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The Family Math Day setting is an ideal way to show parents what "cooperative learning" is all about. I set out to introduce 1st and 2nd graders and their parents to the literal meaning of geometry, that is, "geo" as in "earth," "metry" or meter as in "measure. I began by challenging the children to connect 2 dots (points) on the blackboard with thelongestline they could make.After a few children performed this task, we observed that the

shortestline between 2 points is astraightline. We observed that the straightest straight line was made with astraight edge. Solving this challenge, I said, required a skill calledvisualization.I showed a drawing by Saul Steinberg, which, I explained, began with a geometric line made with a straight edge. The artist made three "visualizations" of what the straight line could become. Parents and children were invited to draw a horizontal straight line with straight edges provided for them and see what they came up with.

Everyone created a picture from the single straight edge line they made. It was pointed out that everyone has powers of imagination, and each and every person shows his uniqueness in making visualizations. The straight line can have different meanings, depending on the configuration of the whole picture.

* * * * * I then asked the children if they could visualize: (a) A shape hidden in a pattern; (b) A way to make a loop of rope into a geometric line or shape.

After we observed a child stretch the rope between 2 points, we saw that it was a straight line, the shortest line, and that changing its position did not change its property as to being straight. We observed 3 positions:

vertical

We observed that the loop of rope held at three points had more space inside than at two. I asked children to come up and measure the interior of the triangle by seeing how many children could stand inside without bending the straight lines.

horizontal, and

diagonal.

Given the same loop of rope, could more children fit inside if there were four points?We required some parents to step inside because we exhausted the number of children. Would there be more space inside with a five-sided geometric shape (

polygon)? We found out that there was! We saw that the more points and sides, and the closer the shape came to a circle, the roomier it got.

* * * * * The question was then posed: If we are to use a straight edge to measure anything, how do we choose a

unit of measure? It was observed that we needed to agree on a length of straight line that we could all agree was a ONE.In ancient Egypt the Pharaoh made a length of string stretching from his elbow to his middle finger a unit of measure. He called it One Cubit. We observed that the cubit was a bit too long for measuring little things or differences in size that didn't come out exactly to a cubit.

Is there a way to divide the cubit into smaller and equal parts?On each desk was a strip of paper 2" X 11". We decided to call the length of these strips a "Unit of One," since as a unit of measure all of them were the same size. We observed that, as with the rope, we could fold the paper strip into equal parts: halves, quarters, eighths, and even sixteenths. A look at any ruler verifies that the unit of one inch is divided in the same way. Is this how the inch historically got its divisions?

We observed that we could determine, by matching straight line segment units, whether a quadrilateral might actually be a square! Some rectangles looked square but were not. We talked about the ancient Egyptians using knotted rope to lay out the foundation for a pyramid.

* * * * * On every desk was graph paper (a grid of squares). I demonstrated that I could color in squares to visualize a square made of squares.

What was the largest square the children and parents could lay out with their straight edges on the graph paper grid of squares?With so large a square, how could we be sure it was truly a square and not a rectangle?

One method to find out is to cut out and fold it along two diagonals. By doing so we observe whether or not all four sides are equal in length.

Next we explored the idea of square numbers. We colored in squares to find out if there were such a thing as certain numbers of [unit] squares that will make a [composite] square. We colored in a "one square square," then a "four square square," etc. Some numbers of squares formed rectangles. Some numbers of [unit] squares (like 7) didn't make a [composite] square or a rectangle. We discovered:

- In all squares, there are as many vertical squares as horizontal squares.

- Whether we count by horizontal rows or vertical rows of squares, the sum of the squares is the same as the product of the number of squares seen either vertically or horizontally.
We inducted into the

Secret Square Societychildren who were able to provide the secret "square number" as their password. This process of discovering numbers that yield squares and rectangles provided us with another very important concept:Measurements on paper or any flat surface (a plane) are made in at least 2 directions - vertical and horizontal.

* * * * * In the finale of the family day one-hour session, parent and child collaborated in the task of enlarging a picture given on a square grid of dots. They were given a picture in a "25 square square" and asked to color and enlarge it in a "100 square square." This culminating activity would demonstrate how children and their older family members could combine their abilities to work out a problem together.

How were they going to match the horizontal and vertical number labels so that the large and small squares would correspond? The proof of their understanding would be not only in successfully copying the given drawing to scale, but afterward in creating a drawing of their own and enlarging it.

All were given extra copies of the dot grid to be taken home.

* * * * * In summarizing, I reviewed a few key ideas for the parents so they could appreciate the pedagogical theory behind the lesson I had given:

Norman Shapiro

- Concrete experiences sequenced in an orderly way develop children's powers of perception and visualization.

- Words and vocabulary grow out of the hands-on activities. They are given on a need-to-know basis.

- Art activities provide a priceless, non-threatening way to motivate and activate children to learn. The end product is the one the children can most identify with. Student art gives the teacher an easy-to-read tool to diagnose achievement and levels of learning. Most importantly, art encourages children to learn by finding it out for themselves. When it comes to art, most children will want to "get it right," and will think it is within their power to do so.

May 1993

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