Teaching Phase: How will the teacher present the concept or skill to students?
The teacher will display the Walking Home document (from the Uploaded Files section.) The teacher will ask students to imagine walking through a neighborhood along the path shown on the Walking Home attachment. It models a boy walking from his house to his friend's house where the houses are arranged linearly. It shows that the boy would have to walk past 4 houses heading east, then turn 90 degrees and walk past 3 houses heading south, for a total of walking past 7 houses. However, if the boy could go the most direct route, say, fly over houses, it would only take the equivalent of 5 houses to get from his house to his friend's house. It would be a shorter distance from one point to another if you could fly "as the crow flies." Unfortunately, we don't always have the option to fly as the crow flies!
This introduces the concept of the Pythagorean Theorem. Explain to the students we are fortunate that for centuries, mathematicians have studied the relationship between the sides of right triangles and the side opposite the right angle (hypotenuse.) Pythagoras is arguably the most famous mathematician using this process for finding the lengths of sides of right triangles and that is what we will be studying, but instead of looking at distance past houses, we will finish up this lesson by looking at points on a coordinate plane and show how we can use any two points to form a right triangle so we can find the distance between the points.
Introduce the terms: "leg," "hypotenuse," "adjacent." Label a right triangle with the terms relating to sides and ask, "How could you describe this right triangle over the phone to someone who did not have the picture in front of them?" (possible responses: the right angle is in between the two legs and never touches the hypotenuse; the hypotenuse connects the two shorter sides together; the hypotenuse is the leg opposite the right angle.)
**Emphasize that the term "hypotenuse" only applies to right triangles.
Introduce the variables used to represent the appropriate legs, and letters that could be used for the lengths of the sides; "a" and "b" are used for legs adjacent to the right angle and hypotenuse = "c" most commonly.
Label a right triangle with a, b, and c appropriately and demonstrate how if a = 3 and b = 4, then we can find the length of "c" by using the Pythagorean Theorem:
Replace the variables with the numbers they represent:
Compute the left side of the equation:
Simplify the left side of the equation:
Ask students if you have found the missing measure. Ask students if a length of 25 is reasonable with a triangle of legs 3 and 4. Hopefully, at least one student will recognize that you have found what the squared measure is. If not, guide them into this reasoning. Ask students for the inverse operation of squaring a number (finding the square root.) If it hasn't been brought up, remind students that when solving equations, you must continue to balance both sides of the equation, so what is done on one side of the equal sign, must be done on the other side. If you take the square root of 25, you must also take the square root of c squared.
What times itself equals 25? What times itself equals c squared?
The hypotenuse is 5.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Draw a right triangle with short leg = 5 and long leg = 12. Ask students to use the Pythagorean Theorem to find the measure of the hypotenuse on their dry erase boards. Encourage them to copy the picture of the triangle down and use the given measures. Then copy the Pythagorean Theorem on their boards and make the proper substitutions.
Give the students at most 2 minutes to work through the problem. Students should be able to produce the minimum of .
This would be acceptable initial steps, and as the teacher circulates, encourage students to compute the squares and find the sum of the left side of the equation.
The teacher will bring the class back as a whole group and demonstrate the process for finding "c" to ensure all students have the correct process.
Say to the students, "Look at the lengths of the legs. Does 13 seem reasonable?"
Draw a right triangle with a short leg measure of 8 and a long leg measure of 15. Ask the students to find the measure of the hypotenuse using their dry erase boards. (The hypotenuse will be 17.)
Circulate to ensure students are copying the triangle correctly, the Pythagorean Theorem correctly, and substituting the correct measures in the equation correctly. Encourage any struggling students to compute the left side of the equation so they can find the square root, and thus, the measure of the hypotenuse.
The teacher should present the last example where a leg and hypotenuse are given, and the third leg must be found. This requires the students to solve for a value other than "c" and the teacher may choose to have the students substitute values in for given sides and work backwards to solve for either "a" or "b" or demonstrate solving for a different leg by using inverse operations.
, however, if you need to find "a" or "b", it might be helpful to have the formula set up with all terms that will be substituted on the same side of the equation. For example:
subtracting from each side
The teacher should have confidence that the students are able to comfortably use the Pythagorean Theorem before moving to finding the distance between two points on a coordinate plane. The following activity provides more opportunity to become familiar with the Pythagorean Triples.
Students will be paired to complete a Pythagorean Triple Sort activity. The teacher should use discretion when partnering students, and it is suggested that a high student be paired with a medium student and a medium student be paired with a low student. This avoids having high achievers doing all the work with a low achieving student not able to contribute to the task at hand.
The cards should be printed on card stock and pre-cut (laminated if possible for repeated use) to save class time when possible. Students will be asked to find matches of three possible side lengths that would create a right triangle. This activity is designed to acquaint students with the most frequently used Pythagorean Triples (set of three numbers that when the smaller two numbers are each squared, then summed; the sum is equal to the third number squared.) There are a total of 13 sets of Pythagorean Triples in the sort activity. The teacher should circulate as the pairs are sorting their cards and guide the students into matching sides by using the theorem to assist. Since some triangles have already been used as examples in the teaching phase, the students should have some recollection of 3-4-5 triangles, 5-12-13 triangles, 7-24-25 triangles, and 8-15-17. This activity also allows students to recognize that multiples of triples will also work with the Pythagorean Theorem.
The solutions to the Sort are: 2.5-6-6.5; 3-4-5; 6-8-10; 5-12-13; 15-36-39; 16-30-34; 9-12-15; 7-24-25; 12-16-20; 8-15-17; 15-20-25; 10-24-26; 60-80-100
Continue with the Independent Practice section for Day 1.
After providing the students with the correct answers for the independent practice sheet, the teacher should spend no more than 5 minutes to answer any misconceptions.
The teacher will have a coordinate graph with a range of -10<x
Ask, "Can anyone plot the points (1,3) and (1, 7) on the board for us?" Allow a student to graph the points, and think out loud the process used.
Ask the students "What is the distance between the two points (4,3) and (1,3)?"
Demonstrate counting the 4 units between 3 and 7 as you draw a line to represent the distance.
Show the students that it is easy to determine the distance between the two points, because we are able to count units when the distance is vertical on a coordinate plane.
Ask a student to plot the coordinate pair (4, 3) on the same graph. Ask the students, "What is the distance between the two points, (4,3) and (1,3)?"
Demonstrate counting the 3 units between 1 and 4 as you draw a line to represent the distance.
Ask the students, "Why was it easy to find the distance between the points?" (possible responses: you can count; you can subtract the numbers.)
Ask the students, "What is the distance between (1, 7) and (4, 3)?" (possible responses: 7 because 3 + 4 = 7; 4 because when you connect the points, that is how many unit boxes the line segment cuts through.)
Use a straight-edge to demonstrate that the distance is shorter than the sum of the two sides already known.
Ask the students, "What do the two known sides PLUS the third side form when their endpoints are connected together?" (possible responses: triangle; right triangle.)
Ask the students, "How can a right triangle help us find the missing distance/length?" (possible response: Pythagorean Theorem; use a squared + b squared = c squared.)
Have students use the Pythagorean Theorem to find the missing hypotenuse on their dry erase boards. Check for understanding by having the students hold their solutions up.
Project a new coordinate graph on the board. Ask a student to come to the board and plot the coordinates (-6, 2) and (6, 7). Ask students, "How can we use the Pythagorean Theorem to find the distance between the two points, if we don't have a right triangle?" (possible responses: draw a right triangle; find the vertical distance up and the horizontal distance over.)
Have another student come to the board and draw the right triangle on the coordinate plane to provide the visual needed to find the leg measures. Have another student provide the lengths of the legs. Ask students to work the problem on their dry erase boards. (13)
**Possible student errors to watch for: finding the correct horizontal distance as it crosses through two quadrants. This may be an opportunity to review absolute value; students may forget to square the side lengths; students may forget to find the square root when they find the sum of the squares of the two side lengths.
Use the following sets of coordinate points (5, 3) and (-1, 1/2); (-9, 1) and (6, 9); the next set of points (10, 10) and (3, -14) will require a grid that has a range of -15<y
Solutions: (5, 3) and (-1, 1/2) will provide side lengths of 6 and 2.5 which will find a hypotenuse of 6.5 (a multiple of 5-12-13); (-9, 1) and (6, 9) will provide side lengths of 8 and 15 which will find a hypotenuse of 17; and (10, 10) and (3, -14) will provide sides lengths of 7 and 24 which will find a hypotenuse of 25.