I wanted to quickly share an awesome activity that is usable in a variety of classes. A website I use occasionally to browse teacher resources is TES. This UK based site is a treasure trove of shared lessons and activities. This gem, Amelie’s Fashion Mystery starts with a simple question: Will Amelie make it to the catwalk in time? Students work through mathematical clues in order to determine if this supermodel makes it on time to the fashion show. The task requires students to utilize a huge range of math skills and is differentiated with two versions. Thank you Jake Mansell for a great activity. Here are the files: AmelieFashionMysteryintroslideandvisibilitygraph AmelieFashionMystery(visibilityfromgraph) AmelieFashionMystery(visibilityfromformula) (1)
Author: Megan Schmidt
Stripping Down the Stock Photo
Smack dab in the middle of all of the awesomeness coming out of Rafranz Davis’s blog was a gem that stuck in my brain: Addressing the Edu Stock Photo. In short, Rafranz challenged the twitter/blogging teaching community to take a reflective opportunity to address a difficult issue in your school or classroom. Taking on this challenge made me feel a sense of freedom from what’s frustrating in my classroom by taking off the shiny bow and acknowledging what I could do more effectively in my classroom. Today’s Algebra class ended up being a great opportunity to reflect on what hasn’t been working in my classroom. It started very typically by doing some estimation. I walked around the room and noticed who was jotting down an estimate milliseconds before I wandered past their desk. I saw who was more interested in their snap chat than participating in sharing estimates and reasoning. I let the frustration build and boil over a little with my raised voice. The breaking point came when a student literally talked over me in a regular conversation-volume voice as if I weren’t leading a class in an objective. I sat down at my desk and felt in that brief moment like I was never going to get these kids to care about math. Didn’t they know how much time and effort I put into figuring out how to help them learn? Why didn’t they appreciate how much I cared about their learning. I kept putting the estimations on the board, not really saying anything. It would have been very easy to shut down at that point. There were about 25 minutes left in the week, and learning meaningful mathematics seemed out of the question at this point. Then I had a profound realization that changed my whole view of my class in an instant. I was angry and frustrated at the wrong thing. The kids in this class are stuck in the same cycle of schooling that they have been in for years. They know that they are tracked in the “low level” math class, and they have come to accept that math is not something they’ll be expected to be good at. And they also have come to expect the same cycle of student/teacher frustration: kids will talk and goof off, the teacher will get angry, yell, punish, and send kids out. Things will be calm for a few days and then they can begin the cycle again. It’s not the student’s fault, they don’t know any better. And it’s always worked before for them because they got themselves this far. I know this cycle is playing out in my classroom because these are nice, likeable kids. They’re creative and interesting. They’re emotional and sometimes dramatic. And I love them. I have loved the opportunity to get to teach them. But I could do a better job than I am. I could complain about the size of the class. And I do. Or I can change what I actually have control of, which is helping these students learn mathematics. I do have control over giving them opportunities to interact positively with a discipline that they have been fearful of going all the way back to timed-arithmetic.
Much of what impacts our memory of particular events as positive or negative is rooted in how the story ends. I believe the same can be true for education. My incurable optimism tells me that something else will work for these kids and I believe in them and in myself. To end that class period on Friday, I called on a good friend, Nrich.
One hundred percent of them participated, 100% of them engaged and wanted to be the one closest to 1000. It was a small victory, but was absolutely essential in ending the week pointed in the right direction again. Mathematical curiosity ensued for a brief moment (Why did one person get 1008 and another 992? Who is the winner?) Now Monday won’t feel like more of the same for my students and me. It’s a new chance to bring them together with mathematics and hopefully have some fun in the process.
Alright, Mr. Stadel. We’ve Got Some Bacon Questions
Greetings, Mr. Stadel. We know that you are very busy. We appreciate your brief attention. Rather than bombard you with tweets, we decided to bloggly address our questions and comments about your Bacon Estimates.
First of all, bravo. You dedicated an entire section of your estimation180 blog to a culinary wonder some refer to as “meat candy.” Even our vegan teacher felt compelled to engage us with these estimates. (She says it is for the sake of the learning.)
Second, the time lapse videos of the cooking are pretty sweet. Too bad the school internet wouldn’t stop buffering. But nice touch, Mr. Stadel. Nice touch.
A question: Did you know that the percent decrease in length of bacon is 38% after cooking, but the percent decrease in width is only 23%? We figured that out adapting your “percent error” formula to the uncooked/cooked bacon. Do you have any initial thoughts about that discrepancy? Is it bacon’s “fibrous” fat/meat striped makeup that allows it to shrink more in length than width, inch for inch?
Also, did you know that the percent decrease in time from the cold skillet to the pre-heated skillet is 29%? That one was a little harder for us to calculate, because we figured out that we needed to convert the cooking times to seconds rather than minutes and seconds.
To summarize, we wanted to thank you, Mr. Stadel. Our teacher tells us that you dedicate your time and energy to the estimation180 site so that WE don’t have to learn math out of a textbook. We wanted to tell you that we appreciate it. And the bacon. We appreciate the homage paid to bacon.
Sincerely,
Mrs. Schmidt’s Math Class
St. Francis, MN
When the Answer is E: He Falls Off the Roof and Breaks His Neck
Our annual state testing season is almost here. The juniors will partake in the Minnesota Comprehensive Assessments in Mathematics a week from Tuesday. Our department decided issuing a practice test to all of our juniors would help re-familiarize them with long lost skills. After distributing copies during our monthly staff meeting, I’m always curious if any teachers in other disciplines look at the practice materials. Much to my delight, the choir director approached me at lunch on Friday, test in hand.
Mr. Warren: Is this test just like the MCAs?
Me: Most likely similar. Why?
Mr. Warren: Ok, well look at this one.
Mr. Warren: I think the answer is E, Xai s going to fall and break his neck.
The conversation went on for another few minutes, with me agreeing that what’s been called “math education” includes ignoring the context of situations and focusing on a procedure. In fact, I was curious how many juniors who completed this practice test even noticed that the situation was outrageous.
Since we were running on a 2-hr delay schedule Friday, I thought it would be the perfect opportunity to present the problem to my algebra class. They are mostly juniors who have been continually frustrated with a mathematics curriculum that doesn’t make any sense in the real world.
Me: Read through this problem. Does it make sense?
Student: ok, it looks like 32.
I didn’t expect any of them to apply any trigonometry, so I thought we needed to approach the problem differently. In fact, I wasn’t even concerned about the angle measure. I wanted them to look at the scenario itself.
Me: Imagine this scenario. We’ve done a lot of estimating in here. We need to envision a 20-foot ladder, three feet away from a house. Does this seem reasonable?
Unfortunately, it did seem reasonable to most of them. I needed another approach.
Me: ok, how could we simulate this in classroom-scaled size?
Student: Get a ruler.
Me: Perfect. How close does it need to be to the wall?
Students: (a chorus of answers)
After exploring multiple methods of calculating exactly how far, we arrived at 1.8 inches. With as much drama as possible, I set the ruler against the wall, exactly 1.8 inches away.
Me: Does this look like a ladder that any of you would want to stand on? (of course, a few did). Keep in mind, this is a TWENTY foot ladder, not a 12 inch ruler.
Student: Yea, I don’t think anyone is climbing up that ladder and coming down in one piece.
Another Student: What if they had a spotter?
A spotter! Now we’re talking. To be honest, I have no idea if a spotter could hold a 20-foot ladder so that it could be placed three feet from the wall. But now I’m interested to find out!
I know Mathalicious investigated a similar scenario using a claim from Governor Janet Napolitano.
In my mind, these are the questions that should be circulating Facebook and aggravating parents. This is the kind of math that should rile up Glenn Beck and company. Our state of Minnesota opted not to adopt the Common Core State Standards in Mathematics, but requiring this kind of math instead is what is actually dumbing down the curriculum. It assumes that the real world doesn’t apply, only rote procedure does. “Just figure out the answer, don’t question the situation,” is what kids read and do over and over when problems like this are solved without real context. A richer classroom experience for both teachers and students comes when we ask students to assess the reasonableness of situations, create new scenarios that are more appropriate, and solve the new problems they develop. The CCSS Standards for Mathematical Practice tell students that it’s vital that they “construct viable arguments and critique the reasoning of others.” I don’t think “critique the reasoning of others” should be reserved for only reasoning created in the classroom. I’d like my students to critique the reasoning of the creator of these types of problems and others like it that have been deemed a necessary component of high school math success.
Thank you, Mr. Warren for igniting the exciting conversation in my classroom.
Creative Craziness
I teach a lot of 9th graders this trimester. We offer a class called probability and statistics 9 and it is open to 9th grade students who also will have had the quadratic portion of algebra 1 this year. I really enjoy this class for multiple reasons. First, it lends itself very well to applying math to real-world scenarios. Secondly, the hands-on opportunities are endless.
One of the issues I have been committed to improving with my own professional demeanor is the way I deal with 9th grade boys. Nothing brings out my sarcastic, short-tempered, disagreeable side like the antics of freshman boys. There’s something about the decision to play soccer with a recycling bin that just invokes the my inpatient side. Regardless, I need to develop more patience with this demographic. Boys are unique, both in the way that they act and the way that they perceive acceptable behavior. I’m not talking about “I’m bored” acting out. I’m talking about the “I really need to see if this eraser will fit in this kids ear” kind of acting out. I think that my short fuse has more to do with my failure on my part to fully understand them rather than gross misbehavior on their part. What I’m really trying to grasp here is not “why can’t these kids sit still?” But more “when they can’t sit still, what makes them want to kick a recycle bin around the room or toss magnets at the learning target?” I think if I had a better understanding of what drives those behaviors, I could deal with them more productively. Suggestions?
Class Commences – an hour I won’t soon forget
Recently, Michael Pershan unearthed a Shell Centre gem straight from the 80’s (literally). This collection of materials is fantastic, and hopefully demonstrates to both students and teachers that engaging in rich tasks and high-level thinking is timeless.
I decided to give the function unit a shot in my Algebra 2 class today. Some background on this group of students: there are 38 juniors and seniors, last hour of the day, in a class geared toward lower-level students. So far though, the only thing that’s been “lower” in this class is the number of empty desks I have. I handed out this task, gave minimal directions and let them go for a few minutes on their own:
from: Shell Centre for Mathematical Education, University of Nottingham, 1985
It was so interesting to watch the different ways each of them started. Some began with 7, since that was the first you saw when reading the graph from left to right. Others insisted to work from 1 to 7, identifying the corresponding people along the way. A few worked the other way around, from the people to the graph.
I walked around to make sure each student was able to get started and that those who thought they had determined a solution also supported their claims. Then, I wrote the numbers 1 – 7 on the dry-erase board, stepped back, and let these kids amaze me.
One student volunteered an answer, and then handed the marker off to another. I intervened only briefly to make sure that every student had an opportunity to contribute if he or she wanted. Once 7 names were completed, I knew a couple of them were out of place. I sat and said nothing, and this entire class showed me what they are capable of. Here was a class full of students labeled mathematical underachievers completely nailing SMP #3. Their arguments were viable, their critiques constructive, their discussion productive. It bothered a few of them that I wouldn’t let them know if/when they were correct. But most of them are starting to understand that my main focus here is not the correct answer, but the incredibly rich and interesting process they used on their journey to finding it. They came up with multiple ways to support their answers and noticed tiny details about the people that supported their findings. For example, did you notice that Alice is wearing heels? According to my students, that is perhaps why she appears slightly taller than Errol.
I had a heart-to-heart with this group when we were done about how proud I was at how they conducted themselves throughout this task. I’m really thoroughly looking forward to a fantastic trimester with this special group of kids. Their work on this task gives both of us the confidence that they can tackle something more difficult next time, and they are capable of mastering high-level mathematics this trimester.
Notice and Wonder with Gusto
My daughter was very content on the airplane ride from Fort Meyers to Minneapolis watching Frozen for the 102nd time. I took this opportunity to read the Noticing and Wondering chapter of Powerful Problem Solving, the superb new publication from Max Ray and the Math Forum crew. I took so many notes on this chapter since this is a strategy that I think every teacher can implement, no matter their apprehension about new strategies. It is such an easy set of questions to ask: What do you notice? What does that make you wonder? Those two questions can open up an entire class period of rich discussion and mathematical exploration. No one explains this classroom strategy better than Annie Fetter of the Math Forum in her Ignite Talk. (Seriously, if you have not seen this 5 minute, dynamite, game-changing video, stop reading and go there now. )
Last Thursday was day 1 of our high school’s third trimester. The first day of the slide into the end of the year. Regardless, the first day of the trimester always seems like the first day of school: the anticipation of a scenario that’s been played over and over in the minds of teachers and students becomes reality. For me, this day meant the last hour of the day I would be met with 38 (you read that right) “lower level” Algebra 2 students. My class is most likely the last high school math class that these juniors and seniors will take, and many of them do not like math or are convinced they are not any good at it.
This class has been in the forefront of my mind most of the year for a lot of reasons. One of those reasons being that after Jo Boaler’s class this summer, I know that a huge barrier to raising the achievement levels of students in this class is the students’ beliefs that they are capable of doing high level mathematics. And I also know that a key component to getting these kids to perform better is to give them feedback that allows them to believe that they are capable of it in the first place.
Because of the structure of some of our high school courses, most of these students have not had experience with higher degree graphs, equations, or functions. They may have seen something similar in their science coursework, but quadratics have not formally been introduced.
I gave them the following graph along with the scenario and let the noticing and wondering begin: Mrs. Bergman likes to golf and her golf shot can be modeled by the equation: y= -0.0015x(x-280).
A couple of them stuck to non-math related Noticings (the graph is in black and white), but almost all of them noted multiple key characteristics of the equation and/or the graph. Some highlights:
- The graph doesn’t have a title and it needs one.
- Both heights are in yards
- Horizontal distance goes up by 80. Height by 5.
- The peak is in the middle of the graph.
- The graph is symmetrical
- The maximum height is about 28 – 29 yards
- The distance at the maximum height was about 120 yards
- She hit the ball 280 yards.
- The number in front of x is negative
- The graph curves downward
- It has an increase in height and then a decrease in height.
- As the ball reaches the peak height, the rate the ball climbs slows.
The list of Wonderings was even more impressive to me. A lot of them wondered things like what kind of club she was using, if the wind was a factor, did she have a golf glove, how much power she used to hit the ball, the brand of her tees, clubs, glove, ball, etc. Then one student laid out something so profound, it made the entire class stop and and acknowledge the excellent contribution:
“What distance would the ball have traveled if the maximum height were 20 yards rather than 28?” (audible ooo’s here)
After this student said that, the floodgates opened with great questions from others:
- What was her average height for the shot?
- What is the maximum height that she is capable of hitting the ball?
- Is this a typical shot for this golfer?
- If the maximum height was higher, like 35 yards, how far would she hit the ball?
- What is the exact maximum height that she hit the ball and how far did she hit it when it reaches that maximum
There were still a few that couldn’t get passed what kind of glove she was wearing or tee she was using, but most of the students stepped up their Wonder Game when one single student demonstrated a rich example.
What I really love about this strategy is that it is so easy to implement into your classroom routine with the resources you already have. For example, rather than starting with a procedure for solving quadratic equations, simply ask the students what they notice about the structure of the problem. How is it the same or different from problems they have done recently? Ask them to list attributes of the equation. I have found most often, the noticing of one student triggers the noticings of others and the list becomes progressively more sophisticated.
I have heard from some teachers that they do not use try this strategy out of fear of students making a list of trivial noticings (like, the graph is black and white). They will include those every time; expect it. But by acknowledging those seemingly trivial items, that student, who would not have dreamt of entering the conversation before now has received validation of his or her contribution to the discussion. And when students feel heard and their opinions valued, their contributions will start to become more profound.
I’m very proud of this class. I’m really looking forward to the creative perspective that their noticing and wondering will bring.
Block Talk With My Kid
I often wonder if my daughter will view the occupational status of her parents through eyes of appreciation or resentment. Will she loathe the fact that teacher parents are more aware of the goings-on of their child’s academic life? Or perhaps she will more often appreciate that math homework help will be easy to come by at home? Both of these scenarios have the potential to have a positive impact on her achievement. However, I had a profound realization today that what actually will help my daughter be successful is maybe neither of those things. While she may not be able to get away with teenage class antics as easily as her peers whose parents are not teachers, I do not think that what my husband and I provide for our daughter’s intellectual growth is something that only teachers can give. ANY parent can engage in rich conversations with their children and see a growth mindset at work.
I want to give proper credit to Christopher Danielson for his Talking Math with Your Kids website and book that drew my attention to how I was interacting with my child mathematically. As a secondary teacher, I am grateful to have gotten a better understanding of how number sense develops in young children and how I can help foster that development. Danielson’s website, Talking Math with Kids has been invaluable in recognizing the everyday math conversation opportunities to have with my daughter.
Her preschool days have been spent at a Montessori school. Although the noodle necklaces, punch cards, and snips of paper are an everyday treat, I was particularly excited when Maria came home with this gem the other day:
Of course my immediate excitement was over the fact that the correct number of squares were colored for each number. But then I realized that there might be a rich math conversation potential involving that worksheet. But, I had no idea what to do with it. I threw it out on Twitter and got many great ideas, including:
Friday was Parent Day at Montessori Central, so I took this as an opportunity to talk some math with my kid. Luckily, I talked her into this same worksheet. We explored all kinds of great number-driven curiosities: which ones make squares, which ones make rectangles, which ones make neither? How many squares are left over after it’s colored? How are the rectangles for 2, 4, 6, and 8 alike?
I realized after that experience the limitations of the number worksheet. For instance, the numbers at the top, while providing an opportunity for “tracing,” don’t allow the learner to explore a particular number any further. I wanted Maria to explore more ways to color 6 boxes, but her 3.5 year old brain saw the “7” above the next 10-block and would not allow it. Some worksheet surgery might be in my future.
I do not think that being a teacher makes these conversations natural. I saw something was mathematically correct, but I didn’t have much experience with turning that into a real mathematically rich conversation with my daughter. I’m thankful for both the math teacher and parent twitter community for throwing ideas my way. Any parent is capable of taking something they like and turning it into a teachable moment. When interacting with our kids, we need to do less showing and more asking; less telling and more listening. She seems happy about it, doesn’t she?
A Speedy Makeover for the Intermediate Value Theorem
As a college algebra teacher, I was not satisfied with the way I presented the intermediate value theorem last trimester. I felt the lesson was somewhat isolated from other concepts we had studied and definitely was disconnected from the real world. My approach lacked a hook and was laddened with procedure. Committed to teaching the concept better this trimester, I recorded the following video while (someone else) was driving:
I know, not a high quality masterpiece, but I think I captured what I needed to illustrate the theorem.
I ask the students to draw a graph of the speed of the car with respect to time. After playing the video a number of times, I had them share their graphs with their seat partner. As I circulated the room, I noticed their results fell into one of these three categories:
Graph A
Graph B
Graph C
After examining the options, I had them choose which graph they felt represented the situation most accurately. Spoiler Alert: The overwhelming majority of them chose Graph B. Their reasoning: it’s unclear what happened to the speed between seconds 10 and 15 therefore, there should be a space in the graph. Those vying for Graph C cleverly argued that there was no audible “revving of the engine,” indicating that the car continued to slow. Others supporting C claimed that even though we could not see the speed, they know how a speedometer works and can make a reasonable assumption about what happened in that time frame.
Enter this student’s graph and the Intermediate Value Theorem (trumpets):
I liked this students “shading” through unknown speed region, so I projected it for everyone to discuss. They were able to determine the value of the function at ten seconds, f(10), was approximately 45 miles per hour and the value of the function at fifteen seconds, f(15), was approximately 35 miles per hour. They also knew that the car must have reached 40 miles per hour sometime in between 10 and 15 seconds. “How do you know that?” I pryed. Gem response of the day: “Well, speed is continuous and I can’t go from 45 mph to 35 mph without going through 44, 43, 42, 41, 40 mph, and so on.” Bingo. Intermediate Value Theorem. No boring procedural explanation necessary.
We applied this “new” knowledge to a polynomial function so that they could get a handle on some of the algebra and notation used. And as a bonus, they also seemed to grasp that this theorem does not only apply to crossing the x-axis, a common misconception students had last trimester.
Moving forward, I’ll definitely work on creating a better video!
Diagnostic Questions – A Tribute
The brilliant Diagnostic Questions website has been live for a few months now. After using it once, I’ve completely convinced of its profound positive potential in my classroom. Craig Barton and Simon Woodhead really have outdone themselves in creating a database of diagnostic questions. If you have struggled with anticipating student responses or identifying sources of errors, this resource is a total winner. The site allows teachers to quickly create quizzes that identify student misconceptions. For example, here is a question from the ‘Probability – Experimental’ section:
All of these answers are carefully crafted so that the teacher can see what students aren’t grasping. Here is an example of student work:
Now I’m able to see how this student got 13, rather than just marking it wrong and moving on. Obviously this isn’t a new phenomenon to have students explain their answers to multiple choice questions. However, two key features of this website make it noteworthy above other sites that feature multiple choice questions:
1. The deliberate multiple choice answers chosen to manifest misconceptions
2. The easy-to-use format of the site allowing teachers to quickly create, administer, and grade these quizzes.
Thank you, Mr. Barton and Mr. Woodhead for this tremendously helpful tool. I look forward to contributing questions to your database in the near future.
Number Loving Beagle 