I prepared a lesson plan to work with a student. I carefully considered how I would introduce the topic, the path that the lesson might take and the questions that I would ask to prompt our discussion. I thought about the manipulatives that we might use to visualize and physically interact with the problem. I had a course carefully laid out.
I started by drawing an irregular, kidney shaped area on the desk and asked the student how he would estimate the area of the shape. I was prepared for a number of different responses that I thought I might hear... but the student didn't follow my carefully plotted course for our lesson. Instead he replied, "I'd use Pick's Theorem."
I grew up sailing on the Columbia River. When changing course on a sailboat, you can either turn the bow (the front of the boat) through the wind (i.e. tacking) or you can turn the stern (the back of the boat) through the wind (i.e. jibing). When tacking, the boom gently moves from from one side of the boat to the other. Jibing on the other hand can be dangerous as the boom suddenly jumps to the other side of the boat. When the student suggested Pick's Theorem, it felt like changing course by jibing instead of by tacking.
After our excursion through Pick's Theorem we found our way back to estimating the area with some manipulatives. First we covered the shape with square tiles and then we covered the shape with pennies.
We found that we could cover the shape with 66 square tiles. I asked the student how the area we found with Pick's Theorem and the area we found with square tiles compared. Through our discussion we decided that we needed a common way to talk about these areas so we converted both to square centimeters. We found that the area from Pick's Theorem was 382.5 cm^2 and the area using square tiles was 412.5 cm^2. Next, we looked at our penny solution. We looked up the diameter of a penny online and found that 135 pennies at 2.85 cm^2 each gave us a total area of 384.75 cm^2. While discussing how this estimate compared to our others, the student started talking about Alex Thue and his theorem on circle packing (this student has a really good memory). The student remembered that the efficiency of hexagonal packed pennies was about 91%. So we used this efficiency to correct our penny estimate to make it even better. This led to another discussion that I hadn't planned on about tesselations and polygons that tile the plane. The student said he had read in a book that there were 14 irregular pentagons that tile the plane. His book was a few years old however so he didn't know that a 15th pentagon had been discovered in 2015 or other recent work in this area.
While the lesson didn't go quite as I had planned, I was really happy to be able to take the student's contributions to the discussion and weave them into the overall narrative of our work. Being flexible, listening to students and incorporating their contributions into a discussion can sometimes throw you off course and you might end up someplace unexpected. The journey along these altered courses however can be incredible.
June is here and the end of the school year in Nova Scotia is rapidly drawing near. In grade 7 math classes, it is time for students to break out their geometry sets and practice geometric constructions. The outcome for this unit (7G01) states, "Students will be expected to perform geometric constructions, including: perpendicular line segments, parallel line segments, perpendicular bisectors and angle bisectors."
The intention of this outcome is for students to be able to describe and demonstrate these constructions using a straight edge and compass. In developing the understanding of these constructions, students are exposed to a variety of methods and tools such as paper folding, Mira, and rulers. I've written about geometric constructions in the past, so I won't get into a discussion of the merits of Euclidean constructions. My concern is that this unit could turn into a series of steps to memorize for a short list of basic geometric constructions that lacks coherence and context. If you really need to construct a heptadecagon using only a straight edge and compass, I'd expect you to look up the steps, not have them memorized. Additionally, in my opinion, the exercises in the student textbook are routine and dull. I think that this is a unit that has a lot of potential for student engagement but could easily become tedious.
Dan Meyer gave an inspiring ignite talk titled "Teaching the Boring Bits" at the 2013 CMC-North conference. In this talk, Dan encourages teachers to engage students by creating an intellectual need for new knowledge. Teachers should strive to provide students with a reason to want to know new mathematical skills and methods.
A grade 7 teacher that I know thought that incorporating Islamic geometric designs into this unit would give a purpose and context for doing geometric constructions. Another factor in selecting this context was that she has a number students who are recent immigrants from the Middle East in her class. We brainstormed some ideas and developed several activities to infuse into this unit that might help give this outcome some coherence and allow students to be creative and artistic. The teacher started by using a template and pattern from Eric Broug's School of Islamic Geometric Design. Students used the template and followed the instructions to create and colour their designs which were then tessellated in a grid to make a group composition on the bulletin board.
Later in the unit, students were challenged to construct eight-pointed stars using geometric constructions without a template (although a template could be used for students that needed additional supports). Creating this design using a straight edge and compass required students to perform the majority of the constructions required by this outcome. Students also had the opportunity to use their creativity to personalize their design and make it unique.
A number of students were very interested in creating designs of increasing complexity. They were able to pursue this to apply their geometric skills to create some very impressive designs. Have you used any creative or artistic activities to teach students geometric constructions? If so, I'd appreciate hearing about it!
This is the second year for the Math Photo Challenge on twitter. Each week a new challenge topic is presented. During the challenge, you can follow @mathphoto16 on twitter to see the weekly prompt. You can also visit https://mathphoto16.wordpress.com/challenges/ to see the weekly prompt. Participants post photos on twitter and use the hashtag #mathphoto16 and the hashtag for the weekly prompt. This year, the challenge is being hosted and organized by Amie Albrecht (@nomad_penguin) and John Rowe (@MrJohnRowe) from Australia. Last year, I sat back and watched as people tweeted photos but this year I decided to participate and encouraging other teachers and students to do so as well. With this post, I'm going to recap the challenges and the photos I tweeted throughout the summer.
The Week #2 (June 19 - 25) challenge was #scale. Scale is a ratio that measures the relative size of two objects. We encounter scale in our daily lives such as longer/shorter, faster/slower and bigger/smaller. I posted two photos from the Halifax Stanfield International Airport baggage carousels. Each of the three baggage carousels in the domestic arrivals area has a scale model built by Anchor Models. The MacKay and Macdonald Bridge models are 47 ft long and have a scale of 1/60. The model of the Halifax Town Clock has a scale of 1/12.
The Week #3 (June 26 - July 2) challenge was #lines, curves and spirals. I posted a photo of a pediment portico and eyebrow dormer from a house in the South End of Halifax. I really like this style of architecture and it is pretty common on the peninsula in Halifax. A few weeks later, I was at the Halifax Central Library and took a photo of the lines and shadows in the buildings central atrium.
The Week #4 (July 3 - 9) challenge was #multiples. Many objects can be arranged in multiples and groups. This helps us count them more easily and share them out equally. I posted two photos for week 4. The first is an arrangement of tea cups from a tea house in Beijing. The second is a picture of the 3 chimneys from the Tufts Cove Generating Station on the Halifax Harbour. I like how each of the 500 ft tall smokestacks are partitioned into alternating white and red bands.
The Week #5 (July 10 - 16) challenge was #zero. Without zero, mathematics would be nothing. Search out the concept of zero. Be creative in your interpretation. I posted a photo of the sign for Exit 0 on Nova Scotia Highway 102. While most Canadian provinces use kilometre based exit numbers, Nova Scotia uses sequential exit numbers. The only other Canadian province which uses sequential exit numbers is Newfoundland and Labrador.
The Week #6 (July 17-23) challenge was #infinity. Infinity characterizes things that never end. Look for interpretations of infinity. This was a difficult concept to capture with a photo and there were relatively few photos submitted on Twitter. I posted two photos. The first depicted how the imagination is infinite. A toy such as a set of blocks can be used for an limitless number of different games and activities. The second photo I posted was from the television show The Big Bang Theory. In the episode titled "The Friendship Algorithm" (S2, E13), Sheldon gets stuck in an infinite loop in his algorithm.
The Week #7 (July 24-30) challenge was #shapes. Shapes enclose the space around us into areas, in both regular and irregular ways. Along with single shapes, look for congruence, similarity, tessellations and more. I posted a photo that I took of the The Beijing National Aquatics Center. The bubble like geometric shapes covering the exterior of the building are based on the Weaire–Phelan structure.
I really enjoyed participating in this photo project and seeing all of the interesting and creative photos that were posted by people from all around the world. It gave me a purpose to view my community through a mathematical lens and connect mathematical ideas to the world around me.
I've been taking a close look at geometric constructions in the Nova Scotia mathematics curriculum. There is one outcome involving geometric constructions in the NS curriculum. Grade 7 outcome G01 states that students, "will be expected to perform geometric constructions, including: perpendicular line segments, parallel line segments, perpendicular bisectors and angle bisectors." The curriculum guide goes on to define constructions as "performed using a compass and straight edge (without markings) as tools." This is the only outcome that I am aware of that has a focus on the tool being used and not on the mathematical understanding to be achieved. I'm not sure I understand this focus on the tool. Especially such an archaic tool. I can almost hear the students...
Student: Why do I have to use a compass and a straight edge to draw this equilateral triangle? Wouldn't it be easier to use a ruler and a protractor?
I worked with a bunch of elementary teachers on these compass and straight edge constructions and they found them frustrating and tedious. When we performed the same constructions using patty paper and dynamic geometry software, the teachers found the constructions less confusing, enjoyed them more, and seemed to get a better understanding. So I asked, "why is this outcome so focused on a specific tool?" If we have efficient and accurate modern tools, why do we still teach students to use compass and straight edge? Is it more "elegant and civilized"... like learning Latin even though no one really speaks Latin anymore? Anyway, I love a good Star Wars analogy, so....
In grade 6, students look at transformations of 2-D shapes to create a design. In this outcomes, G04, pattern blocks are used as a tool to explore tesselations and the geometric concepts that result from these designs. The outcome does not talk about the tool being used or tesselations. These discussions are saved for the background of the outcome and its indicators. I recently read an article talking about how students explore tesselations and how their interest can be sparked by creating aesthetically pleasing designs. The author stated that, "tiling provides one of the richest possible studies of geometric concepts as well as a powerful topic for open-ended mathematical exploration, fostering important mathematical practices. For students, mathematical aesthetics can be a guiding force for mathematical inquiry, just as it is for mathematicians." from R. Scott Eberle. ““I Don't Really Know How I Did That!””. Teaching Children Mathematics 21.7 (2015): 402–411.
Ultimately, this is what I hope teachers focus on when teaching geometric constructions in grade 7. I'd like to see them work with students on open-ended mathematical explorations of geometric shapes and properties using a compass and straight edge as an entry into this exploration. Memorizing the steps to do geometric constructions should not be the focus of this unit. The focus should be the understanding that comes from working with shapes regardless of the specific tool that is used, exempli gratia compass and straight edge, patty paper, or dynamic geometry software (notice how I used a bit of Latin there?).
So why are Euclidean constructions an outcome in the NS mathematics curriculum? That is a good question. Neither the curriculum guide nor the textbook mention the history of compass and straight edge constructions or why students should be practicing them. Here is my hunch... Let's start with some history. Geometry was an integral part of the classical liberal arts education going all the way back to the middle ages and the Qradrivium. Much of the geometry education at this time consisted or reproducing the propositions from Euclid's Elements. Many of the proofs in Euclid's Elements were proven with geometric constructions using a compass and straight edge. Because of this, constructions were an integral part of the curriculum. It was the tool by which Euclid's postulates were both reproduced and proven. Over time, many parts of the classical curriculum have disappeared from modern education. Euclidean constructions however have long had their staunch supporters and remain. Here is a quote from D.E. Smith's book The Teaching of Geometry (1911), "Constructions excite students’ interest, guard against the slovenly figures that so often lead to erroneous conclusions, has a genuine value for the future artisan, and shows that geometry is something besides mere theory."
Here is a quote from "Ask Dr. Math" at the Math Forum, that I think best elucidates the reason for constructions in our curriculum.
"Certainly learning how to use the tools is useful. Some of the techniques are useful in construction (of buildings, furniture, and so on), though in fact sometimes there are simpler techniques builders use that we forget to teach. But I think the main reason for learning constructions is their close connection to axiomatic logic. If you haven't heard that term, I'm talking about the whole idea of proofs and careful thinking that we often use geometry to teach. Euclid, the Greek mathematician who wrote the geometry text used for centuries, stated many of his theorems in terms of constructions. His axioms are closely related to the tools he used for construction. Just as axioms and postulates let us prove everything with a minimum of assumptions, a compass and straightedge let us construct everything precisely with a minimum of tools. There are no approximations, no guesses. So the skills you need to figure out how to construct, say, a square without a protractor, are closely related to the thinking skills you need to prove theorems about squares."
If you have any insight into Euclidean constructions and their place in the mathematics curriculum, I would love to hear your opinions.
Nova Scotia Mathematics Curriculum Outcomes
Grade 6 G04 - Students will be expected to perform a combination of successive transformations of 2-D shapes to create a design and identify and describe the transformations.
Grade 7 G01 - Students will be expected to perform geometric constructions, including: perpendicular line segments, parallel line segments, perpendicular bisectors and angle bisectors.
I created the above WODB question for a group of elementary teachers. I spent quite a bit of time creating and revising it (you can see my drafts below). Making a good WODB problem takes a bit of practice I've found. I'm a bit embarrassed about how poorly designed (and sometimes goofy) some of my earlier attempts were.
Here are some suggestions why each image might not belong:
Top Left: It is the only shape with more than one interior shape (a triangle and two trapezoids). It is the only shape that is not composed of congruent shapes. The only shape without rotational symmetry.
Top Right: It is the only shape with 4 interior shapes. It is the only shape whose interior shapes are all similar to its exterior shape. It is the only shape with a single angle measure (60 degrees).
Bottom Left: It is the only shape with right angles in it. It is the only triangle that is point down (it tips over).
Bottom Right: It is the only shape which includes concave shapes. It is the only shape with interior reflex angles in it. The only shape without reflective symmetry.
Updated 9/2/2016 - Added some additional suggestions regarding symmetry from Jennifer Bruce.
I've been working with Van Hiele's levels of geometric thinking lately and how it relates to the geometry outcomes in the NS Mathematics curriculum. This got me thinking about how a student's level of geometric thinking relates to WODB problems. In designing the problem above, I was intentionally trying to create something aimed at students at Van Hiele's descriptive level vice the visual level. Given a student's level of geometric thinking, how might they respond to this problem? Are some WODB problems better suited for students at a visual or descriptive level or are WODB problems inherently differentiated and students at all levels can relate meaningfully to these style of problems.
Another question that I've been thinking about recently is how to use WODB questions. Should I use them to introduce new terminology or to review terminology that students have already been exposed to. For example, in the problem above, I made sure to include a concave shape. Would it be more effective to wait until students learn concave and convex polygons and then reinforce it with this problem or should I introduce the vocabulary here and then talk about its definition? These are questions that I'm still exploring.
Nova Scotia Mathematics Curriculum Outcomes
Grade 2 G03 - Students will be expected to recognize, name, describe, compare and build 2-D shapes, including triangles, squares, rectangles, and circles.
Grade 3 G02 - Students will be expected to name, describe, compare, create, and sort regular and irregular polygons, including triangles, quadrilaterals, pentagons, hexagons, and octagons according to the number of sides.
Grade 4 G02 - Students will be expected to demonstrate an understanding of congruency, concretely and pictorially.
Grade 5 G02 - Students will be expected to name, identify, and sort quadrilaterals, including rectangles, squares, trapezoids, parallelograms, and rhombi, according to their attributes
Grade 6 G01 - Students will be expected to construct and compare triangles, including scalene, isosceles, equilateral, right, obtuse, or acute in different orientations.
Grade 6 G02 - Students will be expected to describe and compare the sides and angles of regular and irregular polygons.
Grade 9 G02 - Students will be expected to demonstrate an understanding of similarity of polygons.