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The Pythagorean Theorem

Reducing Radicals - Students may or may not have spent much time reducing radicals in previous math courses. You may need to review how to do this and how to perform operations with radicals. Students may need to see more than one way to reduce radicals. The most common is to take out the greatest factor which is a perfect square, but this can be difficult for many students with weak mental math skills. Another method is to make a factor tree and find the prime factorization of the number and identify doubles that can be taken out. There is not necessarily a “best” way, it is more important to figure out which way your students can best reduce radicals correctly.

Method 1: \sqrt{72}=\sqrt{36 \times 2}=6\sqrt{2}

Method 2: \sqrt{72}=\sqrt{9 \times 8}=\sqrt{3 \times 3 \times 2 \times 2 \times 2}=3 \times 2\sqrt{2}=6\sqrt{2}

The following are some examples that will help students review the basic properties of radicals and practice reducing radicals.

Example 1: \sqrt{112}

Answer: \sqrt{16 \times 7}=4\sqrt{7}

Students may not recognize right away that 16 is a factor of 112. This problem can also be solved by completely factoring 112 (method 2).

Example 2: 4\sqrt{192}

Answer: 4 \sqrt{64 \times 3}=4 \times 8 \sqrt{3}=32 \sqrt{3}

Example 3: 2\sqrt{5}+\sqrt{45}

Answer: 2\sqrt{5}+3\sqrt{5}=5\sqrt{5}

Review with students that in order to add or subtract radicals, the radical must be identical. Sometimes they can meet this condition by simplifying one or both terms and sometimes they just won’t be able to add them together.

Example 4: \sqrt{6} \times \sqrt{18}

Answer: \sqrt{108}=\sqrt{36 \times 3}=6\sqrt{3}

Students may have forgotten that radicals can be multiplied or divided to form a new radical which they may then be able to reduce.

Example 5: \left(\sqrt{11}\right)^2

Answer: 11

Review the definition of a square root with students to help them understand this one. The square root and the square cancel each other out - they are inverse operations.

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. You should follow the link given in the text to see additional proofs of the Pythagorean Theorem. 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. 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 A=\frac{1}{2} bh. 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 that 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.

Squaring a Negative - One of the most common errors students make when using the distance formula is that they use their calculator incorrectly when squaring negative numbers. Remind students that when they square a negative number, the result is positive. Describe the difference between -8^2 and (-8)^2 and review the correct of order of operations. You may also want to stress the connection between the distance formula and the Pythagorean theorem and encourage them to think of the difference between the x’s and y’s as lengths and therefore they can just use the absolute value of the difference (i.e. ignore the negative).

Converse of the Pythagorean Theorem

Acute and Obtuse Triangles - Many students have trouble remembering that the inequality with the greater than is true when the triangle is acute, and that the equation with the less than is true for obtuse triangles. It seems 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 sign; 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.

Is It Really a Triangle? - I have found that once students start using the Pythagorean theorem to determine whether lengths form a right, acute or obtuse triangle that they forget completely that the sum of two sides must be greater than the remaining side in order for a triangle to exist at all. The following example illustrates this misunderstanding. You may want to put it on the board and ask your students what kind of triangle is formed.

Example: What kind of triangle is formed by lengths 3, 4, 7?

Answer: None! There is no triangle at all. 3+4=7.

If students used the Pythagorean theorem and didn’t check to make sure there was a triangle at all then they would have said that the triangle is obtuse. This in incorrect.

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. The students need to remember 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.

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 they 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 30-60-90 triangle, the ratio of the sides is 1:2:\sqrt{3}, and if they do not really think about it, they sometimes put the \sqrt{3} as the hypotenuse because it might seem bigger than 2. Using the opposite relationship is a good method to use when working with these triangles. Just bring to the students’ attention that 2>\sqrt{3}.

How Do I Find the Short Leg Again? - While students may quickly memorize the two special right triangle ratios, they may have trouble applying the ratios to find the unknown sides. One way to help students with this process is to have them write the ratios with a variable. For example the 30-60-90 triangle ratio would be x:2x:x\sqrt{3}. Next, have them identify which side they are given and use the appropriate part of the ratio to determine x and the other side.

Example 1: Find the other two sides in a 30-60-90 triangle given that the hypotenuse is 8.

Answer: First set 2x=8 and solve to get x=4 which is the short leg. The long leg is then 4\sqrt{3}.

Example 2: Find the length of a leg in an isosceles right triangle with hypotenuse 3.

Answer: The ratio for an isosceles right triangle or 45-45-90 triangle is x:x:x\sqrt{2}. Since we are given the hypotenuse here, x\sqrt{2}=3. Now we must solve for x as shown below.

\frac{x\sqrt{2}}{\sqrt{2}} &= \frac{3}{\sqrt{2}}\\x &= \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}=\frac{3\sqrt{2}}{2}

Rationalizing the Denominator - Sometimes student will not recognize that \frac{1}{\sqrt{2}} and \frac{\sqrt{2}}{2} 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 special right triangles.

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 \sqrt{2} into a calculator and get 1.414213562, that this decimal is only an approximation of \sqrt{2}. They also do not realize that when arithmetic is done with an approximation, that the error usually grown. If 3.2 is rounded to 3, the error is only 0.2, but if the three is now multiplied by five, the result is 15, instead of the 16 it would have been if original the original number had not been rounded. The error has grown to 1.0. 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.

Tangent, Sine and Cosine

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 90^\circ 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 0^\circ <m<90^\circ.

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. 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. It may be a good activity to have students make right triangles with a particular acute angle measure and compare the ratios of the sides. They should see that no matter how big or small they made their triangle, they get the same ratios as their classmates. You could also refer to the special right triangles to make this connection. Explaining that the names sine, cosine and tangent were given to these ratios and the values were recorded in tables by angle measure may help them understand the idea a little better. You may even want to show them a trig table of values and explain that their calculator is simply looking up a value in a table when they type in the trig function and a particular angle measure.

Here is a Sketchpad activity that may further enhance student understanding:

  1. Students can construct similar right triangles using dilation from the transformation menu.
  2. After choosing a specific angle they should measure the corresponding angle in all the triangles. Each of these measurements should be equal.
  3. The legs of all the right triangles can be measured.
  4. Then the tangents can be calculated.
  5. Student should observe that all of ratios are the same.

Trig Errors are Hard to Catch - The math of trigonometry is, at this point, not difficult. Not much computation is necessary to chose two numbers and put them in a ratio. 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.

SOHCAHTOA - This pneumonic device has been around for a while because it helps students keep the ratios straight. Another way to write it that makes it even more clear is: S \frac{O}{H} \  C \frac{A}{H} \ T \frac{O}{A}.

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 and cosine ratios 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.

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.

Example: \Delta ABC is a right triangle with the right angle at vertex C. AC=3 \ cm,BC=4 \ cm. What is the sine of \angle A?

Answer: AB=5 \ cm by the Pythagorean theorem, therefore \sin A = \frac{4}{5}.

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.

It may also help to explain that the inverse or “arc” trigonometric function does the reverse of the original. Each of the original three trig functions essentially go to a table, look up an angle measure and find the corresponding ratio. The “arc” functions will look up the ratio in the table and give back the corresponding angle measure. Understanding what it is that the calculator does when they use these functions may help reinforce student understanding.

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.

  1. 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.
  2. 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 can decide between opposite and adjacent for the remaining two sides.
  3. Now, they need to look at the two sides they have chosen, and decide if they need to use sine, cosine, or tangent.

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