Chords and Secants and Tangents, oh my!

Once you get to circles, chords, tangents, secants, arcs, and angles in Geometry, it's all just a nightmare of things to memorize — which means things to forget or confuse.

Rather than have students memorize this mishmash of formulas, I decided to have them focus on recognizing situations and relationships. And a good way to do that is with a four-door foldable.

Make it and use it.

Humans are tool-using animals. So let's do this thing!

Doors #1, 2, 3, and 4 each have only a diagram on their front door with color-coding because, as my brilliant colleague Alex Wilson always says, "Color tells the story."

Inside the door, we wrote as little as we possibly could. At the top we wrote our own description of the situation depicted on the door of the foldable. What's the situation? Intersecting chords. What is significant? Angle measures and arc lengths. What's the relationship? Color-code the formula.

Move on to Door #2.

We did the same thing with the chord, secant, and tangent segment theorems.

These two foldables are the only tools students can use (plus a calculator) for all of the investigations we have done with this section. They are learning to use it like a field guide — a field guide to angles and arcs, chords, secants, tangents. I hear conversation snippets like, "No, that can't be right because this is an intersecting secant and tangent situation."

My favorite is the "two tangents from an external point" set-up, which we have dubbed the "party hat situation" in which n = m (all tangents from an external point being congruent).

A big part of the success of this activity seems to come from teaching students how to make tools and to advocate for their own learning. Help them make their tools, set them loose, and get out of their way.

That's a pretty good strategy for teaching anything, I think.

New strategy for introducing INBs: complex instruction approach

After months of not feeling like my best teacher self in the classroom, I got fed up and spent all weekend tearing stuff down and rebuilding from the ground up.

INBs are something I know well — something that work for students. So I decided to take what I had available and, as Sam would say, turn what I DON'T know into what I DO know. Love those Calculus mottos.

So I rebuilt my version of the exponential functions unit in terms of INBs. But that meant, I would have to introduce INBs.

As one girl said, "New marking period, new me!" The kids just went with it and really took to it.

Here is what I did.

ON EACH GROUP TABLE: I placed a sample INB that began with a single-sheet Table of Contents (p. 1), an Exponential Functions pocket page (p. 3), and had pages numbered through page 7. There were TOC sheets and glue sticks on the table.

SMART BOARD: on the projector, I put a countdown timer (set for 15 minutes) and an agenda slide that said,

• New seats!
• Choose a notebook! Good colors still available!
• Make your notebook look like the sample notebook on your table 

As soon as the bell rang, I hit Start on the timer, which counted down like a bomb in a James Bond movie.

Alfred Hitchcock once said, if you want to create suspense, place a ticking time bomb under a card table at which four people are playing bridge. This seemed like good advice for introducing INBs to my students.

I think because it was a familiar, group work task approach to an unfamiliar problem, all the kids simply went went with it. "How did you make the pocket? Do you fold it this way? Where does the table of contents go? What does 'TOC' mean? What goes on page 5?" And so on and so on.

I circulated, taking attendance and making notes about participation. When students would ask me a question about how to do something, I would ask them first, "Is this a group question?" If not, they knew what was going to happen. If it was, I was happy to help them get unstuck.

Then came the acid test: the actual note-taking.

I was concerned, but they were riveted. They felt a lot more ownership over their own learning process.

There are still plenty of groupworthy tasks coming up, but at least now they have a container for their notes and reflection process.

I'm going to do a "Five Things" reflection (trace your hand on a RHS page and write down five important things from the day's lesson or group work) and notes for a "Four Summary Statements" poster, but I finally feel like I have a framework to help kids organize their learning.

I've even created a web site with links to photos of my master INB in case they miss class and need to copy the notes. Here's a link to the Box.com photo files, along with a picture of page 5:

We only got through half as much as I wanted us to get through, but they were amazed at how many notes we had in such a small and convenient space.

It feels good to be back!

Building concept maps is harder than it looks

I'm having students create a concept map as a summative assessment for our Complex Numbers unit and... w o w — there is all kinds of learning going on.

 

Students are working in groups and can use all their notes and assignments from the unit. Some kids jumped right in and started hacking away. Others whined and asked why we couldn't just have a normal test.

We are using Post-Its, scissors, pencils, and paper to do our constructions.

In-process projects range from amazing to struggling, but what impresses me most is how much the work reveals about what students are figuring out and how each student is understanding and constructing meaning in their learning. It also demands that learners own their own learning.

Because this is so revealing, I am probably going to use concept maps both as formative assessments before and during the unit as well as using them as a 'ways of understanding' tool to help them consolidate their learning.

BREAKING: OK, this activity is definitely a keeper. Students are really digesting their learning, talking about it, debating how to represent it, and clarifying areas of confusion for themselves. Here is an outstanding example from today:

Because it always pays to follow Sam around like a duckling #Made4Math

Last week of school before the break — and it's FINALS WEEK.

Because I'm so late to the game and don't really know what my kids have or haven't learned over the semester, all of the Algebra 2 teachers gave our classes two periods to work on the final. Last Thursday/Friday (our second block period of the week), they had the whole period to work on the exam. Then this week (our actual Finals Week), they'll have their whole block period to work on test corrections/finishing — using my markings as a guide.

So I've got about 150 finals to score preliminarily, which is why it made SO MUCH SENSE for me to use much of this precious Sunday to make my own version of Sam's amazing personalized planner.

Here is Sam's planner:

And here is my version:

I think this even counts as a #Made4Math entry, although it should probably be listed as a #MadeInsteadOfMath submission. :)

Using Exeter problems as an intro to algebra tiles

We are an Apple 1-1 school, so I am always happy to figure out lessons my students can use their laptops for.

I also like to use manipulatives in Algebra 1. It's not easy to get all students to accept the need to use multiple representations (such as an area model), but they help enormously to extend kids' conceptual understanding of the distributive property — plus they make a return appearance a lot when we get to the Festival of Factoring in the late winter.

So the National Library of Virtual Manipulatives seemed like a natural fit. But what problems to use to introduce them?

Enter the Exeter Math 1 problem sets.

I have been using the Exeter problems with my advanced 8th grade students taking Algebra 1 almost every week during our Problem-Solving Workshops on block days. Each page is a self-contained "problem set" that builds from simplest principles and often loops back on itself later in the page. This gives students a chance to give themselves a pat on the back for having discovered and developed an intuition for activating their own prior knowledge. I then have them write up one of the problems they solved as a problem of the week to give them practice in blending symbolic and graphical representations with verbal representations (don't forget the verbal representations!). So much Common Core math in such a small span of time!

I will write more about using the Exeter problems as a resource for long-form problem-wrestling with my students, but here I just want to talk about the specifics of introducing algebra tiles.

One of the features of the Exeter problems that we do not get to take much advantage of is the way they build page over page. They will introduce part of a concept or skill in a problem on page 5, say, then introduce the next part of the concept in a problem on, say, page 8. This makes so much sense if you are teaching using one page per day and working through all of them. But for those of us who dip in and nick out once a week, that isn't really possible.

But... by the time we got to around page 21, it dawned on me that I could collect the five problems they use that introduce algebra tiles and put them on a single sheet of paper.

Then I could give that to students during Problem-Solving Workshop, along with a quick intro to the NLVM, to let them teach themselves how to use algebra tiles!

So that is what I did this week.  :)

A couple of programming notes if you want to try this yourself with your students:
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1. Start at the very beginning with NLVM and PREPARE FOR TECH HICCUPS
NLVM can be extremely persnickety. This is probably due to some perverse desire to help us cultivate CC standards of mathematical practice #1. Encourage yourself and others to persevere.

Your network may have restrictions on how students can use Java-enabled apps on school equipment. We had some hiccups getting NLVM to run on everybody's system at first. Firefox seems to be the most reliable browser for NLVM. Also, you need to have the most up-to-date version of Java on the student's system.

On our network and systems, students can only update Java by logging out and in again or by restarting their computers. No matter how many times I explained this, some kids still didn't quite figure it out. So much for being "digital natives." Plan to go around to each kid the first time to help them get their systems up and running.

Our system throws up a modal "Security Warning" dialog that forces you to check "I accept the risk" and "Run" before NLVM will load in the browser window. Again, a minor pain in the butt, but you do need to make sure that every kid gets through the security gauntlet to use the system.

Refresh the browser window if need be and be patient which Java and the applet cooperate in loading.
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2. Get everybody to the *FIRST* page of the Algebra Tiles site on NLVM
For reasons that pass my understanding, NLVM dumps you into the sixth page of the algebra tiles site (the activity panel on the right, which loads as "Multiplying Binomials - 1").

You need to have students click the leftward-ho button at the top of this right-hand panel SIX TIMES to get back to the first page, which is called "Distributive Law - 1."

This is stupid but necessary because on the first two pages of this site, you can do things you need at the beginning that will quickly drop away as students gain fluency.

For example, the Distributive Law pages are the only ones where you can easily represent both multiplication over addition AND an area addition model in the same window.

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3. Familiarize yourself with the syntax of the NLVM Algebra Tiles pages
You'll need to tinker with this a bit, to get comfortable with the syntax of the applet, but there are two essential features of the Algebra Tiles distributive property pages:

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4. Click to CREATE tiles in the workspace; drag to MOVE tiles
You can create instances of any of the area blocks that are possible by CLICKING them in the menu bar along the bottom of the workspace. When you click the "x" button, for example, NOTICE that NLVM creates a single instance of a 1-by-x rectangle in the workspace. You can create as many "instantiations" of any of these blocks as you need for any expression you want to represent.

NOTICE that you can drag these critters around in the workspace and add them up, like LEGOs. Or you can drag them into the x-axis tray or the y-axis tray to represent lengths and widths of various area blocks of multiplication.

ALSO NOTICE that you can mouse over the corner of a block in the workspace to rotate it into the position you need.

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5. How to show multiplication over addition (i.e.,  how to show  x (y+2) :
The x- and y-axis are basically x- and y-axis "trays" that students can drag tiles into. Drag a 1-by-x tile into the y-axis tray and it creates an x coefficient. Drag a 1-b-y tile and two unit blocks into the x-axis tray and they become the quantity in parenthesis over which your x coefficient will drape itself in multiplication.

NOTICE that as blocks snap into place in the second axis tray you fill, a red area outline appears in the main workspace between the x-axis tray and the y-axis tray.

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6. How students can confirm for themselves that area addition and multiplication over addition produce equivalent area values (i.e.,  how to show that  x (y+2) = xy + 2x :
In this window, students can create blocks to fill in this red outline and verify for themselves that the area they get using the distributive property is equivalent to the area they can get using the area addition postulate approach.

Have students click to create blocks and then drag them around to fill the red outlined area perfectly.
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MATERIALS

My bastardized worksheet of the five Exeter Math 1 problems that introduce algebra tiles and an area model can be found here on the Math Teacher's Wiki.

NLVM Distributive Property pages are here and 6 pages to the left.

Enjoy!

Intro to Quadratics — from "drab" to "fab" (or at least, to something less drab)

Recently, I created a new anchor lesson for my Algebra 1 quadratics unit. I found that, while I really liked the sequencing of activities and questioning in the NCTM Illuminations lesson on "Patterns and Functions," I found their situation and set-up simultaneously boring, contrived, and inane.

Actual photograph of San Francisco monkeys

hosting a tea party in the wild

As is so often the case, I find that a certain, judicious sprinkling of silliness and fun in the set-up can really liven up the lesson. A certain amount of contrivance is necessary in many activities, even those that are based on "real-world situations." So why not stretch the real world to make it conform to the needs of my algebra students?

The Made To Stick elements are all here: multiple access points are provided through manipulatives, storytelling, and humor.

My student investigation sheet, Table for Eighteen... Monkeys is available on Box.com. A PDF of the Table Tiles master is available here on Box.com
here.

Tiny plastic monkeys sold separately. :)

UPDATE: Worksheets now also on the Math Teacher's Wiki, at http://msmathwiki.pbworks.com/w/page/55614036/Algebra%201#view=page

Go graph yourself!

Yesterday I used masking tape to turn the floor of my classroom into a coordinate plane. 

Students had to graph themselves, then find the slope of the line between themselves and various other points in the room. A good time was had by all, and a few insights were had.

Today I think we will also graph all the bits of trash that usually get left on the floor by lunch time. That will give us time to set up for a fierce game of Coordinate Plane Battleship.

Oh, the things we do to promote a deeper conceptual understanding! :)

Life on the Number Line - board game for real numbers #made4math

UPDATE: Here is a working link to the zip file: https://drive.google.com/open?id=0B8XS5HkHe5eNNy10MWZVSDNKNnc

Last year I blogged about my work on a Number Sense Boot Camp, so I won't rehash all of that here. This year I want to give the follow-up on how I used it last year, what I learned, and how I'm going to use it this year in Algebra 1.

This was my breakthrough unit last year with my students. It anchored our entire Chapter 2 - Real Numbers unit and really solidified both conceptual understanding and procedural fluency in working with real numbers, the real number line, operations on real numbers, and both talking and writing about working with real numbers. We named it Life on the Number Line.

Here's how the actual gameboards, cards, and blank worksheets looks in action (sans students):

I sure hope I didn't make a bonehead mistake in my example problem!

The most effective thing about this activity was that it compressed a great deal of different dimensions of learning into the same activity, requiring learners to work simultaneously with the same material in multiple dimensions. So for example, they had to think about positive and negative numbers directionally in addition to using them computationally. They had to translate from words into math and then calculate (and sometimes reason) their way to a conclusion. They had to represent ideas in visual, verbal, and oral ways. And they had to check their own work to confirm whether or not they could move on, as no external answer key was provided.

Since they played Life on the Number Line for multiple days in groups of three or four players comprising a team who were "competing" in our class standings, learners felt that the game gave them an enormous amount of practice in a very short amount of time. Students also said afterwards that they had liked this activity because it helped them feel very confident about working with the number line and with negative numbers in different contexts.

I also introduced the idea of working toward extra credit as a form of "self-investment" with this game. For each team that completed and checked some large number of problems, I allowed them to earn five extra-credit points that they could "bank" toward the upcoming chapter test. Everyone had to work every problem, and I collected worksheets each day to confirm the work done and the class standings.

What I loved about this idea was that students won either way — either they had the security blanket of knowing they could screw up a test question without it signifying the end of the world, or they got so much practice during class activities that they didn't end up actually needing the five extra credit points!

Students reported that they felt this system gave them an added incentive to find their own intrinsic motivation in playing the game at each new level because it gave them feelings of autonomy, mastery, and purpose in their practice work.

The game boards were beautifully laminated by our fabulous office aide but do not have to be mounted or laminated. The generic/blank worksheets gave students (and me) a clear way of tracking and analyzing their work. And the game cards progressed each day to present a new set of tasks and challenges.

All of these materials are now also posted on the Math Teacher Wiki.

Let me know how these work for you!

UPDATE 10/27/2016: Here is a working link to a zip file of all the components for this: https://drive.google.com/open?id=0B8XS5HkHe5eNNy10MWZVSDNKNnc

#made4math | Words into Math - Taming Troublesome Phrases with an interactive foldable translator

It's been busy here in the Intergalactic Cheesemonkeysf R&D Laboratories
(see trusty assistant hard at work, right). Ever since Twitter Math Camp 12, I've been working on implementing all the lessons and activities I learned about in person from my fabulous math teacher tweeps!

I'm using the Interactive Notebook structure that Megan Golding-Hayes showed us, and I'm also incorporating a lot of Julie Reulbach's foldables. The most helpful insight (out of many) I received from Julie was the idea of using a foldable as a way of getting kids to SLOW DOWN and trust the steps of the process as they're working on word problems. So I've made a nifty little foldable like hers that will go into an INB pocket the first week and will be usable on all quizzes and tests.

One of the reasons I like having students develop tools they can use on tests is that many of the discouraged math learners just don't trust their own learning. They have a habit of "collapsing" when they encounter a first speed bump. So from the perspective of encouraging students' courage in problem-solving, it is good to allow them to have tools they can use, even if the tools are sometimes nothing more than a security blanket — a talisman or a good-luck charm they can touch as a tangible reminder of their own courage and resourcefulness. So a four-step problem-solving foldable serves double duty: it acts both as a checklist (as in Atul Gawande's New Yorker piece and book) and as a reminder to have courage and perseverance in working through problems.

However many students have a habit of either not using the tools or finding the tools too complicated or frustrating. Nowhere has this been more evident than when I've given them approved lists of words and phrases they should stop, consider, and look up if need be. The charts and lists seem to turn into giant floating word clouds that signify nothing. So I wanted to come up with a slightly more interactive than usual foldable that students could use as a way of isolating and decoding some of the most troublesome words and phrases they get hung up on. Not only does it slow them down, it gives them a focal task that redirects an anxious mind.

After a lot of research on both blogs and on Pinterest ("PINTEREST!" #drinkinggame), I came up with the idea of a folded sleeve with a sliding chart insert, containing the phrases that often confuse kids or cause them to second-guess their translations from words into math. Here's what the finished product looks like:

Here is a close-up:

I used OmniGraffle to make the sleeve template and I used Pages, Preview, and Adobe Acrobat to make the insert. I'm linking to the Troublesome Phrase Translator sleeve, a generic sleeve you can customize for your own fiendish purposes, and a PDF of my exact insert (Troublesome Phrase Translator INSERT). 

If you want to make your own inserts, you'll need to set up your own table (Word, Pages, Excel, etc) making sure that your row height is exactly 1/4 inch. Your LHS cells should be 1 9/16" wide and your RHS cells should be 1/2 inch wide. You can have about 19 or 20 rows, depending on what you put in them.

Sometimes a little magical thinking is just the thing to displace a discouraged learner's anxiety (or freaked-out-ness) for that extra second it might take to recommit to the process of solving a problem. If that helps me hang onto just one extra student a day, it's a win. But usually I find that a tool like this will encourage multiple students to encourage each other's confidence as well, which is an even bigger win in my book!