Creating Technologies to Support Measurement

I worked with a student teacher who had been an undergraduate research assistant for the Strengthening Tomorrow’s Education in Measurement project, and was a recent graduate of Michigan State’s Teacher Education program. She is currently interning in a nearby classroom. She agreed, along with her mentor teacher, to design and implement a lesson with my support. Two redesigned applets resulted, and a third is possible. The link below connects to her lesson.

The student teacher had taught a lesson that incorporated van de Walle’s Crooked Path and Broken Ruler activities two weeks before the technology lesson (Van de Walle, 2013). She created crooked paths in her classroom using masking tape on the floor and on some tables. Students measured them, moving from path to path as they completed each measurement.

The student teacher felt that the STEM Jagged Path applets (see link above) would support the learning goals of her previous lesson by encouraging students to practice skills they had developed as well as deepening their earlier thinking. She asked all students to explore Activity 1, using Activity 3 as an extension for students who finished quickly.

The student teacher and I talked above another applet that could support students’ thinking. She said that her mentor teacher worked hard to encourage her students to confront misconceptions and change their thinking. We designed and created a third applet (shown in Activity 2, using the site link above) that would show the student two “wrong” ways to measure. We asked the students to explain what they thought was the thinking behind each method, and to explain which method they thought was correct – or if neither was correct, to explain what a correct method would be.

Supporting spatial measurement tasks with technology

Session presented at the Michigan Council of Teachers of Mathematics annual conference in Traverse City, MI.

Download slides here.

In this hands-on session, we introduced several sample tasks that included use of computer applets to strengthen understanding of length and area concepts. Many students struggle with length and area, using only pen and paper or physical manipulatives. In this session, we provided materials to support their learning (that will be available for use during and after the session).

We asked participants to use their devices to work through one sample lesson (Area of Rectangular Regions) including Launch / Explore / Summarize components. We gave participants two sample lessons that they could in their classrooms, along with applets designed and created by Strengthening Tomorrow’s Education in Measurement project.

Stehr, E.M., Gönülateş, F., & Siebers, K. (2013, August) Supporting spatial measurement tasks with technology. Paper presented at the Michigan Council of Teachers of Mathematics Conference, Traverse City, MI.

GeoGebra – Malfatti’s Problem

In Axiomatic Geometry, I and a partner explored Malfatti’s problem (see references below). I created a GeoGebra worksheet to support the presentation we later gave. I share some details here.


Malfatti originally asked this question is 1803 and the question, “How to arrange in a given triangle three non-overlapping circles of greatest area?” remained open until 1994.


As far back as 1384, in a manuscript of Italian mathematician Gilio de Cecco da Montepulciano a problem is presented of how to construct 3 circles, inside a triangle, that are tangent to each other and tangent to 2 edges of triangle. The same problem was posed by 18th century Japanese mathematician Ajima Naonubu. In 1803, Malfatti refined the problem to include minimizing the amount of area in the triangle “wasted”, i.e., constructing the largest circles possible. In 1826, Steiner (that sly old fox) developed a construction of three circles, tangent to each other and touching 2 edges of the triangle.  This was a solution to the problems posed by Montepulciano and Naonubu, but not Malfatti.  Stiener demonstrated 3 circles in a triangle but did not minimize the area “wasted”. In 1930, Lob and Richmond showed Malfatti’s circles do not produce the largest circles (i.e., minimal waste).  Optimal circles are obtained using greedy algorithm. In 1967, Goldberg showed that the Mafatti circles will never produce greatest area and the Lob and Richmond ‘greedy algorithm’ always produces the largest total area.  In 1994, Zalgaller and Los classified all the different ways 3 circles can be arranged in a triangle and developed a formula for determining which method of arranging the circles is optimal for a given triangle.  They also proved that there are 14 combinations of circles in a triangle that will yield optimal area.


Essentially, if there is room for a circle to ‘grow’ then the system is NOT rigid.  The two examples on the left are rigid, the one on the right is not, the middle circle could change radius or ‘grow’.


Zalgaller and Los proved the result systematically, reducing all possibilities to 14 combinatorially, non-identical, rigid configurations of three circles in a triangle. See cases below.


Malfatti’s Marble Problem asks:

  • How can we arrange in a given triangle three non-overlapping circles of greatest area?
  • For an Isosceles triangle, the first two circles are drawn as K1 and K2. Then we choose K3a or K3b depending on the relationship between sin(alpha) and tan(beta/2).
  • Play around with the triangle and tell me –
    • Do we use K3a or K3b when sin(alpha) > tan(beta/2)?
    • Do we use K3a or K3b when sin(alpha) < tan(beta/2)? W
    • hat do we do when sin(alpha) > tan(beta/2)?
    • Also, are the three circles always symmetric for an Isosceles triangle?



Andreatta, M, Bezdek, A, & Boronski, J (2010) The Problem of Malfatti: Two Centuries of Debate. The Mathematical Intelligencer.

Coolidge, J. L. (1923). Some Unsolved Problems in Solid Geometry. The American Mathematical Monthly, 30(4), 174-180.

Goldberg, M. (1967). On the original Malfatti problem. Mathematics Magazine, 40(5), 241-247.

Zalgaller, V. A., & Los’, G. A. (1994). The Solution Of Malfatti’s Problem. Journal Of Mathematical Sciences, 72(4), 3163-3177.

Presentation Reference

Humes, W.R. & Stehr, E.M. (2013). Malfatti’s Marble Problem. Presentation prepared for final project in D.S. Durosoy’s MTH 432: Axiomatic Geometry course.

Virtual Manipulatives and Dynamic Representations in Length and Area Measurement.

Session presented at the Math in Action annual conference at Grand Valley State University in Allendale, MI. We explored measurement concepts using virtual manipulatives and dynamic simulations to encourage reasoning and sense-making that can support robust understanding of measurement.

Download session slides here.

We created three tasks for participants to explore during the session, each considering area: one as a paper and pencil task, one using physical manipulatives, and the third using an online tool from NCTM Illuminations.

Download task sheet here.

Pencil and Paper Task – We asked participants to determine the area of parallelograms and to mark those with the same area.

Physical Manipulatives Task – We asked participants to create a parallelogram with the same base and height as a given rectangle. We asked: How do the areas of the original shape and your new shape compare? Explain.

Virtual Manipulatives Task – We sent participants to the NCTM Illuminations page:  We asked them to find as many parallelograms as they could, all with area of 88 square units. We asked them to discuss patterns they saw.

We ended by asked: What could the learning goal for these three tasks be? How do the three tasks support learning in different ways? How does each block learning in different ways?

Possible benefits of simulations:

Stehr, E.M. & Siebers, K. (2013, February) Fractions as lengths. Paper presented at the Math in Action Conference, Grand Valley State University, Allendale, MI

In the beginning, it is always dark. (Neverending Story)

I taught mathematics at the university level for several years and spent quite a bit of time learning neat stuff that I could use in my classes.  Some of it failed spectacularly but some was useful.  It can be overwhelming to find the right tool to do the thing that needs to be done – something that is easy to use, easy to learn, and that actually supports the students’ learning in a different way.  That is, something that provides affordances that aren’t granted by paper, physical objects or discussion.

It is difficult to find the time to be really thoughtful about choice of technology and its implementation without neglecting the day-to-day work that teaching requires.  When I taught, I spent too much time learning how to use technologies that I then only used once or that didn’t support learning in the way I had expected.

When I taught, I created PowerPoints that the students could print out and write on.  I printed the notes to Windows Journal (and later OneNote) and then used them in class to write on and work out examples. I used classroom response systems: First, eInstruction Crickets and then students’ own cell phones with PollEverywhere. I used interactive website and presentation creators, such as SoftChalk and Articulate Engage, along with screen capture software, such as Camtasia and Jing, to create websites for my students and to provide them with online activities and videos as resources and study aids.

I used mathematics software such as Mathematica (and Wolfram Demonstrations Project and TI-84 SmartView). I created online homework using Desire2Learn (D2L) quizzing tools, videos of homework solutions, dynamic interactions with Wolfram Demonstrations, animations with TI-84 SmartView, and even made cardstock manipulatives that students could cut out and use at home.

Looking back, I realize that often my efforts supported development of procedural fluency without attending also to developing conceptual understanding. My view of technology is that it is a tool that should be used thoughtfully and to support learning that may be difficult (or impossible) without that particular tool.

Creating a Number Line in PowerPoint

Making a nice number line in PowerPoint is quick and easy – it should only take about 10-15 minutes. These instructions are really just a frame for learning a few neat formatting tricks that will make using PowerPoint for graphical interfaces like posters (or even presentations!) a snap.The first thing is to “Insert” a “Shape” (in this case a line) and choose the color, thickness, and other effects using “Format” in “Drawing Tools”. To make sure the line is straight, hold the “Shift” key down as you drag it out.PowerPoint-NumberLine-01Make a smaller line for the tick marks, and copy paste however many times you’d like. The position doesn’t matter – we’ll straighten those out in a minute.PowerPoint-NumberLine-02Left-click and drag the mouse  to highlight all of the ticks fully. When you release the mouse button, all of the ticks will be selected. PowerPoint will only select shapes (or images) if it is entirely within the highlighted area, so you can be very selective by leaving out even a tiny particle of a shape.PowerPoint-NumberLine-03If you select objects that you didn’t want to select, just hold the <CTRL> key down and click on them to deselect.PowerPoint-NumberLine-04Once you’ve selected everything, go to the “Format” window and look to the far right for the “Align” menu.  This menu gives you many options and, if you haven’t yet,  you should definitely experiment and figure out what they all do and think about how they can be useful.PowerPoint-NumberLine-05These tickmarks need to be aligned horizontally, so choose “Align Middle” (although really top or bottom would also work fine in this case.)PowerPoint-NumberLine-06Now, click anywhere else on the slide to deselect all of the ticks. Choose one and drag it all the way to one side so that the tick on the left and the tick on the right are in the desired positions.  (You can drag it around without ruining alignment by holding the <Shift> button down.PowerPoint-NumberLine-07Once they are in place, go back to the “Align” menu and choose “Distribute Horizontally”.  Congratulations! You have a number line (albeit blank).PowerPoint-NumberLine-08Adding the numbers is easy too – just use text boxes and the align menu to get them straight and in place.

Microsoft Mathematics and OneNote

(Note: Microsoft Mathematics can also be used independently of OneNote)

Microsoft OneNote has many excellent features that support note-taking and searchability, but there are too many features to list in one post. This post will focus on Microsoft Math and OneNote. First of all, there is a built in calculator in OneNote that can be used when typing.  Just type the calculations, add an equals sign and press enter:


(Note that trigonometric functions in OneNote are in degrees rather than radians.)  For a list of mathematical operations and supported functions, check out the informational page at

Microsoft OneNote makes it easy to ink mathematics using the “Ink to Math” option:

MSMath-02It does pretty well at guessing what you mean – but the more information it has, the  better it can guess.  So expect to see it adjust frequently as you write.


Once you are comfortable with the Ink to Math feature, using Microsoft Mathematics can give many more options.  Microsoft Mathematics is a free download and can be used independently, but once you have installed it then you can also install the add-in for OneNote. This gives many options.  You can use the add-in to graph equations.


You can also use it to factor or expand polynomial expressions, find derivatives or integrals, or even plot graphs in 2D or 3D:


The best part about this is that you can use this in class as a substitute for chalk and blackboard, and then print the notes to pdf so the students can use them as a resource, or to create their own activities in OneNote.