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Ratio and Proportion

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, especially if they are looking at the male to female student ratio at their top three college choices.

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 is 16 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 usually given in the correct order.

Key Example:

1. Junior got a new hybrid. He went 525 \;\mathrm{miles} on the first five gallons that came with the car. He just put 12 \;\mathrm{gallons} in the tank. How far can he expect to go on that amount of gas?

Answer:

\frac{525}{5} & = \frac{x}{12} && \text{He can expect to go}\ 1,260\  \text{miles}.\\x & = 1260

Note: Students will 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}

Properties of Proportions

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 rations are equal to each other. It is not appropriate to cross-multiply when two fractions are being added or subtracted.

Examples:

Cross Multiply Don’t Cross Multiply
\frac{3}{4}=\frac{10}{x} \frac{3}{4}+\frac{10}{x}
\frac{x+3}{4} = \frac{10}{x} \frac{x+3}{4} - \frac{10}{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 Can’t be Reduced
\frac{3 \cdot 2}{5 \cdot 2} \frac{3+2}{5+2}
\frac{3(x-4)}{3 \cdot 2} \frac{x-4}{4}

Color-Code the Proofs – The proof in this section requires many substitutions of similar looking expressions. It is difficult to see where everything is coming from and moving to. When presenting the proof in class use colors so the variables will be easier to follow. Another option is to have the students do the color-coding. Once they understand the mechanics of the proof in the lesson, they will be able to do the similar proof in the exercises.

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, like 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 identity 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.

Who’s in the Numerator – When writing a proportion students sometimes carelessly switch which polygon’s measurements are in the numerator. To combat this I tell the students to choose right from the beginning and BE CONNSISTENT 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.

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 rule 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 a about trigonometry, and that the AA Triangle Similarity Postulate is what make trigonometry possible. If the students know what an important postulate this is, they will be motivated to understand and learn how to apply it. 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.

Key Exercises:

1. \triangle {ABC} is a right triangle with right angle C, and \triangle {ABC} \sim \triangle {XYZ}.

Which angle in \triangle {XYZ} is the right angle?

Answer: \angle Z

2. \triangle {CAT} \sim \triangle {DOG}, \angle A is a right angle

CA = 5 \;\mathrm{cm}, CT = 13 \;\mathrm{cm}

What is DG?

Answer: DG = 13 \;\mathrm{cm},

Students must use the Pythagorean theorem and the definition of similar polygons.

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'\mathrm{s} and A'\mathrm{s} just like the congruence postulates and theorems. The A'\mathrm{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'\mathrm{s} represent a different requirement in similarity postulates then they did in congruence postulates and theorems. Congruent triangles have congruent sides, but similar triangles have proportional sides. Each “S” is 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
“S”\ \leftrightarrow congruent sides “S”\ \leftrightarrow proportional sides
SSS AA
SAS SSS
ASA SAS
AAS
HL

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.

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 can not be used to find the lengths of the parallel segments.

Ideally student 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 problem. 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.

Additional Exercises:

1. \triangle {ABC} has point E on \overline{AB}, and F on \overline{BC} such that \overline{EF} is parallel to \overline{AC}.

AE = 5 \;\mathrm{cm}, EB = 3 \;\mathrm{cm}, BF = 4 \;\mathrm{cm}, AC = 10 \;\mathrm{cm}

Find EF and FC.

(Hint: Draw and label a picture, then draw another figure where the two triangles are shown separately.)

Answer:

\frac{3}{8} & = \frac{EF}{10} && \frac{3}{5} = \frac{4}{FC} \ \ \ \text{or}\ \ \ \frac{3}{8} = \frac{4}{FC+4} \\EF & =3 \frac{3}{4}\ \text{cm} && FC = 6\frac{2}{3}\ \text{cm}

Similarity Transformations

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. Student frequently forget to square the scale factor when working with the ratios of a figure and its image. This is an important concept that is frequently used on the SAT and on other standardized tests.

Key Exercises:

\triangle {ABC} has coordinates at the following vertices. A(1, 13), B(6, 1), and C(1,1).

1. Graph \triangle {ABC}.

2. Use the distance formula to find the length of each side of \triangle {ABC}.

3. Calculate the perimeter of \triangle {ABC}.

4. Calculate the area of \triangle {ABC}.

\triangle A’B’C’ is the image of \triangle {ABC} under a dilation centered at the origin with scale factor 3.

5. Graph \triangle A’B’C’.

6. Use the distance formula to find the length of each side of \triangle A’B’C’.

7. Calculate the perimeter of \triangle A’B’C’.

8. Calculate the area of \triangle A’B’C’.

Compare \triangle A’B’C’ to \triangle {ABC}.

9. 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.

2. AC = 12, BC = 5, AB = 13

3. 30

4. 30

5.

6. A’C’ = 36, B’C’ = 15, A’B’ = 39

7. 90

8. 270

9. The ratios of the side lengths and the perimeter are 3:1 the same as the scale factor. The ratio of the areas is 9:1, the square of the scale factor.

Self-Similarity (Fractals)

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 fractal 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 it gives the instructor some precious time to catch-up on paperwork. It is the best approach for all.

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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