Perimeter, Area and Pythagorus 

Introduction:

• This is an example of a third grade teacher trying to review a lesson on perimeter, while presenting a new lesson on area.
• There was a problem with a square and two adjacent halves, cut diagonally, slanting away from the top corners of the square. It was designed to show two square units.
• The teacher also wanted them to find the perimeter, forgetting that the hypotenuse would be longer.
• Then he came up with a way to "solve" the problem, using LOGO and a large monitor, as a way of presenting the "proof" to the whole class.

LOGO activity to prove a² + b² = c²

Grade Level: 4-5

Prior knowledge:

• Some knowledge of perimeter, area, and multiplication
• Ability to use a calculator, to find sums, products, and square root.
• Also, the ability to put values for variables into a formula.
• Knowledge about right angles, and the concept that all angles in a triangle will add up to 180 degrees.

Task:

• Download the appropriate public domain LOGO software from the website.
• Do the worksheet using LOGO (commands/ hints on separate linked web page).
• Go to the Pythagorus web page to see their solution.

Resources:

• LOGO foundation main page
• Get software (public domain LOGO programming language)
• Picture of Pythagorus
• Worksheet for "Perimeter, Area and Pythagorus (html) / (pdf)

The Process:

1. Do the LOGO procedure. (BACK 100 LEFT 90 FORWARD 100 RT 90 FORWARD 100 RT 135 FORWARD 100)
2. The lines will not meet. They are all 100 units long. Moving the decimal two spaces makes each one equal 1.00 units.
3. (HIDETURTLE) move the line forward in small increments until the lines almost meet, then move only one (turtle) unit at a time until they do touch. Add all of those to the initial 100. (The total should be 141.) Move the decimal place also so that it is 1.41 units. a² + b² = c² is the formula so put in the values for "a" and "b".
4. 1*1 +1*1 = 2 1.41 is the square root of 2. (Do this on the calculator.) So, the formula holds true (in this case anyway).

Learning Advice:

Have students learn simple calculations with calculators:

• Add a series of numbers
• Subtract numbers from other larger numbers
• Multiply like numbers (single digit), and:
• find the product
• find the square root of each product
• Multiply unlike numbers (single digit), and:
• find the square root of those products
• then re-multiply that amount by itself and look at the product.

Evaluation:

Look at the LOGO design. Check the "c" amount. Look at the worksheet.

Extensions:

1. Redraw new triangles, using a similar procedure, but change the "a" and "b" sides to 80, 90, 120, etc.
2. Put these values into the formula and predict how many turtle spaces will be needed to close the figure. Remember to move the decimal place over to the left to eliminate the zero in the units place.

Conclusion:

This was designed to show, as low as third grade, that the diagonal side of half of a square is MORE than the length of each side. So, if you are trying to review perimeter, DON'T use half squares until you are prepared to use this lesson!