Carpenters do the best geometry

A carpenter’s measuring tricks fascinate me. These guys have figured out ways to do math without all that pesky math getting in the way.

The most recent trick I learned is a great geometry class problem: how to divide anything into equally-sized sections. For example, suppose I want to place the handles exactly 1/4 the way from each end on this drawer front.

I’ve also read that cabinetmakers use this trick to set equally-spaced dovetail joints.

You can prove the “trick” using a triangle congruence theorem1.

Better yet, there’s a tool to do the same division. Again, you can prove the tool works with a triangle congruence theorem2.

Edit: I took this in to my 9th grade students who are repeating Georgia’s Math 1 class (it’s a combination of algebra and geometry) and they crushed my excitement. Perhaps it’s just the crowd but those guys shot me down in flames. Zero interest in a cool trick. Zero interest in why it works. Honestly, I had higher hopes for “hey guys, you can divide anything — any size — into equal segments with only mental math.”

Georgia Performance Standard: MM1G3. Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons. c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, HL).

1 ASA, as shown here
2 SSS

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10 thoughts on “Carpenters do the best geometry

  1. Zero interest in a cool trick. Zero interest in why it works.

    *sigh* Sounds familiar. Switched a class from doing an interesting project to Algebra problems, and they were happier doing the problems.

  2. Glad to hear I’m not alone, Jason. I should know by now that “interesting” is in the eye of the beholder.

  3. I don’t have a solution to this problem, but I *just* this minute re-read an old post at dy/dan, where people were talking about how students behave better when they’re asked to do things that don’t involve thinking. I”m guessing these two problems are related.

    I’d love to get everyone who comments over there thinking about this as a case study. What would it take to get the students interested in something like this?

  4. Geometry _is_ doing math without doing math.

    So what does interest these kids?

  5. “Promoting Purposeful Discourse”, a new book from NCTM, gets at the heart of this. Students really do learn best through collaboration and conversation, but these skills must be purposefully taught. And it takes a lot of work and time. They need to be able to draw and interpret representations, listen to each other, ask questions, clarify, build on concepts. It takes at least 6 weeks, all of which can involve progress on standards, but too often, teachers give up way before they can reasonably expect to see results and also underestimate the planning they need to do for successful discussions.

    You can start with teacher-created work that students can analyze or critique–often a favorite activity for the math-haters–to begin to develop these skills. Which definitions can they disprove with examples? Which equations really match this representation? Which calculation method is generalizable? Etc.

  6. I run into the same problem in History classes – students would much rather be spoonfed names and dates for a “memorize & regurgitate” test than look at the ebb and flow of history. It usually takes about a month of me not teaching names and dates for them to get into the deeper way of looking at things. Once that happens, I find I get a lot more “I understand this now” than when I was just teaching the rote facts.

  7. @Jane – Thank you for the book reference. The Georgia curriculum relies heavily on the students discovering relationships and I agree with you and the author that it must be purposefully taught. I haven’t taught my kids how to do this. No one has. It’s high time I/we do start. I will begin with your suggestion.

    @Ian – Great point! About a month, eh? I think I feel a plan starting to form.

    @phil – It’s a really tough crowd to interest. They’ve failed 1st semester 9th grade math at least once, failed the 8th grade standardized test, and a not insignificant number will drop out.

    This class turns my philosophy of teaching on its head. For example, I don’t think student interest has to be about hitting something they’re naturally interested in. Instead, I think that if I make a connection with a kid, on any level, then he’s more likely to hear me out and learn something. I’ve made connections yet still struggle with engagement of any sort.

    @Sue – What does it take to get students interested? Depends so highly on the crowd. I had no problem interesting other kids with a really dull problem.

    I think the real nut of my struggle is how to engage the “repeaters”. The rough demographics are: You’ve failed math at least once, you hate math, you know you’ll never need this again.

  8. It think it’s a fascinating idea to throw something out to the class that they can trash as a way to get them engaging with the content. If they care about proving you wrong, that is. 🙂

    Promoting Purposeful Discourse sounds like Haim Ginott For Adults.

  9. I’ve been thinking about how to incorporate this into a lesson/project/investigation for a couple days now. Here’s what I’m going to try for non-academic 10th graders who have to study similar figures.
    1. I will challenge them to try to divide a piece of paper into three equal sections using only a ruler. I’ll let them flounder with trying to divide 11” by 3. I hope they’ll struggle and compare with classmates.
    2. I will show them this process.
    3. I will ask them to use it to divide a paper into four equal sections (pretending it’s a drawer so it wouldn’t be twice foldable.
    4. I will give them a long strip of cardstock called a sentence strip (ask an elementary teacher) and ask them to come up with a process to divide the width of their desks into 8 equal sections. From this, I hope that they will figure out that they have to fold the strip in half three times so that they have a relatively sturdy “ruler” of unknown length divided into 8 equal sections. Then they can draw the lines on their desk (in pencil), and get the 7 marks they need to divide it into 8 equal sections.

  10. I take Geometry right now, in the eigth grade, and let me tell you: Geometry is not doing math without doing math. You Algebra 1 teachers lied to us, telling us Geometry would be just about shapes…yes, I realize that a triangle is a shape, but I don’t need to know whether two triangles are congruent to get through life. Honestly, I agree with your 9th graders…

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