I've put the elevator speeches to rest. Today's two minute speech went well... I think. As one event concludes, I'm excited to resume my ongoing thoughts about number sense and students.
Working with teachers and students, I can't help but be inundated with thoughts about things I miss, look forward to, and will be challenged with this year in and out of the classroom. However, there's one thing I crave more than anything else, and that's working with students, which usually entails having number sense conversations with students.
Today, I had rich number sense conversations with students in Math 7, Math 8, and high school Algebra classes. As I sit here and put the finishing touches on the slides for my upcoming conference workshop, Get Students to Argue in Class With Number Sense Activities, I can't put to words how valuable it is to allow students to talk about math in math class. That's an oversimplification, but we seriously need to provide our students with opportunities to talk to each other, even argue with each other. Mathematical Practice 3:
Construct viable arguments and critique the reasoning of others.
These nine words may be the most important words in the entire Common Core effort.
Last December at CMC North, I was honored to give an Ignite talk about something I'm passionate about: number sense and student conversations. It's titled: Number Sense: I Don't Like This Game Anymore.
Students are hungry for number sense conversations in math class. I really do believe. If you don't believe me, just put up this picture in your class and ask your students, "How long would it take to use all of that?" Then ask them to convince you of their conclusion.
As October quickly approaches, I look forward to seeing you at an upcoming conference this school year. In the meantime, I hope to post more about number sense.
This is my final elevator speech in this series. My 120-second elevator speech for tomorrow is to focus on rigor. Yes, that's difficult to capture in 120 seconds. However, this is what I'll be going with.
I love how teachers are hungry for modeling with mathematics. That in itself, can be one of the most vital elements to rigor in mathematics.
Did you check out Steve's elevator speech on Day 5? Pretty awesome, right? Here is the second one he emailed me.
"Appealing to an audience that recognizes school math isn't working well enough."
Regardless of what you may think of the Common Core, you must recognize that school mathematics hasn’t been working for far too many students. You’ve probably heard that the K–12 mathematics program in the United States has been aptly characterized in many rather uncomplimentary ways: underperforming, incoherent, fragmented, poorly aligned, unteachable, unfair, narrow in focus, skill-based, and, of course, “a mile wide and an inch deep.” Most teachers are well aware that there have been far too many objectives for each grade or course, few of them rigorous or conceptually oriented, and too many of them misplaced as we prematurely ram far too much computation down too many throats. It’s not a pretty picture and helps to explain why so many teachers and students have been set up to fail and why we’ve created the need for much of the intervention that test results seem to require.
These are realities that the Common Core has been designed to fix. How? First, the new standards are common. No longer will publishers cater to a few large states and stuff their books with the union of fifty sets of demands. No longer will our assessments be developed by the lowest bidder and overwhelmingly comprised of low-level, multiple-choice items.
Instead, the prospects of a Common Core set of standards are for shorter, more web-based, better-focused instructional materials and for computer-adaptive, computer-delivered, and instantaneously-scorable constructed response-item assessments. Second, ignore the misrepresentations and take heart in the fact that the Common Core standards are coherent. These standards replace the vagueness of strands (number, measurement, geometry, statistics, and algebra) with domains, clusters, and well-conceived grade-to-grade progressions of standards. Moreover, they are fair. Many procedures that we have come to teach at grade x, have been moved to grade x + 1, giving us all a chance to build prerequisite knowledge and slow down what has become a drag race through the curriculum. And, lastly, they are teachable. There are only about thirty standards—of varying sizes and depth—at each grade level, resulting in a far more manageable teaching load than the forty to fifty objectives per year that many of us now face. If you care about your children, if you care about readiness for citizenship and the workplace, and if you care about our future leaders making informed decisions, you should be fighting for, not against, the Common Core.
~ Steve Leinwand
Again, I want to thank Steve for taking the time to prepare two awesome elevator speeches. Hopefully, they've inspired you as much as they've inspired me. Maybe you can use parts in your own elevator speech when the time presents itself. It's not too late to add your own in the comments. Tomorrow, I'll post the last and final elevator speech, which happens to be my two minute speech for my district.
I have three more days of elevator speeches to share. Now would be a good time to say why I initiated this whole elevator speech series. I was asked by my district to give a 90 second presentation on rigorous mathematics standards. I will be giving this 90 second presentation on Monday at our district's State of the Schools breakfast where many community members, parents, board members, teachers, and administrators will be present. You know, all the stakeholders. In preparing those 90 seconds, I needed to push myself to come up with elevator speeches related to Common Core math standards and rigor. So Day 7 will be my 90 second elevator speech from the State of the Schools.
For Day 5 and Day 6, I'm honored to share two elevator speeches from Steve Leinwand. It was so cool to see an email from Steve in my Inbox, with him saying, "OK - challenge accepted!"
If you've followed my blog, you know I have great admiration for Steve and have been inspired by him numerous times. It won't surprise you that I thoroughly enjoy (and support) his first speech.
"Appealing to an audience that wants more for their children."
It’s only one of eight Common Core Standards for Mathematical Practice, but we can change schools and change lives if we truly implement Mathematical Practice 3: “Construct viable arguments and critique the reasoning of others.” In many ways, these nine words may be the most important words in the entire Common Core effort. We can’t expect students to construct viable arguments unless we ask them “why?” and “how do you know?” and “can you convince us?” When we ask such questions we are laying the foundation for the reasoning and justifying that represent the thinking that schools need to develop in all students. Similarly, we can’t expect students to critique the reasoning of others unless we create classrooms where student thinking is valued and students contribute to their own learning within communities of learners. Moreover, this isn’t just mathematics, but what needs to happen in English language arts, social studies and science as well. So when one cuts through all of the misrepresentations and politics that surround the Common Core, these powerful nine words transcend our differences and capture what every parent and every citizen should be demanding from their schools and for their children.
~Steve Leinwand
Thank you Steve for sharing your wisdom and fervor. I look forward to sharing your next elevator speech on Day 6.
When I started this idea of preparing an elevator speech about the Common Core, I figured I'd get as far as I can on my own. I wanted to stretch myself and avoid tapping into resources or my Jedi masters. It's day 4 and I find myself at that point where I need a pick-me-up. Who better to do that than Steve Leinwand and his super math gang of leaders at NCSM? All 2014 NCSM attendees received the following framework:
It's TIME: Themes and Imperatives for Mathematics Education.
I figured I would snag a few lines from the section titledSupport an Understanding of the Breadth and Depth of Mathematics Content Knowledge (found on pages 20-21). Today's elevator speech is is from the pros (all 15 writers):
The CCSSM promotes teaching few concepts, but teaching them in more depth, with deeper understanding as the goal. However, teachers must have a deep understanding of the content themselves to teach for deeper understanding.
Mathematics involves more than just recalling facts and performing routine procedures. Mathematics needs to be understood as an integrated collection of knowledge and skills, not as a series of discrete procedures. Mathematics must also be understood as connected to other disciplines and to the world in which we live. A technology- and information-based society requires citizens to be able to think, reason, and analyze. Knowing mathematics means being able to adapt and apply mathematical ideas to new situations and to a variety of problems.
You can find this on page 21. The framework is a quick read at 58 pages and appendices full of strategies and resources. Get your own copy, you can't have mine.
I had a break between events this afternoon, so I stopped at In-N-Out for an iced tea and a chance to hang out for a few minutes. If you've been to a recent training or presentation of mine, I briefly share my admiration for the In-N-Out business model. Although Barry Schwartz doesn't talk about In-N-Out in his TED talk, The Paradox of Choice, he tells a parallel story extremely well.
A few notes from this afternoon:
Today's elevator speech isn't one I'd consider using, but something to chew on:
I enjoy In-N-Out Burger because of the experience, convenience, service, affordability, and great taste. When I think of the experience at a deeper level, every detail is important: the ingredients are fresh, the service is friendly and efficient, and the menu is simple yet customizable. Common Core and the 8 Math practices can be a rich experience that encapsulates critical thinking, conceptual understanding, and applied math. When you decompose Common Core, every part is important because both standards and practices demand our teachers and students to explore concepts in depth, obtain procedural fluency, and apply said skills to real-world situations using mathematical modeling.
Today, I'm going to take a different direction. Enter the elevator and press your floor number. Here we go:
I believe the Common Core standards are a tool for both students and teachers to explore problem-solving and increase mathematical conceptual understanding. Picture the first cell phone you saw or owned as a tool used to communicate. Compare it to any of today's cell (smart) phones and how the tool has evolved and improved. As a tool, the Common Core standards will help students evolve to be better, more robust math students in order to think critically and communicate.
For the next week, I will challenge myself (and you) to work on an elevator speech each day about Common Core Math. I will try to be as fluid as possible in my thinking as if I were describing Common Core to a complete stranger as we rode an elevator together. And with that, here's today's elevator speech:
For me, Common Core is a tool to help students see how the world around us can be explained using critical thinking and mathematical properties. My favorite part of Common Core are the 8 Standards for Mathematical Practice. No matter what grade level, the practices emphasize that both teachers and students are problem solvers and how important it is to understand why mathematical properties and procedures exist, and not just the answer. I get to see my students thinking and focusing on finding solutions through hard work and collaboration instead of me just standing up at the front of the room, telling them a procedure they will most likely forget in a day or two. I get to see students be creative in their mathematical thinking and I help guide them when they need teacher support.
Come along for the ride and share your elevator speech in the comments. Take a minute or two to compose an elevator speech. It's a work in progress...
Not sure I made the best teaching move today, but I had to try it. We explored Dan Meyer's "Will it hit the hoop?" task(s).
Act 1: Roll "Take 1"
Agree on the question, "Will he make the basketball shot?"
Ask students to make a series of guesses for a total of six takes.
Act 2: Ask for information I typically ask students to think of information they would find useful in answering the question. Today, I went somewhere else with Mathematical Practice 5. I asked students to make two lists:
List 1: Math tools that would be UNhelpful.
List 2: Math tools that would be helpful.
This is the fourth and final week of the summer academy. My students have been exploring many math tools. I'll list the activity/task with the prevailing tool(s):
Class height: Measurement, mean, median, mode, range
Vroom Vroom: Measuring, Data collecting, Desmos, function of best bit, quadratic equations
As you can see, many of our tasks were dominated by slope-intercept and Desmos. I didn't find their lists surprising.
I love how some students thought Desmos would be helpful, while others thought it'd be unhelpful. Those that found it unhelpful, wished you could insert images into Desmos so they could use sliders to find the path of Dan's shots. Boy, were they happy when they discovered you could import images. My first class was split down the middle: half thought slope-intercept might be useful and half didn't. It took a few convincing students to explain why Vroom Vroom was an example where a linear function was unhelpful.
Overall, I'm pleased with this approach, but I wouldn't do it with every task. It might confuse students that there's only one way to solve a task and detract from the importance of MP 5. I thought this was a fitting opportunity for students to mainly see the difference between a linear function and quadratic function. Specifically, I wanted them to see the advantages of using sliders in Desmos with a quadratic function instead of a linear function. I think students need to shuffle through their tool belt often and pick the right tools for the right task. I think today it was necessary. Dan has written about this or breaking students' tools. Moving forward, it's a matter of using this strategy at relevant times and not overusing it. However, I might be wrong altogether. That's where it's your turn to chime in...
California is an SBAC (Smarter Balanced Assessment Consortium) state. This last week my school started the SBAC Field Tests and I was a Test Administrator for my 7th grade classes. Before I continue, let me post part of the Security Affidavit I had to sign.
That's right, I will not divulge the contents of the field test. However, I will first refer you to last year's post where I made a video comparing released CST questions and SBAC practice questions. Here's a reminder (screen shot), comparing just two questions.
This week, I felt like my students were looking at SBAC practice questions that were on steroids. Since I can't speak about the SBAC Field Test questions, I took my Deodorant 3 Act task and put what I think the SBAC steroid version might look like. I have nothing against SBAC. I tried to create a similar task that had rigor, complexity, and mathematical modeling.
First, my Deodorant task goes like this:
Act 1: How long will it take to use all of that deodorant?
Act 2: Data from the first 4 sticks.
Act 3: The answer is still in the works.
Sequel: How many sticks of deodorant would a person use in one lifetime?
Here's how I'd see this same task presented SBAC-on-steroids-style.
I walked away this week, thinking our students need to do many things.
Read the story.
Decode the text.
Understand the question.
Organize the data.
Retrieve and access the correct skill(s) or skill set.
Apply the necessary skills.
Perform the correct operations with the above skills.
Interpret their answer.
Explain (and articulate) their answer.
As a teacher of many ELD students, I can safely say that the following steps are already challenging; 1, 2, 3, 8, and 9. Don't get me wrong. I believe in literacy, but I wouldn't want language to be a barrier when assessing a student's mathematical abilities.
Hear this though: Students must make sense of the problem before they can use mathematical modeling to predict the answer. Then, they must articulate how they got their answer. I would consider this expectation the new norm.
I'm not done. I could totally see SBAC taking this deodorant task and creating an additional question that would complete my 3 Act. Check out this doozy.
We're looking for students to drag numbers to both axes, use a line of best fit, make a mathematical prediction, and explain everything again. The only thing I left out of this question was for students to write an equation for the line they draw.
I have more to say about this, but that's enough for now. I'm already thinking about how to better prepare my students for these types of questions, which should be my next post. If you have any thoughts, please share. If you've made it this far, here's a preview of Act 3 for my deodorant task. Don't worry, I keep my shirt on!
I recently submitted my speaker proposals for both 2014 CMC conferences. One of my proposals is for the following session: Title: "Get Students Arguing in Math Class with Number Sense Activities." Description: Get students to productively argue about math situations. Participate in number sense activities requiring students to construct viable arguments, critique the reasoning of others, and use sense-making. Get ready to throw down.
I also had to answer a few questions justifying the session and connecting it to the CCSS and 8 Mathematical Practices. I provided the following connection: The presenter will use number sense activities to get participants to construct viable arguments and share their reasoning like students. Using presenter-made tasks (Estimation 180) and other online resources appropriate for grades 3-8, attendees will be able to see the importance of student reasoning and creating productive discourse in the classroom. Teachers will also be provided with sentence frames and stems for all students, especially English language learners.
I'm really excited at the thought of this session getting accepted so I figured I would jot down a few ideas here and see what you all have to contribute. Even if I'm not accepted, I think every math class has to have students productively arguing at times. Doing estimation challenges with my students has been so beneficial for them to get better at the art of arguing. However, I know it could be better. I'm not sure if you've ever experienced it before, but it's a treat to stand off to the side or in back of a group of students arguing about a question in math. They have no idea you're nearby because they are so caught up in the argument. Don't get me wrong, it's not like they're swearing at each other and calling each other names. They are having a rich discussion, sharing conjectures, examples, counterexamples, etc. and I have the pleasure of spectating. I usually turn to an innocent bystander (nearby student) and whisper, "Awesome, look at them arguing. Isn't it great?" The student usually looks shocked that I'm happy their classmates are arguing. I love it!
I'd like this session to place a big emphasis on two mathematical practices: MP 3: Construct viable arguments and critique the reasoning of others. MP 1: Make sense of problems and persevere in solving them.
I plan to break these two practices down more in my session. For now, I feel I need to focus on three major parts to arguments: having excellent content, capturing the arguments, and indirect facilitation.
Content: There's a wealth of content available, but I think the more controversial the point of contention, the better. One of my favorite moments in math class was when we did Mathalicious' Datelines lesson. Students were arguing about which celebrity shouldn't date another celebrity because of the age discrepancy. Some students disagreed about the rule of (n/2) + 7, especially since I was teaching 14 year-olds at the time. It was awesome. Here's my current list of resources that have given my students great things to argue about:
My kids went nuts arguing about a similar question to this "Would You Rather" found here.
These don't have to be full-on lessons. They can be warm-ups, math talks, used during classroom transitions or to break up your direct instruction, etc. I'm really looking forward to using @MathCurmudgeon's site MathArguments180.com Imagine your students arguing about which student should pack your parachute based on this data.
Capture: I need to capture these arguments for a few reasons. Students need to listen to other students argue, especially from different classes. My memory is very porous, and I can't remember what students say verbatim. Students can listen to the recordings and pick a side, or provide their own agreement or dissent. I'd also love to share student arguments with other teachers, especially at this session. How do I capture this?
I just downloaded Voice Memos for iPad onto my school iPad. I will test it out next week with students. Wish me luck. Here are the features I'm optimistic about:
It will record in the background while another app is running.
It was $1.
I can pause the recording.
I can trim audio clips.
I can sync with Dropbox.
Have any tips for capturing student arguments?
Facilitation: Here's where I need to do a better job. For many of my students, English can get in the way of them articulating their point. I'd like for students to listen better to each other and respond accordingly. I want to hear what they have to say. I want them to be a contender in their disagreement, but I don't want them to be held back because of language deficiencies. Therefore, I need to provide them with sentence starters and stems. Fortunately, these can be used with any student. Here's a few:
My opinion about this is _____________.
I could argue that _____________.
I disagree with your statement that _____________ because _____________.
Have any stems or sentence frames you're already using with students to help them articulate their thoughts?
Me: I need two volunteers. You have no idea what you're doing. Thanks Brianna and Jesus. Go stand in front of the whiteboard on the side of the room. You are the two contestants in today's Spelling Bee.
This is how I opened today's lesson. Wait. A Spelling Bee in math class? I address the audience:
Me: I need your help. I am going to ask you a question. The answer is a number. I am not interested in any categories like gender, height, age, birthday, first name, last name, etc. For my Spelling Bee, I need you to take my contestants and order them for me. What's the maximum amount of ways I could order these two contestants?
Students have time to think and some quickly raise their hand to say, "Two."
Me: Show me. Tell us what they are.
Student: Right now Brianna is first. Jesus is second. We could switch them and Jesus goes first.
Me: [looking at Brianna and Jesus] Do what she said.
Brianna and Jesus switch order.
Me: Have I maxed out all the possible combinations for ordering Brianna and Jesus?
Class: Yes!
For a little comic relief, I toss Jesus an easy word to spell.
Me: Jesus, spell "cat".
Jesus: C-A-T
Me: Wait. What?
I learned today that most kids don't know how a spelling bee works, so I call on a few kids to explain the three steps:
Say the word.
Spell the word.
Repeat the word.
Me: Jesus, let's try this again. Spell "cat".
Jesus: Cat. C-A-T. Cat.
Me: Bri, spell "discombobulate".
Brianna: Ughhhhhhh. What?!
Me: Okay, can I get a third contestant for our spelling bee? Jesus, since you're the winner, please pick someone.
Standing in front of the audience, I now have Jesus, Brianna, and Garry.
Me: Okay, let's say their current order is one possible combination. Let's keep Jesus first. Can you get any other combinations with Jesus being first?
Student: Yea, switch Bri and Garry.
I look at Bri and Garry.
Me: Do it! Okay we now have two possible combinations. Have we maxed out the possible combinations with Jesus being first or can we get more?
Class: We're maxed out.
Me: Okay, someone give me a new combination.
Student: Put Brianna first this time. Then Jesus. Then Garry.
Me: Okay, we now have three combinations. Can we get more where Brianna is first?
I repeat this process until the class has agreed we maxed out our combinations with six total. Great. I toss this information in a table like this to keep track of it.
Me: So what if I add a fourth contestant to the spelling bee?
Sarah: No!
Me: Really Sarah? What? Are we going to have more or less combinations?
Sarah: More.
Me: Gimme some guesses everyone. Toss something out there for fun. How many combinations could we get with four people in the spelling contest?
Students tell me 8, 10, 9, 12, 16, 13 and I write all of them up on the board. I ask for some quick reasoning behind the guesses.
Me: Ok, thanks. You all can't be right. Instead of moving people around, let's do this instead.
I gave each group a sandwich bag with four different colored snap cubes: red, green, blue, yellow. Students were to work in their groups to figure out all the possible combinations of four colors. They were to write it down in their notes for the day. I circulated the room, noticing student work.
For groups that think they're done, but wrong (like only 12 combinations): I zone in on one combination and keep their two colors fixed, "Have you maxed out all the combinations with these two at the front?" Usually this is the only nudge they need to get closer to the correct number of combinations.
For groups that are on track: I make it obvious I note their work, or ask for a quick explanation, or I quickly move to another group.
Groups that finish and have the correct answer: I have them explain their work, organization, process, and reasoning. I ask if they feel confident and usually they do. I'm not going to string them along. I respond, "That makes sense to me." followed by:
Me: So what if I gave you a fifth color?
Student: [typical response] Ughhh.
Me: Oh, what's wrong?
Student: That's a lot of work.
Me: I know, right? I'm right there with ya. I wouldn't want to write out all those possible combinations either. So, your job is to try and figure out a shortcut. In other words, if I just gave you four colors right now, how could we quickly get 24 combinations without writing them all out. If I'm now giving you five colors, what would be a quick way to figure out all the possible combinations?
Once I see that most groups have reached the magic number (24), I show them this and have them count.
Me: One clap on three for the closest guess.
1-2-3 CLAP! Many kids see that 4 groups of six combinations yields 24 combinations. I toss 24 into our table and ask the whole class about finding the possible combinations for five colors. Typically, the students want to avoid this nonsense and express some noise of rebellion.
Me: What's wrong? You guys don't want to write out all the combinations? Well, let's try and find a shortcut. Do we see anything from our table that might help us?
To my pleasant surprise, at least one kid in each of the three participating classes found the following relationship:
Abraham, Brianna, and Daisy: You take the previous "Combos" result and multiply it by the diagonal "Colors" amount to get the new amount of "Combos."
Me: Let's see if that works.
It does. Great!
Me: Okay hot shots! This is a great shortcut. What if our principal walked in and gave us 13 colors. How would I quickly figure out the total number of combinations since I don't have the number of combinations from 12 colors?
Here's where I introduced the use of factorials. Yes, I could have spent time getting the kids to look for this pattern, but I simply didn't have or make the time. I felt it was a good place to show them that putting the factorial symbol after a number means to multiply it by all of the natural numbers less than the given number.
4! = 4 x 3 x 2 x 1 = 24
Me: So if our principal walked in and said, "Find all the combinations of 13 colors." we'd go thirteen...
Class: ...times twelve, times eleven, times ten, times nine...
In reflection, this lesson created more successes for my students than I anticipated. Some include:
Discovering patterns and relationships within a table,
Creating a need for the factorial of a number,
Adding another vocabulary term to our tool belt, and
Finding combinations more efficiently.
This lesson started with a low-entry of two students and two combinations. We built in the next part by finding six combinations for 3 students. We built in a guess for the combinations of four students so they can invest in the question and look for patterns. We manipulated four colors, organized our combinations, made conjectures, and arrived at a reasonable answer that maxed out the combinations. We pushed those students who finished early to discover a shortcut on their own. We created a need for avoiding excessive work with larger numbers and a need for some type of formula (factorials) that will get us the same result.
I came into this lesson with a rusty understanding of factorials, probability, and combinations. Anyone who is against Common Core State Standards, think again! It's making math teachers know their content better, so they can better serve their students. It's opening the door for students to reason their way in math class. I'm not blogging to get into the importance of CCSS right now. However, I'm convinced this was way better than me standing in front of the students telling them to put an exclamation point after 4 (like this 4!) and to just multiply 4 by 3 by 2 by 1 to get all the possible combinations of four somethings. Instead, the students discovered the relationship (pattern) within the table and felt confident in discovering the total combinations of five colors without drawing them all out.
Last month I added the Lessons page to Estimation 180 so you can quickly access lessons I've made. I will continue to add lessons as I make them and host them in that space.
This month, I'm adding a Presentations & Workshops page to the site. Since last November, I've been fortunate to work with some amazing math teachers at conferences and workshops. I've learned a lot and have truly enjoyed doing math with teachers as we share instructional strategies and lessons. My goal is to help support math teachers in strengthening their instructional tool belt for the Common Core classroom.
I'm excited about this new chapter. Drop me a line if you're interested. PD, 945
Head over to Estimation 180 and you'll see this lovely new option in the menu bar.
LESSONS!
That's right!
LESSONS!
I've added a "Lessons" page with many lessons I've created, sorting them by their CCSS. I'd like to thank Dan Meyer and Robert Kaplinsky for their friendly suggestions (nudging) to tag my lessons in an attempt to make it easier for other teachers to find and use. Plus, I'm tired of my lessons collecting digital dust and hope that teachers can find and use them.
I was honored to give a workshop for teachers in my district today. The workshop became the motivating factor for making this Lessons page. Right now, most of the lessons are 3 Act lessons that can be found at Dan's 101qs.com A few other lessons are ones I've blogged about. However, I have added two test pages at Estimation 180 where the entire lesson is available for teachers to use. Right now. At Estimation 180.
These two full-on lessons are ready for you and your students. You'll see all three acts, teacher notes, student work, student handout (if you like/need), and downloadable videos. Let me know if you have any thoughts, advice, or questions.
I hope this "Lessons" page is useful and/or better than that silly unorganized spreadsheet I've got lingering. You'll notice a few links are under construction, but many links deliver the goods. Check in often for updates.
What I like: I like the placement of the x-axis along the ground to represent zero height.
I like how this task reminded me of the low arching bridges along George Washington Memorial Parkway in Alexandria, Virginia.
What I dislike: I dislike that the x-axis and the y-axis were already placed for us. The students have no say in this. I dislike how the arch is already "modeled" by the given function. There isn't any chance for students to explore this on their own, especially if they had no say in the placement of the y-axis. I dislike the answer to this question. It's hilarious. Get this: The truck has to be dead center so that it will allow 0.23 feet of clearance on each side of the truck. Regarding number sense, what is twenty-three hundredths of a foot? No one talks like that, do they? After converting this answer, I could see myself telling the driver, “You have less than 3 inches to spare on each side. And that’s ONLY if you center the truck with the middle of the bridge." Let's look for an alternate route or someone might have to get out of the truck [not it] to guide the driver.
Things I'm intrigued by: What was the reasoning behind the placement of the y-axis? Why isn't it dead center or along the right wall? Why isn't there any sign on this bridge that says the maximum height and/or width of trucks allowed? Is this a "one way" road?
Here's what I did: *Disclaimer: I'm not pretending to nail this Makeover: I think it can be better. That's your job: so let's get it on and help me in the comments. I'll admit, the Makeover was more work than I anticipated and I'm tapped, but I'm happy to do it now during the summer. Thanks Dan for the Makeover challenge!
I found an accident report for a coach bus that crashed into this exact bridge (below) in 2004. There are many of these low arched bridges located along George Washington Memorial Parkway in Alexandria, Virginia. I've seen a few of them when we've taken our 8th graders to visit Mt. Vernon. I remember our bus driver telling us about this specific collision.
1) Show your students this picture, but don't tell them about the collision:
Allow students to make observations and ask questions (maybe Notice and Wonder). Tell them where this bridge is located if they ask. Don't tell them what the signs say. Have a discussion.
2) Now show your students this picture and ask:
Which of these (six) vehicles would safely pass under the arched bridge?
3) Have students make guesses and write it down. You're taking a chance, but at least one student should notice that some vehicles might pass safely using the left lane, but not when the same vehicle is traveling in the right lane.
4) Ask your students what information or tools they might need to help determine which vehicles can safely pass through this arched bridge.
Bridge height(s)
Vehicle height(s)
Width of road
Width of lanes
5) Find the vehicle heights we'll be working with. Depending on the time you have, students can use the internet for finding the average height of each vehicle. I did the grunt work for you with this slide:
6) Show students three heights of the bridge and street dimensions. They probably want to know what those yellow signs on the bridge say. Too bad! The picture is low quality and very pixelated. I'll admit, this might feel like we're now stringing the kids along, but let's offer them measurable dimensions, not some arbitrary equation that "models" the arch. Share the following: Height of the bridge on the left side Height of the bridge in the center Height of the bridge on the right side
Width of the entire road (including space for lane lines and shoulder) and width of two lanes.
7) Offer your students Desmos or Geogebra. Plot the three heights. Use sliders to find an equation that models this low arching bridge. Here are three four scenarios I came up with in Desmos. I'm still not sure which I like best. You decide. I've linked the Desmos files for you to mess around with. Where do you fancy the y-axis?
Okay, I like both the center and the justified right. Placing the y-axis in the center of the bridge made it a lot easier to find an equation that modeled the bridge. Placing the y-axis on the right side of the bridge might produce negative x-values, but since distance is never negative, the absolute value of the domain will tell me how many feet away from the right side of the road the vehicle must be.
8) Give students time to explore the functions, quadratics, sliders, domain, range, and so on. There's more. This task requires students to apply the heights of the vehicles in a specific manner. Sure, students can click and drag on the graphs in Desmos to find the heights of vehicles and determine if it safely passes, but what part of the car "safely passes"? The top left? Top center? Top right? Therefore, students have to now take into account the width of the vehicle. Let's go back to the original question:
Which of these (six) vehicles would safely pass under the arched bridge? And in what lane?
Which vehicle(s) will pass safely in both lanes?
Which vehicle(s) will only pass safely in the left lane?
Which vehicles(s) would have to go into the oncoming traffic lanes?
Which vehicle(s) need to stop and turn around?
Ask how far the vehicles will be from the right side curb when "passing safely"?
9) Tell students to look for a little more clearance than 0.23 feet (2.76 inches). You can read the accident report for all the details about the street and bridge. You'll find the clearance heights posted on the bridge and about 1,500 feet before the bridge.
Unfortunately, the accident report will also show the bus that collided with the bridge while the driver was talking on his cell phone. The bus ran into the bridge without even applying the brakes.
What you did or suggested: Amy Zimmer emailed: "Is it the new Daniel Craig James Bond that has the train scene where he has to duck just before he is about to run into the bridge when the good guy and the bad guy are fighting on a speeding train?" followed by "I would give lots of trucks and see which ones fit."
If you've made it this far. I appreciate your determination and perseverance. Thanks for tuning in. I know this task can be better, so let's get it on in the comments.
Up next, Global Math Department presentation on August 13, 2013: Back to School Night: Ignite. Join the fun.
We decided to get Netflix recently and I was excited to see that Cheers episodes are available. I occasionally put an episode on in the background while I get work done. I came across this episode that literally snuck in some math (money, raises, time, rate) right before the end of the episode. Sam Malone, the owner of the bar in the tie (played by Ted Danson), is talking with Woody Boyd, a bartender (played by Woody Harrelson), about a raise. Roll Act 1:
After consulting with my man, Nathan Kraft, I bleeped out a part of Woody's last line. The two of us discussed the tendency a bleep can have in implying some profanity was removed. So if this lesson goes horribly wrong, blame Nathan! All those toothpicks finally caught up with him. Here's how the exchange goes between Sam and Woody:
Sam: We were talking about your 50 dollar a month raise.
Woody: Sam, it was a hundred a month.
Sam is caught for trying to pull a fast one on Woody. Woody appears to let it slide, but something occurred to Woody. He turns to Sam and the exchange continues:
Woody: I think a hundred a month is too steep. I'll settle for [BLEEP] a week.
Sam (without blinking): You got it!
I anticipate students noticing that the amount was bleeped out and wondering what was bleeped. I anticipate students not sure if Woody said, "[BLEEP] a week" or something inaudible? I anticipate students noticing that the studio crowd laughs while wondering if Sam was just made a fool by Woody. I would love to first have a leisurely conversation with students about who they think just got the better deal in this exchange, Sam or Woody? Or was there even a better deal to be had? If you've ever watched an episode of Cheers, you know that neither character has a strong IQ. If anything, Woody is portrayed as a real naive, gullible, and takes-you-at-face-value type of character. Sam is about a handful of points above Woody. So what about Act 2 after you take some guesses from the class on who just got the better deal from this exchange?
This might be the first 3 Act lesson in which I don't have any additional information for Act 2. In all fairness, this might not fit my previous rant on measurable acts, but I think the 8 Standards for Mathematical Practice are rubbing off on me (in a good way), especially Practice 4: Model with Mathematics.
I posted the Woody's Raise lesson on 101qs.com with very little in Act 2 because I'd love to know where the teacher would take this with his/her class. This type of teacher discretion can't be packaged in an online portal or catalog of video instruction. Here's what I threw out there for Act 2 (the first edition):
At what "raise" amount per week would Woody "settle" for the:
Better deal
Equivalent deal
Worse deal
I have many questions when thinking about Act 2. Here's a few: Over time, when does Sam or Woody begin to benefit or suffer from this deal, compared to the $100 raise per month? Do all months have exactly four weeks? Does that matter or should we use 52 weeks in a year? How would you anticipate students representing Woody's better deal versus the worse deal? What would this look like graphically? What would this look like organized in a table? What equations could you anticipate students writing? If any? How does this deal apply to Woody's hourly rate? In what classroom could you talk about the tips Woody might make? Remember this takes place in a bar. Middle school students? High school students? College? A workshop with teachers? I think there's a lot of fun to be had with this video clip. Let's Roll Act 3 and see what Woody would "settle" for instead of the $100 a month raise:
I'm posting this lesson because I'm thinking out loud. More importantly, I'm curious what you would do in between Act 1 and Act 3 with your students. How would it be different in an elementary classroom? Middle school classroom? High school classroom? Teacher workshop? What would your Act 2 be?Where would you take this lesson with your students? I believe this is a multi-dimensional lesson that can take on some great mathematics. Bleeping out that weekly rate in Act 1 really opens up Act 2 for some rich mathematical discussions and modeling. Toss your Act 2 in the comments. Thanks!
Man, I love a good Ignite talk. 5 minutes. 15 seconds per slide. 20 slides. Concise. Succinct. Compelling.
Why not do my own version of an Ignite talk at Back to School Night next year? I get 10 minutes with parents and would love to change it up a little this coming year. Trust me, after surviving last year, I think the parents deserve a better, improved, and more reassuring version of Mr. Stadel. I'll explain that last sentence in some upcoming blog posts that I'll use to debrief about the 2012-2013 school year. If you're not sure what an Ignite talk is, let me introduce you to my man, Steve Leinwand.
If you like that, check out more Ignite talks by Annie Fetter, Dan Meyer, Max Ray, and Phil Daro. These are my go-to talks when I need a math pick-me-up. Do the math, that will be a little over 20 minutes well spent, being inspired by some key people in our math community. Seriously, check out those four talks.
I'm brainstorming in this space, so feel free to share some input please. At Back to School Night, I'll start by giving a brief 30-60 second introduction of what an Ignite talk is and how they work. I'll give an Ignite talk for 5 minutes, covering any of the following things:
This leaves approximately 4 minutes for parents to ask questions or something else... Have any suggestions for those last 240 seconds?
Who's with me? Does anyone else want to do a Back to School Ignite talk? There's already been some interest generated on Twitter and I started a Back to School Ignite list. Shout at me if you're in. Or is this a really foolish idea? Seth Leavitt, my new online colleague and EnCoMPASS Fellow asked if I'll post it online. I don't see why not. Maybe we can create a space for Back to School Ignite talks.
*UPDATE: Each item listed above does not correspond to its own slide. I simply listed ideas that could possibly work their way into the presentation. Some support each other. For example, when talking about the importance of problem solving, I would mention resources such as 3 Act lessons and The Math Forum's PoWs. Feel free to add to or subtract from the list.
I compared a few questions from CST and SBAC. CST stands for California Standards Tests and these are questions that were on previous STAR (Standardized Testing and Reporting) tests given to students in California and have been released to the public. I'm using recently released practice questions from SBAC (Smarter Balanced Assessment Consortium) that have been designed with Math 6 and Math 7 Common Core State Standards (CCSS) in mind.
I'm curious what thoughts or questions you might have. Leave them below. Thanks.
I hope you'll allow me to vent for a bit. I have been encouraging my students to be in tune with the 8 Mathematical Practices by Standard of the CCSS for some time now. It's pretty safe to say that my students know I really favor Mathematical Practice Standard 6, Attend to Precision. However, some of the resources I occasionally use in class are beginning to play tricks with everyone's minds, including mine. Here's a resource I have, Cooperative Learning and Geometry by Becky Bride.
Don't get me wrong, I like this book. It has some great explorative exercises that have appropriately challenged my students. For example, look at this exercise to help students derive the 30-60-90 triangle relationships. Take an equilateral triangle, its altitude, and the Pythagorean Theorem to find out the special relationships between the shorter leg, longer leg, and hypotenuse. Great.
Here's where I start to beat my head against the wall. The book uses diagrams that simply shouldn't be used, especially in the context of 30-60-90 triangles. Look closely...
That's right, the 30 degree angle is opposite the longer (drawn) leg for questions 1, 3, and 4. My students get bothered by this contradiction. I do too. I have no problem admitting this to them. I'm honest with them saying, "I know guys. It goes against everything we strive to do in here. I encourage you guys to attend to precision and check for reasonableness. Yet, I give you this. I'm sorry. It says at the top 'not drawn to scale', but they should be drawn to scale. Right guys?!"
I think this about sums it up. Students will come up and ask about the dimensions they've solved for and whether or not they're reasonable. I'm proud of my students for making sense of their answers and checking for reasonableness. I know something is a skew when my response to those students is,
"I never assume those things are drawn to scale."
I feel rotten saying this to students. I feel like I've just provided them with a worthless and menial task. I've let them down. I feel dirty. Mr. Stadel's quality control group hasn't done their job. What message are we sending students? Do they think we're out to trick them? Do the directions read, "Find the mistakes?" They should. It's times like these that force me to (gladly) keep a closer eye on the content I provide my students with. Don't just throw some triangles at them with random angles and units. Make sure they're reasonable.
Have you ever felt this way? Have you ever been caught in this situation? What did you do? How do we avoid these situations again? How do we demand better quality content from publishers? How do we make sure we provide our students with content that matches the CCSS and Mathematical Practices? Maybe you're okay with these types of diagrams, so please explain why. I want to hear from you all on this.
