CyberMuse Teachers - Lesson Plans
Seeing Math in Art
Lesson Plan Activity:
Triples, Triples, Triples: Grade 7-8
Students will look at the different pieces of the installation With Reference to Pythagoras, including the initial sketches and designs, and then consider the different types of right angle triangles that would work in the dimensions presented in the sketch accompanying the sculpture.
Students will learn the Pythagorean Theorem, both geometrically and numerically.
Students will represent transformations using the Cartesian coordinate plane, and make connections between transformations and the real world.
Students will solve problems involving right triangles geometrically, using the Pythagorean theorem.
Cross Curriculum Links:
One 50-minute lesson, including discussion and reflection (2 if they will be building the triples).
Look & Discuss
Present and discuss Henry Saxe?s With Reference to Pythagoras. (Tabs will provide you with information on the theme, composition, interpretation and the artist.)
A downloadable Presentation that you can add to or manipulate will also help share these images in your classroom.
- Blank paper
- Square pieces ? different colours if possible
- Graph paper
- Prepare copies of the painting, ideally one for each student.
- Assemble the necessary materials, particularly the square pieces.
- Photocopy questions and examples, if desired.
1) Identify the various component pieces of the installation. Be sure to look at the preliminary studies for this work (Notebook First Sketch, Maquette, Studies) in order to look at all the components.
2) Identify the title?s reference and why do you think the artist chose this title?
3) Review the statement ?all is number?. How has this been interpreted and what do you think this phrase means?
4) What is a Carpenter?s Square and what was it used for? Is it still used today? Hint: Look at a store bought math geometry set ? you will be surprised by what you see!
Notebook and First Sketch for "With Reference to Pythagoras", 1974
© Henry Saxe
In the sketch from the Notebook, the artist draws an initial Carpenter?s Square. The dimensions for the two legs are 57? and 98?. The hypotenuse is labeled 112?.
a) Check using Pythagorean Theorem how far those measurements are off, being as accurate as possible.
b) Express your answer as a percentage error of the true measure.
Maquette for "With Reference to Pythagoras", 1974
© Henry Saxe
Looking at the supportive studies for this work, you can see that the slats on the bottom of the piece begin with a Carpenter?s Square. Imagine that the largest of the Carpenter?s Squares is the one illustrated in the Notebook.
In mathematics, a Pythagorean Triple is a set of three positive integers a, b, and c, such that a right-angled triangle exists with legs a, b, and hypotenuse c. Another way of saying this is that there are three positive numbers a, b, c that satisfies Pythagorean Theorem. The hard part of these is making sure that they are integers (that is they work out ?cleanly? with no need for decimals or rounding).
How many Pythagorean Triples are there with legs less than 57 and 98? Imagine these Pythagorean triangles working as Carpenter?s Squares in the pile of Notebook One.
Hint: One common triple is (3, 4, 5) and from this one you can get a whole bunch more including (6, 8, 10) and (30, 40, 50). Using a similar strategy can help you find the basic ones and then multiply them until you run out of space.
Study for "With Reference to Pythagoras", 1974
© Henry Saxe
One side of the installation has a grid pattern. Indicate using integer translations, how many ways there are to get from the bottom left corner to the top right using the grid pattern as indicated there. You may want to use the graph paper to illustrate that you have them all.
Study for Column Section "With Reference to Pythagoras", 1974
© Henry Saxe
Take it Further
Students can build the Pythagorean triples actually with paper, based on the triples that students have created. Students can also consider which other transformations can occur on the grid in order to get from one corner to another.
The student demonstrates limited understanding of Pythagorean Theorem.
The student demonstrated some understanding of Pythagorean Theorem.
The student demonstrates a thorough understanding of Pythagorean Theorem.
The student illustrates some paths using Cartesian coordinate plane, but does not indicate any path properly.
The student illustrates most paths using Cartesian coordinate plane, but has a few errors recording the paths.
The student illustrates all paths using Cartesian coordinate plane, and has only minimal errors recording the paths.
The student has difficulty applying Pythagorean theorem to problems.
The student applies Pythagorean theorem, with some consistency, to problems.
The student applies Pythagorean theorem correctly in all problems.