The Pythagorean Theorem
Presenting the Proof – The proof of the Pythagorean theorem given in this lesson provides a wonderful review of, and use for, what the students just learned about similar triangles. Sometimes it is difficult for students to see the three right triangles contained in the figure and how the sides correspond. It is helpful to make an additional drawing of the three triangles so that they are separate, and oriented in the same direction. Using both figures for reference students can more easily verify the proportions used in this proof.
Skipping Around – Not all texts present material in the same order, and many instructors have a preferred way to develop concepts that is not always the same as the one used in the text. The Pythagorean theorem is frequently moved from place to place. If the students have not done similar figures yet, or if area has already been covered, the proof of the Pythagorean theorem given in the exercises may be the better place to focus the students’ attention. Proofs are hard for most students to understand. It is important to choose one that the students can feel good about. Don’t limit the possibilities to these two, research other methods, and pick the one that is most appropriate for your class. Or better yet, pick the best two or three. Different proofs will appeal to different students.
The Height Must be Measured Along a Segment That is Perpendicular to the Base – When given an isosceles triangle where the altitude is not explicitly shown, student will frequently try to use the length of one of the sides of the triangle for the height. The will do this repeatedly, even after you tell them that they must find the length of the altitude that is perpendicular to the segment that’s length is being used for the base in the formula . Sometimes they do not know what to do, and are just trying something, which is, in a way, admirable. The more common explanation though is that they forget. The students have been using this formula for years, they think they know this material, so they just plug and chug, not realizing that the given information has changed. Remind the students that now that they are in Geometry class, there is an extra step. The new challenge is to find the height, and then they can do the easy part and plug it into the formula.
Derive the Distance Formula – After doing an example with numbers to show how the distance formula is basically just the Pythagorean theorem, use variables to derive the distance formula. Most students will understand the proof if they have seen a number example first. Point out to the students that the number example was inductive reasoning, and the proof was deductive reasoning. Taking the time to do this is a good review of logic and algebra as well as great proof practice.
Converse of The Pythagorean Theorem
Mnemonic Devise for Acute and Obtuse Triangles – Many students have trouble remembering that the inequality with the greater tan is true when the triangle is acute, and that the equation with the less than is true for obtuse triangles. It seams backwards to them. One way to present this relationship is to compare the longest side and the angle opposite of it. In a right triangle, the equation has an equal sing; the hypotenuse is the perfect size. When the longest side of the triangle is shorter than what it would be in a right triangle, the angle opposite that side is also smaller, and the triangle is acute. When the longest side of the triangle is longer than what is would be in a right triangle, the angle opposite that side is also larger, and the triangle is obtuse.
Review Operations with Square Roots – Some of the exercises in this section require students to do operations with square roots. This is an essential skill for working with special right triangle which is an important topic that is also covered in this chapter. Many students struggle with using roots in algebra, and they have probably not thought about this topic for a year. Depending on the level of the class, it may be wise to take a day, or half a day, to review operations with square roots. Here are some sample problems of the basic operations with square roots that student will have to know how to do in order to be successful in this chapter.
Add or Subtract:
Using Similar Right Triangles
Separate the Three Triangles – The altitude from the right angle of a triangle divides the triangle into two smaller right triangles that are similar to each other, and to the original triangle. All the relationships among the segments in this figure are based on the similarity of the three triangles. Many students have trouble rotating shapes in their minds, or seeing individual polygons when they are overlapping. It is helpful for these students to draw the triangles separately and oriented in the same direction. After going through the process of turning and redrawing the triangles a few times, they will remember how the triangles fit together, and this step will no longer be necessary.
Color-Coded Flashcards – It is difficult to describe in words which segments to use in the geometric mean to find the desired segment. Labeling the figure with variables and using a formula is the standard method. The relationship is easier to remember if the labeling of the triangles is kept the same every time the figure is drawn. What the students need to remember, is the location of the segments relative to each other. Making color-coded pictures or flashcards will be helpful. For each relationship the figure should be drawn on both sides of the card. The segment whose measure is to be found should be highlighted in one color on the front, and on the back, the two segments that need to be used in the geometric mean should be highlighted with two different colors. Using two colors on the back is important because the segments often overlap. Making these cards will be helpful even if the students never use them. Those that have trouble remembering the relationship will use these cards frequently as a reference.
Add a Step and Find the Areas – The exercises in this section have the students find the base or height of triangles. They have all the information that they need to also calculate the areas of these triangles. Students need practice with multi-step problems. Having them find the area will help them think through a more complex problem, and give them practice laying out organized work for calculations that are more complex. Chose to extend the assignment or not based on how well the students are doing with the material, and how much time there is to work on this section.
1. Refer to the figure used to give the relationship of the altitude as the geometric mean of the lengths of the two segments of the hypotenuse on page 478 of the text.
Let and . What are the values of and ?
Special Right Triangles
Memorize These Ratios – There are some prevalent relationships and formulas in mathematics that need to be committed to long term memory, and the ratios made by the sides of these two special right triangles are definitely among them. Students will use these relationships not only in the rest of this class, but also in trigonometry, and in other future math classes. Students are expected to know these relationships, so the sooner learn to use them and commit them to memory, the better off they will be.
Two is Greater Than the Square Root of Three – One way that students can remember the ratios of the sides of these special right triangles, is to use the fact that in a triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. At this point in the class, students know that the hypotenuse is the longest side in a right triangle. What sometimes confuses them is that in the triangle, the ratio of the sides is , and if they do not really think about it, they sometimes put the as the hypotenuse because it might seem bigger than . Using the opposite relationship is a good method to use when working with these triangles. Just bring to the students’ attention that .
Derive with Variables – The beginning of the last chapter offers students a good amount of experience with ratios. If they did well on those sections, it would benefit them to see the derivation of the ratios done with variable expressions. It would give them practice with a rigorous derivation, review and apply the algebra they have learned, and help them see how the triangles can change in size.
Exact vs. Decimal Approximation – Many students do not realize that when they enter into a calculator and get , that this decimal is only an approximation of . They also do not realize that when arithmetic is done with an approximation, that the error usually grown. If is rounded to , the error is only , but if the three is now multiplied by five, the result is , instead of the it would have been if original the original number had not been rounded. The error has grown to . Most students find it more difficult to do operations with radical expressions than to put the numbers into their calculator. Making them aware of error magnification will motivate them to learn how to do operations with radicals. In the last step, it may be nice to have a decimal approximation so that the number can be easily compared with other numbers. It is always good to have an exact form for the answer so that the person using your work can round the number to the desired degree of accuracy. Less accuracy is needed for building a deck than sending a robot to Mars?
Trig Thinking – Students sometimes have a difficult time understanding trigonometry when they are first introduced to this new branch of mathematics. It is quite a different way of thinking when compared to algebra or even geometry. Let them know that as they begin their study of trigonometry in the next few sections the calculations won’t be difficult, the challenge will be to understand what is being asked. Sometimes students have trouble because they think it must be more difficult than it appears to be. Most students find they like trigonometry once they get the feel of it.
Ratios for a Right Angle – Students will sometimes try to take the sine, cosine or tangent of the right angle in a right triangle. They should soon see that something is amiss since the opposite leg is the hypotenuse. Let them know that there are other methods of finding the tangent of angles or more. The triangle based definitions of the trigonometric functions that the students are learning in this chapter only apply to angles in the interval .
The Ratios of an Angle – The sine, cosine, and tangent are ratios that are associated with a specific angle. Emphasize that there is a pairing between an acute angle measure, and a ratio of side lengths. Sine, cosine, and tangent is best described as functions. If the students’ grasp of functions is such that introducing the concept will only confuse matters, the one-to-one correspondence between acute angle and ratio can be taught without getting into the full function definition. When students understand this, they will have an easier time using the notation and understanding that the sine, cosine, and tangent for a specific angle are the same, no matter what right triangle it is being used because all right triangles with that angle will be similar.
Use Similar Triangles – Many students have trouble understanding that the sine, cosine, and tangent of a specific angle measure do not depend on the size of the right triangle used to take the ratio. Take some time to go back and explain why this is true using what the students know about similar triangle. It will be a great review and application.
- Students can construct similar right triangles using dilation from the transformation menu.
- After choosing a specific angle they should measure the corresponding angle in all the triangles. Each of these measurements should be equal.
- The legs of all the right triangles can be measured.
- Then the tangents can be calculated.
- Student should observe that all of ratios are the same.
Remind the students that if the right triangles have one set of congruent acute angles, then they are similar by the AA Triangle Similarly Postulate. Once the triangles are known to be similar it follows that their sides are proportional. The ratios are written using two sides of one triangle and compared to the ratios of the corresponding sides in the other triangle. This is different but equivalent to the ratios students probably used to find missing sides of similar triangles in previous sections.
Sine and Cosine Ratio
Trig Errors are Hard to Catch – The math of trigonometry is, at the point, not difficult. Not much computation is necessary to chose two number and put them in a ratios. What students need to be aware of is how easy it is to make a little mistake and not realize that there is an error. When solving an equation the answer can be substituted back into the original equation to be checked. The sine and cosine for acute angles do not have a wide range. It is extremely easy to mistakenly use the sine instead of the cosine in an application and. The difference often is small enough to seem reasonable, but still definitely wrong. Ask the student to focus on accuracy as they work with these new concepts. Remind them to be slow and careful.
Something to Consider – Ask the students to combine their knowledge of side-angle relationships in a triangle with the definition of sine. How does the length of the hypotenuse compare to the lengths of the legs of a right triangle? What does that mean about the types of numbers that can be sine ratios? With leading questions like these students should be able to see that the sine ratio for an acute angle will always be less than one. This type of analysis will prepare them for future math classes and increase their analytical thinking skills. It will also be a good review of previous material and help them check there work when they first start writing sine and cosine ratios.
Rationalizing the Denominator – Sometimes student will not recognize that and are equivalent. Most likely, they learned how to rationalize denominators in algebra, but it is nice to do a short review before using these types of ratios in trigonometry. Student will have to be able to easily switch between the two forms of the number when working with the unit circle in later classes.
Two-Step Problems – Having the students write sine, cosine, and tangent ratios as part of two-step problems will help them connect the new material that they have learned to other geometry they know. They will remember it longer, and be better able to see where it can be applied.
1. is a right triangle with the right angle at vertex .
What is the sign of ?
Answer: by the Pythagorean theorem, therefore .
Note: The sine of an angle does not have units. The units will cancel out in the ratio.
Inverse Trigonometric Ratios
Regular or Arc – Students will sometimes be confused about when to use the regular trigonometric function and when to use the inverse. They understand to concepts, but do not want to go through the entire thought process each time they must make the decision. I give them this short rule of thumb to help them remember: When looking for a ratio or side length, use regular and when looking for an angle use arc. They can associate “angle” and “arc” in their minds. Use the alliteration.
Which Trig Ratio – A common mistake students make when using the inverse trigonometric functions to find angles in right triangles is to use the wrong function. They may use arcsine instead of arccosine for example. There is a process that students can use to reduce the number of these kinds of errors.
- First, the students should mark the angle whose measure is to be found. With the angle in question highlighted, it is easier for the students to see the relationship the sides have to that angle. It is fun for the students to use colored pencils, pens, or highlighters.
- Next, the students should look at the sides with known side measures and determine their relationship to the angle. They can make notes on the triangle, labeling the hypotenuse, the adjacent leg and the opposite let. If they are having trouble with this I have them look for the hypotenuse first and always highlight it green, then they and decide between opposite and adjacent for the remaining to sides.
- Now, they need to look at the two sides they have chosen, and decide if they need to use sine, cosine, or tangent. It might help to have a mnemonic device to help them remember the definitions of the trigonometric functions. A common one is soh-cah-toa. The student can write this abbreviation on the top of every paper and refer to it when necessary. For example, in an exercise, if they decide it is the adjacent leg to the angle, and the hypotenuse that they have measures for, that is the “ah” portion of cah. They will know to use cosine, and be reminded that the length of the adjacent leg will e in the numerator of the ratio.
Make a Graph – Sometimes student will have a hard time seeing a pattern in a list of numbers. One way to help them remember the general trends in the trigonometric ratios is two have them make a graph. They can put the angle measure on the horizontal axis and the ratio, in decimal form, on the vertical axis. They will have to use different scales, of course. Now they can use their calculators to find the trig values of different angles at every five or ten degrees between zero and ninety and plot points on their graph. The comparison would be most meaningful if they put all three on the same set of axes with different colors. The process of making the graph and the visual representation of the pattern will form an impression in the students’ minds that will be useful and lasting.
Acute and Obtuse Triangles
Law of Sines or Law of Cosines – At first, it may be difficult for student to determine if they need to use the Law of Sines or the Law of Cosines to find a measure in a particular situation. Here is a good thought process for them to use.
- First have them look for the two, fairly easy to recognize, Law of Cosines situations. They have all three sides and are looking for an angle, or have two angles and the included side and are looking for the third side.
- If it is not one of these, then they need to try to set-up a Law of Sines proportion.
The Third Angle – Remind students that the three angles of a triangle have a sum of , and that this fact is often helpful when applying the Law of Sines or Cosines. Sometimes they may not be able to fine the angle they want directly, but if they find the third angle, they can use the Triangle Sum Theorem to get the measure they need.
Two Exercises in One - Sometimes the students will have to use both the Law of Sines and the Law of Cosines to find a measure in a specific triangle. For instance, let’s say they have two sides and the included angle of a triangle, but they do not want to find the third side, they want to find the other angles. It will be necessary to use the Law of Cosines to get the third side, and then use that third side to get the ratio for the triangle so the Law of Sines could be used. Remind students to be creative when solving exercises. They should use all of their mathematical knowledge to figure out the solution.
Triangle Labeling – Stress the labeling convention of using a capital letter for a vertex and the same letter in lower case for the opposite side. It is especially important when using the Law of Cosines to fine an angle. Students need to verify that they start the Law of Cosines with the side opposite of the angle they are finding.
Multiplication Before Addition – At this level, students usually faithfully follow this application of the order of operations except, when using the Law of Cosines to fine an angle. The last term of the Law of Cosines is . This is four values being multiplied together. The cosine function is new to students. They will not see it as the representation of a number and will separate it from the other terms. Frequently, they will subtract the from the or add it to the on the other side of the equation. Use a big multiplication symbol when writing and using the formula. It can be written, , to remind students to use the proper order of operations.