Math Worth Thinking About: The Concepts that Underlie the Formula
Watch this video to see:
Watch this video to see:
- how students can create models to help them discover mathematical concepts upon which formulas are based, and
- a rich application of the Pythagorean Relationship that offers several avenues for robust mathematical investigation.
The CT/Coding Advantage: Establishing Universal Applicability
Watch this video to see:
- how the development and use of a computational model allows students to see the universal applicability of mathematical concepts, and
- how the Computational Thinking Pedagogical Framework guides educator integration of CT/Coding activities with mathematics instruction.
Minds On: A Hypotenuse Investigation
1. Present students with the following problem:
A man is trying to zombie-proof his house. He wants to brace his front door as shown in the photo to the right. The following conditions exist:
- The door is perfectly perpendicular to the floor.
- The centre of the doorknob is exactly 1 metre from the floor.
- The bottom stair is exactly 2 metres from the door.
2. Have students work on this question for 5 to 10 minutes. Then, invite students to suggest approaches for finding an answer. At an appropriate moment, tell the students that they will return to this problem after the next investigation.
Action: A Closer Look at Relationships within Triangles
3. Provide students with the following instructions:
Many years ago, a Greek mathematician named Pythagoras made an interesting discovery about the geometry of triangles. Follow the steps below to uncover Pythagoras’ findings for yourself.
3. Provide students with the following instructions:
Many years ago, a Greek mathematician named Pythagoras made an interesting discovery about the geometry of triangles. Follow the steps below to uncover Pythagoras’ findings for yourself.
- Work with a partner. Each of you should create one right triangle on a Geoboard. Make sure the two resulting triangles are not congruent.
- Record your own triangle near the center of a sheet of dot paper.
- Use a ruler to draw squares on the sides of your triangle. The sides of each square should be equal to the side of the triangle on which it is built.
- Using the area of one dot-paper square as a unit of measurement, find the areas of your three squares.
- Share your work with your partner.
- Repeat the activity with obtuse triangles and then acute triangles.
- Examine your results for patterns and/or relationships that appear consistent.
4. Facilitate a class discussion to allow students to share their observations and conjectures about the areas of the squares built onto the sides of different types of triangles. The following questions (taken from the lesson Pythagoras Delivers the Mail in The Super Source – Patterns/Functions – Grade 7-8) support student thinking and discussion around this topic:
- What method(s) did you use to find the areas of your squares?
- For which squares was it hard to find the area? For which was it easy? Why?
- What patterns did you discover?
- What generalizations can you make about the areas of the three squares built on the sides of a right triangle? an obtuse triangle? an acute triangle?
- Do you think your findings would be consistent for other triangles of the same type? How do you know?
5. In pairs, have students explore a wide range of right triangles in the Pythagorean Relationship Scratch code linked here. Upon receiving input for sides a and b, the code generates the specified right triangle and the squares on all three sides. Additionally, the square on the hypotenuse is partitioned into 4 congruent triangles and (in most cases) a square in the centre. This partitioning makes it easier for students to determine the square’s area and to see that in all right triangles, the sum of the squares of the two shorter sides is equal to the area of the square of the hypotenuse.
Consolidating the Learning 6. Invite students to discuss their findings from the Scratch code. Elicit the fact that the Pythagorean Relationship applies to all right triangles and that knowing the area of the square of the hypotenuse is the key to determining the length of the hypotenuse itself. |
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7. Direct the students’ attention to the problem they encountered in the Minds On phase of the lesson. Ask them to apply their new understanding to the problem to determine the length of the plank. Consider the following questions for consolidation purposes:
- Where is the right triangle in this situation?
- How does the Pythagorean Relationship apply to this situation?
- How does knowing the height of the doorknob and the distance from the door to the stairs help you determine the length of the plank?
Action: The Wheel of Theodorus
8. Present an image of the Wheel of Theodorus and describe it as a design constructed entirely of increasingly larger right triangles arranged to resemble a nautilus shell.
8. Present an image of the Wheel of Theodorus and describe it as a design constructed entirely of increasingly larger right triangles arranged to resemble a nautilus shell.
9. Provide students with the Wheel of Theodorus instructions (linked here) and set them to the task of drawing one by hand. As they work, ask them to make mental notes of the steps they take including times when they re-enact same sets of steps with different values/lengths. Also prompt them to consider how the Pythgorean Relationship is present in the design.
10. Once students have completed their paper and pencil versions, ask them to develop a code in Scratch that will result in the automated creation of the Wheel of Theodorus. There are many paths a student can take to accomplish this task, but seeing as the goal for this math learning experience is for students to develop a deep understanding of the Pythagorean Relationship, they should be encouraged to use their newfound knowledge to some degree in their codes. An example Scratch code for the Wheel of Theodorus is linked here. 11. Ask students to work in pairs to determine the lengths of all the hypotenuses in their Wheels of Theodorus. |
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Consolidating the Learning
12. Invite students to share their Scratch codes for the Wheel of Theodorus. Select at least a few who have embedded the Pythagorean Relationship in their codes and highlight the manner in which they have used and represented this concept. Also, discuss the lengths of the hypotenuses on the Wheel of Theodorus with particular focus on the use of the Pythagorean Relationship.
Extension: Beyond Squares
13. Prompt students to investigate whether the Pythagorean Relationship holds when non-square shapes (e.g., equilateral triangles, semi-circles, similar rectangles) are built onto the sides of a right triangle.
Curriculum Expectations
The Pythagorean Relationship is planted firmly within the grade 8 mathematics curriculum in Ontario. However, there is reason to consider making this concept available to grade 7 students (without formal assessment, of course, as they cannot be made responsible for knowing content beyond their grade level). There is quite a lot of triangle measurement and geometry written into the grade 7 math curriculum, but some investigations and activities that address these expectations quite well actually require some familiarity with the Pythagorean Relationship. Ultimately, the artificial fragmentation of mathematics by strand and grade into bite-sized components is at fault in this case, as well as others. That being said, some numeration concepts associated with the Pythagorean Relationship (i.e., exponents, square roots) are not part of the grade 6 curriculum, and so this geometrical concept is best saved for the intermediate grades.
The Wheel of Theodorus addresses the following curriculum expectations:
12. Invite students to share their Scratch codes for the Wheel of Theodorus. Select at least a few who have embedded the Pythagorean Relationship in their codes and highlight the manner in which they have used and represented this concept. Also, discuss the lengths of the hypotenuses on the Wheel of Theodorus with particular focus on the use of the Pythagorean Relationship.
Extension: Beyond Squares
13. Prompt students to investigate whether the Pythagorean Relationship holds when non-square shapes (e.g., equilateral triangles, semi-circles, similar rectangles) are built onto the sides of a right triangle.
Curriculum Expectations
The Pythagorean Relationship is planted firmly within the grade 8 mathematics curriculum in Ontario. However, there is reason to consider making this concept available to grade 7 students (without formal assessment, of course, as they cannot be made responsible for knowing content beyond their grade level). There is quite a lot of triangle measurement and geometry written into the grade 7 math curriculum, but some investigations and activities that address these expectations quite well actually require some familiarity with the Pythagorean Relationship. Ultimately, the artificial fragmentation of mathematics by strand and grade into bite-sized components is at fault in this case, as well as others. That being said, some numeration concepts associated with the Pythagorean Relationship (i.e., exponents, square roots) are not part of the grade 6 curriculum, and so this geometrical concept is best saved for the intermediate grades.
The Wheel of Theodorus addresses the following curriculum expectations:
Acknowledgements
The Wheel of Theodorus is partially based on "Pythagoras Delivers the Mail" from The Super Source: Patterns/Functions – Grades 7-8. The original lesson is found here.
References
[Barred door]. (n.d.). Retrieved July 2, 2018, from http://preparedcitizenwsg.blogspot.com/2011/08/how-to-fortify-your-home.html
Gadanidis, G. (2015). Coding for young mathematicians: 2nd ed. London: Western University.
[Nautilus]. (n.d.). Retrieved July 3, 2018, from https://en.wikipedia.org/wiki/Nautilus
Kotsopoulos, D., Floyd, L., Khan, S., Namukasa, I. K., Somanath, S., Weber, J., & Yiu, C. (2017). A Pedagogical Framework for Computational Thinking. Digital Experiences in Mathematics Education, 3(2), 154-171.
Lesson 10: Applications of the Pythagorean Theorem. (n.d.). Retrieved July 3, 2018, from https://im.openupresources.org/8/teachers/materials/8/10/8-8-10-practice_problems.pdf
Mathematics: Revised: The Ontario curriculum: Grades 1-8. (2005). Toronto: Ontario Ministry of Education.
Spiral of Theodorus. (2018, July 03). Retrieved from https://en.wikipedia.org/wiki/Spiral_of_Theodorus
The super source: Patterns/functions - Grades 7-8. (2007). Vernon Hils, IL: ETA/Cuisenaire.
Unit 10 Visualizing Geometric Relationships - EduGAINs. (n.d.). Retrieved July 3, 2018, from http://www.edugains.ca/resourcesMath/CE/LessonsSupports/TIPS4RM/Grade8English/Unit10_VisualizingGeometricRelationships.pdf
The Wheel of Theodorus is partially based on "Pythagoras Delivers the Mail" from The Super Source: Patterns/Functions – Grades 7-8. The original lesson is found here.
References
[Barred door]. (n.d.). Retrieved July 2, 2018, from http://preparedcitizenwsg.blogspot.com/2011/08/how-to-fortify-your-home.html
Gadanidis, G. (2015). Coding for young mathematicians: 2nd ed. London: Western University.
[Nautilus]. (n.d.). Retrieved July 3, 2018, from https://en.wikipedia.org/wiki/Nautilus
Kotsopoulos, D., Floyd, L., Khan, S., Namukasa, I. K., Somanath, S., Weber, J., & Yiu, C. (2017). A Pedagogical Framework for Computational Thinking. Digital Experiences in Mathematics Education, 3(2), 154-171.
Lesson 10: Applications of the Pythagorean Theorem. (n.d.). Retrieved July 3, 2018, from https://im.openupresources.org/8/teachers/materials/8/10/8-8-10-practice_problems.pdf
Mathematics: Revised: The Ontario curriculum: Grades 1-8. (2005). Toronto: Ontario Ministry of Education.
Spiral of Theodorus. (2018, July 03). Retrieved from https://en.wikipedia.org/wiki/Spiral_of_Theodorus
The super source: Patterns/functions - Grades 7-8. (2007). Vernon Hils, IL: ETA/Cuisenaire.
Unit 10 Visualizing Geometric Relationships - EduGAINs. (n.d.). Retrieved July 3, 2018, from http://www.edugains.ca/resourcesMath/CE/LessonsSupports/TIPS4RM/Grade8English/Unit10_VisualizingGeometricRelationships.pdf