This twoday lesson teaches students to use the Pythagorean Theorem with simple right triangles on the first day, then progresses to using the theorem to find the distance between two points on a coordinate graph.
General Information
Freely Available: Yes
Attachments
CPALMS the shortest distance between two points.notebookCPALMS the shortest distance between two points.pptx
distance btwnptsquiz.docx
distance between two points pdf lessonfile.pdf
distance between two points lessonslides.pdf
distance btwn pts hwksheetv2.docx
distance btwn pts hwk sheet v2.pdf
AtCF WalkingHomeDemo.docx
Pythagorean TriplesSorts.docx
AtCF Pythagorean TheoremIndPrac.docx
AtCF Pythagorean Theorem Ind Prac (m).docx
AtCF SummativeAssessment.docx
AtCF Summative AssessmentKey.docx
AtCF Pythagorean Theorem Ind Prac (m).docx
AtCF Ind Prac 2 Distance Bt PointsonaGraph.docx
AtCF Ind Prac 2 Distance Bt PointsonaGraphKey.docx
graphingcoordinateplane.pdf
Lesson Content

Lesson Plan Template:
General Lesson Plan 
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will be able to apply the Pythagorean Theorem to:
 determine unknown side lengths in right triangles in realworld problems and mathematical problems in twodimensions (**Threedimensional problems are not covered in this lesson.)
 recognize sides of a right triangle as Pythagorean Triples.
 find the distance between two points on the coordinate plane.

Prior Knowledge: What prior knowledge should students have for this lesson?
Students should enter the lesson:
 nanospelltypo">MAFS.5.OA.2.3  Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.
 nanospelltypo">MAFS.8.EE.1.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² ×==1/3³=1/27
 nanospelltypo">MAFS.8.EE.1.2  Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
 nanospelltypo">MAFS.8.G.2.6  Explain a proof of the Pythagorean Theorem and its converse.
 nanospelltypo">MAFS.5.OA.2.3  Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

Guiding Questions: What are the guiding questions for this lesson?
What does a crow have to do with the Pythagorean Theorem?
How can right triangles help find the distance between two points on a coordinate plane?
Why do we need special names for different sides of right triangles and how do you decide which ones are which? 
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?
Day 1:
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 precut (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 345 triangles, 51213 triangles, 72425 triangles, and 81517. This activity also allows students to recognize that multiples of triples will also work with the Pythagorean Theorem.
The solutions to the Sort are: 2.566.5; 345; 6810; 51213; 153639; 163034; 91215; 72425; 121620; 81517; 152025; 102426; 6080100
Continue with the Independent Practice section for Day 1.
Day 2:
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 straightedge 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<ySolutions: (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 51213); (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.

Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?
Day 1: As the Crow Flies Independent Practice Sheet will be independent practice for the first part of this lesson. This sheet will allow the students an opportunity to practice using the Pythagorean Theorem to solve for the measure of the hypotenuse when given the measures of the legs of a right triangle. The last two problems demonstrate that not all right triangles will have whole number lengths and require students to round answers when finding the square root of a nonperfect square.
Day 2: As the Crow Flies on a Graph Independent Practice Sheet will be completed by students independently at their seats or as a homework assignment. The nanospelltypo">summative assessment should be given only after the independent practice has been completed, checked, and corrected by students. The teacher should determine if the students are ready for a nanospelltypo">summative assessment or whether more review is necessary.

Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
The teacher should bring the class together for an end review each day and ask the following:
Day 1:
 What does the Pythagorean Theorem have to do with a crow?
 How can you tell which side of the triangle is the hypotenuse?
 What do we call the other two sides of a right triangle?
 Can we use these three labels on every triangle?
 Will the hypotenuse always be a whole number?
 What do you do if you are already given the hypotenuse and one leg?
 Can you tell if you have a right triangle from 2 side lengths, before you use the Pythagorean Theorem?
 Think (and then share) one example where you can use the Pythagorean Theorem to solve a problem in the realworld.
Day 2:
 How can we use a right triangle to find the distance between two points on a coordinate graph?
 Is it possible to find the distance between two points on a graph without drawing a right triangle? (This will determine the students who would benefit from working on the extension activity.)
 Is it more likely or less likely to find a whole number length when using the Pythagorean Theorem to find a distance between two points, and why?
 What information, besides the two points, should someone have before they try to find the distance between two points on a coordinate graph?
 Think of (and then share) a realworld example of needing to find the distance between two points that are provided to you.

Summative Assessment
The students will complete the As the Crow Flies assessment (see attachment) that will provide the teacher with information on whether the students have mastered the concept of finding the distance between two points on a coordinate graph by using the Pythagorean Theorem.

Formative Assessment
Before teaching the lesson, students should have been taught nanospelltypo">MAFS.8.G.2.7 (see Prior Knowledge), and the teacher will ask students to answer the following problems (with or without a calculator):
(solutions 58; 8)Students who are unsuccessful with this task will need support with exponents and square roots during the lesson.
Throughout the lesson, the teacher will do a formative assessment by observing students' work, listening, and being mindful of misconceptions. The teacher will ask questions to probe students' thinking and provide guidance. Students will be encouraged to revise their work.
The teacher's observations related to these and other questions will yield information about what the students appear to know and are able to do and will provide guidance in making instructional decisions.
Students will find the missing lengths of right triangles on dry erase boards at their seats during the teaching phase and the guided practice phase. They will also find the side lengths of a right triangle on graph paper when given two points on a coordinate graph, and then use the drawn leg lengths to find the hypotenuse using the Pythagorean Theorem.

Feedback to Students
The teacher will provide verbal feedback and demonstrate the correct method to find the distance during the teaching phase. The students will receive feedback on their independent practice for day one when they check their work against the correct solutions at the start of the second day's lesson.
Assessment
 Feedback to Students:
The teacher will provide verbal feedback and demonstrate the correct method to find the distance during the teaching phase. The students will receive feedback on their independent practice for day one when they check their work against the correct solutions at the start of the second day's lesson.  Summative Assessment:
The students will complete the As the Crow Flies assessment (see attachment) that will provide the teacher with information on whether the students have mastered the concept of finding the distance between two points on a coordinate graph by using the Pythagorean Theorem.
Accommodations & Recommendations
Accommodations:
Provide struggling students with the As the Crow Flies Independent Practice (m) modified sheet that contains steps, provided measures, and blanks for students to fill in. The students can scaffold up to generating the entire problem. However, as a stepping stone, this accommodation provides an entry point to the Pythagorean Theorem.
Provide support for English Language Learners with translations, dictionaries, and/or examples of unfamiliar vocabulary words.Extensions:
Students that are able to deduce that the leg lengths are generated by finding the difference between the xcoordinates and the ycoordinates are ready to be introduced to the distance formula: Students can use the independent practice sheet 2 and find the hypotenuse without graphing the points, drawing the triangles, and using the Pythagorean Theorem with the side lengths (as the side lengths will be found through the distance formula.)

Suggested Technology: Computer for Presenter, LCD Projector, Scientific Calculator Special Materials Needed:
Teacher Materials:
 Coordinate graph to project with an xaxis from 10 to 10 and yaxis from 15 to 10
 Worksheet Keys
Student Materials:
 SortActivity Cards (a set in baggies for each pair of studentslaminated if at all possible for multiple uses)
Class sets of the following worksheets:
 Independent Practice
 Independent Practice 2
 Lesson Assessment
 Independent Practice (m)  modified worksheets as needed
General Supplies:
 Graph paper
 Straight edges
 Dry Erase Boards for desks
 Dry Erase Markers/Erasers
 Scientific Calculators (or a calculator with square root capabilities)
Further Recommendations:
Conviction with An Angle Is Upheld by the Court of Appeals is an article that appeared in the New York Times in November 2005 by Michael Cooper. The article relates a reallife situation in which a prosecutor used the Pythagorean Theorem to convict a drug dealer. While it is a valuable article, teachers should be discretionary when choosing to share it with students and may wish to simply share it as an anecdotal account.
Additional Information/Instructions
By Author/SubmitterThis lesson may align with the following Mathematical Practices:
 MAFS.K12.MP.4. Model with mathematics.
 MAFS.K12.MP.5. Use appropriate tools strategically.
Source and Access Information
Aligned Standards
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