tag:blogger.com,1999:blog-16974716106860077302017-11-22T00:47:41.038-05:00f(t)Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.comBlogger326125tag:blogger.com,1999:blog-1697471610686007730.post-55595930377313442032017-11-22T00:47:00.003-05:002017-11-22T00:47:41.049-05:00Respecting the Intellectual Work of the GradeA thing that I think we did really well in <a href="http://im.openupresources.org/" target="_blank">Illustrative Mathematics 6–8 Math</a> was attend carefully to really deep, important things that adults that already know math can easily overlook. For example, what does an equation mean? What does it mean for a number to be a solution to an equation? What does it mean for two expressions to be equivalent? (This is an example of the crucially important foundational understanding that gets short shrift when we rush kids through middle school math.)<br /><br />Confusingly, the symbol = can mean a few different things.<br /><br /><ul><li>If we represent the quantities in a word problem with <i>x</i> + 2 = 3<i>x</i>, we might mean, "What value in place of <i>x</i>, if any, makes each side have the same value?" </li><li>If we decide to see what happens when <i>x</i> is 4, we might write down <i>x</i> = 4. In this case, the = symbol means, "At this moment, we assign the value 4 to <i>x</i>." </li><li>If we represent two quantities that we <i>suspect</i> are equal no matter their value of <i>x</i> with 4 + 6<i>x</i> – 12 = 2(-4 + 3<i>x</i>) and use properties of operations to rewrite each expression until they are identical to each other, = means an entirely different thing: "I think these expressions are equal no matter what I substitute for <i>x</i>, and I would like to know for sure."</li></ul><br /><br />If you're reading this you probably realized all of that a long time ago, but none of this is at all obvious to your average 6th grader, or even 11th grader, and 6th grade is where in the common core math standards we are supposed to make a big pivot from arithmetic to algebra.<br /><br />This is just one task in a 6.EE arc designed to foster deep understanding, but I think it exemplifies the careful approach that we take for the sake of sense making. [<a href="https://im.openupresources.org/6/students/6/8.html" target="_blank">student materials</a>] [<a href="https://im.openupresources.org/6/teachers/6/8.html#activity-2" target="_blank">teacher materials</a> (requires free registration)]<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-u2ypH1DMY7M/WhM0ijGZPdI/AAAAAAAANvY/9Bym75SVbsAgUSEwwmZfqCZECbSM-cvgACLcBGAs/s1600/Screenshot%2B2017-11-20%2B23.33.54.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="921" data-original-width="793" height="640" src="https://4.bp.blogspot.com/-u2ypH1DMY7M/WhM0ijGZPdI/AAAAAAAANvY/9Bym75SVbsAgUSEwwmZfqCZECbSM-cvgACLcBGAs/s640/Screenshot%2B2017-11-20%2B23.33.54.png" width="548" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">The purpose of this task is to understand that two expressions that are equal for every value of their variable are called equivalent expressions. After the teacher eases students into it, students have a chance to work through the task. The entire point is to contrast <i>x</i>+2 and 3<i>x</i>, which are only equal (the same length) when <i>x</i> is 1, with <i>x</i>+3 and 3+<i>x</i>, which are equal no matter the value you substitute for <i>x</i>. Hey we could have told you <i>x</i>+3 and 3+<i>x</i> are equal no matter the value of <i>x</i> because of the commutative property. I wonder what other properties we can make clever use of. That's the purpose of the next task.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-FCsoO3haSJU/WhM3SdG9v9I/AAAAAAAANvk/iuSDHRnkFDENKgTStf3KARYJh3so0mDhwCLcBGAs/s1600/Screenshot%2B2017-11-21%2B00.12.37.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="147" data-original-width="847" height="108" src="https://2.bp.blogspot.com/-FCsoO3haSJU/WhM3SdG9v9I/AAAAAAAANvk/iuSDHRnkFDENKgTStf3KARYJh3so0mDhwCLcBGAs/s640/Screenshot%2B2017-11-21%2B00.12.37.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Now, it was hard to choose activities from one lesson to share. This curriculum is multi-faceted and has some super cool stuff. (Akshully, the entirety of unit 6 in grade 6 is one of my favorite things in the world, along with the continuation of focusing on the EE standards in grade 7, unit 6. Check it out.) And this lesson is all steak and no sizzle. But getting the unsexy but necessary bits right is something I'm really proud of.</div><div class="separator" style="clear: both; text-align: left;"><br /></div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-46094593075030528982017-07-31T08:20:00.000-04:002017-08-02T16:53:26.168-04:00FAQ: What Can We Change?<span style="background-color: white; color: #444444; font-family: "verdana" , "geneva" , sans-serif; font-size: 13px;">We are putting the finishing touches on the <a href="http://openupresources.org/math-curriculum/" target="_blank">Illustrative Mathematics Middle School Curriculum</a>.</span><span style="background-color: white; color: #444444; font-family: "verdana" , "geneva" , sans-serif; font-size: 13px;"> (For early access to sample units in the pilot, you'll have to share your contact info with us </span><a href="https://www.illustrativemathematics.org/lead_forms/new" style="background-color: white; color: #4d469c; font-family: Verdana, Geneva, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank">here</a><span style="background-color: white; color: #444444; font-family: "verdana" , "geneva" , sans-serif; font-size: 13px;">, but version 1 will be released any day now.) </span><span style="background-color: white; color: #444444; font-family: "verdana" , "geneva" , sans-serif; font-size: 13px;">I'm putting together a FAQ for people in our organization so they are prepared for questions we know they will get. This is the second in a series; <a href="http://function-of-time.blogspot.com/2017/06/faq-so-when-do-i-teach.html" target="_blank">here's the first one</a>.</span><br /><span style="background-color: white; color: #444444; font-family: "verdana" , "geneva" , sans-serif; font-size: 13px;"><br /></span><span style="color: #444444; font-family: "verdana" , "geneva" , sans-serif;"><span style="background-color: white; font-size: 13px;">Today's Q can come in many forms: "Do I have to do it this way?" "How much freedom is there to change things?" "Can I still use my favorite activities?"</span></span><br /><span style="color: #444444; font-family: "verdana" , "geneva" , sans-serif;"><span style="background-color: white; font-size: 13px;"><br /></span></span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-T0pQitleulo/WX8htSe4ceI/AAAAAAAAL-Q/v-qjtzBmcootrjhqpSWlQjj8UE2e1gE9QCLcBGAs/s1600/chefs-749563_960_720.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="637" data-original-width="960" height="212" src="https://1.bp.blogspot.com/-T0pQitleulo/WX8htSe4ceI/AAAAAAAAL-Q/v-qjtzBmcootrjhqpSWlQjj8UE2e1gE9QCLcBGAs/s320/chefs-749563_960_720.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Source: https://pixabay.com/en/chefs-competition-cooking-749563/</td></tr></tbody></table><span style="background-color: white; color: #444444; font-family: "verdana" , "geneva" , sans-serif; font-size: 13px;">This is an analogy I learned from someone at Louisiana Department of Education, where they are </span><a href="http://blogs.edweek.org/edweek/curriculum/2017/01/study_finds_louisiana_leads_the_way_in_understanding_teaching_state_standards%20.html" style="font-family: verdana, geneva, sans-serif; font-size: 13px;" target="_blank">getting impressive results</a><span style="background-color: white; color: #444444; font-family: "verdana" , "geneva" , sans-serif; font-size: 13px;"> by incentivizing schools to </span><a href="https://www.louisianabelieves.com/academics/ONLINE-INSTRUCTIONAL-MATERIALS-REVIEWS/curricular-resources-annotated-reviews" style="font-family: verdana, geneva, sans-serif; font-size: 13px;" target="_blank">choose well-aligned curricula</a><span style="background-color: white; color: #444444; font-family: "verdana" , "geneva" , sans-serif; font-size: 13px;">. If you were to try and cook a new, complicated recipe, you would probably make it as it's written the first few times you make it. You don't know what all the ingredients are for, you don't know the rationale behind all of the instructions, you don't really understand how it works, yet, before you cook it a few times. Once you start to understand the recipe, though, you can make smart choices to modify it to suit your tastes and needs: substitute green beans for eggplant, leave out the almonds, or take it out of the oven a little earlier, for example.</span><br /><span style="color: #444444; font-family: "verdana" , "geneva" , sans-serif;"><span style="background-color: white; font-size: 13px;"><br /></span></span><span style="color: #444444; font-family: "verdana" , "geneva" , sans-serif;"><span style="background-color: white; font-size: 13px;">Just like a dish you want to eat is a cohesive whole, people need to think of a curriculum as a coherent, connected, fairly complicated whole, with dependencies. Standards are one thing—they are a statement of what kids should know at the end. A curriculum makes choices, and choices have consequences. We set up pins in October that we knock down in February. After students have a well-designed opportunity to learn a term, idea, or skill in one unit, we believe that they will be able to remember it in a later unit. This is what you <i>want</i> out of a curriculum. You want kids to be able to make connections between ideas.</span></span><br /><span style="color: #444444; font-family: "verdana" , "geneva" , sans-serif;"><span style="background-color: white; font-size: 13px;"><br /></span></span><span style="color: #444444; font-family: "verdana" , "geneva" , sans-serif;"><span style="background-color: white; font-size: 13px;">The starkest example of this is a question we got from one of our pilot schools: "The word slope just shows up in grade 8, unit 3, as if the kids are already supposed to know what it means. This is terrible! What is going on here?" What was going on was, they skipped units 1 and 2, which were about transformations, thinking transformations were less important, and jumped right to the unit called "linear relations." The end of unit 2 takes a transformational approach to understanding the meaning of slope. (We use dilations to understand what it means for polygons to be similar, learn properties of similar figures, and then use slope triangles (similar right triangles with their hypotenuses lying on the same line) to show why we are allowed to refer to <i>the</i> slope of a line.)</span></span><br /><span style="color: #444444; font-family: "verdana" , "geneva" , sans-serif;"><span style="background-color: white; font-size: 13px;"><br /></span></span><span style="color: #444444; font-family: "verdana" , "geneva" , sans-serif;"><span style="background-color: white; font-size: 13px;">Just like a new recipe, you kind of have to teach a coherent curriculum the way it is written<i> for a couple years</i> before you really understand what is in there. Then, you are in a position to understand what it is safe to substitute or rearrange. </span></span>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-28929324062161570932017-07-29T16:00:00.000-04:002017-07-29T16:00:08.937-04:00Your Opinion of #MTBoS Has More to Do with You Than It Does with #MTBoS"Someone's opinion of you has way more to do with them than it does with you." I have a smart mouth and also get upset when other people are upset with me, so I've likely heard this aphorism more than the average person. It's been floating into my head lately, not because I think someone is upset with me (for once) but because of thunderstorms on Twitter over use of a hashtag. I'd like to propose that what someone thinks of MTBoS (Math Twitter Blogosphere) has more to do with them than it does with MTBoS. Consider:<br /><br /><ul><li>a mid-career math teacher who checks out Twitter, finds a hashtag he doesn't understand and conversations under that hashtag he doesn't understand</li><li>an organizer who has poured immeasurable energy into welcoming first-time attendees to TMC under the banner of MTBoS</li><li>a popular blogger and speaker who wants his ideas to have a broad and lasting impact on the way mathematics is taught, and has evidence that #MTBoS is a barrier to interested people accessing those ideas</li><li>an early-career math teacher who figured out what #MTBoS means by asking someone or google and periodically checks out the hashtag for inspiration</li><li>an early adopter of blogging and twitter who found many friends for life in MTBoS who make up a part of her support network and social circle</li><li>a math teacher who discovers #MTBoS, tries asking a question on twitter with that hashtag, and gets no response</li><li>a math teacher who had good results with resources found through MTBoS, but doesn't feel like a member of the club because she doesn't want to start a blog</li></ul><div>Here is me anticipating people getting upset and trying to head that off: I'm not trying to characterize any of these as selfishly motivated. All of these archetypes exist only because they want what is best for their students, all of humanity, or both. Also, all of these people's feelings are legitimate, because of course they are, because they are having them, and I'm not suggesting otherwise. Finally, if none of these describe you, I'm sorry and you still matter. This isn't an exhaustive list, it's my musings over breakfast.</div><div><br /></div><div>My prediction is that #MTBoS isn't going anywhere anytime soon. At least until the current crop of organizers of all things MTBoS retire, or as long as they remain good at generating energy among newcomers. </div><div><br /></div><div>My other prediction is that other hashtags will grow and fade in popularity. Easier to interpret hashtags are appealing because there is a lower barrier to entry, but they also tend to get diluted by spammy marketers, and then people stop paying attention to them. One possible explanation for the longevity and strength of #MTBoS as a hashtag is that it's a bit of a secret handshake.<br /><br />Here is one idea I have: when you use MTBoS not as a hashtag, but in longer form (on a blog post or while speaking), always follow it with "Math Twitter Blogosphere." The way Rachel Ray always said "E-V-O-O extra virgin olive oil." Clue the noobs in. It's a kindness.</div><div><br /></div><div>I'm looking forward to meeting and learning from new people on whatever hashtag we come up with and maintaining my enthusiasm for MTBoS and all we have accomplished and all of the good work yet to come.</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-81041666304206712382017-06-09T21:49:00.002-04:002017-08-02T16:54:21.662-04:00FAQ: So When Do I Teach?We are putting the finishing touches on the <a href="http://openupresources.org/math-curriculum/" target="_blank">Illustrative Mathematics Middle School Curriculum</a>. (For early access to sample units in the pilot, you'll have to share your contact info with us <a href="https://www.illustrativemathematics.org/lead_forms/new" target="_blank">here</a>, but we're looking at mid-July for the release of version 1.)<br /><br />We're often in the position of talking to teachers who have heard about the materials and are evaluating them, or whose district has adopted them and they are just learning about them. I'm putting together a FAQ for people in our organization so they are prepared for questions we know they will get. I am thinking to hash some of the Q's out in blog form, first. So theoretically this one in the first in a series. If you want to fight with me on anything I have to say, please speak up!<br /><br />Imagine this scenario: you demonstrate a problem-based activity with a group of teachers. You let them know that this is a grade 6 task where students have already learned to use double number lines and tables to represent a set of equivalent ratios. By this point, students are also familiar with recipe contexts; they know that an equivalent ratio of a recipe tastes the same. Here is the task:<br /><br /><div class="" style="background-color: white; box-sizing: border-box; color: #3e3f3a; font-family: Roboto, "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 10px;">Lin and Noah each have their own recipe for making sparkling orange juice.</div><ul class="compressed-list" style="background-color: white; box-sizing: border-box; color: #3e3f3a; font-family: Roboto, "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 10px; margin-top: 0px;"><li class="" style="box-sizing: border-box;">Lin mixes 3 liters of orange juice with 4 liters of soda water.</li><li style="box-sizing: border-box;">Noah mixes 4 liters of orange juice with 5 liters of soda water.</li></ul><div class="whitespace-small" style="background-color: white; box-sizing: border-box; color: #3e3f3a; font-family: Roboto, "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 10px;">How do the two mixtures compare in taste? Explain your reasoning.</div><br />The task is launched with a notice and wonder, they start happily working away, and you monitor what they are doing. You invite a few of them to make their reasoning visible to everyone, deliberately selecting them to share in a way that highlights a particular nuance you want to make sure everyone will understand, making mathematical connections between their approaches. (If you're savvy, you'll recognize this structure as Smith and Stein's <a href="http://amzn.to/2rbpwbY" target="_blank">5 Practices</a>, though my short description here isn't really doing it justice.) After conducting this discussion, many voices have contributed. Earlier in the day, you did another activity that loosely followed this same structure. You think, hey, I've done a pretty good job demonstrating the basics of how a problem-based classroom is meant to operate.<br /><br />Then you get the question, maybe timid but very curious, "So, when do I <i>teach</i>?"<br /><br />So here is a response that I'm turning over.<br /><blockquote class="tr_bq">Can you say a little more about what it looks like when you teach, as it looks in your mind, here? Okay, it sounds like synonyms for what you are describing might be <i>telling</i> or <i>explaining</i>. Is that fair? Okay. It's expected that you'll do some telling and explaining when using our stuff as it's meant to be used. The difference is in the timing. Let's think about what we did in the sparkling orange juice activity. You had a chance to work on a task, a few people shared their approaches, and then we made some observations about their approaches. What do you think the mathematical learning goal of that activity was? </blockquote><blockquote class="tr_bq">"Well, I remember seeing two sets of equivalent ratios represented with a double number line and with a table, and then so-and-so explained how she computed how much orange juice for 1 liter of soda water for both mixtures. It seemed like the point was that when you want to know which mixture tastes stronger, you need to create equivalent ratios so that one of the quantities is the same for each mixture. For example if orange juice to soda water is expressed as $15:20$ and $16:20$, you know that the second recipe tastes stronger." </blockquote><blockquote class="tr_bq">Okay cool. Do you think you got out of that activity what was intended? Does that mean you learned something? Does that mean teaching happened? </blockquote><blockquote class="tr_bq">There's still telling and explaining. Mathematical playtime is awesome, but a problem-based classroom is not just about mathematical playtime. We have clear learning goals for the course, each instructional unit, each lesson, and each activity. </blockquote><blockquote class="tr_bq">The way it's different than you might be used to is <i>when</i> the explaining happens. Perhaps you are used to first explaining something, and then kids do some work on the thing you just explained. In problem-based instruction, this is reversed. Kids have a chance to try and figure some stuff out first, you see what they come up with, and then after they've had a chance to get good and familiar with the context, the question being asked, the constraints, and they at least make some progress. . . then you take steps to make sure the relevant learning goals are made visible. Sometimes this part looks like explaining or telling.</blockquote><blockquote class="tr_bq">I'd suggest that teaching is a really broad and complex set of skills and behaviors, and telling or explaining is just one of them, and that telling or explaining isn't the only way to help kids understand something. In fact, does that approach work well for every student? How much do your students remember of what you explained the next day, or the next week? If you're completely satisfied with how things are going, awesome, but I bet you're here because either you or someone in your school endeavored to look for ways of conducting a math class that might work better for more kids, so that things made sense to them and the learning stuck around. </blockquote><br />Did I miss anything to address this particular question? (Please note that this is one vignette from two days of learning, and we spend time on a whole <i>bunch</i> of other things as well.) Does any of that come across badly? I want to acknowledge the person's completely understandable discomfort but also not shy away from asserting that teaching and learning happen in a problem-based classroom, and that we did it this way because we think <i>better</i> teaching and learning happen.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-80965850056718910452017-03-03T13:07:00.002-05:002017-03-03T13:08:34.154-05:00Anyone Want to Classroom Test Something? (grade 7)Hi! We are field testing all of our new materials in pilot schools, but I have one activity where the first draft was unworkable, and we have to come up with something totally new, and since the pilot schools are past this point I can't throw another version back to them. So...Internet... want to try something out for me? This is working toward the CCSS standard 7.EE.B.4a, so it's for seventh graders or students working on grade 7 material. The assumption is that they already have some strategies for reasoning about and solving equations of the form p(x+q)=r and px+q=r but that throwing negative numbers into the mix is relatively new.<br /><div><br /></div><div>Mainly what I am worried about here is that question 2 will go awry and students will go overboard and way far away from equation types they know about. And I also don't know whether that would be a good thing that students and teachers can just roll with, or if it's going to present challenges that are too much for too many people.</div><div><br /></div><div>So, if (and only if) this fits in with your plans, please try it out and let me know how it goes! Thanks in advance!</div><div><br /></div><div>Okay here's the task:</div><div><span style="font-family: "arial"; font-size: 11pt; white-space: pre-wrap;"><br /></span></div><div><span style="font-family: "arial"; font-size: 11pt; white-space: pre-wrap;"><br /></span></div><div>1. <span style="font-family: "arial"; font-size: 11pt; white-space: pre-wrap;">Here are some equations that all have the same solution. Explain how you know that each equation has the same solution as the previous equation. Pause for discussion before moving to the next question.</span></div><div><span id="docs-internal-guid-8f3129cc-9557-5f87-972c-e8bd7dbf6de7"><br /></span><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><div style="text-align: center;"><span id="docs-internal-guid-8f3129cc-9557-5f87-972c-e8bd7dbf6de7"><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><i>x</i> = -2</span></span></div></div><span id="docs-internal-guid-8f3129cc-9557-5f87-972c-e8bd7dbf6de7"></span><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><div style="text-align: center;"><span id="docs-internal-guid-8f3129cc-9557-5f87-972c-e8bd7dbf6de7"><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><i>x</i> - 3 = -5</span></span></div></div><span id="docs-internal-guid-8f3129cc-9557-5f87-972c-e8bd7dbf6de7"><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><div style="text-align: center;"><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">-5 = <i>x</i> - 3</span></div></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><div style="text-align: center;"><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">500 = -100(<i>x</i> - 3)</span></div></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><div style="text-align: center;"><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">500 = (<i>x</i> - 3) ᐧ -100</span></div></div><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><div style="text-align: center;"><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">500 = -100<i>x</i> + 300</span></div></div><br /><div dir="ltr" style="line-height: 1.38; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">2. Keep your work secret from your partner. Start with the equation -5 = <i>x</i>. Do the same thing to each side at least three times to create an equation that has the same solution as the starting equation.</span></div><br /><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">3. Write the equation you ended up with on a slip of paper, and trade equations with your partner. See if you can figure out what steps they used to transform -5 = <i>x</i> into their equation. When you think you know, check with them to see if you are right.</span></span></div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-30163338895200041522017-02-25T11:29:00.002-05:002017-02-25T11:29:41.570-05:00Is This Thing On?Hello, Blogoworld! I'm not sure if anyone is still listening, but if you are, I have a short assignment for you. I'm preparing a talk where I'll show different people's sample work to the same problem. So I'd like to collect a bunch of different responses. Here is the problem:<br /><br />A sloth can go 50 feet in 7 and a half minutes. How far can it go in an hour and a half?<br /><br />If you'd like to participate, I need a good photo of your hand-written work. Upload it wherever, and share in the comments of this post. Bonus points for use of representations with more structure than dividing and multiplying. If you have access to a young person, it would be cool to have some samples that are in little kid handwriting. Thank you!Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-45891314097304716102016-01-20T18:37:00.000-05:002016-01-20T22:23:56.730-05:00What I Think a Rate Is Right Now<h2>Stating a few assumptions before I get into this:</h2><div>I'm going to explain how I use the word "rate" and the phrase "unit rate" (and also throw around the word "ratio" somewhat recklessly) and it might not match what's in your textbook or how you use the words in your classroom. Some textbooks proclaim that ratios may only involve like units whereas rates use unlike units. In the physical sciences they typically use "rate" to refer to a measurement with respect to time, specifically. All names for things are conventions. I'm not trying to say that you or your textbook or the physics teacher are wrong. Here is a complete list of the arbiters of correctness when it comes to conventions: </div><div><ol><li>mathematical consistency</li><li>people in the act of communicating about the same situation understand each other</li><li>you're not setting up a person for massive confusion later on </li></ol></div><div>The definition of a trapezoid is a good example. Is a trapezoid a quadrilateral with one and only one pair of parallel sides, or is it a quadrilateral with at least one pair of parallel sides? Said another way, is a parallelogram a special type of trapezoid, or is a parallelogram by definition never also a trapezoid? Answer: <span style="background-color: white; font-family: "lyon display" , "georgia" , "times" , serif; line-height: 1.04;">¯\_(ツ)_/¯</span> It depends on a choice made by a person. Textbooks often present definitions like, "This is what the word means!" when they really mean something more like, "This is a choice we made in order to move forward." </div><div><br /></div><div>I'm working on a common core aligned math curriculum for sixth grade. So something to understand as a consequence of that: I'm thinking about how to make these ideas make sense to kids in middle school. So I don't want to write a post about mathematically ironclad definitions that would pass muster with research mathematicians; I want to write a post about stuff that it would be wonderful for kids age 11-13 to understand and is also flexible and useful to build on in later studies.</div><div><br /></div><div>And one last preliminary: sometimes it's important for teachers to understand some nuances and it's not as important for students to understand them at the same level of detail. So, I'm not suggesting that any of this post is appropriate for instructional or assessment purposes with students. For example, an appropriate question for a student might be "In a fruit punch, the ratio of cups of grape juice to cups of soda water is 2:5. How many cups of grape juice for every cup of soda water?" But this question would not be appropriate for sixth graders: "In the ratio 2:5, what is the unit rate?" Because, ew.</div><div><h2>Okay so here we go</h2></div><div>A tortoise travels 10 inches in 3 minutes. A snail travels 8 inches in 3 minutes. Are they traveling at the same rate? (Assuming they're both traveling at a constant rate.)</div><div><br /></div><div>No they are not traveling at the same rate, but I hope you didn't need to compute anything to know that. You can tell because they traveled different distances in the same amount of time. In this context, you have their distances traveled, you have the time it took, but then you have this <i>third thing</i> that means something concrete in the context -- how fast they are going. Their rates. We can express the tortoise's rate as 10 inches in 3 minutes or around 3.33 inches per minute or a foot-and-a-quarter every 270 seconds but the real live concrete in-context rate is the concept of how fast (or in this case, slow) he is moving.</div><div><br /></div><div>(Note for curriculum nerds: at some point you have to make it explicit to students that "are these happening at the same rate?" is structurally the same question as "are these equivalent ratios?" Not super relevant to this discussion but it seems worth mentioning.)</div><div><br /></div><div>At one store, 2 pounds of M&M's cost $14. At a different store, 2 pounds of M&M's cost $16.95. Which is a better deal? Did you have to compute anything to know that? No, you can compare the good-deal-ness without computing the cost of 1 pound or how many pounds you can get for $1. The rate is a <i>third thing</i> going on here capturing how-good-is-this-deal that could be expressed in different ways, one of which is a unit price.</div><div><br /></div><div>Those examples were different types of quantities (distance and time, weight and cost) but we can talk about rates with same quantities like volume and volume. </div><div><br /></div><div>Kate mixes 2 oz of gin with 5 oz of tonic water. Ashli mixes 3 oz of gin with 7 oz of tonic water. Are they sipping the same beverage? This is not so easy because none of the quantities match up. So now we need to find how many ounces of gin for one ounce of tonic water for each beverage right? We could, for sure, but we don't have to. We just have to compare equivalent ratios for the same amounts in the different concoctions:</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-fvMjrpbBth4/VqAHDrcxzfI/AAAAAAAAFGU/ruGNDkd2MH0/s1600/gt.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="123" src="http://1.bp.blogspot.com/-fvMjrpbBth4/VqAHDrcxzfI/AAAAAAAAFGU/ruGNDkd2MH0/s320/gt.PNG" width="320" /></a></div><div class="" style="clear: both; text-align: left;"><br /></div><div class="" style="clear: both; text-align: left;">If I used 3 oz of gin to mix a beverage that tastes the same as my original drink, I would need 7.5 oz of tonic water. Ashli only mixed her 3 oz of gin with 7 oz of tonic water, so Ashli's was a bit stronger than mine. Here, that third hidden thing going on is the potency of the beverage, and I'm still asserting that it's a rate, and I still haven't figured out how many of anything per one of anything.</div><div class="" style="clear: both; text-align: left;">So, let's sum up what we have so far: in any set of equivalent ratios that represents a context, there is a third thing that characterizes something meaningful about those two things happening at the same time. It could be land speed, how much of a good deal you are getting, beverage strength, the tempo of a song (number of beats to number of minutes), how crowded my neighborhood feels (number of people to square miles)... This third thing hidden within a set of equivalent ratios is a concept I'm calling a rate.</div><div><br /></div><div>But then, it's often convenient to refer to the special equivalent ratio that is something-paired-with-a-one: "how many of these for every <i>one</i> of those?" It is convenient for at least two reasons (and probably more). First, it helps you solve equivalent ratio problems pretty quickly. For example, I know that I get a certain lovely shade of orange acrylic paint if I mix 3 teaspoons of yellow paint with 2 teaspoons of red paint, but I want to make the same shade and I need alot of it so I want to use up the 9 teaspoons of red paint I have on hand. How much yellow paint should I mix it with? I might approach that problem in any number of ways, but a good way is to reason that 2:3 is equivalent to 1:1.5, so to solve 9:? I just need to multiply 1.5 by 9. (This explanation would be clearer if I drew you a ratio table or another double number line but I am getting tired and it's almost cocktail hour.) It's convenient to use a word to name the 1.5, and "unit rate" is as good a name as any. I like how the "unit" part reinforces that it has something to do with 1. A question kids should be able to answer as part of their process is, "What does the 1.5 mean in this context?" and they should be able to say "there are 1.5 teaspoons of yellow paint for every 1 teaspoon of red paint."<br /><br />Second, it's a way to express that third thing in a set of equivalent ratios with just a single value which can be algorithmetized (like if you want to tell a computer how to do it.) In the gin-and-tonic example above, we could have computed that Kate's drink had 2/5 oz gin for every ounce of tonic water, and Ashli's drink had 3/7 ounce of gin per ounce of tonic water, and since 3/7 is greater than 2/5, Ashli's was stronger.<br /><br />Then actually later in seventh grade I could write an equation for the relationship that is my recipe, <i>g</i> = 2/5 <i>t</i>, where <i>t</i> is volume of tonic water and <i>g</i> is volume of gin, and 2/5 is re-named the constant of proportionality for the set of all gins and tonics of that particular strength, and I could graph this equation and an equation representing Ashli's recipe and note that the line representing her recipe is steeper, but we're really getting ahead of ourselves here.<br /><br />Okay, this was a long post, but we're almost done. I believe that my interpretation is supported by the CCSS standards and the RP progression document, although I also believe that those documents also allow you to conclude that rate only means "how many of these for every one of those" (because the only examples they give for "rate" are quantities per 1). But if you're going to use rate to mean how much of this for every one of that, I think you need to come up with another word for that third-thing physical quantity that I am calling rate.<br /><br />Alright. Comments are on. Come at me, nerds.</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-48492846777666252092016-01-04T14:00:00.001-05:002016-03-21T17:38:02.809-04:00In Defense of UnsexyAt IM we're writing a sixth grade curriculum, and much of my time is spent writing, reviewing, and begging other people to write and review new <a href="https://www.illustrativemathematics.org/search?query=6." target="_blank">grade 6 tasks</a> that really just hit the fundamental stuff. After five-ish years of accepting task submissions, we have some holes. <i>Because nobody wants to write easy questions.</i><br /><br />Do you know what teachers have the hardest time finding?<br /><br />Quality, basic stuff.<br /><br />It is getting relatively easy to find the rather-complicated application problems, the projects, the whole new grading systems, the elegant warmups, the pinterest-worthy graphic organizers. Many people have invented and shared some very sexy, awesome stuff, and they are changing many teachers' and kids' experience of math. Hooray for that!<br /><br />But so many teachers aren't helped by sexy stuff. I think one reason is that they don't think the payoff is worth the time investment. Or maybe that changing the whole way they run class is too intimidating. I'd be happy to entertain alternative theories.<br /><br />Here's a cooking show analogy: in the early-mid 2000's people enjoyed Mario Batali's homemade gnocchi and Bobby Flay's 90-ingredient curries and Alton Brown's coconut cake that takes THREE DAYS (I'm not kidding. Three days.) But you know who I watched every day at 4:30? Rachel Ray. I suffered through her saying "yummo" and "EVOO-that-means-extra-virgin-olive-oil" approximately 19 times per episode, and she taught me how to get a reasonable meal on a plate in 30 minutes and how to chop a damn onion.<br /><br />This blog grew in popularity (and stays relatively popular even though I neglect it so) not because I invented something big and sexy but because it offered relatively easy swaps for practice worksheets and ugly, fresh-off-the-smartboard rewrites of high school lessons that made the kids do a tiny bit more thinking than usual.<br /><br />So I'm starting to hear my low-level angst echoed <a href="https://ihati.wordpress.com/2016/01/04/solving-teacher-problems-making-it-easier-to-be-better/" target="_blank">elsewhere</a> and it's bubbling over. I have a request for you if you are a math teacher and you have a blog.<br /><br />Share your kinda-borderline boring stuff. Your small tweaks that unloaded the right amount of cognitive lifting onto the kiddos. Your rather-basic task or set of tasks that don't seem that exciting, but your kids always seem to readily grasp that topic. Your snippets of classroom dialog where everybody ended up going OHHHH. Your artful arrangement of pieces of instructional units you found lying around. How you took that cool instructional idea you read in that book and figured out how to do it in a congruent triangles lesson.<br /><br />We need you. Your kinda-lame-but-seems-to-do-the-trick exponent rule investigation is going to make you somebody's superhero. If you share them with me (add a comment on this post, tweet them at me, whatever), I'll re-share them and compile them in new posts. (And probably they will get added to some of those wonderful virtual filing cabinets and wikis.)<br /><br /><h3>Update: Some gems in the comments. And some shared on Twitter. </h3><br /><ul><li><a href="http://limitsdne.blogspot.com/2015/11/constant-difference-building.html" target="_blank">Constant Difference: Building Understanding</a>, Bryan Dickinson's students engage in some fierce MP2 and MP8 extending a visual pattern.</li><li><a href="https://fractionfanatic.wordpress.com/" target="_blank">Intro to Integration</a>, Julie Morgan nails the productive struggle and the connections to prior learning by asking a backwards question.</li><li><a href="http://mathteachernerds.blogspot.com/2016/01/intro-to-angles-in-degrees-and-radians.html" target="_blank">Intro to Angles in Degrees and Radians</a>, Danielle Reycer increases mathematical coherence by reorganizing a series of lessons.</li><li><a href="https://twitter.com/MathMinds" target="_blank">Kristin Gray</a> likes to combine <a href="https://teacher.desmos.com/polygraph/custom/560aa8df9e65da561507a5ce" target="_blank">Polygraph: Points</a> by Robert Kaplinsky with <a href="https://www.illustrativemathematics.org/content-standards/6/NS/C/6/tasks/2227" target="_blank">this task</a> to orient students to the coordinate plane.</li></ul>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-52324112636448810702015-11-21T13:45:00.004-05:002015-11-21T13:45:48.106-05:00NCTM Nashville PresentationI had the pleasure of attending the NCTM regional meeting in Nashville this week. I learned some cool stuff that I'm still processing, and I got to do a presentation. In the presentation I tried to explore whether the way I would rewrite and rework lessons when I was a high school teacher can be generalized and communicated to other people. I was, I think, marginally successful.<br /><br />NCTM is trying this cool pilot where participants can engage with presenters after the conference. So instead of sharing stuff about my presentation here, I'm going to <a href="http://regionals.nctm.org/nashville/plan-a-killer-lesson-today/" target="_blank">send you over to the presentation page on their site</a>.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-9323694218862632352015-10-02T11:11:00.000-04:002015-10-03T11:19:55.380-04:00Friday Favorites 7Happy Friday! (It's really Saturday but I'm going to backdate this post and pretend it's Friday. Ha! Technology!) My reading and favoriting has slowed down because I have made the decision to limit my Twitter time, which is exceptionally mature of me, I think. (Using <a href="http://www.stayfocusd.com/" target="_blank">Stay Focusd</a>, which is a chrome plugin that yells at you for not working. It's brilliant.) What I'm mostly doing these days is a zillion math problems, which is pretty fun, actually... You know how when professional chefs see a bag of onions, they get excited because they get to chop a bag of onions? That's how I feel about doing a bunch of math problems. It's a little bit drudgery, but satisfying. Still and all, when something gets a little mentally difficult it can't be too easy to distract myself. Twitter needs to not be an option in those moments.<br /><br />This is not a favorite because I made it myself, but it's public, so I might as well share it. It's a place to <a href="https://www.tumblr.com/blog/mathspo" target="_blank">stash mathematically interesting artifacts</a> that I might turn into tasks or assessment questions or lessons. There's nothing worse than needing to write a question in a context and googling for hours. You're welcome, future Kate.<br /><br />Now here are real favorites:<br /><br /><h4><a href="http://cheesemonkeysf.blogspot.com/2015/09/proportional-reasoning-capture.html" target="_blank">Capture Recapture with Goldfish</a></h4>I did this lab in an Algebra 1 class ages ago. It reminds me of that illustration of statistics vs probability: If you know what's in the bag, reach in and grab a handful, and want to predict what's in your hand, that's probability. If you <i>don't </i>know what's in the bag, reach in and grab a handful, and use the handful to predict what's in the bag, that's statistics. It's a good activity, but my first or second year teacher self probably didn't do such a great job with it. Because, obviously, I didn't have Elizabeth and Julie's helpful writeups. I like the way Elizabeth frames how it fits into a bigger Algebra 1 picture. I could also see using it in a stats lab in a way that emphasizes sampling and sample proportions just as easily as a 7th grade-ish solving proportions lab.<br /><br /><h4><a href="https://crazymathteacherlady.wordpress.com/2015/09/16/constructions-was-much-better-this-year/" target="_blank">Problematizing Geometry Constructions</a></h4>I love everything about this. Using a popsicle stick as a straight edge: pro move.<br /><br /><h4><a href="http://www.brilliant-insane.com/2015/09/schooling-terrible-teacher-10-things-parents-never.html?utm_content=bufferc8e00&utm_medium=social&utm_source=twitter.com&utm_campaign=buffer" target="_blank">How Parents and Students and Teachers Can Work Better Together</a></h4>...is a better headline than the clickbaitey one they gave this article. Which is empathetic and treats everyone involved as a professional and a human. Forward anonymously to those parents whose first move is calling the Principal.<br /><br /><h4><a href="https://tle.soe.umich.edu/" target="_blank">Michigan's Teaching and Learning Exploratory</a></h4>Don't let the boring name fool you - Michigan has done an awesome thing here by posting hours and hours of unedited classroom footage. I learned in the last chapter of <a href="http://amzn.to/1M0Q96g" target="_blank">Why Don't Students Like School?</a> that looking at video of yourself or someone you know is too scary a place to start, and it's easier to watch and practice constructively critiquing someone you don't know. This resource makes that a whole lot easier.<br /><br />Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-56405876551426594382015-10-01T14:04:00.000-04:002015-10-01T14:04:57.327-04:00Every Bit of This<a href="http://www.stolaf.edu/people/steen/Papers/07carnegie.pdf" target="_blank">Link</a><br /><blockquote class="tr_bq">High schools focus on elementary applications of advanced mathematics whereas most people really make more use of sophisticated applications of elementary mathematics. … Many who master high school mathematics cannot think clearly about percentages or ratios.</blockquote>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-15887489652417997082015-09-30T09:22:00.004-04:002015-09-30T11:49:17.115-04:00Exponential Functions and also Area of a TriangleThat title is confusing, right? I know! I just wanted to alert y'all to some tasks that recently went up on Illustrative Mathematics that might address some of your needs, if you are teaching these things.<br /><div><br /></div><div><b>Exponential Functions</b>: These tasks involve negative exponents in a functional relationship in a context and are aligned with F-LE.</div><div><ul><li><a href="https://www.illustrativemathematics.org/content-standards/HSF/LE/A/2/tasks/2130" target="_blank">Decaying Dice</a> (It's like the penny lab for modeling half-life that kids often do in Earth Science... except with dice.)</li><li><a href="https://www.illustrativemathematics.org/content-standards/HSF/LE/A/2/tasks/2127" target="_blank">Predicting the Past</a> (Making sense of negative integers in the domain of a simple exponential growth function.)</li><li><a href="https://www.illustrativemathematics.org/content-standards/tasks/2129" target="_blank">All Your Base are Belong to Us</a> (Exponential decay and negative exponents, together at last. Bonus points if you get the reference.)</li><li><a href="https://www.illustrativemathematics.org/content-standards/HSF/LE/B/5/tasks/2128" target="_blank">DDT-Cay</a> (Interpreting the exponent in a half-life equation.)</li></ul></div><div><br /></div><div><b>Area</b>: These are meant to be used to build understanding as you're working toward a formula for area of a triangle in sixth grade (6-G.1). But they could be useful to reactivate knowledge at the beginning of a study of area in a later Geometry course.</div><div><ul><li><a href="https://www.illustrativemathematics.org/content-standards/6/G/A/1/tasks/2131" target="_blank">24 Unit Squares</a> (To remind kids what area means and stymie their attempts to use formulas they don't understand.)</li><li><a href="https://www.illustrativemathematics.org/content-standards/6/G/A/1/tasks/2132" target="_blank">Areas of Right Triangles</a> (Depending on how you approach area of any triangle, this might be a necessary precursor.)</li><li><a href="https://www.illustrativemathematics.org/content-standards/6/G/A/1/tasks/2133" target="_blank">Areas of Special Quadrilaterals</a> (Emphasizes decomposition into familiar figures.)</li></ul></div><div>And, hey, it is non-trivial for me to test stuff out with kids these days, so if YOU try them out and you notice stuff or have suggestions, you can comment here or better yet, right on the task on the IM site. (Please let me know if you do that - I don't think I get a notification. And thanks!)</div><div><br /></div><div>I did draft the initial versions but I can't take credit for these. Tasks published on IM are very much a team effort. Many thanks to <a href="https://twitter.com/mythagon" target="_blank">Ashli Black</a> who is an ace reviewer and helped me make these a ton better.</div><div><br /></div><div><br /></div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-57738796912872919652015-08-28T09:52:00.003-04:002015-08-28T09:52:41.881-04:00Friday Favorites 6<div>Happy Friday! I am elbows deep in Trello, of all things, but the cat is good company. Here we go...<br /><br /></div><div><h4><a href="https://twitter.com/Desmos/status/635081996704813056" target="_blank">Team Desmos</a></h4></div><div>I took a stab at Activity Builder with <a href="https://teacher.desmos.com/activitybuilder/custom/55d76883772721050e71663e" target="_blank">an activity</a> that deals with discovering pi and thinking of circumference vs diameter as a proportional relationship. And wow, it's so much better because of their Twitter interaction. I don't know if my favorite part is setting a table to make points draggable only vertically, or their suggestion to share Teacher Notes in a <a href="https://docs.google.com/document/d/1esXIrESWgJpIz_eVmiM-awZelNqb3ZeJHCTLniysH0Y" target="_blank">linked google document</a>.<br /><br />In case you haven't heard, there's also a repository <a href="https://sites.google.com/site/desmosbank/" target="_blank">of user-created Desmos activities here</a>. Mileage may vary.<br /><br /></div><h4><a href="https://drive.google.com/open?id=0B8XS5HkHe5eNfmNVSjYzXzRtTWVfUm1xWE9uRHdJbWZ6U05OdW9XLTc3ejV2OHdXYlQtSnM" target="_blank">All the Math Talking Points</a></h4><div>Are in this shared google folder. If you haven't grokked the magic of Talking Points yet, go read you <a href="http://cheesemonkeysf.blogspot.com/search/label/TalkingPoints" target="_blank">some cheesemonkey wonders</a>.<br /><br /></div><h4>OER Curricula and Curricular Outlines</h4><div>In case I haven't talked your ear off about it yet, I'm of the strong opinion that a school's math department should Decide on a Coherent Curriculum and riff off of that, rather than expecting their teachers to create a curriculum on the fly using random resources they find on the Internet. Some textbook series are good, and there are also decent OER (Open Educational Resource) ones are already out there, and too many people don't know about them.<br /><br /><ul><li><a href="http://math.newvisions.org/" target="_blank">New Visions for Public Schools (High School)</a></li><li><a href="https://www.carnegielearning.com/learning-solutions/curricula/middle-school/" target="_blank">Carnegie Learning (Middle School)</a></li><li><a href="http://collegeready.gatesfoundation.org/student-success/high-standards/literacy-tools/mathematics-design-collaborative/" target="_blank">BMGF Mathematics Design Collaborative (Middle and High School)</a></li></ul><div><br /></div><br /><h4><a href="http://www.doingmathematics.com/blog/the-teacher-partnership-origins-and-goals" target="_blank">This Coaching Model</a></h4></div><div>Where your team gets a Teacher Partner - someone who teaches a few classes but also coordinates your collaborative teacher learning. I love this.<br /><br /><div><h4><a href="https://algebrainiac.wordpress.com/2015/08/12/2015-2016-new-year-goalsexpectations/" target="_blank">Jessica's Practice is Wide Open</a></h4></div>I might be a little obsessed with other people's planning documents.<br /><br /><h4><a href="http://www.cultofpedagogy.com/classroom-icebreakers/" target="_blank">Icebreakers That Won't Make you Cringe</a></h4><div>You know what I'm talking about. <a href="https://twitter.com/tchmathculture/status/635242825261559808" target="_blank">h/t Lani</a>.<br /><br /></div></div><h4><a href="http://mathhombre.blogspot.com/2015/08/where-do-i-start.html" target="_blank">John's Exhaustive Tour of the Good Stuff</a> </h4><div>Where do I start? Here.</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-31441876342744457892015-08-19T22:13:00.000-04:002015-08-19T22:13:13.286-04:00And Then There Was Not Teaching Some MoreWaddup, nerds. Just a quick note about what is going on around here, which is that I've joined forces with <a href="https://www.illustrativemathematics.org/" target="_blank">Illustrative Mathematics</a> to do some very exciting curriculum work. I'll keep y'all posted here as I am able. <div><br /></div><div>Practically that means it's not a new school year for me, which sucks, because I love the first day of school. There's something so inspiring about a fresh start. And also because there won't be as much to report here. </div><div><br /></div><div>But it also means I work at home, which, I'm not going to lie, is pretty boss. I can get all the work done with a cat in my lap and also throw in a load of laundry and also prepare real food for dinner. </div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-69L2pVywGzQ/VdU3QZyxDjI/AAAAAAAAEx0/c8_-4eRjY_E/s1600/2015-08-14%2B11.06.35.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="180" src="http://1.bp.blogspot.com/-69L2pVywGzQ/VdU3QZyxDjI/AAAAAAAAEx0/c8_-4eRjY_E/s320/2015-08-14%2B11.06.35.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">My rig. I realize the television is dominant in this photo, but I haven't actually turned it on yet. It's just extra. That fridge is full of fizzy water.</td></tr></tbody></table><div><br /></div><div>I'm around, on the Internet, of course, and I want to keep doing the Friday Favorite thing. You are welcome to yell at me when I slack off.</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-58417483458052509132015-07-24T12:01:00.000-04:002015-07-24T12:01:31.369-04:00Friday Favorites 5<div>Happy TMC, everybody! I know all the TMC-ers are busy TMC-ing right now, but it's Friday! Here we go...</div><div><br /></div><h4><a href="http://www.fishing4tech.com/fishin-solo-blog/the-mtbos-search-engine">John's </a><a href="http://www.fishing4tech.com/fishin-solo-blog/the-mtbos-search-engine">MTBOS search engine</a></h4>What a good idea. I don't know how I missed this.<br /><br /><h4><a href="https://docs.google.com/document/d/1zfRgtBc4n3Pf2xDhiEIwWwymhBEPi0OiINmhDALf7Pc/edit" target="_blank">Tracy's Proof Games</a></h4><div>Here at camp there's a "Games and Strategies" class running this week, and kids keep running up to staff proposing games like "We start at zero, take turns adding 1, 2, or 3, first one to 19 loses." These are kind of addicting, is what I'm saying, and motivate and "Support Generalizing, Conjecturing, Strategy, and Proof-Like Reasoning," as the title suggests. And here are a zillion of them in one document!</div><div><br /></div><h4><a href="http://edushyster.com/i-am-not-tom-brady/" target="_blank">I Am Not Tom Brady</a></h4><div>Just putting this out as a public service announcement that schools that pull shit like this exist, so you can walk away quickly if you get a whiff of it in an interview. h/t Lani for the share.</div><div><br /></div><h4><a href="http://mathbabe.org/2015/07/22/the-17-armed-spiral-within-a-spiral/" target="_blank">Cathy's Write-up of a 17-Armed Spiral</a></h4><div>Here's some recreational math for you, in the spirit of math camp.</div><div><br /></div><h4><a href="http://statteacher.blogspot.com/2015/07/teacher-binder-2015.html" target="_blank">Shelli's Teacher Binder</a></h4><div>Back in the days of student-ing, my life was all about my paper organizer. I had very specific requirements and shopped and shopped until I found it. These days I'm a more scattered leaving-digital-detritus-in-my-wake kind of organizer, but this makes me think maybe it's not too late. </div><div><br /></div><h4><a href="https://www.flickr.com/photos/133462526@N05/" target="_blank">Look How Pretty</a></h4><div>The #mathphoto15 Flickr stream. </div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-91073895123835320002015-07-17T17:46:00.000-04:002015-08-14T08:44:20.986-04:00Summer Problem-Solving CourseThis summer I have the privilege of teaching a problem solving class to mathematically-inclined rising eighth graders. The course is called Math Team Strategies because a big goal is to get kids more ready for contests like MATHCOUNTS and the AMC contests. But we are also looking to highlight problem solving strategies that are broadly useful, whether kids decide to participate in contests or not.<br /><br />I'm going to make this post pretty nuts and bolts just the facts ma'am - it's the nitty gritty details for people who want the ideas.<br /><br />I lovingly plucked from the work of, and want to give tons of credit to:<br /><ul><li>Matt Weber, who is teaching this same course at this program's other site</li><li><i>Crossing the River with Dogs</i> by Johnson, Herr and Kysh [<a href="http://amzn.to/1VaQ17w" target="_blank">Amazon</a> <a href="https://books.google.com/books/about/Crossing_the_River_With_Dogs.html?id=_JEcNfARtfUC" target="_blank">Google Books</a>]</li><li>MATHCOUNTS <a href="http://www.mathcounts.org/past-competitions" target="_blank">Past Competitions</a> and <a href="http://www.mathcounts.org/resources/school-handbook" target="_blank">School Handbook</a></li></ul><h2>Pacing</h2>Eight days, two hours a day, one focus strategy per day. On the final day, instead of a new strategy, students experience a somewhat-complete MATHCOUNTS contest.<br /><h2>The Strategies</h2>(Most of these are chapter titles in <i>Crossing the River with Dogs</i> - but that book has many, many more chapters. It's awesome. You should check it out.)<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-ajftYFUV0Oo/ValosnVgABI/AAAAAAAAEuo/k7DHUdfVeSQ/s1600/2015-07-17%2B12.54.28.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-ajftYFUV0Oo/ValosnVgABI/AAAAAAAAEuo/k7DHUdfVeSQ/s320/2015-07-17%2B12.54.28.jpg" width="180" /></a></div><h2>The Lesson Flow</h2>For each day, I selected problems that lent themselves to that day's strategy. Some problems are from <i>Crossing the River</i>, some are from old MATHCOUNTS contests, and some I made up. Additionally, we developed a few mathematical shortcuts over the course of a few days, like counting permutations with repetition and the length of a diagonal of a square. I cut the problems up onto slips, so students would only have one problem at a time. (For a longer course, or perhaps for older students, I'd probably elect to use <i>Crossing the River</i> as a text.)<br /><br />All the students worked on the same problem at the same time, standing at chalkboards. I had anywhere from 6 to 12 students in a class, so this was manageable. I also had a TA who was a math-major undergrad. Nirvana. Before I left home I grabbed a handful of fridge magnets, thinking they might be useful for something, and we used them so students could stick the current problem to the chalkboard.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-oxeuEM8OLag/Valpd45XlCI/AAAAAAAAEu8/pf8Bi-5L9pw/s1600/2015-07-14%2B14.51.26.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://2.bp.blogspot.com/-oxeuEM8OLag/Valpd45XlCI/AAAAAAAAEu8/pf8Bi-5L9pw/s320/2015-07-14%2B14.51.26.jpg" width="320" /></a></div><br /><h2>The Posters</h2>The intention was for the whole class to go over each problem before everyone started the next one. (See <a href="http://function-of-time.blogspot.com/2015/07/a-magical-incantation.html" target="_blank">this post</a> about group discussions.) Of course, some students took longer and needed support. When I am helping, I tend to make the same suggestions and ask the same questions over and over. This poster was for students to refer to if both the TA and I were busy when they got stuck.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-CbjIZINOQnM/VakwfXlMmMI/AAAAAAAAEuQ/ya22L6DErfU/s1600/2015-07-07%2B11.57.20.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-CbjIZINOQnM/VakwfXlMmMI/AAAAAAAAEuQ/ya22L6DErfU/s320/2015-07-07%2B11.57.20.jpg" width="180" /></a></div><br />Also, of course, some students finished more quickly than the group. I also tend to always make the same suggestions when students say they are "done," so I made this poster for them, too.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-oQGAKfjhj7I/VakwfgZwY0I/AAAAAAAAEuM/uYapzVwc0M8/s1600/2015-07-07%2B11.57.27.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-oQGAKfjhj7I/VakwfgZwY0I/AAAAAAAAEuM/uYapzVwc0M8/s320/2015-07-07%2B11.57.27.jpg" width="180" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><h2 style="clear: both; text-align: left;">The Self-Assessment</h2><div class="separator" style="clear: both; text-align: left;">Before we went over each problem, I asked the students to turn in their problem slip with their name and a rating of the problem from 1 through 4. I did compile this data in a spreadsheet, but I'm not sure what to do with it. But I thought the self-assessment couldn't hurt.</div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-rkP-ROswAK4/VakwfcfLjzI/AAAAAAAAEuI/bpOzPCKE4pw/s1600/2015-07-07%2B11.57.06.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-rkP-ROswAK4/VakwfcfLjzI/AAAAAAAAEuI/bpOzPCKE4pw/s320/2015-07-07%2B11.57.06.jpg" width="180" /></a></div><br /><h2>The Resources</h2><a href="https://drive.google.com/folderview?id=0Bz4S-NJpJht7fmd5dC0tdDJPX3VjQXMybEpxLWUyWHNtdmk5M1ZUckVma0YzaUU0ejBvN2s" target="_blank">Will be here</a> until someone holding a copyright yells at me to take them down. Or maybe this is fair use. I dunno. I hope it's good advertising for the publications cited above. Some of the problems turned out to be too easy, and I'll be changing them if I'm back next year. Some were too hard, but I thought it was okay to give kids at most one problem a day that was a big stretch for them. When that happened, I invited the TA to share their solution.<br /><h2>And That's about That</h2>This was a really rewarding course. The kids loved it, I loved it, we all just had a grand old time talking about math for two hours a day! It was refreshing to not feel pressure to cover content at a breakneck speed, or sell kids on math (these kids already like math), or have to assign grades. (This morning when we did a sample MATHCOUNTS Sprint, a girl asked "Does this count? Oh, wait. We don't have grades." And she worked hard on it anyway.)<br /><br />Questions, feel free to throw them in the comments.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-961369125664493552015-07-17T06:00:00.000-04:002015-07-18T14:07:06.759-04:00Friday Favorites 4It's the second week of Math Camp... that means I have a little time to post. Yay! Things are still a tad chaotic here - long days, tween drama, field trips, little sleep, etc etc, but I finally have the class I'm teaching all planned out through the end of this week. Phew! Time to write and reflect and observe some great teachers in action.<br /><br />Also I taught some 13 year old boys how to juggle yesterday. Before I signed up for that duty, I did not consider how many times I would have to say the word "balls." The first time was awkward, but then we naturally took it to a ridiculous extreme. "MALACHI! CONTROL YOUR BALLS!" (Normally I wouldn't post photos of student faces, but this one is on <a href="http://spmps.weebly.com/july-141.html" target="_blank">the program's website</a>.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-YuT_U9UDAXQ/VaZ2tvRSEBI/AAAAAAAAEtQ/9YzT38fCoLE/s1600/2015-07-14%2B15.20.47.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://1.bp.blogspot.com/-YuT_U9UDAXQ/VaZ2tvRSEBI/AAAAAAAAEtQ/9YzT38fCoLE/s320/2015-07-14%2B15.20.47.jpg" width="320" /></a></div><br />And now for some fresh faves...<br /><h4><a href="http://kalamitykat.com/2015/07/05/ifttt-improves-my-daily-blogging-habit/" target="_blank">Megan's Easy Way to Start Blogging</a></h4>Although it's really valuable professional learning for lots of people, keeping up a blog during the school year can be a daunting proposition. An on-ramp can be a 180 blog - just take and publish one photo a day from your classroom. This practice has less overhead in terms of time, but gets you in the habit of noticing things to share. <a href="http://kalamitykat.com/2015/07/05/ifttt-improves-my-daily-blogging-habit/" target="_blank">This recent post by Megan Hayes-Golding</a> suggests one way to set this up using Instagram, IFTTT, and Wordpress to make it low friction so that you are more likely to stick with it. If you're unfamiliar with those platforms, don't worry - they are all pretty easy to get started. You could have this up and running in a few hours if you're new to it (a few minutes if you're not). Also, IFTTT works with lots of different services.<br /><br /><h4><a href="https://fivetwelvethirteen.wordpress.com/2015/06/30/teaching-and-intuition/" target="_blank">Dylan Builds His Intuition</a></h4>Dylan Kane has been chronicling his growth as an early-career teacher. If you haven't been following along, you should plug into that. I really enjoyed <a href="https://fivetwelvethirteen.wordpress.com/2015/06/30/teaching-and-intuition/" target="_blank">his post</a> about the ways he has to be attentive to avoiding pitfalls of bias and developing intuition that will be productive in his practice, because they paralleled some of the things I realized along the way (although he has articulated them much better).<br /><br /><h4><a href="http://www.megcraig.org/?p=703" target="_blank">Meg Encourages MTBoS Users to Make It Work for Them</a></h4>Much as I love our spirited army of awesome, folks can get a tad dogmatic and judgey from time to time. It can be a turn-off, when you come across some strident prose that makes you feel like you're doing everything wrong. <a href="http://www.megcraig.org/?p=703" target="_blank">Meg Craig's</a> post speaks to two audiences: seekers of resources and conversations, who are reminded to stick with it and make it work for them. Also sharers of resources and initiators of conversations, who she gently offers ways to phrase your sharing so that it's a bit more inviting and inclusive. <br /><br /><h4><a href="http://blog.mrmeyer.com/2015/your-conference-session-is-the-appetizer-the-internet-is-the-main-dish/" target="_blank">Dan Meyer is going to fix NCTM for Us</a></h4><a href="http://blog.mrmeyer.com/2015/your-conference-session-is-the-appetizer-the-internet-is-the-main-dish/" target="_blank">Here's how.</a> Thanks, Dan. (Adding some clarification here because I'm afraid this sounded snarky - I'm totally sincere. I'm really excited about the prospect of NCTM taking up the recommendations of the ShadowCon organizers. I think we all of us NCTM members realize that NCTM is not working well for many members and prospective members, and I wholly support these concrete proposals.)<br /><br /><h4><a href="http://mathmamawrites.blogspot.com/2015/07/playing-with-math-can-you-write-review.html" target="_blank">Please Review Our Book</a></h4>Have you read <a href="http://amzn.to/1M6BkhW" target="_blank">Playing with Math</a>? Are you going to? (You should! It's so awesome.) It's <a href="http://amzn.to/1M6BkhW" target="_blank">on Amazon now</a>, and it would be great to get some more reviews. (Since I wrote one of the essays I'm ambivalent about writing one myself.)Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-17950000220321736662015-07-15T19:18:00.000-04:002015-07-17T12:08:35.594-04:00A Magical IncantationSo this week I'm basically the luckiest girl in the world, because <a href="https://researchinpractice.wordpress.com/" target="_blank">Ben Blum-Smith</a> is on staff at <a href="http://www.artofproblemsolving.org/spmps/about.html" target="_blank">SPMPS</a>, and he observed me teach and then we had a conversation about it. (I know. Be jealous.)<br /><br />He offered a concrete suggestion enabling student dialog which I want to share. I am pretty good at getting kids to talk to each other about math in pairs or triples...<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-t1H7J0TuEYw/Vabo9dOQP0I/AAAAAAAAEts/WU-2KQihCtY/s1600/chalktalk.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://4.bp.blogspot.com/-t1H7J0TuEYw/Vabo9dOQP0I/AAAAAAAAEts/WU-2KQihCtY/s320/chalktalk.png" width="320" /></a></div><br />but I've always struggled with conducting good conversations with the whole group -- getting kids to talk to each other about math <i>in front of everybody</i>. (Aside from the two kids in every class who always raise their hand for everything.)<br /><br />What we have been doing in this class is having everyone work out solutions to a task on the board. (Classes are small enough (7-11 for my classes), I've partaken of the vertical-non-permanent-surfaces kool aid, and kids at camp are exhausted because it's a three week slumber party, so keeping them on their feet helps with the awakeness.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-dVZR2uNC6Z4/VabpZiSL5KI/AAAAAAAAEt0/aGTgi4xONfA/s1600/2015-07-10%2B17.24.43.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://2.bp.blogspot.com/-dVZR2uNC6Z4/VabpZiSL5KI/AAAAAAAAEt0/aGTgi4xONfA/s320/2015-07-10%2B17.24.43.jpg" width="320" /></a></div><br /><br />When everyone is done-ish, we gather around someone's solution and they walk us through it.<br /><br />So let's say that a student presents her solution or approach to a task to the whole group. Generally they speak too fast, and gloss over important bits. When they are finished, I have uncovered many, many unproductive questions to ask the rest of the class:<br /><br /><ul><li>Does anyone have any questions for Bianca? (crickets)</li><li>Miguel, what do you think of Bianca's solution? ("I don't know. It's fine.")</li><li>Does anyone have anything to add? (more crickets)</li><li>Did anyone approach it a different way? (Actually I never ask this anymore, because I pick usually two students with different approaches to present.)</li></ul><br />But here is the magical incantation that can pick this lock:<br /><br /><div style="text-align: center;"><i>Hey, so-and-so, would you explain your understanding of Bianca's solution?</i></div><br />This is a lovely question. I tried it out at every available opportunity today. Interpreting another student's written and verbal solution requires all kinds of nice cognitive work. I imagine that as kids come to expect that they might be asked this question, they're more likely to be more attentive to others' explanations. And, it offers a nonthreatening invitation into the conversation where a student is immediately clear on what she's expected to say.<br /><br />Ben mentioned that he didn't really grok the power of this move until well into his classroom experience, and I think I'm kind of in the same boat. I'm sure I've heard of it before, but now I'm in a place where I can really deploy it surgically. Well I mean today I deployed it in kind of carpet bomb fashion. It's like a new toy I can't put down. But I'm going to enjoy the process of integrating it.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-10934611250252529162015-07-03T06:00:00.000-04:002015-07-04T15:36:37.608-04:00Friday Favorites 3Hey! You thought I forgot, didn't you? DIDN'T YOU?! (Excusable. That would be totally in character.) I just arrived a few days ago at the best mathematical summer thing in the world, the <a href="http://www.artofproblemsolving.org/spmps/" target="_blank">Summer Program in Mathematical Problem Solving</a>, where I'm teaching a course called Math Team Strategies. It's so great, you guys. The staff is the bomb. The campers get here tomorrow. Now for some faves:<br /><br /><h4><a href="https://docs.google.com/document/d/1G0OMlzxSLCVtbf2XpjCquRAkefwb6JMY9Eq7ouXnVUs/edit#" target="_blank">Get Your Mathematical Modeling On</a></h4>...starting here. These are ideas for data students can easily collect, organized by function type. Compiled by <a href="https://learningandphysics.wordpress.com/" target="_blank">Casey Rutherford</a>.<br /><br /><h4><a href="https://www.youtube.com/watch?v=DhSjD5nLkjY" target="_blank">Sean Sweeney's New Video</a></h4><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/DhSjD5nLkjY" width="560"></iframe><br /><br />Who doesn't love a sweet math song? Okay there are people but you have to admit that this is delightful, even if you're not a singing-in-math-class type. Also if you need to catch up on Sean's (and other members of his school's) previous works of art: here's <a href="https://www.youtube.com/watch?v=TTYKcHJyLN4" target="_blank">Graph Shop</a>, <a href="https://www.youtube.com/watch?v=-chXvU4pza4" target="_blank">f(u)</a>, and the classic <a href="https://www.youtube.com/watch?v=IyfpM-ruafo" target="_blank">Slope Rider</a>.<br /><br /><h4><a href="http://nicoraplaca.com/pd-a-math-task-for-teachers/" target="_blank">Nicora Placa's Math Tasks for Teachers</a></h4>Nicora breaks down how she chooses or adapts mathematical tasks to use for teacher learning. This one is maybe a bit specialized for folks who work with groups of teachers, but if you are looking for good PD to sign up for as a math teacher, this kind of learning has had a huge impact on my practice.<br /><br /><h4><a href="https://medium.com/@PearDeck/pear-deck-and-google-classroom-715c9b109428" target="_blank">Google Classroom + Peardeck</a></h4>If you're at a GAFE school, and you haven't checked out what Google Classroom can do in the past three months or so, you really should get on that. (Especially if you're still using Doctopus. GC is way easier.) And Pear Deck is, of course, the money. And now they're more integrated. Go make some cool shit happen.<br /><br />(My favorite use of Pear Deck is asking kids to find numbers for (<i>x</i>, <i>y</i>) that make an equation true, and then each student plots that point on the Pear Deck slide. The collective points lie along a line, or a circle, or whatever. Connection between multiple representations: made. I used that move on them like a dozen times this year and it never got old. <a href="http://mrorr-isageek.com/logarithmic-warm-up/" target="_blank">Here's a straightforward post by Jon Orr</a> showing what this can look like.)<br /><br /><br />Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-34654083800066762532015-06-19T09:21:00.003-04:002015-06-19T09:26:11.224-04:00Friday Favorites 2Hey there! Two Fridays in a row! Whaddup! Here are some things that got my attention in a good way this week:<br /><br /><h4><a href="http://emergentmath.com/2015/06/06/necessary-and-sufficient-conditions-for-school-improvement/" target="_blank">Geoff Krall's Minimal Conditions</a></h4>Geoff Krall (of <a href="http://emergentmath.com/my-problem-based-curriculum-maps/" target="_blank">PBL Curriculum Map</a> fame) gives an excellent wide-angle view of <a href="http://emergentmath.com/2015/06/06/necessary-and-sufficient-conditions-for-school-improvement/" target="_blank">practices school staff should engage in</a> when they get serious about improving instruction. My favorite thing about this is it seems so do-able. There are things small groups of teachers can start doing with the PD time that's in their control, or if that time isn't yet in their control, suggests some concrete practices to start advocating for.<br /><br /><h4><a href="https://picrust.wordpress.com/2015/06/10/tinkering-with-virtual-patty-paper/" target="_blank">Allison Krasnow's Virtual Patty Paper</a></h4><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-yGcrQRqW_dA/VYQYOL06wgI/AAAAAAAAErA/SevQZ0gABk4/s1600/ss2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="196" src="http://4.bp.blogspot.com/-yGcrQRqW_dA/VYQYOL06wgI/AAAAAAAAErA/SevQZ0gABk4/s200/ss2.png" width="200" /></a></div><br />Allison rediscovered a great <a href="http://amzn.to/1I1EnXq" target="_blank">patty paper book by Michael Serra</a>, and noticed that all of the activities could be recreated on Geogebra. I love this! It demonstrates that ways for students to tinker with ideas -- the important part -- is somewhat independent of choice of technology. Use the patty paper, create a Geogebra version, use both, or give students a choice.<br /><br /><h4><a href="https://www.teachingchannel.org/videos/illustrative-mathematics-sbac" target="_blank">What Collaborating Looks Like</a></h4>Many of us know that we should be collaborating with building colleagues on the nuts and bolts of planning and instruction, but if you've never done this before, it can be hard to imagine what it looks like. <a href="https://www.teachingchannel.org/videos/illustrative-mathematics-sbac" target="_blank">This video series</a> (a collaboration between Teaching Channel, Illustrative Mathematics, and Smarter Balanced) is a really excellent resource including teachers working in elementary, middle, and high school math before, during, and after instruction.<br /><br /><h4><a href="http://jhhs.d214.org/staff_resources/default.aspx" target="_blank">Jackie Ballarini's School's Starting Page</a></h4>Hey, if you haven't put all the stuff your new teachers need to know in one place, like this, you should! <a href="http://jhhs.d214.org/staff_resources/default.aspx" target="_blank">This page</a> was shared during a conversation initiated by <a href="https://sonatamathematique.wordpress.com/" target="_blank">Rachel </a>about supporting new teachers, and everybody drooled over it.<br /><br /><h4><a href="http://infinitesums.com/commentary/2015/youll-have-to-drag-me-out" target="_blank">Jonathan Claydon is Not Leaving</a></h4>I really enjoyed reading <a href="http://infinitesums.com/commentary/2015/youll-have-to-drag-me-out" target="_blank">Jonathan's piece</a> about why he intends to remain a classroom teacher. In this environment it's contrary to so many other articles coming out about folks throwing in the towel, and I think Jonathan shares important sentiments that usually go unarticulated, or at least don't go viral. But should.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-14447134549824247532015-06-18T09:18:00.002-04:002015-06-18T21:59:59.285-04:00Surprises in Scatter PlotsOn Derby Day, in my living room:<br /><br />"Do you think horse races have gotten faster over time, like people races?"<br /><br />"We can find out!"<br /><br /><a href="https://en.wikipedia.org/wiki/Kentucky_Derby#Winners" target="_blank">Heads to wikipedia</a>. Does <a href="https://docs.google.com/spreadsheets/d/1nY7qPT0O1XzUV7jC31RtNxv_3wUvOOXDOlGXve5eXrg/edit?usp=sharing" target="_blank">some fancy footwork in drive</a> to convert units of time from M:SS.SS to seconds. Heads to <a href="https://plot.ly/~k8nowak/3" target="_blank">plot.ly</a>.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-tLahV8_5a6g/VYLB5cJOYFI/AAAAAAAAEqY/uWWph4bNnEM/s1600/Winning%2BKentucky%2BDerby%2BTimes.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="256" src="http://1.bp.blogspot.com/-tLahV8_5a6g/VYLB5cJOYFI/AAAAAAAAEqY/uWWph4bNnEM/s400/Winning%2BKentucky%2BDerby%2BTimes.png" width="400" /></a></div><br />"Whoa, something weird happened in 1896."<br /><br />"Is that when they figured out jockeys should be tiny?"<br /><br />"Oh, <a href="https://en.wikipedia.org/wiki/Kentucky_Derby#History" target="_blank">look</a>, they made the track shorter."<br /><br />"Oohhhhhh."<br /><br />"Let's only look at times since 1896."<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-FvhAanrSmA4/VYLDdM_ldAI/AAAAAAAAEqk/tE9fIemDejc/s1600/Winning%2BKentucky%2BDerby%2BTimes%2Bsince%2B1896.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="241" src="http://4.bp.blogspot.com/-FvhAanrSmA4/VYLDdM_ldAI/AAAAAAAAEqk/tE9fIemDejc/s400/Winning%2BKentucky%2BDerby%2BTimes%2Bsince%2B1896.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">"So, yes, but it's leveling off?"</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">"Looks that way."</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">"Huh."</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">This data could be fun to build out for an activity to get kids using whatever scatterplot-creating tools you want them to use. It's also nice for interpreting plots -- it smacks you in the face that something changed in 1896, and there's a quick and satisfying explanation. Enjoy!</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-89130131241365811962015-06-12T07:49:00.002-04:002015-06-12T07:49:29.552-04:00Favorites Fridays 1Hi! Welcome to Favorites Fridays. Instead of just sharing or retweeting on Twitter, which is ephemeral and misses lots of people, I'm going to start collecting my favorite stuff from the mathematical educational Internet from the week here. I've never been one for regular publishing or weekly series-es, but we're going to give this a try. (This may be a dumb time to start this because I'm heading off on vacation next week and I promised my boi-freeeen I'd give Twitter a rest, so I'll skip a week soon but anyway.) I hope you find it useful, but this is also for my personal archival use too. Here goes!<br /><br /><h4><a href="http://recursiveprocess.com/mathprojects/" target="_blank">Dandersod's Calculus Projects</a></h4><div><blockquote class="twitter-tweet" data-cards="hidden" lang="en"><div dir="ltr" lang="en">Calculus Project Day 2! "The Math of Twitter" <a href="http://t.co/ncqFwJpE7u">http://t.co/ncqFwJpE7u</a> Benford's law and the Friendship Paradox</div>— Dan Anderson (@dandersod) <a href="https://twitter.com/dandersod/status/608264630956376064">June 9, 2015</a></blockquote>Dan Anderson (<a href="https://twitter.com/dandersod" target="_blank">@dandersod</a>) (does anyone else just think of him in their head as "dandersod?") set a project for his calculus kids, live-tweeted it, and <a href="http://recursiveprocess.com/mathprojects/" target="_blank">published their reports</a>. You might have mixed emotions about the phrase "calculus projects," but I found these to be super fun, interesting, entertaining reading.<br /><br /><h4><a href="https://teachingmathculture.wordpress.com/2015/06/09/facilitating-conversations-about-student-data/" target="_blank">Lani's Memo</a></h4><blockquote class="tr_bq">This memo focuses on research-based ideas on how to support common planning time so that it has the greatest potential for teacher learning about ambitious mathematics teaching. To that end, we provide a framework for effective conversations about mathematics teaching and learning. We develop the framework by using vignettes that show examples of stronger and weaker teacher collaboration.</blockquote>"Sometimes, you ask and the internet answers." Lani Horn came through with what <a href="https://twitter.com/JuliaTsygan/status/608260045252534272" target="_blank">Julie, and many teachers are looking for</a>: nuts and bolts direction for teachers hungry for useful professional conversations. We're tired of wasting collaboration time and "PLC time" (a now-meaningless name if there ever was one) on aimless, unhelpful activities that don't have an impact on our practice, and we know there's a better way. <a href="https://teachingmathculture.wordpress.com/2015/06/09/facilitating-conversations-about-student-data/" target="_blank">This post is going to be a huge help</a>. Bonus: <a href="https://teachingmathculture.wordpress.com/2015/04/28/making-sense-of-student-performance-data/" target="_blank">a summary on research about using student performance data.</a><br /><br /></div><h4><a href="https://vimeo.com/129522026" target="_blank">Tracy Zager's ShadowCon Talk</a></h4><div><blockquote class="twitter-tweet" data-conversation="none" lang="en"><div dir="ltr" lang="en"><a href="https://twitter.com/k8nowak">@k8nowak</a> <a href="https://twitter.com/TracyZager">@TracyZager</a> "In a class where math is taught in an authentic way, confusion is a good thing." Boom.</div>— Matt Enlow (@CmonMattTHINK) <a href="https://twitter.com/CmonMattTHINK/status/606933431608557568">June 5, 2015</a></blockquote><script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script><br /></div><div>It will blow your doors off. Tracy is dazzling. <a href="https://vimeo.com/129522026" target="_blank">Just go watch it</a>. Best use of word clouds in history.<br /><br /><h4><a href="http://www.michaelkaechele.com/how-to-build-a-pbl-culture/" target="_blank">Mike's How to Build a PBL Culture</a></h4><blockquote class="twitter-tweet" data-cards="hidden" lang="en"><div dir="ltr" lang="en">How to build a PBL Culture -my compilation of activities to start the year with students new to <a href="https://twitter.com/hashtag/PBL?src=hash">#PBL</a> <a href="http://t.co/7YRDKkyQc5">http://t.co/7YRDKkyQc5</a> <a href="https://twitter.com/hashtag/edchat?src=hash">#edchat</a></div>— Mike (@mikekaechele) <a href="https://twitter.com/mikekaechele/status/609316678510493696">June 12, 2015</a></blockquote><script async="" charset="utf-8" src="//platform.twitter.com/widgets.js"></script>Mike's PBL is Project Based, but I think this fab collection of activities and recommendations for kicking off a school year would work just as nicely if your PBL is Problem Based.<br /><br />And that's a wrap! Somebody hold me accountable for doing this next Friday! </div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-87093756092368544882015-05-07T15:24:00.000-04:002015-06-11T18:12:10.312-04:00Pretty Painless GamificationToday I was at a loss for something fun-ish to review circles in Geometry. I hastily searched my Evernote for "review games" and came across <a href="http://mathtalesfromthespring.blogspot.com/2009/10/ghosts-in-graveyard.html" target="_blank">this gem from 2009 from Kim</a>. She called it Ghosts in the Graveyard for Halloween, but since it's springtime I went with a garden theme. I modified the activity slightly.<br /><br />1. Set up a Smartboard file like so, for six groups to play. The ten objects to populate their gardens are infinitely cloned, and the fences are locked in place so they can't be accidentally moved. (This screenshot shows the "gardens" in the middle of a class.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-zI47e_jzECg/VUu4nHyaGJI/AAAAAAAAEmc/PV3QZhOzgyc/s1600/Capture.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="310" src="http://1.bp.blogspot.com/-zI47e_jzECg/VUu4nHyaGJI/AAAAAAAAEmc/PV3QZhOzgyc/s400/Capture.PNG" width="400" /></a></div><br /><br />2. Students in groups of 3-4. I wrote students' initials (in red) next to their garden.<br />3. Every student gets a copy <a href="https://www.dropbox.com/s/x2c3erjgijtgkj4/Quiz%20Review%20Questions.pdf?dl=0" target="_blank">of ten problems</a>.<br />4. When all group members understand a problem, they call me over. I randomly choose one student to explain how they did it.<br />5. If she can explain their process sufficiently, she can go up to the Smartboard and add the corresponding item to their garden. (If not, I just say okay, I'll be back in a couple minutes.)<br />6. They were instructed to use the review problems to help them study for the quiz tomorrow, so if they didn't get to all the problems in class, it was okay.<br /><br /><b>Why I liked it:</b><br /><br /><ul><li>It did not take forever to set up. I used the review questions I was planning on giving them anyway, and just had to whip up a smartboard page which took all of 5 minutes.</li><li>You wouldn't think that the state of an illustrated garden on a smartboard file would be very motivating, but they all worked diligently for the entire 30-ish minutes we did this. Thanks, Zynga.</li><li>I heard lots of good discussion as they made sure all of their group members understood a problem before calling me over.</li><li>Nobody could slack off and nobody got bored.</li></ul>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-34919352447379020262015-03-02T19:49:00.000-05:002015-06-11T18:12:46.789-04:00Kicking Some Serious Triangle BootyThe children understand that sin, cos, and tan are side ratios. The children! They understand! They are not making ridiculous mistakes, and they can answer deeper understanding questions like, "Explain why sin(11) = cos(79)." I think right triangle trig is a frequent victim of the "First ya do this, then ya do this" treatment -- where kids <a href="http://mathwithbaddrawings.com/2015/02/11/the-church-of-the-right-answer/" target="_blank">can solve problems but have no idea what is going on</a>. There's often not a ton of time for it, and it responds well to memorized procedures (in the short term). So, if your Day One of right triangle trig involves defining sine, cosine, and tangent, read on! I have a better way, and it doesn't take any longer.<br /><br />First, build on what students have already learned about similar triangles. Ideally, this unit immediately follows that one. On Day One, I assign each pair of students an angle. (You guys have 20 degrees. You all have 25. etc etc, all around the room, so each pair of students is responsible for a different angle.) They work through this document (<a href="https://www.dropbox.com/s/jn1qw86o71flq2d/Measuring%20Angles%202015.docx?dl=0" target="_blank">docx</a> <a href="https://www.dropbox.com/s/nc5p1e6nb47xgd7/Measuring%20Angles%202015.pdf?dl=0" target="_blank">pdf</a>), using <a href="http://tube.geogebra.org/student/m686897" target="_blank">Geogebra to do the measuring</a>. They write down the length of the side opposite and adjacent their angle, for triangles of five different sizes. It's important that they write down the lengths, divide them with a calculator, and experience surprise and wonder why they are all exactly the same. (Geogebra made this soooo much better and easier than when I did this with rulers and protractors. So much better. In fact, one of my Matt colleagues basically deserves a medal for all the times he's said "Why don't we just do this with Geogebra?" this year.)<br /><br />On this day, they just do opposite/adjacent ratio, share the ratio for their angle in a shared spreadsheet, and then everyone has access to the shared spreadsheet (an opp/adj-only trig table) to solve some problems (in that same document). The thing is, they are figuring out how to use what they have learned to solve the problems; they're not just repeating a procedure that was demonstrated. This took one 45-minute period, including checking Chromebooks out and in. I collect the sheets and look for students who had a strategy for #11 (how to solve when the variable is in the bottom of the ratio) so they can share their strategy next class.<br /><br />Next two classes, I provide them with <a href="https://www.dropbox.com/s/uahqj9aeq0et8mo/Table%20of%20Side%20Ratios%20-%20Google%20Sheets.pdf?dl=0" target="_blank">a table of all three ratios</a> (to tape in their notebook) for angles of 5-degree increments, and they work their way through <a href="https://www.dropbox.com/s/6qz93lbxu41t9tr/7.7%20Exploring%20ratios%20in%20right%20triangles.pdf?dl=0" target="_blank">this page</a> with appropriate help. For example, in the first set of problems, I just had them label the sides first. Then choose the ratio for all the problems, then solve for an answer. This particular document is not terribly pretty, because I had limited time to put it together. In every class, someone wanted to know why they couldn't use like hyp/adj if that equation was easier to solve. For those that asked, I pointed out that we couldn't look up hyp/adj in the table, BUT, they could use the other angle in the triangle. (And yes I'm aware they could use 1-over the ratio in the table, but that seemed like an overly complicated strategy to suggest.) I gave them a few find-sides and find-angles problems (limited to the angles in their table) to practice for homework. They did not all get to the back, but the kids to catch on/work quicker had something to do after the basic problems.<br /><br />Today I spilled the beans that these ratios have special names, and we could look them up in our calculator. We mostly <a href="https://www.dropbox.com/s/g60iwfeclqh7fsq/7.8%20Notes.pdf?dl=0" target="_blank">spent the period getting used to looking stuff up in the calculator</a> including some <a href="https://plickers.com/" target="_blank">hot Plickr action</a>, and <a href="https://www.dropbox.com/s/re1kc1d1cd32z7p/Find%20someone%20who%20can%20Bingo%20-%20Google%20Docs.pdf?dl=0" target="_blank">working on these problems</a> which they are finishing for homework. I told them they only had to do one "Explain why," but they had to complete all the rest.<br /><br />Tomorrow on our block day, we are going to go outside and figure out the heights of some really tall things (<a href="https://www.dropbox.com/s/9c3xeaf2aps4a52/Measuring%20a%20Really%20Tall%20Thing.docx?dl=0" target="_blank">docx</a> <a href="https://www.dropbox.com/s/dfycvyve71uc7yr/Measuring%20a%20Really%20Tall%20Thing.pdf?dl=0" target="_blank">pdf</a>). There are lots of "measuring tall things" activities out there, but I heavily adapted <a href="http://www.wmich.edu/engineer/ceee/edcsl/pdf/Indirect%20height.Tangents.pdf" target="_blank">this document</a>, so thanks to Christopher Conrad for posting it.Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0tag:blogger.com,1999:blog-1697471610686007730.post-86174664632238230902015-01-23T14:36:00.000-05:002015-06-11T18:13:04.261-04:00On Making Them Figure Something OutOften when I don't really know a great way to teach something, I end up defaulting to Making Them Do Something and Making Them Notice Something, and then finally Making Them Practice Something. I think lots of people do, and it's not bad. It's loads better then Telling Them Something, and it's certainly better than Children with Nothing To Do. But lots of times, the learning that comes out of MTDS and MTNS doesn't really stick that great. They can maybe do an exit ticket, but ask them a question that relies on The Thing in a week, and you just get a bunch of blank stares. So in my planning, I'm taking as a signpost Making Them Figure Something Out or MTFSO. I think the best existing curricula <i>depend</i> on MTFSO, and it must be nice to be working with one of those.<br /><br />So here's an example: the discriminant in Algebra 2, or said another way "What kind of roots does this quadratic equation have?" We're not particularly concerned in Virginia with them being able to define the word "discriminant," but they should be able to recognize whether the solutions are rational or irrational, real or non-real. Given a quadratic equation, they should be able to figure out what kind of solutions it has, and know how to describe those kinds of numbers. Here is a sample released item (<a href="http://www.doe.virginia.gov/testing/sol/released_tests/2013/algebra_2_released_in_spring_2014.pdf" target="_blank">question 50 of 50 in this document</a>).<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-G-9xtsIZ488/VMKYv-go_BI/AAAAAAAAEJI/L3L83EjhsOI/s1600/Screenshot%2B2015-01-23%2Bat%2B1.53.12%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="135" src="http://1.bp.blogspot.com/-G-9xtsIZ488/VMKYv-go_BI/AAAAAAAAEJI/L3L83EjhsOI/s1600/Screenshot%2B2015-01-23%2Bat%2B1.53.12%2BPM.png" width="400" /></a></div><br />You can find all kinds of MTDS/MTNS lessons about the discriminant. <a href="http://hchsmath.weebly.com/uploads/2/4/9/3/24930428/day_1_-_10.7_notes_key_.pdf" target="_blank">Here</a> and <a href="http://roensmath.edublogs.org/files/2013/04/MHF4U-Unit-2-6-disc-inv-spring-2013-1t1ddlg.pdf" target="_blank">here</a>, for example. But here's what I came up with to turn it into MTFSO. We had already spent a day on simplifying radicals (including with imaginary results), a couple days on solving by undoing and solving with the quadratic formula. Then we made a big map of different kinds of numbers with special attention to recognizing rationals vs irrational and real vs non-real.<br /><br />For this particular lesson, they first solved four equations using the quadratic formula: x<sup>2</sup> - 10x + 9 =0, x<sup>2</sup> - 6x + 9 = 0, x<sup>2</sup> - 7x + 9 = 0, and x<sup>2</sup> - 4x + 9 = 0. When we debriefed their solutions, we spent time describing the types of solutions, but we did not belabor the point about why the roots came out each way.<br /><br />Then, they sat in groups of 2-3 at big whiteboards and got one of these sets of questions:<br /><blockquote class="tr_bq">A1) Come up with a new, original quadratic equation whose roots are <b>real and irrational</b>. Demonstrate that your equation works by using the quadratic formula to solve it.<br />A2) Come up with a new, original quadratic equation whose roots are <b>real, rational, and unequal</b>. Demonstrate that your equation works by using the quadratic formula to solve it. </blockquote><blockquote class="tr_bq">B1) Come up with a new, original quadratic equation whose roots are <b>imaginary</b>. Demonstrate that your equation works by using the quadratic formula to solve it.<br />B2) Come up with a new, original quadratic equation whose roots are <b>real, rational, and equal</b>. Demonstrate that your equation works by using the quadratic formula to solve it.</blockquote>(In the first class to do this, I gave out the tasks haphazardly. But from that experience, learned that "rational" and "equal" are harder to find than "irrational" and "imaginary." So I adjusted accordingly for the second class. Each set above consists of one of the easier ones, and then one of the harder ones.)<br /><br />I saw different approaches... start with a desired answer and try to work backwards. Write out the quadratic formula with blank spots and repeatedly fill in and try values. Start with an equation very close to one we had just worked with and see how it worked out. And of course the bane of all group work reared its head - one kid grabs a marker and takes over while everyone else is happy to let them.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-JY5p8w51GjU/VMKgkf2xy6I/AAAAAAAAEJo/AH85VKipLFI/s1600/20150122_135425.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="http://1.bp.blogspot.com/-JY5p8w51GjU/VMKgkf2xy6I/AAAAAAAAEJo/AH85VKipLFI/s1600/20150122_135425.jpg" width="320" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-7zI66pdZ2B0/VMKgfrwe2uI/AAAAAAAAEJg/OXuE6iqClHo/s1600/20150122_135405.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://3.bp.blogspot.com/-7zI66pdZ2B0/VMKgfrwe2uI/AAAAAAAAEJg/OXuE6iqClHo/s1600/20150122_135405.jpg" width="180" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-EkIYiNLjQnU/VMKgbAVf8lI/AAAAAAAAEJY/AahC9rFHFLI/s1600/20150122_135355.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-EkIYiNLjQnU/VMKgbAVf8lI/AAAAAAAAEJY/AahC9rFHFLI/s1600/20150122_135355.jpg" width="180" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">After groups had a chance to figure stuff out and explain how they did it (it all came down to paying attention to what kind of number was under the radical, of course), we summarized with some notes:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-Lwc4AkyXM2U/VMKhpBxNk2I/AAAAAAAAEJs/MiSdO53kk8w/s1600/Screenshot%2B2015-01-23%2Bat%2B2.30.14%2BPM.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="258" src="http://2.bp.blogspot.com/-Lwc4AkyXM2U/VMKhpBxNk2I/AAAAAAAAEJs/MiSdO53kk8w/s1600/Screenshot%2B2015-01-23%2Bat%2B2.30.14%2BPM.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: left;">And that is that. A general understanding of what was going on persisted to the next day... We'll see how Monday goes.</div>Kate Nowakhttps://plus.google.com/116597620145081274111noreply@blogger.com0