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Ratios and Proportions

Keep it in Order - When writing a ratio, the order of the numbers is important. When the ratio is written in fraction form the amount mentioned first goes in the numerator, and the second number goes in the denominator. Remind the students it is important to keep the values straight to avoid confusion or misunderstandings. For example, if they are looking at a college and see that the male to female ratio is 11 to 12, it is important to know which one comes first to interpret the ratio correctly.

To Reduce or Not to Reduce - When a ratio is written in fraction form it can be reduced like any other fraction. This will often make the arithmetic simpler and is frequently required by instructors for fractions in general. But when reducing a ratio, useful information can be lost. If the ratio of girls to boys in a classroom is16 to 14, it may be best to use the fraction \frac{16}{14} because it gives the total number of students in the class where the reduced ratio \frac{8}{7} does not.

Consistent Proportions - A proportion can be correctly written in many ways. As long as the student sets up the ratios in a consistent, orderly fashion, they will most likely have written a correct proportion. There should be a common tie between the two numerators, the two denominators, the numbers in the first ratio, and the numbers in the second ratio. They should think about what the numbers represent, and not just use them in the order given in the exercise, although the numbers are often given in the correct order.

Example: Victor got a new hybrid. He went 525 gallons on the first five gallons that came with the car. He just put 12 gallons in the tank. How far can he expect to go on that amount of gas?

Answer: \frac{25}{5}=\frac{x}{12}, so x=\frac{12*25}{5} and Victor can expect to go 1,260 miles.

Note: Students may be tempted to put the 12 in the numerator of the second ratio because it was the third number given in the exercise, but it should go in the denominator with the other amount of gas.

The Fraction Bar is a Grouping Symbol - Students know that parenthesis are a grouping symbol and that they need to distribute when multiplying a number with a sum or difference. A fraction bar is a more subtle grouping symbol that students frequently overlook, causing them to forget to distribute. To help them remember have them put parenthesis around sums and differences in proportions before they cross-multiply.

Example: \frac{x+3}{5}=\frac{x-8}{7} becomes \frac{(x+3)}{5}=\frac{(x-8)}{7}

Everybody Loves to Cross-Multiply - There is something satisfying about cross-multiplying and students are prone to overusing this method. Remind them that cross-multiplication can only be used in proportions, when two ratios are equal to each other. It is not appropriate to cross-multiply when two fractions are being added, subtracted, multiplied or divided. It might be helpful to do some example of these to illustrate the difference and discuss the difference between “cross multiplication” and “multiplying across.”

Examples: Cross multiply here: \frac{3}{4}=\frac{10}{x} or \frac{x+1}{7}=\frac{8}{x}

Don’t cross multiply here: \frac{3}{4}+\frac{10}{x} or \frac{x+1}{7} \div \frac{8}{x}

Only Cancel Common Factors - When reducing a fraction or putting a ratio in simplest terms, students often try to cancel over an addition or subtraction sign. This problem occurs most frequently when students work with fractions that contain variable expressions. To combat this error, go back to numerical examples. Students will see that what they are doing does not make sense when the variables are removed. Then go back to example with variables. Hopefully the students will be able to carry over the concept.

Examples: Can be reduced: \frac{3  \cdot 2}{5 \cdot 2}=\frac{3}{5} and \frac{3(x-4)}{3 \cdot 2}=\frac{(x-4)}{2}

Can’t be reduced: \frac{3+2}{5+2} \neq \frac{3}{5} and \frac{(x-4)}{4} \neq x-1

Similar Polygons

A Common Vocabulary Error - Students frequently interchange the words proportional and similar. Remind them that proportional describes a relationship between numbers, and similar describes a relationship between figures. You can relate this difference in definition back to the difference between the terms equal and congruent.

Compare and Contrast Similar with Congruent - If your students have already learned about congruent figures, now would be a good time to review. The definitions of congruent and similar are very close. Ask the students if they can identify the difference; it’s only one word. You can also point out that congruent is a subset of similar like square is a subset of rectangle, or mother is a subset of women. Understanding the differences between congruent and similar will be important in upcoming lessons when proving triangles similar.

Use that Similarity Statement - In some figures, which sides of similar polygons correspond is obvious, but when the polygons are almost congruent, or oriented differently, the figure can be misleading. Students usually begin by using the figure and then forget to use the similarity statement when necessary. Remind them about this information as they start working on more complicated problems. The similarity statement is particularly useful for students that have a hard time with visual-spatial processing. It is a good idea to do several examples in which students are “forced” to use the similarity statement to align the correct sides and angles to get them in the habit of using the similarity statement rather than their “eyes.”

Who’s in the Numerator - When writing a proportion students sometimes carelessly switch which polygon’s measurements are in the numerator. To help students avoid this pitfall, I tell the students to choose right from the beginning and BE CONSISTENT throughout the problem. When it comes to writing proportions if the students focus on being orderly and consistent, they will usually come up with a correct setup.

Bigger or Smaller - After completing a problem it is always a good idea to take a minute to decide if the answer makes sense. This is hard to get students to do. When using a scale factor, a good way to check that the correct ratio was used is to notice if the number got bigger or smaller. Is that what we expected to happen?

Update the List of Symbols - In previous lessons it has been recommended that students create a reference page in their note books that contains a list of all the symbols and how they are being used in this class. Students should add the symbol for similar to the list, and take few minutes to compare it to the symbols they already know. Sometimes students will read the similarity symbol as “approximately equal”. It is standard to use two wavy lines for approximately equal and one wavy line for similar, but this is not always the case.

Similarity by AA

Definition of Similar Triangles vs. AA Shortcut - Let the students know what a deal they are getting with the AA Triangle Similarity Postulate. The definition of similar polygons requires that all three corresponding pairs of angles be congruent, and that all three pairs of corresponding sides are proportional. This is a significant amount of information to verify, especially when writing a proof. The AA postulate is a significant shortcut; only two piece of information need to be verified and all the rest comes for free. When students see how much this reduces the work, they will be motivated to understand the proof and will enjoy using the postulate. Everybody likes to use a tricky shortcut. A fun activity to explore this concept would be to have students create a triangle with three given angle measures. Then, in pairs, compare the lengths of corresponding sides to discover that they are indeed similar. Then ask students if they really needed to be told the third angle measure? This process should help them remember AA Similarity and help reinforce the idea that AA does not ensure congruent triangles.

Get Some Sun - It is always a good idea to create some variety in the class. It will keep students’ minds active. Although it is time consuming, get some yard sticks and take the students outside to measure a tree or a flagpole using their shadows and similar triangles. Have them evaluate their accuracy. They will have to measure carefully if they are to get a reasonable numbers. This will give them some practice using a ruler and converting units. The experience will also help them put what they are learning about similar triangles into their long term memory.

Trigonometry - Let the students know that the next chapter is all about trigonometry, and that the AA Triangle Similarity Postulate is what make trigonometry possible. Mentioning what is to come will start to prepare their minds and make learning the material in the next chapter that much easier. Here are some problems that involve similar right triangles to accustom the students to this new branch of mathematics.

Example 1: \Delta ABC is a right triangle with right angle C and \Delta ABC \sim \Delta XYZ. Which angle in \Delta XYZ is the right angle?

Answer: \angle Z

Example 2: \Delta CAT \sim \Delta DOG with right angle at A. If CA=5 \ cm, CT=13 \ cm and DO=15 \ cm, what is the length of \overline{OG}?

Answer: 36 cm

There is more than one step required to solve this problem. Students must use the Pythagorean theorem and the definition of similar polygons. First, the ratio between the figures is determined by \frac{CA}{DO}=\frac{5}{15}=\frac{1}{3}. Next, we need the length of \overline{AT} since it corresponds to \overline{OG}. Using the Pythagorean Theorem gives us AT=12 \ cm. Now we can set up the proportion \frac{1}{3}=\frac{12}{x} and solve it to get x=36.

Similarity by SSS and SAS

The S of a Triangle Similarity Postulate - At this point in the class, students have shown that a significant number of triangles are congruent. They have learned the process well. When teaching them to show that triangles are similar, it is helpful to build on what they have learned. The similarity postulates have S’s and A’s just like the congruence postulates and theorems. The A’s are treated exactly the same in similarity postulates as they were in congruence theorems. Each A in a similarity shortcut stands for one pair of congruent corresponding angles in the triangles.

The S’s represent a different requirement in similarity postulates than they did in congruence postulates and theorems. Congruent triangles have congruent sides, but similar triangles have proportional sides. Each S in a similarity postulate represents a ratio of corresponding sides. Once the ratios (two for SAS and three for SSS) are written, equality of the ratios must be verified. If the ratios are equal, the sides in question are proportional, and the postulate can be applied.

It is sometimes hard for student to adjust to this new side requirement. They have done so much work with congruent triangles that it is easy for them to slip back into congruent mode. Warn them not to fall into the old way of thinking.

Triangle Congruence Postulates and Theorems Triangle Similarity Postulates

\begin{aligned}&\text{S} \ \rightarrow \ \text{congruent sides} && \text{S} \ \rightarrow \ \text{proportional sides}\\&\text{SSS} && \text{AA}\\&\text{SAS} && \text{SSS}\\&\text{ASA} && \text{SAS} \\&\text{AAS} && \\&\text{HL} \end{aligned}

Only Three Similarity Postulates - Students will sometimes try to use ASA, or other congruence theorems to show that two triangles are similar. Bring it to their attention that there are only three postulates for similarity, and that they do not all have the same side and angle combinations as congruence postulates or theorems. It may help them to show them that the Congruence Postulates ASA and AAS are both “represented” by the Similarity Postulate AA.

Proportionality Relationships

Similar Triangles Formed by an Interior Parallel Segment - Students frequently are presented with a triangle that contains a segment that is parallel to one side of the triangle and intersects the other two sides. This segment creates a smaller triangle in the tip of the original triangle. There are two ways to consider this situation. The two triangles can be considered separately, or the Triangle Proportionality Theorem can be applied.

(1) Consider the two triangles separately.

The original triangle and the smaller triangle created by the parallel segment are similar as seen in the proof of the Triangle Proportionality theorem. One way students can tackle this situation is to draw the triangles separately and use proportions to solve for missing sides. The strength of this method is that it can be used for all three sides of the triangles. Students need to be careful when labeling the sides of the larger triangle; often the lengths will be labeled as two separate segments and the students will have to add to get the total length.

(2) Use the Triangle Proportionality theorem.

When using this theorem it is much easier to setup the proportions, but there is the limitation that the theorem cannot be used to find the lengths of the parallel segments.

Ideally, students will be able to identify the situations where each method is the most efficient, and apply it. This may not happen until the students have had some experience with these types of problems. It is best to have students use method (1) at first, then after they have worked a few exercises on their own, they can use (2) as a shortcut in the appropriate situations.

Make sure to do many examples where students need to use the similar triangles to solve for one of the parallel segments. Combining this requirement in the same problem where the shortcut can be utilized is especially helpful for them to begin recognizing the difference.

Example: \Delta ABC has point E on \overline{AB} and F on \overline{BC} such that \overline{EF} \ || \ \overline{AC}. Given AE=5 \ cm, EB=3 \ cm, BF=4 \ cm and AC=10 \ cm, find EF and FC.

Answer: \frac{3}{8}=\frac{EF}{10}, EF=3 \frac{3}{4} \ cm

\frac{3}{5}=\frac{4}{FC} or \frac{3}{8}=\frac{4}{FC+4}, FC=6 \frac{2}{3} \ cm

Encourage students to draw a diagram and label it with the given lengths. Next, they may wish to draw the two similar triangles separately to better visualize which parts correspond.

Proportions with Angle Bisectors - Students have a hard time with this one because the two triangles formed by the angle bisector are not similar. Students need to see lots of examples to get this one straight. Remind them that this proportion is true only when the segment is an angle bisector- it is not necessarily true for altitudes or medians in the triangle.

Similarity Transformations

Apostrophe vs Prime - When a geometric figure is transformed (by translation, rotation, dilation, etc.) the image is denoted using the “prime” marking. For example, if \Delta ABC is transformed, the image is denoted by \Delta A'B'C'.

Scale Factor Compared to Segment and Area Ratios - When a polygon is dilated using scale factor, k, the ratio of the image of the segment to the original segment is k. This is true for the sides of the polygon, all the special segments of triangles studied in chapter five, and the perimeter of the polygon. The relationship holds for any linear measurement. Area is not a linear measurement and has a different scale factor. The ratio of the area of the image to the area of the original polygons is k^2. Students frequently forget to square the scale factor when working with the ratios of a figure and its image. This concept will be explored further in Chapter 10: Perimeter and Area, but the idea can be introduced here.

Example: \Delta ABC has coordinates A(1, 13), B(6, 1) and C(1, 1). Complete the following:

  1. Graph \Delta ABC.
  2. Use the distance formula to find the length of each side of \Delta ABC.
  3. Calculate the perimeter of \Delta ABC.
  4. Calculate the area of \Delta ABC.
  5. \Delta A' B' C' is the image of \Delta ABC under a dilation centered at the origin with scale factor 3. Graph \Delta A'B'C'.
  6. Calculate the perimeter of \Delta ABC.
  7. Calculate the area of \Delta A'B'C'.
  8. Compare \Delta A' B' C' to \Delta ABC. What is the ratio of each set of corresponding side lengths, the perimeters and the areas? What do you notice when these ratios are compared to the scale factor?

Answers:

  1. Graph
  2. AC=12, BC=5, AB=13
  3. 30
  4. 30
  5. Graph
  6. AC=36, BC=15, AB=39
  7. 90
  8. 270
  9. The ratios of the side lengths and the perimeter are 3:1, the same ratio as the scale factor. The ratio of the areas is 9:1, the square of the scale factor.

Extension: Self-Similarity

More Complex Fractals – Students need to begin learning about fractals with the simple examples given in the text. Once they have taken some time to work with, and understand the self-similar relationship, it is amazing to see how complex and beautiful fractals can become. Numerous examples of exquisite fractals can be found on-line. If you are lucky enough to have access to computers and a projector, have the students search for fractals and choose their favorite to share with the class. Student will begin to realize the importance of what there are learning when they see what a huge ocean they are dipping their toe into.

Applications - Many students need to know how a subject is useful before they are motivated to spend time and energy learning about it. Throughout the text there have been references to modeling and how mathematical concepts often need to be adjusted to fit the world around us. Fractals are used to model many aspects of nature including tree branches, shells, and the coast line. Knowing of the applications of fractals motivates students. If time permits give a more in-depth explanation, or use this topic to assign research projects.

Video Time - Self-similarity and fractals make up an extremely complex visual topic. There are many videos in common use that can give a much more exciting and attention grabbing explanation than most teachers can deliver while standing in front of the classroom. These videos are not hard to come by, and they give an excellent explanation of the material. It is a nice change of pace for the students, and make help them develop some genuine interest in the subject of mathematics.

Create Your Own Fractal - Having the students create their own fractal outside of class is a fun, creative project. This gives the more artistically minded students an opportunity to shine in the class, and the products make beautiful wall decorations. Here are some guidelines for the assignment.

  1. The fractal should fill the top half of a piece of 8 \frac{1}{2} \times 11 inch plain white paper turned vertically. To give them more space, provide them with legal size paper. Be aware that each student will probably require more than one piece before they create their final product.
  2. The fractal should be boldly colored to accentuate the self-similarity.
  3. The students should be encouraged to be creative and original in their design.
  4. The bottom half of the paper will have a paragraph explaining the self-similarity in the fractal. They should explain why their design is a fractal.
  5. Create a rubric to give to the students at the time the project is assigned so that they will feel like they are being graded fairly. It is hard to evaluate artwork in a way that everyone feels is objective.

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