1.7: Similarity
Pacing
Day 1 | Day 2 | Day 3 | Day 4 | Day 5 |
---|---|---|---|---|
Ratios & Proportions | Similar Polygons |
Quiz 1 Start Similarity by AA |
Finish Similarity by AA Investigation 7-1 |
Similarity by SSS and SAS Investigation 7-2 |
Day 6 | Day 7 | Day 8 | Day 9 | Day 10 |
Finish Similarity by SSS and SAS Investigation 7-3 |
Quiz 2 Start Proportionality Relationships |
Finish Proportionality Relationships | Similarity Transformations |
Quiz 3 \begin{align*}*\end{align*}Extension: Self-Similarity |
Day 11 | Day 12 | Day 13 | Day 14 | Day 15 |
\begin{align*}*\end{align*}Finish Extension: Self-Similarity Start Review of Chapter 7 |
\begin{align*}*\end{align*}Extension Quiz More Review of Chapter 7 |
Finish Review of Chapter 7 |
Chapter 7 Test |
Start Chapter 8 |
\begin{align*}*\end{align*}Without the Extension lesson, this chapter will take 12-13 days.
Ratios and Proportions
Goal
The purpose of this lesson is to reinforce the algebraic concept of ratios and proportions. Proportions are necessary when discussing similarity of geometric objects.
Teaching Strategies
The Know What? for this lesson is a little different than the ones in in previous lessons. Have students take this activity one step further and apply the concept of a scale drawing to their own room. Students can draw scale representations of their rooms (or even their whole house) and the furniture for an alternative assessment or a chapter project.
Ratios can be written three different ways. This text primarily uses fractions or the colon notation. Remind students that a ratio has no units and can be reduced just like a fraction. With rates, to contrast to with ratios, there are units. Rates are also fractions. Have students compare the similarities and differences between ratios and rates. Ask them to brainstorm types of ratios and rates.
Proportions are two ratios that are set equal to each other. The most common way to solve a proportion is cross-multiplication. Show students the proof of the Cross-Multiplication Theorem in the FlexBook. This proof makes the denominators the same, so that the numerators can be set equal to each other. Students can go through this whole process each time or they can use cross-multiplication, which can be considered a shortcut.
All of the corollaries in this lesson are considered the same as \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*} because when cross-multiplied you would end up with \begin{align*}ad = bc\end{align*}. Therefore, the placement of the \begin{align*}a, b, c,\end{align*} and \begin{align*}d\end{align*} is important. When in doubt about if two proportions are the same, always have students cross-multiply.
Corollaries 7-4 and 7-5 are a little different from the previous three. To show that these proportions are true, it might be helpful for students to see the proofs. Each corollary is proven true when the last step is \begin{align*}ad = bc\end{align*} because this is the same as the cross-multiplication of \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}.
Proof of Corollary 7-4
\begin{align*}\frac{a+b}{b} &= \frac{c+d}{d}\\ d(a + b) &= b(c + d)\\ ad + bd &= bc + bd\\ ad &= bc\end{align*}
Proof of Corollary 7-5
\begin{align*}\frac{a-b}{b} &= \frac{c-d}{d}\\ d(a - b) &= b(c - d)\\ ad - bd &= bc - bd\\ ad &= bc\end{align*}
Brainstorm other possible reconfigurations of \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*} with students to see if there are any other possible “corollaries.” Have students cross-multiply their “true” proportions to check to see if they are valid.
Similar Polygons
Goal
This lesson connects the properties of proportions to similar polygons. An introduction to scale factors is also presented within this lesson.
Teaching Strategies
To see if students understand the definition of similar polygons, have students come up to the front board and draw pictures of two similar polygons or two non-similar polygons. You could also do this several times, before you give students the formal definition, and then students can generate one as a class.
After doing Examples 1 and 2, brainstorm with the class as to which specific types of triangles, quadrilaterals or polygons are always similar. For example, the text cites equilateral triangles and squares. This leads to the fact that all regular polygons are similar. Explore why just equilateral (rhombuses) or equiangular (rectangles) polygons are not always similar.
Students might wonder which value goes on top when finding the scale factor. Remind students that the scale factor is a ratio. In Example 4, it says “\begin{align*}\Delta ABC\end{align*} to \begin{align*}\Delta XYZ\end{align*}.” This is a ratio too. So, the side of \begin{align*}\Delta ABC\end{align*} should go in the numerator and the sides of \begin{align*}\Delta XYZ\end{align*} should go in the denominator. The triangle or figure listed first in the written-out ratio goes in the numerator and the figure listed second goes in the denominator.
In Example 5 we show that the scale factor could be \begin{align*}\frac{2}{3}\end{align*} or \begin{align*}\frac{3}{2}\end{align*}. It is useful for students to know that either can technically be the scale factor. The important thing is that they know when to use which fraction. Ask students which fraction is larger. Then explain that to find the sides of the larger rectangle we would multiply the sides of the smaller triangle by the larger ratio. To find the sides of the smaller rectangle, we would multiply the sides of the larger rectangle by the smaller ratio.
Similarity by AA
Goal
The purpose of this lesson is to enable students to see the relationship between triangle similarity and proportions. Here we discuss the AA Similarity Postulate and show how it can prove that two triangles are similar.
Teaching Strategies
Investigation 7-1 is designed to be teacher-led while the student also does the activity and follows along. Students can work individually or in pairs. One option is to have half the students make the triangle with a 3 inch side, like in step 1, and the other half make a triangle with a 4 inch side, like in step 3. Then, have each pair of students compare their two different triangles and complete step 4 together.
Students can find the corresponding sides in two similar triangles by identifying the congruent angles in the two triangles. Then, the sides that are opposite the congruent angles are corresponding. Students can also place the two largest sides together as corresponding, the shortest sides are corresponding and the middle sides are corresponding. You must be careful with this option, however. Students can only use this method when they know that the triangles are similar.
Another option for indirect measurement (other than Example 5) utilizes the Law of Reflection. It states that the angle at which a ray of light (ray of incidence) approaches a mirror will be the same angle in which the light bounces off (ray of reflection). This method is the basis of reflecting points in real world applications such as billiards and miniature golf. See the additional example below.
Additional Example: You want to shoot the red ball into the corner pocket, as shown below. How far must the cue ball travel in order to do this?
Solution: The answer is the total amount traveled by the cue ball, which is \begin{align*}36 + x\end{align*}. First we need to find \begin{align*}x\end{align*}. Set up a proportion using similar triangles.
\begin{align*}\frac{48}{x} &= \frac{32}{36}\\ 32x &= 1728\\ x &= 54 \ in\end{align*}
The cue ball must travel \begin{align*}54 + 36 = 100\end{align*} inches to sink the red ball.
Similarity by SSS and SAS
Goal
The purpose of this lesson is to extend the SSS and SAS Congruence Theorems to include similarity.
Teaching Strategies
After completing the Review Queue, introduce the lesson by asking students, “How can triangles be congruent and similar simultaneously?” Have a discussion with students about the answer to this question, including how we can prove that triangles are congruent or similar. This will lead into Investigation 7-2.
Investigation 7-2 uses the construction from Chapter 4 for SSS Congruence. Here, students will construct two similar 3-4-5 triangles and then determine if they actually are similar. Have students work in pairs for this activity and circulate to answer questions. Students might need a review of Investigation 4-2 before beginning this investigation. You can show students the provided link in the FlexBook as a review of this construction.
One could say that that SSS Congruence Theorem is a more specific version of the SSS Similarity Theorem. Discuss this point with your students.
As with the previous lesson, make sure students understand which sides are corresponding. If students do not match up corresponding sides, they may get the wrong answer to a homework problem or on a test. Review the points discussed in the previous lesson about how to match up corresponding sides.
Like with Investigation 7-2, you can split the class in half and have one half draw the triangle in step 1 and the other half draw the triangle in step 2 for Investigation 7-3. Then, have students from each half, pair up with someone from the other half and they can do steps 3 and 4 together.
Proportionality Relationships
Goal
In this lesson, students will learn about proportionality relationships when two parallel lines are split by two transversals.
Teaching Strategies
Example 1 can be done as a mini-investigation before Investigation 7-4. Discuss the properties of a midsegment and how it splits the two sides that it intersects. Find the ratio of the split sides (1:1) and the ratio of the similar triangles (1:2). Point out that these two ratios are always different.
Investigation 7-4 should be teacher-led and student can follow along, writing down any important information in their notes. They can sketch your drawings and write down your measurements from steps 3 and 6.
To make the proof of Triangle Proportionality Theorem easier to understand, you can utilize the technique presented earlier in this text where you would copy the entire proof, cut out the statements and reasons (separately) and place it in an envelope. Then, students (in pairs or groups) can put the proof in the correct order.
Before introducing Theorem 7-7, show students an example, like Example 4. See if they can figure out the answer. In actuality, Example 4 really is not any different from the examples within triangles, just that the transversals do not intersect. Rather than being sides of triangles, now these lines are the transversals passing through parallel lines. Show students several different orientations (like Example 5), so they not confused by the homework problems.
Proportions with Angle Bisectors can be a little tricky for some students. Tell them to set up the proportion like the picture below:
Notice that this set-up is different from the proportion given in Theorem 7-8. Using the letters from the theorem, the proportion would be \begin{align*}\frac{BC}{AB} = \frac{CD}{AD}\end{align*}, which is corollary 7-1.
Encourage students to use this set-up if they are having difficulties with the proportion given in the text.
Similarity Transformations
Goal
Dilations produce similar figures. This lesson introduces the algorithm to produce similar figures using measurements and a scale factor.
Teaching Strategies
To help students with this lesson’s Know What? show them pictures of different types of perspective drawings. Here are three very different examples to discuss.
An easy way to remember enlargements versus reductions is a rhyme. Have your students repeat the rhyme, “A reduction is a proper fraction.” Improper fractions are mixed numbers, and greater than 1, thus creating enlargements. Other texts might use other words for enlargements and reductions. Brainstorm with students synonyms for these words. Possibilities are: stretch and shrink or expansion or contraction.
Dilations can also be clarified using a photograph. School pictures are great examples of dilations. Suppose a typical photograph is \begin{align*}4 \times 6\end{align*}. An \begin{align*}8 \times 10\end{align*} enlargement does not alter the appearance. This is also true for shrinking photos for a \begin{align*}2 \times 3\end{align*}. Using a base picture, bring in several enlargements and reductions to further illustrate this concept.
In Examples 1 and 2, a point is dilated. Recall that a point has no dimension, so it cannot be enlarged or reduced. Therefore, the distance from the center of the dilation is stretched and shrunk. Point out to students that the original distance is 6 units and the distance from \begin{align*}P\end{align*} to \begin{align*}Q^\prime\end{align*} is three times 6 units, or 18 units.
Anytime a figure is dilated, the distance from the center of the dilation is also stretched or shrunk according to the scale factor. The dilated point will always be collinear with the original point as well. Even though the terminology image and preimage (the original image) are not introduced until Chapter 12, feel free to introduce it now.
Extension: Self-Similarity
Goal
This lesson introduces students to popular fractals. Fractals possess self-similarity and maintain the properties of similarity.
Teaching Strategies
This optional lesson is a great mathematical connection to art and nature. Show students examples of fractals in art and graphic design. See the examples below. The last picture is a type of broccoli called romanesco broccoli. You can sometimes find it in specialty grocery stores.
Mathematician Benoit Mandelbrot derived the term “fractal” from the Latin word frangere, meaning to fragment. A fractal is a geometric figure in which its branches are smaller versions of the “parent” figure. Most fractals are explained using higher level mathematics; however, students can create their own fractal patterns easily. The Mandelbrot set is illustrated in the middle picture above.
Additional Example: Follow these instructions to create a cauliflower fractal.
- Hold your paper in landscape format.
- Draw a horizontal segment \begin{align*}\overline{AB}\end{align*}, such that \begin{align*}AB = 8 \ in\end{align*}, in the center of the paper.
- Find and mark \begin{align*}C\end{align*}, the midpoint of \begin{align*}\overline{AB}\end{align*}.
- Find the midpoint from \begin{align*}A\end{align*} to \begin{align*}C\end{align*}. This is \begin{align*}\frac{1}{4}AB\end{align*}.
- Use this quarter-length to be the length of \begin{align*}DC\end{align*}. Draw \begin{align*}\overline{DC}\end{align*} such that it is perpendicular to \begin{align*}\overline{AB}\end{align*} at \begin{align*}C\end{align*}.
- Draw lines \begin{align*}\overline{AD}\end{align*} and \begin{align*}\overline{BD}\end{align*} forming \begin{align*}\Delta ADB\end{align*}.
- Repeat steps 3-6 for \begin{align*}\overline{AD}\end{align*} and \begin{align*}\overline{BD}\end{align*}
- Continue to repeat this process for the legs of each new smaller triangle.
Final image:
The surface of this fractal (as you continue with smaller and smaller triangles) looks like the outside of a head of cauliflower.