Don’t Forget the 12 – In this section there are two types of relationships that the students need to keep in mind when writing equations with variable expressions. The first involves the midpoint. When the expressions represent the two parts of a segment separated by the midpoint they just have to set the expressions equal to each other. The second is when comparing the length of a side of the triangle with the midsegment parallel to it. In this case they need to multiply the expression representing the side of the triangle by 12, and then set it equal to the expression representing the midsegment. They may forget the 12 or forget to use parenthesis and distribute. Remind them that they need to multiply the entire expression by 12, not just the first term. Similarly, they may mess up the distribution going the other way. When given the midsegment, they must multiply the whole expression by 2 to get the third side of the triangle.
Midpoint Formula, Distance Formula and Slope Formula - You may need to review these formulas again. Remind students that the Midpoint Formula produces a point. Also, remind students that the slopes of parallel lines are the same.
Parallel - Students may forget that a midsegment is also parallel to the third side of a triangle. They are apt to focus most on the length relationship since it is used most in the problem sets.
Perpendicular Bisectors and Angle Bisectors in Triangles
Construction Frustrations - Using a compass and straightedge to make clean, accurate constructions takes a bit of practice. Some students will pick up the skill quickly and others will struggle. Practice, practice, practice. Help the students individually to make nice arcs. A few minutes of practicing just making circles will help them to get more comfortable and accurate with the compass. They will know right away if they are changing the size of the arc mid circle because the “ends” will not meet up. What is nice about doing construction in the classroom is that it is often the students that typically struggle with mathematics, the more artistically minded students, that excel and learn from constructing figures.
Here are some other tips for good construction: (1) Hold the compass at an angle to the paper rather than perpendicularly. Suggest that students try not to press down very hard- a light arc is sufficient and will be easier to make. (2) Try rotating the paper while holding the compass steady. (3) Work on a stack of a few papers so that the needle of the compass can really dig into the paper and will not slip. (4) Suggest that students hold the compass by the “circle” or vertex where the two radii meet- often students will try to hold the compass by the needle and pencil and they grip it so tightly that they change the angle in mid arc.
Perpendicular Bisector Quirks - There are two key ways in which the perpendicular bisector of a triangle is different from the other segments in the triangle that the students will learn about in subsequent sections. Since they are learning about the perpendicular bisector first these differences do not become apparent until the end of the chapter.
The perpendicular bisector of the side of a triangle does not have to pass through a vertex. Have the students explore in what situations the perpendicular bisector does pass through the vertex. They should discover that this is true for equilateral triangles and for the vertex angle of isosceles triangles.
The point of concurrency of the three perpendiculars of a triangle, the circumcenter, can be located outside the triangle. This is true for obtuse triangles. The circumcenter will be on the hypotenuse of a right triangle. This is also true for the orthocenter, the point of concurrency of the altitudes.
Circumcenter and Perpendicular Bisector Relationship - Stress the relationship between the points on a perpendicular bisector and the center of a circle. Remind students that the definition of the center of a circle is the point equidistant from every point on the circle. Remind students that every point on a perpendicular bisector is equidistant from the endpoints of a segment. Make the connection that this means that the point of concurrency of the perpendicular bisectors is equidistant from all three vertices of the triangle. This means that the vertices all lay on a circle with the center at this point of concurrency- the circumcenter.
Same Construction for Midpoint and Perpendicular Bisector - The Perpendicular Bisector Theorem is used to construct the perpendicular bisector of a segment and to find the midpoint of a segment. When finding the midpoint, the students should make the arcs, one from each endpoint with the same compass setting, to find two equidistant points, but instead of drawing in the perpendicular bisector, they can just line up their ruler and mark the midpoint. This will keep the drawing from getting overcrowded and confusing.
Incenter and Angle Bisector Relationship - Stress the relationship between the incenter and the angle bisectors of a triangle. Again, discuss the definition of the center of a circle. Also, remind students that all the points on an angle bisector are equidistant from the sides of the angles. In this case, the point of concurrency of the angle bisectors will be equidistant from the three sides of the triangle. Remind students that the distance measured from a point to a line (or segment in this case) is measured along a perpendicular segment. You may wish to demonstrate constructing the perpendiculars from the incenter to each of the sides. This segment is the radius of the inscribed circle. Students may not be able to do this accurately themselves but doing this in a demonstration for them may help make the concept stick.
Adaptation - For practical purposes, you may wish to skip having students construct the perpendiculars to determine the correct compass setting for the inscribed circle. In this case, have students to place the center of the compass at the incenter, choose one side, and adjust the compass setting until the compass brushes by that side of the triangle, without passing through it. The word tangent does not have to be introduced at this point if the students already have enough vocabulary to learn. When the incenter is correctly placed, the compass should also hit the other two sides of the triangle once, creating the inscribed circle.
Check with a Third - When constructing the point of concurrency of the perpendicular bisectors or angle bisectors of a triangle, it is strictly necessary to construct only two of the three segments. The theorems proved in the texts ensure that all three segments meet in one point. It is advisable to construct the third segment as a check of accuracy. Sometimes the compass will slip a bit while the student is doing the construction. If the three segments form a little triangle, instead of meeting at a single point, the student will know that their drawing is not accurate and can go back and check their marks.
Special Triangles - In some special triangles, these segments overlap. The following examples may be used to have students explore these cases.
Example 1: Construct an equilateral triangle. Now construct the perpendicular bisector of one of the sides. Construct the angle bisector from the angle opposite of the side with the perpendicular bisector. What do you notice about these two segments? Will this be true of a scalene triangle? Consider what would happen if you found the circumcenter and the incenter. Where would they be located?
Answer: The segments should coincide on the equilateral triangle, but not on the scalene triangle. The incenter and the circumcenter will be located in the same place. You could extend this to an isosceles triangle to show that the angle bisector of the vertex angle overlaps with the perpendicular bisector of the base.
Example 2: Construct an equilateral triangle. Now construct one of the angle bisectors. This will create two right triangles. Label the measures of the angles of the right triangles. With your compass compare the lengths of the shorter leg to the hypotenuse of either right triangle. What do you notice?
Answer: The hypotenuse should be twice the length of the shorter leg. You may wish to refer back to this example in the next lesson to show that the angle bisector is the same segment as the median in an equilateral triangle.
Medians and Altitudes in Triangles
Vocabulary Overload - So far this chapter has introduced to a large number of vocabulary words, and there will be more to come. This is a good time to stop and review the new words before the students become overwhelmed. Have them make flashcards, or play a vocabulary game in class.
Label the Picture - When using the Concurrency of Medians Theorem to find the measure of segments, it is helpful for the students to copy the figure onto their paper and write the given measures by the appropriate segments. When they see the number in place, it allows them to concentrate on the relationships between the lengths since they no longer have to work on remembering the specific numbers.
Median or Perpendicular Bisector - Students sometimes confuse the median and the perpendicular bisector since they both involve the midpoint of a side of the triangle. The difference is that the perpendicular bisector must be perpendicular to the side of the triangle, and the median must end at the opposite vertex. Show lots of examples of the two of these on the same triangle so students have a visual memory of the difference. Discuss with students (or have them explore and then discuss) that these segments will be the same for each vertex and opposite side in equilateral triangles and for the median drawn from the vertex angle of an isosceles triangle and the perpendicular bisector of the base.
Applications - Students are much more willing to spend time and effort learning about topics when they know of their applications. Questions like the ones below improve student motivation.
In the following situations would it be best to find the circumcenter, incenter, or centroid?
Example 1: The drama club is building a triangular stage. They have supports on all three corners and want to put one in the middle of the triangle.
Answer: Centroid, because it is the center of mass or the balancing point of the triangle
Example 2: A designer wants to fit the largest circular sink possible into a triangular countertop.
Answer: Incenter, because it is equidistant from the sides of the triangle.
Extending the Side - Many students have trouble knowing when and how to extend the sides of a triangle when drawing in an altitude. First, this only needs to be done with obtuse triangles when drawing the altitude that intersects the vertex of one of the acute angles. It is the sides of the triangle that form the obtuse angle that need to be extended. The students should rotate their paper so that the vertex of the acute angle they want to start an altitude from is above the other two, and the segment opposite of this vertex is horizontal. Now they just need to extend the horizontal side until it passes underneath the raised vertex.
The Altitude and Distance - The distance between a point and a line is defined to be the shortest segment with one endpoint on the point and the other on the line. It has been shown that the shortest segment is the one that is perpendicular to the line. So, the altitude is the segment along which the distance between a vertex and the opposite side is measured. Seeing this connection will help students remember and understand why the length of the altitude is the height of a triangle when calculating the triangle’s area using the formula A=12 bh.
Orthocenter - Students need to connect this term to something to help them remember that it is the point of concurrency of the altitudes in a triangle. They are likely unfamiliar with the term orthogonal - which means perpendicular. Telling them this may help them connect the term orthocenter to the altitudes (or perpendicular segments from vertices to the opposite side in triangles).
Explorations - When students discover a property or relationship themselves it will be much more meaningful. They will have an easier time remembering the fact because they remember the process that resulted in it. They will also have a better understanding of why it is true now that they have experience with the situation. Unfortunately, students sometimes become frustrated with explorations. They may not understand the instruction, or they may not be carefully enough and the results are unclear. Some of the difficulties can be alleviated by having the students work in groups. They can work together to understand the directions and interpret the results. Students strong in one area, like construction, can take on that part of the task and help the others with their technique.
Some guidelines for successful group work.
- Groups of three work best.
- The instructor should choose the groups before class.
- Students should work with new groups as often as possible.
- Desks or tables should be arranged so that the members of the group are physically facing each other.
- The first task of the group is to assign jobs: person one reads the directions, person two performs the construction, person three records the results. Students should regularly trade tasks.
- Teachers need to circulate and provide additional assistance so that groups do not get frustrated or go off task.
Inequalities in Triangles
The Opposite Side/Angle - At first it may be difficult for students to recognize what side is opposite a given angle or what angle is opposite a given side. If it is not obvious to them from the picture, obtuse, scalene triangles can be confusing, they should use the names. For ΔABC, the letters are divided up by the opposite relationship, the angle with vertex A is opposite the side with endpoints B and C. Being able to determine these relationships without a figure is important when studying trigonometry.
Small, Medium, and Large - When working with the relationship between the sides and angles of a triangle, students will summarize the theorem to “largest side is opposite largest angle”. They sometimes forget that this comparison only works within one triangle. There can be a small obtuse triangle in the same figure as a large acute triangle. Just because the obtuse angle is the largest in the figure, does not mean the side opposite of it is the longest among all the segments in the figure, just that it is the longest in that obtuse triangle. If the triangles are connected or information is given about the sides of both triangles, a comparison between triangles could be made. See exercise #29 in the text.
Add the Two Smallest - The triangle inequality says that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. In practice it is enough to check that the sum of the lengths of the smaller two sides is larger than the length of the longest side. When given the three sides lengths for a triangle, students who do not fully understand the theorem will add the first two numbers instead of the smallest two. When writing exercises it is easy to always put the numbers in ascending order without thinking much about it. Have the students try to draw a picture of the triangle. After making a few sketches they will understand what they are doing, instead of just blindly following a pattern.
Range of Possible Lengths - Students are so used to finding a single “answer” to a problem that they often struggle with the idea that the third side of a triangle could take on multiple possible lengths and how to figure out this range of values. A shortcut is that the length of the third side is always between the difference and sum of the two know lengths. Students also do not immediately understand why the inequality shows values that the third side cannot assume. For example in a triangle with known sides, 5 and 8, the range of possible values for the third side is 3<x<13. They may question why it isn’t 4<x<12 since the third side cannot be 3 or 13. Remind them that the sides don’t have to have whole number lengths.
SSS and SAS Use Color - When the figures have two triangles instead of just one, they become more complex. The students may need some help sorting out the shapes. A good way for them to begin this process is to draw the figure on their paper and use highlighters to color code the information.
Both of the theorems presented in this section require two pairs of congruent sides. The first step is for student to highlight these four sides in a common color, let’s say yellow. Once they have identified the two pairs of congruent sides, they know the hypothesis of the theorem has been filled and they can apply the conclusion.
The conclusions of these theorems involve the third side and the angle between the two congruent sides. These parts of the triangles can be highlighted in a different color, let’s say pink.
Now the students need to determine if they need to use the SAS Inequality theorem or the SSS inequality theorem. If they know one of the pink angles is bigger than the other, than they will use the SAS Inequality theorem and write an inequality involving the pink sides. If they know that one of the pink sides is bigger than the other, they will apply the SSS Inequality theorem, and write an inequality involving the two pink angles.
Having a step-by-step process is good scaffolding for students as they begin working with new types of problems. After the students have gained some experience, they will no longer need to go through all the steps.
Solving Inequalities - Students learned to solve inequalities in algebra, but a short review may be in order. Solving inequalities involves the same process as solving equations except the equal sign is replaced with an inequality, and there is the added rule that if both sides of the inequality are multiplied or divided by a negative number the direction of the inequality changes. Students frequently want to change the direction of the inequality when it is not required. They might mistakenly change the inequality if they subtract from both sides, or if result of multiplication or division is a negative even if the number used to change the inequality was not negative. In geometry it is most common to be working with all positive numbers, but depending on how the students apply the Properties of Inequalities, they may create some negative values.
Extension: Indirect Proof
Why Learn Indirect Proof - For a statement to be mathematically true it must always be true, no exceptions. This frequently makes it easier to prove that a statement is false than to prove it is true. Indirect proof gives mathematician the choice between proving a statement true or proving a statement false and can therefore greatly simplify some proofs. Letting the students know that indirect proof can be a potential shortcut will motivate them to learn to use this type of logic.
Indirect Proof - Students will not really understand the method of indirect proof the first time they see it. Let them know that this is just the first introduction, and that in subsequent lessons they will be given more examples and opportunities to learn this new method of proof. If students think they are supposed to understand something perfectly the first time they see it, and they don’t, they will become frustrated with themselves and mathematics. Let them know that the brain needs time and multiple exposures to master these challenging concepts.
Review the Contrapositive - Proving a statement using indirect proof is equivalent to proving the contrapositive of the statement. If students are having trouble setting up indirect proofs, or even if they are not, it is a good idea to have them review conditional statements and the contrapositive. The first step to writing an indirect proof, can be to have them write out the contrapositive of the statement they want to prove. This will reduce confusion about what statement to start with, and what statement concludes the proof.
Does This Really Prove Anything? - Even after students have become adept with the mechanics of indirect proof, they may not be convinced that what they are doing really proves the original statement. This is the same as asking if the contrapositive is equivalent to the original statement. Using examples outside the field of mathematics can help students concentrate on the logic.
Start with the equivalence of the contrapositive. Does statement (1) have the same meaning as statement (2)?
- If you attend St. Peter Academy, you must wear a blue uniform.
- If you don’t wear a blue uniform, you don’t attend St. Peter Academy.
Let the students discuss the logic, and have them create and share their own examples.
If a good class discussion ensues, and the students provide many statements on a single topic, it may be possible to write some indirect proofs of statements not concerned with mathematics. This could be a good bonus assignment or project that when presented to the class will make the logic of indirect proof clearer for other students.