Some fun(ish) worksheets

I'm going to try to get my box.net materials updated over this coming week.  In the meantime, here are a couple of decent worksheets that you may find helpful.

First, I made one to practice graphing standard form - I just ripped off Mr. K's idea.  Thanks!  And some of my students actually liked the joke (I googled Laffy Taffy jokes).

For tomorrow, students will be graphing systems of inequalities, so I decided to create a little Ohio Jones adventure (Indiana's lesser known brother).  Here is the full lesson and just the activity in pdf form. 

(UPDATE: Here is the follow-up lesson in word form - Ohio Jones and the Pyramid of Power.  Here is the follow-up lesson in pdf form if you're having trouble seeing the word doc).

Here is what the maze should look like after being solved:

Algebra 1: Systems of Equations

We are finally getting to move beyond basic graphing and finding equations of lines.  It was a long slog, but the skills tests show that the majority of my students are starting to get the hang of it.  I always look forward to the systems of equations unit, because it is a chance for students to synthesize what they have been learning all year - and, in a situated context, no less.  My plan this year is to deepen the emphasis on representational fluency and summarizing, to help build all of those neural bridges we want the students to have.  We started the unit Monday, and I was really blown away by my classes today - all of a sudden, I have students doing algebra!  I had them solving systems in pairs, using mini-whiteboards, where one does the graphical solution and the other does the algebraic solution, and then they compare their answers.  They did a great job, and it wasn't until this activity that many students realized the answers should be the same.  I got a couple of those hilarious, indignant "you should have told us!" comments.  Next week is winter break, which doesn't come a moment too soon; however, I'm worried about how much will be lost over the seven days that nobody is asking them about starting points or rates of change.  No matter, it's worth it to have a rest.  Here are a couple of  examples of what we're doing, and the links to the lesson materials thus far.

Lesson 1 (Intro to Systems of Equations)  doc / GeoGebra files / Keynote / Powerpoint
Lesson 2 (Solving y = mx + b Systems)  doc / Keynote / Powerpoint
Lesson 3 (Practice Solving Systems)  doc

Language and Retention of Math Concepts

I've been thinking lately that one of the reasons my students have such difficulty with long-term retention of mathematical concepts is due to the small number of times I ask them to thoroughly summarize what they have learned.  They do lots of problems, but the language of the problems often does not enter into their brains.  As we learned in Orwell's 1984, without language, there is no thought.  So I am going to start providing more explicit opportunities for the students to summarize and discuss what we are doing in class.

Comic Strips  (Unit 5, Lesson 9:  doc / keynote / powerpoint)
Quite a few students are still struggling with graphing lines.  They know the general process, but don't pay attention to the details - is the slope positive or negative; if a term is missing, is it the slope or the y-intercept, and how does that change the graph?  So, I had all students draw comic strips to summarize the process in these different cases.  I like how this went, but I definitely did not provide them with enough time to do all I asked.  Here are a few good examples.  The first didn't scan that well, but he did an awesome job.

Think-Pair-Share  (Unit 5, Lesson 11: doc / keynote / powerpoint)
This is a tool that our humanities classes tend to use a lot.  I got some advice from them, and will be trying these periodically during the next couple of units.  We did one so far, and it went reasonably well for a first try.  Students need a lot of practice both writing down their ideas and sharing them out.  Here is the handout I gave (it was used immediately after doing a Do Now problem of the type described).

Algebra 1: Representations of Linear Equations

Increasing my students' representational fluency has been something I've been working on for a while.  Our second semester started last Monday, and to start my Algebra 1 students off easy, I had them do a four-fold poster of a linear relationship to review what we did last semester: situation, equation, table, and graph.  They did the work fine overall, but quite a few students had more troubling questions than I had expected (i.e. "how do you make a table?").   I guess it just shows that we have to keep going through these different representations and their connections again and again.

We have started the new unit - working with linear equations - in which students have to write the equation of a line given its slope and a point, or two points, or a point and a parallel line.  In the past, I have done this only algebraically (except for the initial explanation of concepts); this time around, the students will have to practice the problems both algebraically and graphically.  And, more importantly, the skills tests will require them to show mastery with both methods.  Let's build those connections!

Here are the first few lessons in the unit.

Lesson 1 (Representations of Linear Functions)
Lesson 2 (Graphing Practice)
Lesson 3 (Write the Equation of a Line)
Lesson 3: Keynote / Powerpoint

And some snippets from the worksheets to illustrate what I am talking about:

 

Introducing Linear Inequalities

To show that a line is a representation of an infinite number of points, I like to give my algebra 1 classes an equation, like y = 2x + 3, and then give each student a couple of different ordered pairs - some that are solutions and some that aren't.  I have them each work out their points, and then go to the board to plot an open or closed circle, depending.  Once all the students sit back down, we look for patterns and see that all the closed circles fell on a straight line.  Discuss, and voila.

This extends nicely to linear inequalities (and systems of equations and inequalities).  On Tuesday, my algebra 2 students were reviewing linear inequalities so I did this activity with them.  I really like it, because it is engaging, and it helps build a mental picture that they can rely on later on when they are struggling through graphing problems on their own.  My students often get stuck on the "pick a test point" part of the process; but now, I ask them if they would have plotted a closed or open circle based on their result, and to think about what the picture on the board looked like.  This usually helps them see which side of the boundary line to shade, and to be able to explain why.

Here is what the board looks like after students plotted their points:

Then, we looked for patterns.  Usually, a student will come to the board and draw some sort of line after getting frustrated with trying to explain it in words.  Then I reveal the shading:

And there are usually some audible "ahhs" and such.  Another great benefit of this is that the string of open circles on the boundary helps students see what the dotted line is all about, and why changing the inequality to include an equals sign would create a solid line - a string of closed circles.

Here is my lesson that goes with this.  And the keynote.

Algebra 1: Graphing Lines Practice

I just used this worksheet from Mr. K for the first time the other day.  I thought it had a pretty cool setup, but I didn't realize just how effective it would be until I used it in my first class.  The "solve the joke" aspect of it helps draw them in, but the hidden beauty is in its self-checking properties.  Since each line must pass through exactly one number and one letter, a line that doesn't do this must be graphed incorrectly.  Students started realizing this and would go back and find mistakes without having to check with an answer key.  The only bad part (sorry to say) is that they had absolutely no idea what the answer was supposed to mean (see earlier post).

I made up a "balloon pop" homework to go with this that was inspired by Green Globs.  I wish I had the tech access for my students play that game.

Algebra 1: Skills List - Spring Semester

I spent a good deal of time right before break trying to figure out exactly how far I can push my students for the second semester of Algebra 1.   These skill items will be broken down into chunks for the skills tests, and MC-ized for the benchmarks and final exam.  I regret how many concepts I had to leave out due to time pressures; and still, the list seems daunting and endless.

If you're interested, this is what my students will be doing over the coming months.

doc / pdf

Algebra 1: Situation Graphing

I learned a few years back that jumping right into graphing slope-intercept equations never worked. This is one of those concepts that, before I became a math teacher, I never would have guessed would be so hard for students to master. Start at the y-intercept, use the rate of change to plot the next point, and you're done - right? Yeah, not really. So after a couple of years of teaching, reteaching, re-reteaching, and tearing my hair out, I decided to try some other things. Eventually, I realized that a ton of scaffolding of the concept of slope was needed, along with firmly rooting linear functions in situated contexts.

One of the constant problem areas is deciding which way to draw the line for a negative slope. To graph something like y = -(2/3)x + 5, students would often move down 2 and left 3. My old attempts at correcting this focused only on the mathematical explanation: -(2/3) = -2/3 = 2/-3. So, you either go down 2 and right 3, or up 2 and left 3. If you go down 2 and left 3, that means -2/-3, which is 2/3. This is a perfectly reasonable way to explain it, but it didn't really provide much of a lifeline to my lower-skilled students, as it hinges on mastery of the division rules of signs, as well as remembering that a fraction also represents a division problem.

The other common problem was for students to correctly identify the starting point number, but to plot it on the x-axis instead of the y-axis.

The way I run the unit now is to provide numerous opportunities to graph and describe situations, both with and without numbers, in just the first quadrant of the coordinate plane. Distance, income, height, and so on. The quantity being measured is always on the vertical axis, and the horizontal axis always represents time. When we eventually generalize to y = mx + b equations on the full coordinate plane, I use the situated contexts as memory anchors. If a student doesn't remember where to start, I say something like, "Where do we show that the Hare got a two foot head start? On the feet axis or on the seconds axis?" In these situations, a positive rate of change always means "moving up" and a negative rate of change always means "moving down", while time is always passing to the right. This is a much more helpful way for my students to think about how to graph their decontextualized lines. Suddenly, there is a reason for the direction the line is moving in, instead of just a sign rule.

Another benefit to this approach is that my students are now a lot more flexible with the form of the equations. My situated equations typically are in the form y = b + mx, which seems like a more natural connection to the preferred method for graphing. Once they grasp that the number without the variable is always the starting point, then they can handle both y = b + mx and y = mx + b relatively interchangeably. Also, it really helps them to understand the difference between equations like y = 2 and y = 2x. The first shows a starting point of 2, with zero rate of change. What does it look like on a graph if someone is not moving, but time is still passing? Exactly - a straight line! (I'm still working on that one - even my highest skilled students still say straight when they mean horizontal. My "all lines are straight" response doesn't usually clarify the way I'd like it to.) And in the second, the rate of change is 2. Ahh, it's like a graph of someone running 2 feet per second... but where did he start from? Zero? Where is that?

This approach takes a ton more time, of course, but I can't see any way around it for my students. I hope that I am providing them with a long-lasting ability to understand and graph linear functions. The semester is ending this week (final exams start tomorrow!), so the test will be to see how much they recall in January, when we move into the next unit. We'll be doing those oh-so-fun problems where you give them a point and a slope (or a parallel line and a point, or two points) and they have to give you the equation of the line. I'm going to experiment with doing every problem both graphically and algebraically (both in instruction and on assessments) to see if the focus on multiple representations helps them master these problems. I'll post more about that in late January (after I develop and teach it!).

My Slope and Graphing Linear Functions unit (Unit 4) is pretty much up-to-date in my box.com widget in the side bar. Here are a couple of examples (in pdf format) of the kinds of activities that they were doing. The Word and Keynote files are all in my box. I'd love to get feedback on any of this.

Lesson 10: Practice graphing with tables
Lesson 11: Learning to graph without tables

Review game: Trashketball

I know that many teachers out there play some form of Trashketball, so this isn't really groundbreaking. However, I always have problems with these kinds of review games. Structuring them so that the higher-skilled students don't dominate or pressure the other students can be quite difficult. Or, looking at it the other way, there are plenty of lower-skilled students who are happy to sit back and let others on their team get the work done for them.

I developed Tic Tac Toe Battle Royale a couple years ago which addresses some of these concerns pretty well. But you can only do the same game so many times. My experiments with Trashketball in the past haven't been that successful, and so I thought about how I could improve it to work more effectively in my class. This is what I came up with:

1. Break students into groups of 3 or 4 - for me, this yields no more than 6 groups in my Algebra 1 classes. Give each group a letter, and each person in the group a number. Write these in a grid on the board. (If there is an unfilled spot in a group, that spot becomes a wild card - any person can take that number.)
   
2. For each round, create 6 separate problems that all target the same concept, but that are slightly different. This prevents the copying problem found in board races.
3. Hand out a template for doing the work on. My freshmen need an organizer for everything. "Get out a sheet of paper" just doesn't fly.
4. Show the 6 versions of the problem, giving the class enough time to get it done.
5. Call for silence. Block the projector. Randomly (or not) call a number between 1 and 4. The student in each group with that number comes to the board - all 6 at once. Have the board sectioned off so they know where to write. They are allowed to bring their own graphic organizer up with them, but no one on the team may offer help at this point. The idea here, of course, is that students must make sure that all group members have done the work. Students who tend to slack off have to at least write down the work that others in their group are doing. Not ideal, but it's better than spacing out.
6. Have the trashketball basket set up. As students complete their work on the board, tell them if they are right or not (make sure to have answer keys ready!). Right answers get a point, and they get to take a shot for a bonus point. There is less waiting around time this way - some students will still be writing their problems as others are already lining up to shoot.
7. Record the scores and move on. Winning team gets a whatever.

How it looks:


I did this for the first time today, and was amazed by how well they did. There were only 2 students in the class that I couldn't get totally engaged. The rest did all their work, were excited to take their shots, and so on. It takes longer to make this activity due to the multiple problems, but it was really worth it. Man, do they love tossing paper balls into the recycle bin.

I know it kind of breaks my respect class norm, but it really warms my heart to hear a kid (who I can usually barely get to sit down, and who really wanted to win) say to his teammate who hadn't done his work on the board carefully: "Fool! I told you it was negative eleven!"

Trashketball Problems (Keynote) (Powerpoint)
Answer Template (Word)

XKCD based lesson: The Coordinate Plane

Ever since I first saw this xkcd cartoon, I wanted to use it in a lesson. I finally put that together this year. I used the cartoon as a way to help convey the idea that points on a coordinate plane are a way to easily visualize the relationship between two different variables. The purpose of the numbers is simply to quantify those relationships, if such a quantification is necessary. I then had students make their own graphs for homework, with variables of their choice. If I had more time to spare, it would have been nice to do this in class (and the outcome would have been better, I think).

This lesson (Unit 4, Lesson 5) and others can be found in my box.com widget to the left. I recently updated Algebra 1, Units 3 and 4.

Here are some examples of the students' work.

Algebra 1: Solving Equations Puzzle

Here is a puzzle activity for reviewing equation solving. I found that it worked better when I made an answer mat for students to put their pieces onto (I indicated a couple of pieces on the mat to help them align the rest of their pieces).

Here are two files in Pages and Word that you can work from to make your own.

Edit:
A comment from David Wees in a previous post with a similar puzzle I did for quadratics:

Yeah your puzzle is cool. So cool that I've created a random generator in Adobe Flex.

See my algebra puzzle generator.

Awesome!

Edit 2:
There is an app called Formulator Tarsia that will do this, but it only works for Windows (which I don't have access to) so I haven't tried it out. Give it a try!

Putting students in control of their learning

In the last couple of years, I've worked to really clarify exactly what skills I expect my students to learn. The assessment system makes it crystal clear what skills students know and don't know. And then I realized: Oh wait - it's only crystal clear to me. Students focus on their test scores, and come in to retake and improve tests, but they really don't think about what mathematical content they need to develop - only what test number they need to retake. I still have a few students who insist on retaking skills tests even though they haven't done any work to learn the skills that they got wrong the first time. Even when this fails to produce the results they want, they still resist actually working with me to learn the skill.

I think that helping students really understand what the individual skills consist of, and what their personal ability level is on each skill, is really the next step. I want students to understand the connection between their level of numeracy and their success in mastering algebraic concepts. I also want students to make connections between their behaviors in class and their growth (or lack of growth) in the lesson's objectives. Finally, I want to provide students with greater differentiation so that all students can both feel challenged and successful.

So, I put all of that together into a new plan for beginning and ending class. Students will start class with a 10 minute Do Now that has three parts. Part 1 is a Numeracy Skill Builder that targets a specific elementary math concept that is either key to the specific lesson, or something that students have been struggling with. Part 2 consists of one or two algebra concepts that are the lesson objectives. These are broken into basic, proficient, and advanced levels. The proficient level is the form in which the concept will be tested on a skills test. Students are told to solve only one problem in each concept, at the level they feel most comfortable at. Part 3 is a multiple choice test prep question. The purpose of this is obvious, as we need to get students ready for state tests, ACTs, placement tests, and so on.

Students have 10 minutes to complete these problems individually and silently. No helping is permitted here (in general), because the purpose is for students to really get a sense of what they know at the beginning of class on their own. At the end of the 10 minutes, I show the answers so students can see how they did, but we don't spend time actually reviewing these specific problems. I quickly collect the papers.

We have the lesson. Ok.

Now, in the last 5 - 7 minutes, I hand back the papers. On the back, students complete the Exit Slip / Reflection. They are supposed to go back to the Do Now problems, pick one algebra concept, and try a higher level problem. The idea is for them to see how much they can improve in an objective over the course of the class period. So, even if they are only able to accomplish the basic level (when they couldn't before), they can see growth in themselves and feel good about that. Students who already could do the advanced concepts at the beginning of the class have a shot at doing a harder challenge problem, so that they too can push their thinking (my advanced students really like this).

I just started doing this today, so I don't have too much to report about it yet. It seems to have gone well, though it took longer than the 10 minutes because I needed to explain the process a few times until they all got what I was talking about. As it becomes part of the routine, I'll know more about what impact it is really having.

Here is the first one we did, in pdf and word formats.

I'd love to get any feedback on any part of this.

Edit
We decided to make the reflection portion into a progress tracker, instead of copying it individually on the back of each Do Now. This log will be kept in a binder in the class. This will allow students to see how they did in previous classes as they are filling out the current reflection. It will also be a very useful document for discussions during grade conferences.

Distributive Property and Multiplying Binomials

I made a review lesson for my Algebra 2 students on these topics, to make sure they are really ready before we start performing operations on complex numbers.

Some instruction, some board races, and there you go. Hope you like it.

Lesson
Keynote
Quicktime


I also updated my Algebra 1 box with unit 2 files and the first four lessons of unit 3.

Algebra 1: Solving Equations

I am beginning the planning stages of our unit on solving equations in Algebra 1. In my past experiences, some students pick this up very quickly, no matter how you teach it, while other students struggle mightily. I want to try some alternate approaches this year, to really reach those students who have not been able to learn this skill in the past. I remembered an order of operations approach that I read about in the NCTM magazine a few years back. I can't recall the name of the article, but a little google searching found me this document that is even better than what I remembered.

Our students in Numeracy already work with bar modeling to solve word problems, so this seems like a natural extension to solving equations. I like this approach because it helps focus on the idea that the variable is a given quantity that must be determined, instead of focusing on the steps that isolate the variable. It also might help with those difficult to master "converting verbal sentences to algebraic equations" problems. Here are a few examples of how this might look. I know the diagrams are a bit confusing at first, but I think they would make more sense to students as they watch them get created and do them by themselves.



I also like the other representation discussed in the article. This is the original order of operations process that I had been searching for. I like this because it gives a very clear framework for solving equations - reversing the order of operations.


When you look at each stage, you can draw equal signs between the boxes. These would be equivalent to the intermediate statements in the traditional "do the same thing to both sides" approach.

So for the unit, I am thinking that we would spend two or three lessons on bar models to build the concept of what we are actually trying to do (find the value of the unknown amount). Then, spend a couple lessons on the order of operations representation to build an understanding of the process for isolating the variable. Finally, transition to the traditional approach, which is clearly the fastest and cleanest way to solve an equation of the three. This would take more time, of course, but the hope is that it would build a more enduring understanding.

Has anyone tried these methods with their students?

Edit:
I wonder now if it would make more sense to start in with modeling sentence/word problems with the bar model method, and not start by saying that we are "solving equations". That way, more students would be engaged with the material, and we could eventually use the bar models to develop the equations.

This way, the unit doesn't start with the problem "solve (3/5)x = 45", which will stop most kids dead in their tracks, but maybe with something like "It took Sandra 45 minutes to finish 3/5 of her homework. How long will it take her to finish it all?", which kids might have more of an entry to. After we solve it, we can then discuss how to represent it as an equation.

Edit 2:
I also need to think about how to incorporate the balance idea and preserving equality... Kids don't always know what the equal sign really means. Maybe in the transition time from the box method to the traditional method?

Edit 3:
(Written on 10/27 - at the end of the unit)
On reflection, the problem was not having enough time to really devote to the two alternative methods. Both did show a lot of promise, but we weren't able to really practice either enough for it to really stick with students. The bar model method really worked to help students set up and solve word problems, so I think I will stick with that next year. Give it some more time so that it really sinks in and can be used to get a deeper understanding of fractional coefficients. I will probably save the GERMDAS method for individual tutoring with students who are not having success with the traditional balance method. Less fights... more differentiation.

All of these lessons have been added to the box widget on the left.

Algebra 1: Skills List

My goal for this weekend was to complete a rough draft of all the skill items that will be assessed on the first semester final exam. These items are assessed in chunks on the weekly skills tests, and in larger chunks on the 6-week benchmark exams. After each benchmark exam, the plan is to spend a lesson or two on targeted reteaching - any ideas that people have on how to make this effective would be very much appreciated.

I've finished the list, and am interested to hear what other Algebra 1 teachers think about the scope and detail of the items. What would you add? Take away?

Year ten at DCP, off to a strong start! My students this year rock.

Ok, so it was more than a month. But at least I'm finally writing again. I don't think I will be able to post about every lesson like I did last year, but I am still going to put my files in the box.net account for you all to look at, use, critique, etc. This year, I am reworking our Algebra 1 curriculum, so I'll be trying some new things, and hopefully be getting lots of ideas from all you other teacher bloggers. If you are reading this, and you are not blogging your ideas or posting your work online, I highly encourage you to give it a go. You will get a lot out of it, and you'll give a lot to the community.

We just finished our first unit on evaluating expressions, including order of operations and working with square roots. Up next will be simplifying expressions, and we're going to be using Algeblocks to model combining like terms and the distributive property.

Here is a link to the Unit 1 files. Unit 2 hasn't quite been written yet...

Please send me feedback, especially criticism. I get lots of thanks from people for posting my work, but rarely does anyone tell me what they think would make things better. It's my tenth year, but I still have a long way to go until I'll be satisfied with my work. I need your help!

Algebra 1: Introduction to Inequalities

I'm not planning our Algebra 1 classes this year, so I have not been producing much for it. But I did put together a scaffolded introduction to inequalities. The objectives are for students to:

• Compare numbers using a number line (i.e. "<" means "to the left of")
• Understand the difference between open and closed circles
• Graph the solutions of a statement like "x < 3"
  
• Understand graphically why adding/subtracting by any number or multiplying/dividing by a positive number does not change the relative position of two numbers, while multiplying/dividing by a negative number does. In other words, students should understand when and why to "flip the inequality sign" when solving inequalities.
  
• Solve and graph linear inequalities

Here is the file.

Algebra 2: Factoring Trinomials (Part 1)

Ok, so I guess it should really be titled Algebra 1, not 2. But my students always need to review/relearn this topic. We'll go easy for the first lesson - only problems where the GCF = 1 and where the leading coefficient is 1. I made a puzzle for them to put together so that it is more fun than just doing a worksheet. I did something like this in the past with my honors class (but with much harder polynomial equations) and they really enjoyed it. That puzzle, once assembled, instructed them to do push-ups to get some candy. This one only requires that they tell me a joke - I'll add them to my arsenal if they're any good.

Lesson 5 (Factoring Trinomials 1)
Puzzle (doc / pdf)

You may also be interested in the puzzle-based Treasure Hunt I did a couple years ago in Algebra 2.

Algebra 2: Power Equations

I had a great Friday. My Algebra 2 classes went well, though the Keynote took about 10 minutes longer than I wanted. But, the slide where I showed them how you can use the Pythagorean Theorem to locate root 2 on the number line was quite successful (I think). They agreed that we could construct the original square out of a 1' x 1' piece of wood. Then, after we use the compass to map out the length of the diagonal, I showed that we could cut a piece of wood that is exactly root 2 feet long. I'm really trying to drive home the idea that irrational roots are still real amounts, and this slide made their brains hum.

And my single Algebra 1 class is starting to go really well. Over a week, and not a single referral. And, only once did I ask a student to step out of the class to calm down. I've got a TA who helps out by checking and logging homework, and then assisting students during practice time. I taught her a couple years back in Algebra 2 honors, and she is now one of a handful of seniors taking Calculus at a local junior college. Plus, I have another former student senior who has decided to use her free period to come every class and sit with Kate, and Kate is very happy with this arrangement. We learned the first part of the order of operations (aside from parentheses and exponents), and though they all surprisingly had heard of PEMDAS and knew that multiplication and division come before addition and subtraction, only about half knew the "left to right" part of it. So, when we got to that example, a big debate erupted, along with "you wanna bet"s and so forth, but it was all done in a positive way. And when the answer was revealed, the kids who were fighting for the wrong side were gracious about it (though I did make it extra clear that they could have been right too, and mathematicians just had to pick one way to do it). They were my last period of the day, and as a gift to them, when I got home I made a positive phone call home to every kid in the class. It took about an hour or so, but I'm hoping that it will turn out to be a good investment in furthering our class culture. The parents were very grateful to hear from me - even the ones who almost had a heart attack when the math teacher was already calling home. I had to do some quick assurances that "todo esta bien, no hay problema!"

On Monday, we will be solving power equations in Algebra 2. Nothing too fancy, but we will be doing Showdown for the first time - one of my favorite collaborative activities. Here are the files:

Lesson 5 (solving power equations)
Lesson 5 Keynote
Keynote Quicktime

Algebra 1: Intro to square roots

Students often have trouble seeing a square root as a number when the radicand is not a perfect square. The point of this activity is to help students develop this understanding by using a geometric metaphor.

square root intro.doc