Name: Akbas, Hasna
Grade Level(s): 9 - 12
Seating Arrangement: Groups of four at tables (heterogeneous by ability and student demographic factors)
Concept: Geometry: Pythagorean Theorem and Euclidean Equations
Objective(s):
- To see where the Pythagorean Theorem comes from.
- To lead students for making mathematical proof.
- To apply Pythagorean Theorem to solving abstract mathematical problems.
State Standard(s) Addressed:
Mathematics:
- Process Standard: 6.1, 6.2, 6.11, 6.13, 7.1, 8.10, 9.2
- Content Standard: 4.12.6, 4.12.7, 4.12.9
Rationale:
The National Council of Teachers of Mathematics (2000) says, 'Mathematics should make sense to students; they should see it as reasoned and reasonable' (p.342). Giving formulas directly to students without proving is not a way that builds up a reasoning skill. Although the level of making proof changes up to education level, high school is the time for making some mathematical proofs because mathematic knowledge of students is deepest in high school. Allowing students to think about ways of proving a theorem would help them to improve their mathematical thinking and problem solving skills.
Making proof is mostly used while studying on pure mathematics. This lesson may give an idea to students about continuing their educational life on mathematics. They have the chance to use their knowledge they have from proof to abstract problems. This makes students to see if they can also use it efficiently. Additionally, using calculator for verifying their solutions would better their ability of using calculator.
Prerequisite Knowledge:
Students must:
- know playing with manipulative,
- know the area formulas of triangle and square,
- know the properties of triangles, including congruency,
- know solving second degree equations.
Unit Fit:
This lesson on Pythagorean Theorem fits into a larger unit on making proof for a wide mathematical concept. Students would have addressed to prove only one theorem but then they will address to use a couple of theorems which they proved themselves on the same problem. They will be able to make connection between their knowledge.
Subject- Area Content:
- Making proof includes thinking all possible ways to reach what you are looking for with using what you have in your hands, like manipulative and algebraic knowledge.
- Students can reach general results by using symbols instead of numbers in formulas they learned at earlier grades.
Vocabulary:
Hypotenuse: The side of a right (90 degree) triangle that is across from the right angle.
Pythagorean Theorem: If a triangle is a right triangle, then the square of the length of one leg added to the square of the length of the other leg is equal to the square of
the hypotenuse.
Right Triangles: A triangle in which one angle is equal to 90 degrees.
Materials:
Manipulative blocks set (1 box per group)
Calculators (2 per group)
Paper proof set (1 per group-It can be designed in two different ways as attached. Big square can be a colorful paper and small right triangles can be white or vise versa.)
Teacher handout (1 for only teacher)
'Euclidean Equations' worksheet (1 per student)
'Special Triangles' worksheet (1 per student)
'First Level Triangles' worksheet for Gearing Down (1 per student)
'High Level Triangles' worksheet for Gearing Up (1 per student)
'Challenging Question' (1 for teacher)
Prior Preparation:
Put the manipulative block sets, 2 calculators, and 2 proof sets per group. Gather and prepare worksheets as listed above.
Lesson:
A. Discuss the following questions with the class.
1. Did you ever prove any mathematical statement?
2. Which steps do you think you should follow?
B. Tell students that in today’s lesson they will try to prove Pythagorean Theorem and Euclidean Equations by using manipulative block sets and paper sets combining with algebra.
A. Have students work in pairs within their groups of four. Give one block set and one paper proof set per group, and tell students that pairs in groups will study on different proving styles as manipulative and paper set.
B. Pairs who finish their thinking should discuss their way of proof with another pair of their group.
C. Ask for volunteers from groups to share their solution. They can show them on blackboard or on an overhead transparency. Other students should provide feedback on the accuracy of the student’s proving method.
D. Show students 'Teacher Handout' paper to see available other proofs on overhead transparency.
E. Give two calculators to one group and write the sides of special triangles as 3-4-5, 5-12-13, 8-15-17 on board. Ask students to check with their calculators whether those numbers and multiple of those number sets with any number provide Pythagorean Theorem.
F. Distribute 'Euclidean Equations' worksheet per student and give students a short time to think how they can prove these equations with also using Pythagorean Theorem let them know that you will continue this at your next lesson.
A. Distribute 'Special Triangles' worksheet per student. Let them try to solve problems. If there is any student who finishes it earlier, give him/her a gear up worksheet as 'High Level Triangles'. If there is any student who is struggling with questions then give him/her a gear down worksheet as 'First Level Triangles'.
B. Make students remember that you will continue the proofs of Euclidean Equations and solutions of their worksheets in your next lesson.
Assignment:
Students will study on their 'Euclidean Equations' and 'Special Triangles' worksheets.
Assessment:
Informal assessment will occur when students are studying in their groups with their partners and when they are sharing their feedbacks with classmates. Formal assessment will be conducted by giving their 'Euclidean Equations' papers to them as a pop-quiz with two or three selected problems from their worksheets in first ten minutes of next lesson.
Gearing Down:
Show only proves with blocks and papers by rearranging them without using algebraic solutions. Distribute 'First Level Triangles' worksheet and solve them on blackboard one by one, step by step.
Gearing Up:
Distribute 'High level Triangles' worksheet per student and if all students can solve this worksheet in lesson, write the 'Challenging Question' on board and let them try to solve it.
References:
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: 20191-9988
Sources Used:
www.ekolhoca.com : Questions for problem worksheets are chosen from this site.
2007 Tüm Hakları Saklıdır