The NCTM Annual Meeting in San Francisco coming up in a few weeks led me to compile this list because I wanted an easy way to share these activities with the attendees at my Friday session. Hope to see you there!

Zooming Integers: Estimate the location of a tick mark on a number line and then “zoom in” one or more times to identify its integer value precisely. [websketch]

Zooming Decimals: Estimate the location of a tick mark on a number line and then “zoom in” one or more times to identify its value to the nearest tenth, hundredth, thousandth, and beyond. [websketch]

Interactive Dials: Count in base 10 or change the dials to count in base 2 or base 3, or any base of your choice. [websketch]

Odometers (Part 1 and Part 2): Explore place-value concepts with odometers that let you adjust each individual digit on their display. [websketch 1 and 2]

Bunny Times: Develop a conceptual understanding of multiplication through this engaging multi-level game.

Arranging Addends: Arrange and overlap three circles so that the sum of the numbers in each circle matches the desired totals. [websketch]

Color Calculator: Display fractions as decimals with each digit represented as a different color. What color patterns can you find? [websketch]

Open the Lock: As pointers move around the dials of a lock, use your knowledge of multiples to align all the pointers facing straight up. [websketch]

Mystery Number: Deduce the value of a secret number by asking questions about its factors. [websketch]

Factor Rainbows: Find all the different ways for a bunny to hop to a location on a number line. The factor patterns you’ll discover create attractive rainbows. [websketch]

Open the Safe: A grid of lights displays patterns based on multiples. Can you open a lock by creating the desired patterns? [websketch]

Factor Patterns: Explore factor relationships in this dynamic grid that displays the factors of any number. [websketch]

Fraction Identification Game: Match two randomly generated fractions to their area representations. [websketch]

Fraction Multiplication: Explore a visual model for multiplying fractions. [websketch]

Balance Scale: Deduce the numerical values of objects on a scale by maintaining balance as you remove objects from the scale. [websketch]

Circles and Squares: Use logical reasoning to solve for the unknown numerical values of a circle and square. For an extra challenge, solve for three unknowns—a circle, square, and triangle. [websketch]

Sneaky Sums: Deduce the numerical values of the symbols in a grid by examining the sum of the symbols in any row or column. [websketch]

Dancing Unknowns: Solve for 10—yes 10!—unknowns as they dance around the screen. [websketch]

Student Height Puzzles: Create and solve visual logic puzzles relating to the relative heights of students. [websketch]

Find the Hidden Treasure: Uncover the location of a hidden treasure on the coordinate plane by piecing together clues about your distance from the treasure. [websketch]

Construction Tools: Experiment with a set of geometric construction tools to see what you can build. [websketch]

Triangle Area Without a Formula: Enclose triangles in rectangles to obtain their areas through subtraction. [websketch]

Triangle Area: Construct multiple altitudes of a triangle to check that the area formula gives consistent results. [websketch]

Hidden Polygons: Create order out of seeming chaos by connecting points with segments to form squares and equilateral triangles. [websketch]

Tessellations: Create tessellations with regular polygons and then explore whether all triangles and quadrilaterals can tessellate the plane.

]]>The fraction game below presents two random fractions at a time and challenges students to match the fractions to their corresponding area representations. The fractions are shown as parts of a circle, but the divisions of the circle are hidden.

There are four different games, and you can navigate from game to game by using the arrows in the lower-right corner of the model. The first game always presents the two circle sectors in the same vertical orientation, making it easier to compare them. You can also drag one circle onto the other. The remaining three games position the two sectors randomly, adding to the challenge. In the first three games, the denominator of the fraction is less than or equal to 12. In the final game, the denominator is less than or equal to 20.

The first two games include a hint button that when pressed, reveals an additional circle divided into equal parts. Dragging the “split” point clockwise divides the circle into more equal parts and counterclockwise into fewer parts. The circle can be dragged onto either of the other circles to help match them to the fractions.

Below are six examples of the types of reasoning that students may use in solving the fraction identification challenges. There are, of course, multiple ways to think about each problem.

*Example A: Match 7/10 and 4/10 to the circles. *Since both fractions share the same denominator, the numerators are enough to identify the larger fraction.

*Example B: Match 1/9 and 7/8 to the circles. *The visual representation of the two fractions highlights that one is near 0, and the other near 1, pointing the way to identifying them.

*Example C: Match 3/7 and 3/8 to the circles. *This example addresses a common misconception that as the denominator of a fraction grows larger, the fraction itself grows larger as well.

*Example D: Match 3/8 and 4/10 to the circles. *The visual representation of the two fractions highlights that both are near one half. Of the two fractions, 4/10 is closer to one half.

*Example E: Match 4/7 and 7/12 to the circles. *These two fractions are so close in value that finding a common denominator is the only reasonable solution path (other than using the Hint button).

*Example F: Match 2/3 and 8/11 to the circles. *These two fractions are also close, but rather than find a common denominator, students may be able to identify 2/3 visually.

What design challenges did we face?

For starters, we knew that the technology should eliminate any accuracy issues that students face with paper and pencil. While it’s far from easy to divide a sheet of paper into fifths or sevenths, Web Sketchpad can make this process simple and precise. Nonetheless, we didn’t want the software to automate the process entirely—students should still choose the number of vertical and horizontal divisions in their square. And given a problem like 1/2 x 1/3, students should be free to divide their square vertically in halves and horizontally in thirds or vertically in thirds and horizontally in halves.

We also wanted students to decide whether to think of 1/2 x 1/3 as 1/2 of 1/3 (in which case they would shade one third of the square and then shade half of that third) or to regard it as 1/3 of 1/2 (in which case they would shade one half of the square and then shade one third of that half). In retrospect, these sound like straightforward design principles, but when you’re building a model, it’s extremely easy to limit the choices available to students, perhaps unintentionally, because the programming involved is much simpler!

I don’t claim that Scott and I achieved the perfect interactive model, but after many scrapped attempts, I’m ready to say that we gave it a good shot. See what you think and let us know.

The video below the websketch explains how to use it.

We began the session by introducing the coaches to a Web Sketchpad model that allowed them to create regular polygon tessellations. You can try it yourself by clicking the image below to open the accompanying Web Sketchpad model. (A movie at the end of this post demonstrates how it works.)

As they explored the model, the coaches determined which regular polygons tessellated the plane and which ones did not. They posed questions to each other, made conjectures, tested theories, and thought about the big mathematical ideas of the investigation.

The coaches determined that the equilateral triangle, square, and hexagon could tile the plane. They noticed that because six equilateral triangles converged at a single vertex, they could determine the angle measure of each triangle by dividing 360° by 6. They made a similar observation about the angles of hexagons.

The coaches then returned to the Web Sketchpad model and investigated which combinations of regular polygons could tile the plane. Above is an example of a particularly attractive tessellation discovered by a coach.

The coaches then turned their attention to arbitrary triangles. The Web Sketchpad model below starts with a triangle *ABC* and provides tools for snapping copies of Δ*ABC* together in either its initial orientation or as a rotated copy (Again, the movie at the end of this post explains how the tools work.) What’s especially nice about this model is that after you’ve built a tessellation using a specific triangle, you can then drag any vertex of Δ*ABC *and the entire tessellation will reshape itself to match the new triangle.

By experimenting with the tessellated triangles, the coaches noticed that the three angles of the triangle always came together at a vertex to form a straight line—a very elegant and memorable demonstration that the angles of an arbitrary triangle sum to 180º.

Finally, the coaches investigated whether arbitrary quadrilaterals could tessellate. As before, a Web Sketchpad model made it easy to experiment with different quadrilaterals simply by dragging the four vertices of quadrilateral *ABCD*. Building on their observation about triangle angles, the coaches discussed how their quadrilateral tessellations demonstrated that the angles of a quadrilateral sum to 360º.

Time was running short since the coaches needed to plan the afternoon session in which fifth graders would try the regular polygon tessellation websketch. As such, the coaches did not have a chance to use the line tool that accompanied the four quadrilateral tools. But try it yourself, using it to connect vertices in your quadrilateral tessellation that lie on a line. In the illustration below, the lines form a tessellation of parallelograms. You can imagine these parallelograms, each with the identical pattern of four triangles, serving as another way to create the same pattern.

]]>Our article summarizes our curriculum unit, *Connecting Geometry and Algebra Through Functions.* This unit’s Web-Sketchpad-based activities connect functions in geometry (transformations whose variables are points on the plane) with functions in algebra (whose variables are points on the number line).

In particular, students work with geometric transformations as functions that take an input point and produce an output point, and relate these functions to algebra by using them to construct the Cartesian graph of a generalized linear function. They dilate and then translate a point, restrict these points to number lines, and ultimately observe that in the algebraic equation *y* = *mx* + *b*, *m* corresponds to the scale factor for dilation and *b* corresponds to the length of the vector for translation.

]]>

As part of that work, I just completed a websketch that nicely mixes practice with logical reasoning. Students are challenged to find a hidden treasure on the coordinate plane by guessing its location. For each guess, students are told its taxicab distance (traveling horizontally and vertically) to the treasure. Using this information as clues, students deduce where the treasure lies.

The websketch below starts by introducing students to the way distance is measured between each guess and the treasure. The second and third pages (accessible by using the arrows in the lower-right corner) present two different views of the coordinate plane, the first showing only the first quadrant and the second showing all four quadrants.

The game is geared towards elementary-age students, but it’s suitably challenging for older students, too. Personally, I find it addicting to play!

]]>Bunny Times uses carrots in a field to represent the array model of multiplication. In the first few levels, students determine the total number carrots in the field and a single bunny eats them all. Some students may count the carrots one by one, but soon, a patch of fog rolls in, obscuring many of the carrots. In later levels, bunny teams eat carrots in unison, with students able to specify the number of bunnies in each team. These game elements lead students to develop new strategies for thinking about multiplication, such as skip counting, adding on, and deconstructing problems like 7 x 8 into (5 x 8) + (2 x 8).

When we released Bunny Times in 2013 as part of the NSF-funded Dynamic Number project, you needed to either own Sketchpad or proceed through a multi-step installation process on the iPad to play it. Now, however, Web Sketchpad makes everything simple: You can play Bunny Times in your web browser by clicking the picture below.

If you tap the ‘Learn to Play’ sign post on the opening Bunny Times screen, you’ll be taken to a video that gives an overview of the game. If you’d like to read about the pedagogy behind the game and a mathematical overview of each level, download this document.

Have fun and share with us your experience of using Bunny Times with your students or children!

UPDATE: Bunny Times is now available to play on the NCTM Illuminations site as well.

]]>

I was reminded of this theorem while reading the article Using Appropriate Tools Strategically by Milan Sherman and Charity Cayton in the November 2015 issue of NCTM’s *Mathematics Teacher*. Their article addresses the pedagogical potential of Dynamic Geometry technology and focuses on the power of a point theorem as a concrete example of how the software can foster high cognitive demand tasks.

Rather than summarize Sherman and Cayton’s ideas here, I’ll describe a Dynamic Geometry application of the power of a point theorem not covered in their article:* Use the power of a point theorem to build a constant-area rectangle. A constant-area rectangle is one whose perimeter can change but whose area remains fixed.*

Since the product of *AP *and* PB *is constant as point* A *travels around the circumference of the circle, this suggests that we construct a rectangle whose dimensions are* AP *and

Does this construction show us every possible rectangle with an area of 21.3 cm^{2}? No. As point *A* spins, we don’t see rectangles where *AP’*s length is close to zero (like *AP* = 0.001 cm) and we don’t see rectangles where *AP’*s length is very large (like *AP* = 300 cm). Indeed, *AP* can be no larger than the diameter of the circle.

Might dragging point *P* very near the circumference work? Try it—you’ll notice that the constant area of the rectangle changes, and that clashes with our goal: We want to see all possible rectangles with an area of 21.3 cm^{2}. (Dragging point *P* outside the circle has interesting results; results that are beyond the scope of this post.)

Fortunately, there is another method of generating a constant-area rectangle. On page 3 of the websketch, you’ll see a right triangle *BDA*. Altitude *AC* is the geometric mean of *BC* and *CD,* meaning that *BC · CD* = *AC*^{2} = 21.3 cm^{2}. By animating point *C* along its horizontal line, distance *AC* remains constant, so we again create segments whose product is constant. But this time, as point *C* moves to the left, distance *BC* approaches 0 and distance *CD* approaches infinity.

There’s a nice connection to be made between these two methods of creating constant-area rectangles. Page 4 of the websketch again shows right triangle *BDA*, but now we see its circumcircle as well. Segments *AA’* and *BD* are chords of the circle with *BC* *· CD = AC · A’C, or * *BC · CD* = *AC*^{2} since *AC* = *A’C*. Thus the right triangle construction is really just a special case of the circle construction, with one of the two chords being a diameter of the circle.

I first wrote about constant-area rectangles in the April 1996 issue of NCTM’s *Mathematics Teacher*. Sketchpad has come a long way in twenty years, but my article’s conclusion still holds: “A chief pleasure of these investigations comes from taking theorems that may seem like nothing more than geometric curiosities and turning them into devices that perform a desired function. Specifically, the geometry that lies behind the chord theorem and the geometric-mean construction becomes the engine driving the movement of the constant-area rectangles. By setting these theorems in motion, students are able to generalize them and uncover relationships that the static counterparts in a textbook cannot reveal.”

I recently had the opportunity to work with a group of students who were testing activities that treat geometric transformations as functions (what I call *geometric functions*). I got lots of good ideas for improving the activities not only by watching the students, but also but also from their suggestions and the recommendations of their teacher, Jessica Shupik, and of my colleagues Daniel Scher and Mike Pflueger.

After several sessions of exploring reflections, translations, and rotations from a function perspective, we turned to the remaining transformation, dilations. The dilation activity emphasizes the way a function rule relates the independent variable (a point) and a dependent variable (its dilated image).

Ms. Shupik’s school promotes a mastery approach to grading and encourages performance-based assessments. With this in mind, we came up with three dilation games for assessing her students’ understanding:

- Game 1: Given an independent variable (a point) and a function rule (a center and scale factor), find the dependent variable (the dilated point).
- Game 2: Given the independent and dependent variables and the center point, find the scale factor.
- Game 3: Given the dependent variable and the function rule (the center and scale factor), find the independent variable. This third game is not solvable in the general case, but fortunately the dilate function has an inverse.

I later added Game 4: Given both independent and dependent variables and the scale factor, find the center of dilation.

Here’s the websketch containing all four games.

(Note the labeling of the points, which is explained in the activity that precedes the assessment. The independent variable is *x,* to ease the later transition to algebraic functions, and the dependent variable is *D _{C,s}*(

As students progress through the levels of each game, the problems become harder and the hints become fewer, ramping up the challenge bit by bit. (To move from one level to the next, first press Reset to start a new game, and then drag the Level slider.)

Ms. Shupik set a level of achievement for each game, and told students to email her a screenshot when they met that level. In retrospect, it might be even more motivating to students to set a hierarchy of levels of mastery, perhaps something like this:

I was surprised while creating and testing this game to discover how much fun I had playing its higher levels. Without the security of measurements and numbers, I had to develop a feel for dilation, a sense for what a dilation by 7/4, or by –0.625, feels like.

The nice thing about this assessment, from my point of view, is that the students were learning about dilation, getting (like me) an experience-based understanding of it, even as they were trying to achieve a particular level of mastery. The assessment and learning occurred simultaneously.

While the rhombus task worked well, it did expose one of the challenges of using Sketchpad: The software features lots of menu commands and toolbox options and navigating all of the functionality can sometimes distract Sketchpad newcomers from the mathematics at hand.

Web Sketchpad differs from Sketchpad by offering a more streamlined approach to mathematical construction. It allows a teacher or curriculum developer to create and provide only those tools needed for a particular task. We can use this approach to lead students to think about a problem in new mathematical ways, just by limiting them to carefully chosen tools.

Let’s consider, as we did in our professional development classes, the challenge of constructing a rhombus. Below are four different toolsets, each of which focuses students on different mathematical properties of a rhombus. Your task is to construct a rhombus with each toolset that stays a rhombus when its vertices are dragged. These are elegant challenges. Each toolset includes a Quadrilateral tool for indicating the four vertices and interior of your rhombus, but it’s really the other tools—the Compass tool, the Parallel Line tool, the Reflect Point tool, and Perpendicular Bisector tool—that are responsible for ensuring that the quadrilateral you construct is a rhombus.

Try it yourself—use each collection of tools to construct a rhombus that stays a rhombus when you drag any of its vertices. Are some of your rhombi more general than others? How would you compare the behavior of your four rhombi when their vertices are dragged? What characteristics of a rhombus does your construction exploit?

If you need help, watch the movie at the end of this post that demonstrates four rhombus constructions, one for each toolset. And please, let us know if you come up with a different construction in addition to those in the movie.

]]>