TEACHER GUIDE TO CLARIFICATION
Understand and apply the Pythagorean Theorem.
8.G.6 Explain a proof of the Pythagorean Theorem and its converse.
Explain a Proof of the Pythagorean Theorem
Students explain the Pythagorean Theorem as it relates to the area of squares coming off of all sides of a right triangle. Students also understand that given three side lengths with this relationship forms a right triangle.
Students are explaining a proof of the Pythagorean Theorem, not proving it.
Students should verify, using a model, that the sum of the squares of the legs is equal to the square of the hypotenuse in a right triangle. Students should also understand that if the sum of the squares of the 2 smaller legs of a triangle is equal to the square of the third leg, then the triangle is a right triangle.
Previous understanding of triangles, such as the sum of two side measures is greater than the third side measure, angles sum, and area of squares, is furthered by the introduction of unique qualities of right triangles. Students should be given the opportunity to explore right triangles to determine the relationships between the measures of the legs and the measure of the hypotenuse. Experiences should involve using grid paper to draw right triangles from given measures and representing and computing the areas of the squares on each side. Data should be recorded in a chart such as the one below, allowing for students to conjecture about the relationship among the areas within each triangle.
Measure of Leg 1
Area of Square on Leg 1
Measure of Leg 2
Area of Square on Leg 2
Area of Square on Hypotensuse
Students should then test out their conjectures, then explain and discuss their findings. Finally, the Pythagorean Theorem should be introduced and explained as the pattern they have explored. Time should be spent analyzing several proofs of the Pythagorean Theorem to develop a beginning sense of the process of deductive reasoning, the significance of a theorem, and the purpose of a proof. Students should be able to justify a simple proof of the Pythagorean Theorem or its converse.
Time should be spent analyzing several proofs so students develop a sense of deductive reasoning, see the significance of the theorem and understand the purpose of a proof.
Previously, students have discovered that not every combination of side lengths will create a triangle. Now they need situations that explore using the Pythagorean Theorem to test whether or not side lengths represent right triangles. (Recording could include Side length a, Side length b, Sum of
Right triangle? Through these opportunities, students should realize that there are Pythagorean (triangular) triples such as (3, 4, 5), (5, 12, 13), (7, 24, 25), (9, 40, 41) that always create right triangles, and that their multiples also form right triangles.
Students should be aware of the Pythagorean triples and their multiples.
Popular triples are:
3, 4, 5 5, 12, 13 7, 24, 25
Students should see how similar triangles can be used to find additional triples. Students should be able to explain why a triangle is or is not a right triangle using the Pythagorean Theorem.
The Pythagorean Theorem should be applied to finding the lengths of segments on a coordinate grid, especially those segments that do not follow the vertical or horizontal lines, as a means of discussing the determination of distances between points. Contextual situations, created by both the students and the teacher, that apply the Pythagorean theorem and its converse should be provided. For example, apply the concept of similarity to determine the height of a tree using the ratio between the student's height and the length of the student's shadow. From that, determine the distance from the tip of the tree to the end of its shadow and verify by comparing to the computed distance from the top of the student's head to the end of the student's shadow, using the ratio calculated previously. Challenge students to identify additional ways that the Pythagorean Theorem is or can be used in real world situations or mathematical problems, such as finding the height of something that is difficult to physically measure, or the diagonal of a prism.
Kansas Association of Teachers of Mathematics (KATM) Flipbooks. Questions or to send feedback: email@example.com. Retrieved from: http://katm.org/wp/wp-content/uploads/flipbooks/8thFlipFinaledited.pdf
Do not just apply the Pythagorean Theorem.
Bill McCallum says “Seeing that something is true is not the same as seeing why it is true.”
Coherence and Connections: Need to Know
Examples of Major Within-Grade Dependencies
Much of the work of grade 8 involves lines, linear equations, and linear functions (8.EE.5-8; 8.F.3-4; 8.SP.2-3). Irrational numbers, radicals, the Pythagorean theorem, and volume (8.NS.1-2; 8.EE.2; 8.G.6-9) are nonlinear in nature. Curriculum developers might choose to address linear and nonlinear bodies of content somewhat separately. An exception, however, might be that when addressing functions, pervasively treating linear functions as separate from nonlinear functions might obscure the concept of function per se. There should also be sufficient treatment of nonlinear functions to avoid giving students the misleading impression that all functional relationships are linear (see also 7.RP.2a).
Examples of Opportunities for Connecting Mathematical Content and Mathematical Practices
The Pythagorean theorem can provide opportunities for students to construct viable arguments and critique the reasoning of others (e.g., if a student in the class seems to be confusing the theorem with its converse) (MP.3).
PARCC Model Content Frameworks: Mathematics Grades 3-11 (version 5). (2017, November).
Retrieved from https://files.eric.ed.gov/fulltext/ED582070.pdf
Illinois Assessment of Readiness Mathematics Evidence Tables.
Evidence Statement Text
Apply geometric reasoning in a coordinate setting, and/or use coordinates to draw geometric conclusions. Content Scope: Knowledge and skills articulated in 8.G.B
i) Some of tasks require students to use the converse of the Pythagorean Theorem.
Also check out Student Achievement Partners Coherence Map.
MARS, Shell Center, Proofs of the Pythagorean Theorem: This formative assessment lesson gives students an opportunity to produce and evaluate geometric proofs.
This website offers 105 proofs of the Pythagorean Theorem. Students could pick a proof and present it to the class.
1. Can you name some Pythagorean Triples? Prove one by drawing it on grid paper.
Some examples are:
3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25 20, 21, 29
2. If you know any two sides of a triangle, can you always find the third side length without measuring?
Make sure students know that the Pythagorean Theorem is only used with right triangles.
3. Is a triangle with lengths 10, 24 and 25 a right angled triangle?
4. Use the following picture to help explain the proof of the Pythagorean Theorem. Guide students to fill in the blanks.
Each side of the big square is ______(a + b)__________.
Area of the big square is ___(a + b)(a + b)___________.
Adding up the area of the smaller pieces:
Area of the yellow square is _____c2______.
Area of each blue triangle is ___ ½ab ___.
Area of 4 blue triangles is ____ 2ab _____.
Area of big square should equal area of pieces:
*Students will need help expanding (a + b)(a + b)
Subtracting 2ab from each sides leaves DONE!
5. Is the triangle, surrounded by the squares, a right triangle? Explain your answer.
Desmos - Classroom Activity