Project AMP Dr. Antonio R. Quesada – Director, Project AMP
The Corner Problem
Lesson Summary:
In this activity, students will explore various approaches to solving the problem of maximizing the length of a rod that can be carried horizontally around a corner using several techniques on the TI-Nspire CAS calculator.
Key Words:
Pythagorean Theorem, local extrema, regression line, horizontal, vertical, similar triangles, right triangle
Background Knowledge:
This is an activity for exploring the maximum length of a rod that can be carried horizontally around a corner. Students should be familiar with finding the minimum and maximum of functions using several different methods, including graphical and calculus. Students should be familiar with several geometry and trigonometry concepts, such as the Pythagorean Theorem as well as properties of right triangles and how to determine their angle measures. They must have a basic working knowledge of the TI-Nspire CAS handheld such as Calculator, Graphs & Geometry, Lists & Spreadsheet, and Notes.
Materials:
TI-Nspire CAS handheld
Worksheet
Ruler/Paper/Pencil
Chalk Board/Chalk
Suggested Procedure:
Students will be put into groups of two or three. The students will be handed a worksheet that is intended to guide them through the main ideas of the activity and provide a place to record their observations. Remind students that they are going to be looking for the maximum length of the rod. Discuss and review how to use different applications on the TI-Nspire CAS handheld, such as Calculator, Graphs & Geometry, Lists & Spreadsheet, and Notes. Pass out worksheets and have the teams complete the activity.
Standards: Patterns, Functions and Algebra
- Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities.
- Students analyze, model and solve problems using various representations such as tables, graphs and equations.
Benchmark/Grade Level Indicator: Use Patterns, Relations and Functions
- Identify the maximum and minimum points of polynomial, rational and trigonometric functions graphically and with technology.
Assessment:
Check students progress in class on the TI-Nspire CAS handheld. Collect worksheets.
***Note: The TI-Nspire CAS calculator at this time is a new piece of technology for many students. In the future students will be more knowledgeable with the TI-Nspire. Throughout the lesson, comment boxes explaining how to complete procedures are included. In the future, once students are comfortable with the TI-Nspire, the comment boxes may be removed.
Activity 1: Estimating the longest rod by collecting data.
Goal: Students will discover a rough estimated solution to the problem using pictures, paper and pencil, rulers and collecting data.
Student’s Names: ______
Problem: Two corridors 3 feet and 4 feet wide, respectively, meet at a right angle. Find the length of the longest non-bendable rod that can be carried horizontally around the corner.
Project AMP Dr. Antonio R. Quesada – Director, Project AMP
1. Using a ruler, paper and pencil, construct a picture. Label the picture as shown to the right. (Note: For students of a younger age, a picture drawn on the coordinate plane will be provided.) /Project AMP Dr. Antonio R. Quesada – Director, Project AMP
2. The rod runs from point A to point B. Using your ruler, obtain an estimate for the measure of length of AB. What do you obtain? Keep in mind, the rod must stay in contact with the points A, B and C.
3. Now, using your ruler, measure from the point O to point A and from the point O to point B. What distances do you obtain for each?
4. What formula can you use to obtain the length of the rod AB using the two distances you discovered above? Verify the distance of AB using the formula. Explain.
5. Draw several other rods and repeat step 3, keeping in mind, the rod must stay in contact with point A, B, and C. Complete the table (on right) with several different measurements, using the formula from step 5 to obtain AB. Obtain at least 10 different values. / Distance from origin to A / Distance from origin to B / Length AB6. Write the values you obtain in the table on board. After each group of students have placed their values in the table on the board and the values has been put in order from least to greatest, determine a rough estimate of the maximum length of the rod that can be carried horizontally around the corner. What value did you come up with?
7. From your observations, what can you conclude about the maximum length of the rod that can be carried horizontally around the corner?
Activity 2: Using a function to determine the longest rod that can be carried horizontally around a corner.
Goal: Students will use similar triangles to obtain a function they can graph in their TI-Nspire CAS. They will then use the TI-Nspire to locate the local extrema (longest rod).
1. Consider your hand drawn figure from Activity 1. Re-label the figure as shown to the right. /2. We do not know the length of AO nor OB. Using your ruler, construct a horizontal line from point C intersecting AO at E. Also, construct a vertical line from point C intersecting OB at F. What do you observe? What do you notice about triangles ACE and CBF? What type of triangles do we have?
3. We know from the previous activity that we can obtain the length of AB by using the Pythagorean Theorem. What do you notice about the length of CE and the length of CF? What about the length of AE and BF?
4. Using the newly formed triangles, how do you determine the length of AB? Using these triangles, determine the length of AB. What is your function? (Remember you need to make sure that your function is in terms of only one variable. Using your knowledge of similar triangles, solve for one of the variables.).
5. Using the TI-Nspire CAS, graph the function you obtained from step 4. Make sure to determine the best window.
Choose Home, Graphs and Geometry (2). Move the cursor to the function line at the bottom of the screen. Type in the function that you obtained in step 4. Hit Enter. (Make sure to use parenthesis in the appropriate places). To change the window, select Menu, Window (4), Window Settings(1). Hit Enter.
6. Determine the local extrema. What value do you obtain? What is represented by the x and the y values in the problem situation? Notice that there are two curves. What do you know about the length of AB to help you decide which curve to use?
Choose Menu, Points and Lines (6), Point On (2). Move your cursor to any point on the graph and hit Enter. Hit Escape. You will now see a hand on the screen. Move the hand over the point you have just placed on the graph. Hold down the Click Key until the hand grabs the point. Using the arrow keys, move the point on the graph. When you reach the local extrema you will see the lowercase letter m (minimum point) pop up by an ordered pair.
7. Now that you have found the local extrema, what is the maximum length of the rod that can horizontally be carried around the corner?
9. What do you conclude about the maximum length of a rod that can be carried horizontally around the corner? Does your maximum length of the rod from activity 1 come close to the length of the rod you have just obtained? Explain.
Extension:
1. Using the function obtained in step 4, use calculus to find the maximum length of the rod that can be carried horizontally around the corner.
- Determine the derivative. What do you obtain?
Select Control, I to insert a new page. Choose Home, Calculator (1). Choose Calculus (5), Derivative (1), Enter. Determine what variable to solve for in the newly developed equation. Enter the given variable. Enter the equation you obtained in step 4. Hit Enter.
- Determine the critical points. What do you obtain? How do you know which critical point(s) you will use?
Select Menu, Algebra (4), Solve (1). Enter your equation from step 4 and set the equation equal to zero. Enter a Comma, and then the variable you are looking to solve for (x). Hit Enter.
- Evaluate your function with the given value. What do you obtain for the maximum length of a rod that can be horizontally carried around a corner? Is the value you obtained the same as the value you obtained from activities 1 and 2?
Store the correct variable by typing in x =, the value of the correct variable, and press Control, Var (sto). Store the number as z. Evaluate your function for l (z). Type in your equation using z instead of x. Press Enter.
Activity 3: Using the TI-Nspire to construct a figure, gather data, obtain a regression line and determine the local extrema.
Goal: Students will use the TI-Npire to construct the figure according to the problem. They will then automatically collect data, obtain a scatter plot, and determine the best fitting regression line. Students will then be able to determine the longest rod that can be carried horizontally around the corner.
1. Using the TI-Nspire CAS, construct a figure for the problem.
Create a new document. Add Graphs and Geometry (2). Select Menu, View (2), Show Grid (5). Select Menu, View (2), Hide Axes (4). Select Menu, Points and Lines (6), Point On (2). Choose a point on the grid to place point. Hit Click key. Choose Menu, Points and Lines (6), Line (4). Click on the point and use arrow keys to construct a line that is as straight up and down as possible. Hit Escape. Select Menu, Construction (9), Perpendicular (1). Click on your point and then on your line. Select Menu, Construction (9), Perpendicular (1). Move your cursor approximately 3 units to the right. Click twice on the line. You should obtain another perpendicular line. To check the measurement, choose Menu, Measurement (7), Length (1). Click on your first point and then click on the second point. You should obtain a length. Click anywhere on screen to place the length. If the length is not exactly 3, place your cursor on one of the points and move it either left or right until the length is 3. Choose Menu, Construction (9), Perpendicular (1). On the last line you created, move your cursor to the line and down as much as possible. Click twice on the line. Choose Menu, Construction (9), Perpendicular (1). On the last line you created, move your cursor to the line and to the right as far as possible. Click twice on the line. Choose Menu, Points and Lines (6), Point On (2). Select a spot on the last line you created to place a point. Click that spot. Next, choose Menu, Measurement (7), Length (1). If the length is not 4, move one of the points until you reach a length of 4. Choose Menu, Construction (9), Perpendicular (1). Click on the last point created and the then click on the last line created. If your bottom line and leftmost line do not meet, choose Menu, Points and Lines (6), Intersection Point (3). Move your cursor and click once on each line.
2. Define your picture.
Select Menu, Points and Lines (6), Segment (5). Click on one point and draw a segment connecting that point to another point. Repeat this until you have connected all points with a segment. To see your picture, you must now hide the lines you created. Choose Menu, Tools (1), Hide/Show (2). Click on each line you created. Hit Escape. Choose Menu, Tools (1), Hide/Show (2). Click on the segment at the top that is 3 cm long and the segment on the right that is 4 cm long. Then, hide each point created at the end of the corridors. Hit Escape.
3. Create you rod.
Select Menu, Points and Lines (6), Line (4). Choose any point on the bottom line and create a line through the top corner point of the connecting corridors. If your line does not extend to the leftmost wall, choose Menu, Points and Lines (6), Intersection Point (3). Click both lines to obtain a point. On the line just created, connect the two end points using the segment tool. Using the hide/show tool, hide the line.
4. Extend your leftmost and bottom wall.
Choose Menu, Points and Lines (6), Line (4). Click the lower corner point and the point on the leftmost wall. Then Click the point on the bottom wall and the point in the lower corner.
5. Label the points as shown on the figure to the right. /Select Menu, Tools (1), Text (5). Move your cursor to the point at the top of the rod, label it A (for capital letters, select Control, Caps). Move your cursor to the point at the bottom of the rod and label it B. Then label the upper corner point C and the lower corner point O for origin.
6. Determine the length of AB, AO and BO. What do you obtain? Click and hold point A or B. Move the point either left/right. What do you notice?
Choose Menu, Measurement (7), Length (1). Click on the first point and then the second point to obtain the length.
7. Save and store your data to a list. Since our values have been labeled, we can use the label as the variable name (bo, ao and ab).
Select the value to be collected and press the sto key by hitting Control, Var. Press Enter twice. The variable name will appear in boldface.