Leveraging Sketchpad to Gain Insight Into an Algebra Problem

Several weeks ago, my friend Sandra came to me with a problem posed by one of her algebra students:

Find all points that are twice as far from a given point A as they are from a given point B.

Sandra and I chose particular locations for points A and B, and got to work solving the problem algebraically. When we were finished, Sandra confessed that her students would find the algebraic manipulations challenging. Might there be another approach that could sidestep the algebra?

My first thought (of course) was to turn to Sketchpad. I set out to model the problem geometrically, and before long, I had built an interactive sketch that worked nicely. I could have called it quits, but I decided to explore a generalization:

Find all points that are n times as far from a given point A as they are from a given point B.

Once again, Sketchpad gave me the tools I needed to model this interesting challenge.

Below is a movie that walks you through my Sketchpad investigation. I think you’ll not only find the mathematics engaging, but the Sketchpad techniques as well.

Are there similar problems in your algebra curriculum that would be amenable to a geometric approach? Let us know! The connections between algebra and geometry are ripe for exploration.

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What Does Rowing Have to Do with Teaching Mathematics?

DSC_5782
Harry congratulates a rower after
sweeping Yale for his last time.

Harry Parker died this summer, two weeks after coaching the Harvard rowing team to yet another sweep of all four races (varsity, JV, freshmen, and spares) against Yale and two days after accompanying his 1980 Olympic crew on a reunion row. I had the honor of rowing for Harry during the 1966 through 1968 seasons, including the 1968 Olympic Games. (In his 51 seasons Harry coached four other Olympic crews and had 22 undefeated regular seasons, 8 official national championships, 8 unofficial national crowns, and a 44-7 record in the Harvard-Yale four mile regatta including clean sweeps for the last six years.)

Not long after Harry’s 2011 diagnosis of a terminal illness, hundreds of those whose lives he touched gathered to honor him. Speaker after speaker told how Harry had inspired and changed them. Harry, uncomfortable with the praise, had one more bit of wisdom to impart.

“What you think you’ve gotten from me,” Harry said that night, “these are things you learned from rowing.”

The secret of his coaching success, Harry said, came from rowing itself, from the discipline, hard effort, consistency, and commitment that rowing required from us to make the boat go fast. His role was to make it possible for us to learn from the sport.

Which brings me to teaching math, and my realization that what my best teachers did was to enable me to learn from the math. My best teaching has not involved teaching math but rather enabling my students to learn from math.

The Common Core State Standards call for students to persevere, to reason, to argue and critique, to model, to use tools, and to attend to precision, structure, and regularity. Though we can encourage our students in these practices, we cannot enforce the practices. The demand for perseverance, reasoning, and other practices needs to come from the math, not from the teacher.

Our job is to help students recognize that math demands perseverance and argument. To make their mathematical boat go through the water fast and smooth, students must develop and practice good mathematical habits of mind.

Just as an athlete develops physical strength, technical skill, and mental toughness through practice and hard work, so the student herself must reason, master tools, and search for mathematical structure and insight.

Harry’s quiet but authoritative approach emphasized the act of rowing. It made us aware of the level of conditioning and mastery of technique required to make our boat go fast. Similarly, my best teaching presents students with challenging mathematical tasks and prompts them to interact with the mathematical elements of that task to seek out a resolution.

Thinking about Harry’s approach helps me understand why I find Sketchpad so valuable:

Sketchpad enables students to do mathematics.

It puts them in direct hand-eye contact with mathematical objects and requires them to engage in mathematical practices. Instead of reading and hearing about the mathematics, they confront it and learn from it.

They push and the math pushes back.

So the next time someone thanks me for a bit of interesting math they learned in my class or has an aha! moment using Sketchpad, I’ll express my happiness and tell them, “What you think you’ve gotten from me, or from Sketchpad, these are things you learned from the mathematics.”

Olympic Bound small
My 1968 crew, ready for our Yale race. Harry is at the far right, and I’m seated on the right.
CrewToday

Before the 2013 Yale race. At the left Kyle stands in for her husband Mark (our manager), and at the right Kathy stands in for her husband Harry.
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Introducing the Dynamic Number Project

Four years ago, my colleague Scott Steketee and I began brainstorming new Sketchpad activities for a National Science Foundation grant called Dynamic Number. Our goal was to use Sketchpad to make ideas from number, operation, early algebra, and algebra come alive through interactive models that emphasized conceptual understanding.

Dynamic Number

Scott and I had lots of help. We piloted activities in classrooms across the United States, including New York, Pennsylvania, Georgia, and California. Our fantastic team of teachers not only tested and critiqued our draft lessons, but they also offered ideas for entirely new Sketchpad models based upon their understanding of where their students could benefit from a dynamic approach.

Now, four years later, we are ready to debut our materials. As you can see at right, our range of topics spans from addition and subtraction to algebra. The Dynamic Number website offers 70 free activities, and all you need to use them is either The Geometer’s Sketchpad or its companion app, Sketchpad Explorer for the iPad.

I’m especially happy with the level of support we’ve been able to provide. The activities include extensive teacher notes, worksheets, and explicit connections to the Common Core. You’ll also find more than eight hours of professional development movies that describe the mathematics and pedagogy of each lesson as well as the simple mechanics of using the Sketchpad models.

I could write more about the Dynamic Number project, but instead I’ll conclude with the short video below that takes you on a tour of some of our most intriguing Dynamic Number models. If you get a chance to try any of our activities, be sure to send us some feedback. We’d love to hear what you think!

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Starting with Sketchpad Activities on Day One in High School

In my previous two posts, I listed some of the new Dynamic Number project activities (for grades 2-5 and grades 5-8) that engage students in manipulating and investigating dynamic mathematical objects from day one.

In this post I’ll list several similar activities suitable for high school mathematics. The first three address familiar topics: binomial multiplication, factoring of trinomials, and angles in polygons. The remaining four need a bit of explanation.

One of our Dynamic Number focus areas was beginning algebra, which led us not only to develop algebra-readiness activities for younger students, but also to rethink how we introduce variables and functions to students. Common Core standard G-CO2 requires students to “describe transformations as functions that take points in the plane as inputs and give other points as output.” We used this “Geometric Functions” approach to create 12 Sketchpad activities (including the four listed below), with the goal of making variables and functions concrete and engaging for students.

As we developed these Geometric Functions activities, we realized three things about functions in two dimensions. (a) Students find it easy to use a mouse or finger to manipulate input variables on a 2D plane, and they find it easy to observe and trace output variables on a 2D screen. (b) Directly constructing, manipulating, and observing 2D variables makes function concepts more concrete and memorable for students. (c) The activities develop the deep connection between functions and geometric transformations, which is particularly useful in addressing the Common Core’s increased emphasis on both of these topics.

I’ll have more to say about these activities in a future post (I’ll also be presenting at NCTM’s Baltimore Regional Conference in October and at NCTM’s Annual Meeting in April.)

(If you haven’t yet seen it, check out Daniel Scher’s recent post on the teacher and student support built into Sketchpad.)

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Starting with Sketchpad Activities on Day One in Grades 5-8

In the Dynamic Number project, one of our goals has been to create activities in which students actually experience mathematical objects by creating them, manipulating them, and investigating them. (George Lakoff and Rafael Núñez describe, in Where Mathematics Comes From, how students’ abstract mathematical concepts are grounded in their sensory-motor experiences.)

With Sketchpad, such experiences have been available in the geometric realm for many years, but our project’s focus was to make it as easy and effective to experience numeric objects (integers, fractions and decimals) and early algebra objects (variables, equations and functions).

That effort produced all of the activities I listed in my previous post on grades 2-5 and all but one of the grade 5-8 activities I’ve listed below. These include activities to zoom the number line to investigate decimals; to create fractions in using number line, rectangular, and circular representations; to connect common fractions with their decimal representations; to use a balance model to solve equations; and to explore variables and functions using a two-dimensional (geometric) representation.

Every one of these activities is designed for students to manipulate and investigate mathematical objects, and in a number of them students themselves create the mathematical objects that they modify and explore.

In my next post, I’ll recommend some Dynamic Number activities for high school use. (And if you haven’t seen it, check out Daniel Scher’s recent post on the teacher and student support built into Sketchpad.)

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Starting with Sketchpad Activities on Day One in Grades 2-5

As an 18-year veteran teacher in Philadelphia public schools, my initial reaction when I first saw Sketchpad was “This would have completely changed the way I taught geometry.”

I phrase it differently today: “This would have completely changed the way students experience mathematics.” a perspective that differs in two important details.

First, I now understand that the emphasis needs to be not on my teaching but on students’ experiences of mathematical objects. Secondly, I now realize the value dynamic mathematical experiences bring not just to geometry students but to students of all ages in all areas of mathematics.

My more mature perspective is due in no small part to my participation in the Dynamic Number project (http://dynamicnumber.org). Daniel Scher and I created and field tested many activities to bring dynamic experiences of number concepts to students from second grade on. While field testing in elementary schools, I watched students manipulate mathematical objects and observe and react to the feedback they received–feedback that came not from any adult but directly from the behavior of the mathematical objects. Students’ excitement was contagious.

So here’s a list of several Common-Core aligned Dynamic Number activities to use with your students in grades 2-5, in different mathematical areas, to establish from the start the value–and fun–of manipulating and experimenting with dynamic mathematical objects. Every activity comes with an activity sketch and extensive Teacher Notes, and most are compatible with the iPad.

Over the next few days, I’ll post some suggested Dynamic Number activities for middle school and high school use. (See also Daniel’s post on the teacher and student support built into Sketchpad.)

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Getting Started with Sketchpad Before Day One

Welcome back to a new school year!

As you get ready for the first day of class, the thought of learning new software may feel like too big a commitment for your busy preparation schedule. But if you’ve been meaning to get acquainted with Sketchpad, now is a good time to familiarize yourself with the software before students return. Where should you look to for assistance?

I’m going to share with you one of Sketchpad’s best-kept secrets: A world of free training resources awaits you in Sketchpad’s Help menu by choosing “Learning Center.” (You can also access the Learning Center online.) Below are some suggestions for making the most of the Learning Center resources before your students return to school.

  • Check out the Sketchpad in the Classroom videos. These videos, focusing on everything from elementary school topics through advanced mathematics, feature students and teachers describing the ways they’ve used Sketchpad in their mathematics classes.
  • Watch some of the Getting Started Tutorials. These quick-start videos introduce to you all of the basic functionality of Sketchpad and show you, step by step, how to use the software. Each video is accompanied by written directions so that you can refresh your memory about particular steps after you’ve finished watching.
  • Watch the What’s New in Sketchpad 5 video. This video provides a short overview of some of the exciting new capabilities of Sketchpad 5, including Hot Text, the Marker and Polygon tools, and picture transformations.

Another fantastic resource to consult prior to the first day of class is our archived collection of Sketchpad webinars. These free one-hour sessions, viewable at your convenience, are a great way to introduce yourself to Sketchpad—just sit back and watch as actual classroom teachers show you how to use the software to teach everything from addition and subtraction through calculus and fractals. For Sketchpad newbies, I recommend watching either the Sketchpad 101 webinar, the geometry webinar, or the algebra webinar.

In the next post, Scott Steketee will describe some of the easy ways you can use Sketchpad starting on the very first day of class. Time to get cracking!

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Revisiting a Childhood Addition Code with Sketchpad

As a fourth-grader in 1977, I had a love-hate relationship with my Addison-Wesley textbook. Its contents overflowed with arithmetic problems, but every so often an entertaining brainteaser appeared to break the monotony of drill practice. These puzzles were clearly marked: Each appeared in a box set aside from the main text and featured a bespectacled fish to introduce the challenge.

The back pages of the textbook featured still more drill, but below each set of problems came a mysterious code. Here is an example:

Code

There was no fish or other indication to suggest that these codes were there for my consumption, but I didn’t care. I needed to satisfy my curiosity and uncover their meaning. I soon realized that the codes were a way to represent the answer to each addition problem with letters in place of the digits. If the answer to 420 + 189 was given as HDA, for example, then H = 6, D = 0, and A = 9.

I now had powerful technique for breezing through each set of drill problems without solving each and every one. By focusing on just enough problems to determine the values of the letters A through J, I could complete the rest by applying the cracked code.

Continue reading

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Mile High (and much more than an inch deep)

At the NCSM conference in Denver last month, we had our fourth annual Math Ignite event, featuring fast-paced, engaging, 5-minute presentations by a variety of teachers and education leaders. I’ve finally finished editing the videos of those talks, which you can view at our YouTube page. There were many excellent talks, but one stood out for me.

Terry_StudentsEleanor Terry, a teacher in Brooklyn, New York, who is part of the Math for America program, gave an inspiring presentation on how her students conducted exit polls and used software including Fathom to analyze the data. View for yourself how this real-world project has led her students to consider themselves statisticians.

The other presentations were wonderful as well, focusing on a wide variety of themes in math education. Here are the speakers and titles of their talks (which link to the videos).

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Polar Graphing

After writing yesterday’s post on the connections between polar and Cartesian graphs, I realized that I hadn’t said anything about how easy it is to start from scratch and create a polar graph in Sketchpad, so I decided to write this post, and include an instructional video. Here are the steps to create the graph shown on the right below.

Simple Polar Graph

  1. Choose Graph | Plot New Function.
  2. Use the Equation menu to choose r = f(θ).
  3. Type “c” (for “cos”), “2″, and “th” (for “theta”).
  4. Click OK.
  5. If your angle units are degrees, Sketchpad may ask if you want to change to radians. (Don’t worry; the graph will be correct no matter whether you want to use radians or degrees.)

That’s it!

This short video shows how easy it is to add parameters to control the amplitude, period, and phase shift:

It’s also easy to create a family of polar functions. Once you’ve modified your graph to show f(θ) = a·sin(b·(θc)), here are the steps used to graph the family of functions shown below.

Family of Polar Graphs

  1. Select both the graph and parameter a.
  2. Choose Construct | Family of Functions.
  3. Set the domain to go from 1 to 6, and the number of samples to 11. This will create samples for these values of a: {1.0, 1.5, 2.0, …, 6.0}.
  4. Click OK.

You’re done. Now experiment on your own!

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