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!

Thanks for sharing!

ReplyDeletecheesemonkey

ReplyDeleteThanks for the detailed post - especially the warnings about how the link pops up on page 6. Most of our Algebra I is taught in our middle school and currently that group of teachers is wrestling with a developing BYOD program where kids have a variety of devises in their hands. iPads, MacBooks (pro and air), and a handful of PCs as well. Would you encourage this NVLM link in such a setting? I worry about the amount of time spent just walking kids through the various java warnings and such. I will definitely encourage my team to sift through the Exeter problem sets for inspiration.

Hi Mr. Dardy,

DeleteMy feeling is that there will always be SOME glitch or other during instruction, and this is no different. I just consider it an investment, because once you get them through it the first time, each subsequent usage of NLVM gets easier for them. I've found that using physical algebra tiles has its own glitches, so it's really no different.

Given how much the kids like to be able to explore these kinds of things on the computer, I consider it worthwhile anyway. It gets them much more bought into the process and can be used moving forward.

Thanks for commenting!

- Elizabeth (@cheesemonkeysf)

I put most of the Exeter Algebra I problems into an Excel database to better see how different strands develop over the year. Spreadsheet is here:

ReplyDeletehttps://docs.google.com/file/d/0B0LlvF7Dr9cheE9qekVnVmdpYzg/edit?usp=sharing

Very good post about the branch of mathematics.

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