| Below you will see the plan I created for my Pre-Algebra class. It is converted Word file, so some pictures were not showing appropriately, thus they got deleted. If you want to see the original file with comments and blackmasters, click here to open Word file. Backwards Design Unit
Pre-Algebra
Pythagorean Theorem Standards 3.0. Students know the Pythagorean Theorem and deepen their understanding of plane and solid geometric shapes by constructing figures that meet given conditions and by identifying attributes of figures.
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3.3 Know and understand the Pythagorean Theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean Theorem by direct measurement. - Objectives of the lesson unit:
- students will learn how to use Pythagorean Theorem,
- students will learn how to solve right triangle,
- students will learn how to use properties of triangles to solve real-life problems,
- students will learn how to use Pythagorean Theorem to measure indirectly.
Synopsis: the Pythagorean Theorem states that for a right triangle the sum of squares of lengths of two legs equals to the square of the length of hypotenuse. Thus, it is possible to find one side of right triangle by knowing any two other sides. It also helps to find area and perimeter of right triangle. In real life we can see that some streets form right triangles, cities are located in corners of right triangles, tents for camping, buildings, etc, contain right triangles in them. So this theorem can help us to figure out distances, shortest path, make sure the building is standing straight. Day 1 10 min | Vocabulary Self-Inventory (TEP 176). Each student is given the chart where he or she has to mark words he or she knows, heard of, and doesn’t know. Students will become aware of how much they already know about the subject. They will already know some words about triangles and geometry. There will be some unknown words for them which they discover while reading. Students will pay closer attention to new words. | Vocabulary Self-Inventory
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I know the word Heard of it Have no idea _____ Similar Triangles _____ Pythagorean Theorem _____ Right Angle ______ Perimeter ______ Isosceles Triangle | ______ Solving a Right Triangle _____ Pythagorean Triple _____ Hypotenuse ______ Legs ______ Area | |
15 min | KWL (RLS, p.35) is designed to engage students and help them to become active learners. I am using this strategy in this section since students already know something about right triangles, Pythagoras and probably his theorem, thus the strategy activates their prior knowledge. Then they will want to learn more. When they write down their questions and answers, they will organize their thoughts and learn the topic better.
Before students start activities and reading, they have to brainstorm what they already know about the Pythagorean Theorem. They will write it down in the first column of the table under K (Know). After that I will ask students what they want to know about the subject and record their questions under W (Want to know). After the activities that follow, and reading the chapter 9.3 and 9.4, at the end of the unit, students will fill out column L (Learned) with answers students found during the unit and in the textbook. For unanswered questions students will place question marks on the chart. |
K-W-L Pythagorean Theorem | K | W | L | It is about triangles. | Can we apply it to any triangle? | No, we can apply it only to the right triangles. | 30 min | Picture book story (TEP 151). This is the group activity. The table arrangement in my classroom makes five groups. Each group will be given copies of picture book "What is your angle, Pythagoras?" pages with only pictures and no words written on them. Groups will have to organize these pictures in such a way that they tell a continuous story.
After students are done, they will be given a chance to share their stories with the rest of the class by traveling around class looking at other people’s stories. Finally, when everybody gets back to their tables, I will read a story to them so that they can compare what they had and what it really was.
I expect students to pay attention to the story and listen carefully. While they are listening, they can take notes and work on their KWL chart. "What is your angle, Pythagoras?" is a picture book about boy named Pythagoras and his adventure during which he learns about relationship of the sides in a right triangle. He tells about his findings to his family and friends, that’s how the theorem becomes famous. | Exercise 1 (5-10 min)
Antoine is in his car at the intersection of California St. and Market St. in San Francisco. He needs to reach the intersection of Market St. and Van Ness Ave. But he knows that Market St. is closed for construction in several places, so he takes California St. west to Van Ness Ave. and the turns south in Van Ness Ave. until he reaches his destination. How many miles of driving would he have saved if he could have driven down Market St.? Exercise 2 (5-10 min) - If c=13 and b=5. Use the Pythagorean Theorem to find the length of the other leg
- An isosceles right triangle has a hypotenuse of length 12. Use the Pythagorean Theorem to find the length of each leg
Exercise 3 (5-10 min) - When the lengths of the sides of a right triangle are all natural numbers, the lengths are called a Pythagorean triple. Find the hypotenuse if the legs are a=8, b=6.(Students should get c=10, tell them it is a Pythagorean triple.)
- Find as many different Pythagorean triples as you can
Day 2 20 min | Frayer Model (TEP 176). This strategy allows students to comprehend what they learned about Pythagorean Theorem after picture story and after they read the textbook section 9.3.
Students will have to fill out four parts of a handout. First, they will write essential characteristics of a Pythagorean Theorem such as it is applied to right triangles, then they will give examples such as right triangles appear in street intersections, buildings etc. Third part is to write down non-essential characteristics such as the sides of right triangle might be integers. Finally they will complete the chart of giving non-examples such as acute triangles’ lengths.
After students complete working on their handouts, we will have class discussion and write down the most important or interesting points on a transparency. I will be asking students to share their ideas and they will be adding them to their Frayer models. | Exercise 1 (5-10 min)
The city planners have asked you to design another park for the city. This park is also on a right triangular site, with legs measuring 38 ft and 42 ft. You need to know the area and perimeter of the site so that grass seed for the lawn and plants for a hedge can be purchased. What are the area and perimeter? Answer: 798 ft2, about 137 ft Exercise 2 (5-10 min)
You work on a street maintenance crew for a city. Part of the street has caved in. You don’t have any ropes or tape measures that are long enough to stretch across the hole. Explain how you can measure the distance across the hole indirectly.
Suppose the measurements of the right triangle’s legs are 24 ft and 52 ft. Find the distance across the hole. Answer: about 57 ft Reflection
20 min | Quick Write (RLS, p.331) Students will have to write as much as possible about suggested topic: what are the examples of right triangles around us? Students will have to come up with a problem. At the end students have to be ready to discuss what they wrote. They will do it in their groups, and then one person from each group will share the group’s ideas with the rest of the class. Whole class will have to solve suggested problem. |
15 min | Dialectical Journal (RLS, p.357 and TEP 176) I will use the version with two columns: in the first column students will have to record the idea and/or a problem that we learned in class to which they want to respond, and where this idea/problem came from. It might have come from the textbook, picture book, other students’ ideas, class discussion, group discussions etc. In the second column they will write their response and/or solution. In case if it a response student will write why they think it is interesting, why it caught their attention, or why it might be useful for them etc. | Dialectical Journal Lines from Text | Response to Text | "Using the lengths of two sides of a right triangle to find the length of the third side is called solving a right triangle." | I thought that when we talk about solving something, we mean to find a variable in an equation. It is very unusual to talk about solving a triangle, because it is a geometrical shape. | | | References - RLS is "Reading and Learning Strategies. Middle Grades through High School" by S.D. Lenski, M.A. Wham, and J.L. Johns
- TEP 176: "Writing, Reading and Language Instruction", M. Sewall, instructor, UCSD, Summer 2005
- TEP 151: "Teaching the English Language Learner", M. Sewall, instructor, UCSD, Fall 2005
- TEP 174: "Secondary Mathematics Teaching Practices", C. Halter, instructor, UCSD, Summer 2005
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