Sep 6, 2017

#STEM lesson seed: 3D printer + Math = Tangram

I am looking for 3D Printing lesson plans for my 4th graders. I want to go beyond just having them replicate the online models. I just received my 3D printer for the fall school year. I hope that you remember me. You always have fantastic workshops. -- former workshop attendee
Happy new school year! The following is the lesson seed I created for this teacher. It is aligned with her district's requirements for the Minnesota standards. If you can use it with your newly acquired 3D printer, please feel free. If you would like to share with us, we can post so other teachers can use.
prosthetic foot, image by Pongratz Engineering

Tangrams Lesson Seed

Activity Overview
  1. Have students play Tangrams 
  2. Use CAD to model your own Tangram set 
  3. Use 3D printer to create the Tangram pieces 
By completing this activity, students will demonstrate:

Standard Activity Extension
5.3 I can measure angles in geometric figures and in real world objects with a protractor
  • Measure the angles of Tangram shapes to model on paper/cardboard and then in CAD
  • Complete the triangle game: start off with different angles and different leg lengths. Then have students complete the triangle and see what type of angles are made.
  1. Can you ever have two or more acute angles in a triangle? Right angles? Obtuse angles? 
  2. How many different triangles can you make with 3 specific side lengths? 
  3. Use drawing triangles to see how angles “add” together. The 30-60-90 and 45-45-90 triangles together can make angles from 0 to 180 in increments of 15. Obviously, you can do 30, 45, 60, 90 just using the triangles themselves. This shows you how to do 15 and 75 which completes the 0-90 degrees. Then you simply do the same on the other side of 90 for 90-180
(earlier standard: ability to measure with ruler)
  • Measure the lengths to create exact replicas 
  • Learn how to extrude in CAD 
  • Measure the height of the pieces to extrude piece in CAD for 3D printing
  • Scale the pieces (smaller or larger) 
  • Play-Doh Fun Factory extrusion machine might be good demo for kids who don’t know what extrusion is. 
  • Shaped notepad (e.g. circular, elephant, fruit) may be way to show how 3D printing will extrude the shape one layer at a time
5.7 I can identify and apply translations (slides), reflections (flips), or rotations (turns) to figures
  • Use cardboard then CAD to figure out how to arrange pieces tightly together for faster printing
  • How can you arrange everyone’s pieces to get maximum number of pieces per sheet when printing?

By playing Tangrams, students will demonstrate
Standard Activity Extension
5.4 I can compare angles according to size
  • Try / develop different strategies using the angles (adding them together by laying next to each other) to figure out which pieces could be used to crease the shape in the challenge card
5.6 I can find the area of different geometric figures and real-world objects
  • Try the strategy of figuring out where the biggest pieces have to go. Then see how they can be combined with the other pieces to create the shapes in the challenge card
  • Calculate the area of the challenge shape and see that it should always add up to the total area of the pieces if all pieces are used
5.7 I can identify and apply translations (slides), reflections (flips) or rotations (turns) to figures
  • See if transforming the shape through reflections, rotations can help give insight into the pieces that might be used to create parts of the challenge shape
  • Swish game helps students practice this

Levels of Tangrams:

  • See how you can make quadrilaterals from triangles 
    • 5.5 I can find the area of common quadrilaterals (squares and rectangles) 
  • Make your own challenge cards 
    • 5.6 I can find the areas of different geometric figures and real-world objects 
    • 5.7 I can identify and apply translations (slides), reflections (flips), or rotations (turns) to figures 
  • Make your own solution video/instructions (to help others “get it”) 
    • 5.7 I can identify and apply translations (slides), reflections (flips), or rotations (turns) to figures 
  • Make your own Tangram pieces and related challenge cards: can you use non-right triangles or non-square or rectangular pieces? 
    • 5.1 I can identify and describe different types of triangles in various contexts 
    • 5.2 I can draw and describe quadrilaterals in various contexts 
    • 5.3 I can measure angles in geometric figures and in real-world objects with a protractor 
    • 5.4 I can compare angles according to size 
  • Design a case for your pieces and challenge cards

Jul 15, 2017

Interesting article... #STEM related articles of interest

In our line of work, we comb through lots of research, studies, and articles. It all adds to the overall understanding (and strategic development of our "gap" products for STEM development), but there are a few gems that we keep finding ourselves going back to.

Engineer's Playground's Pinterest board just added a collection of such articles: Cleverly called "STEM related research and articles." The research papers may require library access but if a public access link is available, we tried to link to that one.

Apr 10, 2017

Logarithms: The #STEM Story Behind Human Calculators

This is actually from the Historical Interlude for the 4th lesson of LASER Classroom's Bringing STEM to Light. However, with the recent Hidden Figures (#HiddenFigures) movie, it seemed that this interlude seemed timely for math teachers who might want to show how math integrates (#STEM at its best).


image by Y. Ng

The importance of logarithms is not always covered in many of today’s pre-Calculus classes. This “invention” from the 1600s is mathematically useful because it allows us to use a small scale to analyze values that cover a wide range, which permits us to see patterns and relationships better.
Compare these two figures, which show how wide range of data looks when graphed on a regular scale versus a logarithmic one. The regular scale hides the details of the actual y values and for x=1 through x=7, but the log scale allows us to see how those values compare with the y values.


More importantly, in the days before electronic computers, logarithms gave humans a faster and more accurate way to do multiple-digit multiplication and division. In fact, many scientific discoveries and technological tools depended heavily on logarithms, including Kepler’s calculations of planetary orbits, British navigational charts for exploration during colonial times, and artillery trajectory charts for Napoleonic campaigns.

Today, the logarithmic scale is still present in our everyday lives through scales for sound (decibels), acidity (pH), and earthquakes (Richter).


How were logarithms developed? Since ancient Babylonian times (2000 BC), humans have looked for methods to do multiplication and division of large numbers quickly and accurately. Until the relatively recent advent of computers and personal calculators in the 1960s and 1970s, people used methods include used using addition, subtraction, and lookup tables. In 1614, Scottish mathematician John Napier published the logarithm concepts that he noticed from studying arithmetic and geometric series. In 1620, the Swiss mathematician and clockmaker Jost Bürgi published tables of logarithm values that could be used in calculations.

In 1859, a French army officer named Amédée Mannheim created the modern form of the slide rule. This toolkit for multiplication and division of multi-digit numbers, reciprocals, powers, roots, and trigonometric values. It relied on the logarithm property that logx (y) + logx (z) = logx (yz). The slide rule uses two rulers marked with a log scale (rather than a regular scale). By lining up the two rulers, you can add two log values together, as shown in the left-hand side of the equation above. If you then read the answer off the log scale rulers, you get the product of the two values (the right-hand side of the above equation). Thus, the slide rule provides a way to do multiplication of large numbers by using edition.


Today’s “modern” technological advances were first developed using only this simple tool–not the complex electronic computers that we associate with calculations day. Rocket design and the development of the atomic bomb during World War II, determining the shape of DNA in the 1950s, missions into space and to the Moon and design of the aircraft (like the Boeing 747) in the 1960s and 1970s–all of these advances were done on slide rules, using logarithms!

Jan 5, 2017

#STEM and History: Safety and Engineering

A few years back, I wrote a short book of LASER lessons for LASER Classroom called Bringing STEM to Light. It was a more STEMy way of approaching how laser technology is taught in technician classes, and the parts I loved the most were the Historical Interludes. LASER Classroom has given me permission to reprint the interludes for those of you who like to link STEM and history!

This was the first Interlude, describing the history of risk analysis and safety, important concepts in engineering that are often overlooked by science, math, and even business folks.
image by babykrul (Gölin Doorneweerd) via

Lasers are relatively new, but the idea of safety--and how to quantify it--is not. All technology comes with risks, and one of the most ancient sets of law, Hammurabi's Code, showed the consequences of unsafe technology (like buildings) by outlining punishments for builders whose houses collapsed. However, figuring out how to make things safe took time.

Step 1: Make It Work

The first step in designing safe technology was to create working technology: for example, builders had to figure out how to make buildings that actually stayed standing! In ancient Egypt, engineers such as Imhotep (c. 27th century BC) developed new construction methods and materials, eventually earning the title of the royal chief of engineering. Once experienced builders learned what worked, they passed those methods on. Imhotep wrote an encyclopedia of architecture that served as the definitive text for Egyptian builders for thousands of years. Likewise, in ancient Rome, Vitruvius's De Architectura (Ten Books on Architecture, c 15 BC) provided guidelines for buildings, hydraulics, aqueducts, roads, and even construction and war machines, and gave instructions for taking required measurements so that future builders could replicate his results.

How did these early engineers figure out what worked? They started with the Goldilocks method of "too much" or "too little," focusing on a qualitative understanding of construction rather than specific numbers. For example, in ancient China, the Kaogong ji (The Records of Examination of Craftsman, c. 210 BC), an instruction manual for building houses, vehicles, and weapons, described how to make an arrow: "The feathers at the end of the shaft are installed in three directiosn [...] When the feathers are too many, the arrows will slow down; when the feathers are too few, the arrow will become unstable."[1]

Step 2: Do It Consistently

Once engineers figured out how to make things work in general, they were able to start thinking about numbers. By quantifying how to build something, they began to quantify how to build it safely as well. In ancient Greece, records of a building constructed by Socrates (c. 341 BC) show early building requirements: "He shall set the joints against each other, fitting, and before inserting the dowels he shall show the architect all the stones to be fitting, and shall set them true and sound and dowel them with iron dowels, two dowels to each stone..."[2]  In 12th century China, the Yingzao Fashi (Treatise on Architecture Methods or State Building Standards) laid out very specific instructions for building a foundation: "For every square chi, apply two dan or earth; on top of it lay a mixture of broken brick, tile and crushed stones, also two dan. For every five-cun layer of earth, two men, standing face to face, should tamp six times with pestles, each man pounding three times on a dent..."[3]

These specific, quantified safety guidelines are the roots of modern building codes. For instance, after the great fire of London in 1664 destroyed many poorly constructed buildings, Parliament enacted building regulations to ensure that new buildings were constructed properly. Today, we have many regulations around our technology--not just our buildings, but also water and gas pipes, electricity, telephones, and even contact lenses. Each warning label represents research that was done to determine the safe operating parameters for the device or system.

If you'd like to learn a bit more about lasers and light, take a look at the LASER Classroom MOOC course, Bringing STEM to Light: Teaching About Light and Optics. And LASER Classroom has a number of free light lessons as well which you might like.


  1. "Archery in Ancient China." China Archery. 29 November 2009. Web 2 Jan 2017. <>
  2. "History of Building Codes." City of Neenah. Web 2 Jan 2017. <>
  3. "Architecture of the Song Dynasty." Wikipedia. Web 2 Jan 2017. <>