any feedback. Thus, I d idn’t have any exemplars to show the next year’s students nor any advice based on assessment feedback. Simultaneously, the time taken in the beginning of the year always made me anxious about whether we would cover the requisite Collegeboard curriculum by the test date; it is always so close as it is that taking the needed days sometimes worked and sometimes didn’t.
I think the experience is super valuable for my students, but the whole poster-thing is a little archaic. This year, as I was adding to my Pinterest boards, I kept seeing infographics that communicated data analysis for so many things. And then when we had the Career day, part of Veronica Smith’s presentation was about how she used infographics to present complex data in a meaningful yet understandable way for the lay person. And this is when I had an Ah-Ha moment! Instead of the usual end of year project, do all of the same things, but have students present findings in an infographic with the in-depth statistical analysis submitted separately.
Here is my first blush attempt at the end of year project. I am really excited but still a little tentative because its a new approach and there are always glitches. But I have a week to think about it!
How do I make Precalculus relevant to my high school students? We study the mathematics of finance! In particular, we use the Present Value and Future Value formulas for situations that are meaningful to teenagers who are looking forward to becoming independent. In comes the Partner Finance Activity.
Students are introduced to the scenario: You have just graduated from college and managed to land a job (hopefully using the degree you just earned.) You are ready to buy your first dream house or condo. I have the students use the randint( function on their Nspires to determine the range of house values they qualify for. Then they define some terms such as FICO score, points, and ARM. Then they use the internet to search for a home that fits their range.
Needed a catch for practicing binomial probabilities in AP Stats. This summer I saw these fishing rods for children aged 1-3…perfect for the high school!
So I put together 10 problems, cut them apart, folded and put a paperclip on each one. That way the magnet on the “fishing rod” would pick up problems from the “pool.” Dorky, but the boys at least had some fun.
The problems were great and the kids were focused and building confidence in their ability to approach and do problems about probability. Here’s an example (with answers :)):
It has been determined that 5% of drivers checked at a road stop show traces of alcohol and 10% of drivers checked do not wear seat belts. In addition, it has been observed that the two infractions are independent from one another. If an officer stops five drivers at random:
a. Calculate the probability that at least one of the drivers checked has committed at least one of the two offenses. 0.5431
b. Calculate the probability that exactly three of the drivers have committed at least one of the two offenses. 0.3344
How do you create engaging practice opportunities?
One of the ways I am trying to ensure long-term retention as well as student discourse is through engaging opener problems and exit questions. Students submit final documents on Fridays through Schoology (love this feature of LMS). Today, I used a problem I found on Bob Lochel’s blog mathcoachblog. Since he used it with his Algebra 2 class, I figured it would be a good out-of-the-blue, retention/review question and it was! At first my students said things like, “I don’t know how to do this!,” and “We haven’t done a problem like this yet.” Of course I encouraged them, saying “of course you can do this…you know I don’t ask you questions that you can’t figure out with what you know.” Some grabbed their calculators, some expanded the factorials and others noticed the underlying structure. Great paired discussions. I then had students share their approach based on the complexity and eloquence of their solutions. I loved seeing the students listening to each other and offering suggestions and asking clarifying questions And I never did say what the “correct” answer was 🙂 However, I did point out that the underlying structure helped many students to get to the answer with ease and without a calculator…think before you begin a “brute force” approach.
Last year in my Precalculus course, I decided I needed to really emphasize the 8 Practices as a way to approach unfamiliar and,to students, “scary” problems. This year I used a problem I call the Crop Duster problem based on a problem I found online at the University of Washington Math 120 Precalculus site…they actually share their textbook (written by David Collingwood) and it has some really cool (a.k.a rich) problems. I introduced the math practices through a foldable by having the students in groups of four read two of the eight practices to themselves, and then summarize to their group.
Then we focused on “making sense of the problem” and “looking for structures” by having the information and the diagram available to students, but no questions. Students were asked to generate a list of what they knew and what concepts from previous courses might apply to the problem (since we really didn’t know what the question was yet). Love how the student embrace a tough problem easily when there is no question to distract them!! How do you introduce the CCSS math practices to your students