geometry

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Lesson planning: Homeschooling my son

My wife and I are homeschooling my son, and so I work with him on his mathematics. For the first part of the year, our experiences have been fairly informal, but I feel like he and I will benefit from more structure, and so I’m planning on writing lessons and keeping track of the resources we have used, as well as reflecting on those lessons to help me plan what’s next. My son’s number sense and understanding of number operations is fairly strong. Our work here will continue informally, but for at least the next month, I’m going to focus on geometry with him.

The Common Core cluster standard in grade 2 geometry is:

2.G.A. Reason with shapes and their attributes

And the specific standards in this cluster are:

2.G.A.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces (Sizes are compared directly or visually, not compared by measuring.). Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

Example task from Illustrative Mathematics:
http://www.illustrativemathematics.org/illustrations/1506
Example MARS task: Don’s Shapes

2.G.A.2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Example task from Illustrative Mathematics:
http://www.illustrativemathematics.org/illustrations/827
Example MARS task: Half and Half

2.G.A.3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Example task from Illustrative Mathematics:
http://www.illustrativemathematics.org/illustrations/826
Example MARS task: Half and Half

My guess is that my son would find all of these tasks fairly easy to do, but I plan on spending our first few days together doing these tasks, and seeing if I can surface my son’s understandings and misunderstandings related to these standards. In terms of a more open-ended project, I suspect that the creation of mathematical art with polygons may be appropriate. For example, my son and I recently created as many hexagons as we could using his pattern blocks (We made one mistake. Can you find it?).

Hexagons

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Open ended problem: Dots and lines

In this video James Grime examines the “challenging” math problem given in the movie Good Will Hunting and points out that it is not actually all that challenging. Unfortunately he is pressed by the person interviewing him to give all of the solutions to the dots and lines problem given.

This problem could easily be extended to be more open-ended simply by leaving the number of dots open. Are there any patterns when you generate diagrams with 2 dots, 3 dots, 4 dots, 5 dots, and so on? What kinds of diagrams are essentially the same (homeomorphic)? What kinds of diagrams cannot be made more simple without changing the character of the diagram (irreducible)?

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Open-ended investigation: Fold and cut

If you can fold a piece of paper anyway you like and as many times as you like, and then take a pair of scissors and make just one straight cut, what possible shapes can you make?

For example, I’ve folded a piece of paper (shown below) three times.

Three folds

Now I make the following cut…

Single straight cut

What shape will this result in when I unfold the paper and lay it flat?

* I first learned about this idea from the movie “Between The Folds.” Here are a few more examples from Dr. Erik Domaine’s website.

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Geometry book

This submission comes from Malke Rosenfeld. It is work her daughter produced independently, at age 6, while playing with some tangrams.

Here’s what I found my dear child doing with the set of car tangrams (inside for some reason) over the last few days.

By way of setting the scene, I should tell you that the kid generally initiates and works independently on any project she pleases during her afternoon quiet time, usually with the company of an audio book. A few days ago during this time she informed me she was “making a math book.” I had observed her tracing tangrams, but was surprised to find that she was doing it in a deliberate way, and that she was also able to clearly describe her discoveries and observations to me. Here is the book, so far. I am just a scribe here — these are her words as she dictated them to me, except where noted:

Math book 1

“Four of these triangles that you see here can make a square. If you pull these triangles apart you can see that they’re little triangles. But you can see on this page that they make a square.” [I see that she has drawn them as individual shapes, which, over the course of her illustration, merge together into her intended shape.]

math book 2

“This rectangle you see is made up of a parallelogram and two triangles. Really they’re just shapes, but when you put them together they make a rectangle.” [It looks like she’s numbered the inside angles of the individual shapes. It also looks like she is again showing the process individual shapes merging into the intended new shape.]

math book 3

“You see the wheels of this bike as rhombuses but really they’re squares turned so their points are facing up and down, and to the side.” [She was gesturing this first, and at first she used the word ‘flipped’ to describe the orientation of the square wheels. I focused her on the orientation of the corners to describe how the square was turned.]

math book 4

“The square and the rhombus that you see here, their edges are both the same length. The difference is a rhombus is a squished square, squished to its side. The rhombus has two larger angles and two smaller angles than the square. But the square has the same angles on each corner.” [These are actually tracings of shapes from the pattern blocks set we have. The ruler markings was her idea for comparing the two shapes. I supplied some new vocabulary in the form of ‘angles’ and made some observations about the difference by overlapping the two shapes, in an effort to help her observe and further articulate the difference between the two.]

I think this is a terrific example of some early geometric reasoning. Clearly Malke’s daughter has learned some mathematical vocabulary from her parents, and has the time and space available to do this kind of exploration. What other things have to be in place for children to be able to make these observations?

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Open ended problem: Parking lot

Parking Lot

I thought of this investigation as I was walking home and happened to notice an unusually shaped parking lot. It made me wonder if the way the parking spots were arranged within the parking lots was the most efficient way to pack the cars in. Obviously the picture above is a sample, and you are free to choose whatever parking lot you would like to share with your students.

Question: What is the most efficient way of creating parking spots within a parking lot so that as many cars as possible are able to park in the parking lot, and so that each of these cars can leave whenever they want?

Here is a Geogebra applet I designed to allow students to explore different arrangements themselves.