<p>The lesson builds on previous lessons by introducing the concept of similarity, which is closely related to congruence. Students will:&nbsp;<br>- use their knowledge of ratio and proportion, as well as angle measurement, to demonstrate that triangles are similar.&nbsp;<br>- apply their knowledge of similar triangles to build the concept of scale factor.&nbsp;<br>- use their knowledge of similar triangles and the scale factor to calculate the missing side and angle lengths in similar triangles.</p>

<p>- What exactly does the word <i>similar</i> mean in a mathematical context?&nbsp;<br>- What is the difference between similarity and congruence?&nbsp;<br>- How do we prove that two triangles are similar?&nbsp;<br>- How can we apply our understanding of similar triangles to figure out unknown quantities in similar triangles?<br>&nbsp;</p>

<p>- <strong>AA (Angle Similarity Postulate):</strong> If two triangles have two corresponding pairs of angles with the same measure then they are similar.<br>- <strong>Congruent</strong>: Having the same size and shape; congruent figures have corresponding sides and angles congruent.<br>- <strong>Corresponding Angles:</strong> Angles in the same relative position in similar or congruent figures.<br>- <strong>Corresponding Sides:</strong> Sides in the same relative position in similar or congruent figures.<br>- <strong>Proportion:</strong> An equation showing that two ratios are equal.<br>- <strong>Ratio:</strong> A comparison between two numbers using division; ratios are commonly expressed as fractions with positive integer numerators and denominators and as positive integers separated by a colon, e.g., 2:3, meaning 2 to 3, or \(2 \over 3\).<br>- <strong>Scale Factor:</strong> The ratio that relates the lengths of the sides of two similar triangles. Since it is a ratio, it can be written as a fraction \(1 \over 2\), 1:2, or 1 to 2.<br>- <strong>Similar Polygons:</strong> Polygons that have the same shape, but not necessarily the same size. Corresponding sides of similar polygons are proportional and their corresponding angles are congruent.<br>- <strong>Similar Triangles:</strong> Two triangles are similar if the three angles of the first triangle are congruent to the three angles of the second triangle and the corresponding sides are all in the same proportion.</p>

<p>- overhead projector<br>- cutout of triangle (sides approx. 6, 7, and 8 cm)<br>- paper and markers or colored pencils<br>- graph paper<br>- several meter sticks<br>- copies of the Similarity worksheet (M-G-4-3_Similarity Worksheet and KEY)</p>

<p>- Your observations throughout pair and group activities must focus on the accuracy of side length and angle measurement, the identification of corresponding sides and angles, and the appropriate application of proportions. It should be noted that misuse of the correct proportion and accurate computation would yield inaccurate results. When analyzing their own results, remind students to check the reasonableness of the answer. For example, when using a ratio of corresponding sides of similar polygons, ensure that each proportion is consistent with the relative sizes of the polygons.&nbsp;<br>- Pair activities encourage partners to evaluate each other's results. Direct the evaluator to specify the exact outcome under consideration, the reasonableness of the result, and the accuracy of the computation. Identify the problem as precisely as possible, such as incorrect side or angle correspondence or incorrect use of the similarity ratio, such as \(3 \over 2\) instead of \(2 \over 3\).<br>&nbsp;</p>

<p>Active Engagement, Modeling, Explicit Instruction<br>W: Remind students of the definition of <i>congruence</i>, as well as the shortcut methods they have learned to prove that triangles are congruent. Recognize that there are congruent triangles, and triangles that do not look anything like one another, but there is also a third type of triangle: triangles with some parts that are the same and other parts that are different. It is worth noting that some triangles are essentially enlargements (or reductions) of other triangles, and this relationship is referred to as <i>similarity</i>.&nbsp;<br>H: The overhead projector-based introductory activity is visually appealing and encourages students to apply their own knowledge of shadow length and direction. It also examines the relationship between the size and shape of the shadow, the direction of illumination, and the size and shape of the item.&nbsp;<br>E: In Activity 1, students choose two angles measures whose sum is less than 180 degrees. When the two angles are assembled as the first two angles of the triangle to be built, the third angle is defined by the sum of the first two angles, since the angle sum of a triangle is 180 degrees. In this way, the third angle is constrained by the measures of the first two in a similar way that side-angle-side congruence constrains the measures of the remaining sides and angles of congruent triangles.&nbsp;<br>R: Activity 3 uses proportionality of shadow length to compare objects with concurrent measurements. Students with previous experience in solving shadow-length proportions will find the principles plausible. Direct measuring, comparing, and calculating helps them learn the proportionality of the corresponding sides.&nbsp;<br>E: Ask students to think of other related shapes (not just triangles) besides shadows or heights. Examples include models, maps, art (perspective drawings), architectural shapes, scaffolds, ladders, and so on. Notebooks, textbooks, and laptop computers are examples of school resources that students may use. Using a ruler (inches or centimeters), encourage students to discover instances of noncongruent polygons with proportionate sides that are similar to one another. Check that the sides chosen for measurement are correctly matching and that the calculations are accurate. The Similarity worksheet is an extra tool for assessing both individual and group understanding of the proportional relationships involved in polygon similarity.&nbsp;<br>T: This lesson has students working in many different modes, combining traditional teaching approaches (e.g., instructor explanations, notes to copy) with hands-on activities such as measuring heights and other physical properties of the world around them. The lesson is tailored to a range of students with a variety of learning styles by using numerous approaches to the single concept of similarity.&nbsp;<br>O: The lesson begins with a classroom activity that can be closely monitored, followed by an explanation of the main idea to be examined. Students have the opportunity to investigate the concept in the classroom before going on to a more open study that involves measuring students' heights and shadows. During this part of the lesson, students are divided into groups so that each has a task and plays an important role in the exploration process. Finally, students are brought together to generate individual problems (to test their individual comprehension) while still having the support of their classmates (and you).</p>

<p>With the overhead projector and a single volunteer, place the triangle cutout immediately on the overhead and have the student trace it. Then, raise the triangle closer to the overhead projector to cast a larger shadow on it. Have the student trace the outline of the shadow on the overhead. Ask the student to measure both triangles and record the measurements on the overhead projector as you lead the class in a discussion on whether the triangles are congruent or not, and what is the same between the two triangles. The questions below can be used to introduce the concept of similarity:</p><p><strong>"Has the overall shape of the triangle changed?" </strong>(<i>no</i>)&nbsp;<br><strong>"Have the measures of the angles changed?" </strong>(<i>no</i>)&nbsp;<br><strong>"Have the measures of the lengths changed?"</strong> (<i>yes</i>)&nbsp;<br>Direct students to sketch the triangles before and after the shapes change. Label the First Triangle and Second Triangle, then draw them as similar as possible to the projector shapes. They should use a straightedge, but there is no need to measure the sides and angles. The main aspect is to record the size changes of the two triangles while keeping the similarity of their shapes.</p><p>Once the triangle measurements have been recorded, the student can return to his or her seat.</p><p>Instruct students to make observations about the recorded angle measurements (which should be similar) and the recorded side measures. Encourage students to compare the relative lengths of the sizes. Use leading questions like, "Are the sides of the larger triangle twice as large? Three times as large?"</p><p>Ask students to calculate <i>how many times larger</i> the larger triangle is relative to the smaller. Write this number on the overhead and title it <i>scale factor</i>. Inform students that these are the concepts they will be investigating today.</p><p>Please provide the following definitions.</p><p><strong>Similar Triangles:</strong> Two triangles are similar if the three angles of the first triangle are congruent to the three angles of the second triangle, and the corresponding sides have the same proportion.</p><p><strong>Scale Factor: </strong>The ratio between the lengths of the sides of two similar triangles. Because it is a ratio, it can be expressed as a fraction \(1 \over 2\), 1:2, or 1 to 2.</p><p><strong>Activity 1: Initial Exploration of AA</strong></p><p>Divide the class into pairs. Each pair should settle on two angle measurements (the sum should be less than 150 degrees). To make measuring and drawing the triangles as simple as possible, propose that one angle be at least 10°. Each student should use these two angle measurements to create a triangle. Sides can be any length. Ask students:</p><p><strong>"Are the triangles you and your partner constructed congruent?"&nbsp;</strong>(<i>most answers: no</i>)&nbsp;<br><strong>"Do they look similar?"</strong> (<i>yes</i>)&nbsp;<br><strong>"How can you verify that they are similar?"</strong></p><p>Tell students that inaccuracies in measurement, line thickness, and line straightness can result in angle sums for triangles that are not exactly equal to 180 degrees. These errors occur naturally as a result of drawing and are not always caused by improper work. Students should verify that their triangles are equivalent by measuring the third angle and the side lengths. Ask students to calculate the scale factor and create a similarity statement (e.g., Δ<i>ABC</i> is similar to Δ<i>DEF</i>; remind them of their knowledge about writing congruence statements for triangles). Ask students:</p><p><strong>"How much information is needed to know that triangles are similar?" </strong>(<i>AA</i>)&nbsp;<br><strong>"Is it necessary to measure all the sides and all the angles?"</strong> (<i>no</i>)</p><p>Introduce Angle-Angle (AA) Similarity Postulate to the class:</p><p><strong>"If two angles in one triangle are congruent to two angles in another, then the two triangles are similar. Why is this true?" </strong><i>(Because the third angle must also be congruent.)</i></p><figure class="image"><img style="aspect-ratio:193/54;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_170.png" width="193" height="54"></figure><figure class="image"><img style="aspect-ratio:527/296;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_171.png" width="527" height="296"></figure><p><strong>Activity 2: Real-life Similarity</strong></p><p>This activity may take the full class period to complete due to setup, &nbsp;obtaining permissions, and moving location of the group. Divide the class into groups of three or four students. Have the groups work outside or in a large room (gymnasium, etc.) where there are shadows of the sun present. If direct sunlight is not available or possible, use instructional locations with adequate direct lighting to cast measureable shadows. Make sure students have paper and pencils. Have one student hold a meter stick perpendicular to the ground while another student measures the shadow. A third student should measure the lengths and draw a triangle on paper to represent the real-world situation. This practice is also effective with a flashlight.</p><p>Ask students:</p><p><strong>"What determines the length of the shadow?" </strong>(<i>the angle of the sun</i>)</p><p><strong>"What causes a shadow to exist?"</strong> (<i>the absence of light</i>)</p><p>Once students understand that the length of a shadow is dictated by the angle of light, they will see that the angle of light does not change considerably if we perform measurements relatively quickly (and does not change at all if the activity is performed indoors).</p><p>Now, have one student in each group stand straight up while another student in each group measures the shadow. A third student should create a triangle to illustrate the situation.</p><p>Ask students what is common between the two triangles (the one representing the meter stick and the one representing the student). Remind them that the angle of the light has not changed considerably. Help students recognize that the two triangles are similar by AA. Ask students, <strong>"Why is the postulate called AA and not AAA?"</strong> (<i>AA is the minimal criteria for similarity</i>.)</p><p>Have students calculate the <i>scale factor</i> (by comparing the small and large triangles). Each group should use the scale factor and similar triangles to calculate the height of the group member who casts a shadow, and then measure the actual height of the student to see how accurate they were. Each group should perform this exercise for each other member of the group.</p><figure class="image"><img style="aspect-ratio:561/235;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_25.png" width="561" height="235"></figure><p>Ask students if they could have easily measured their heights at the start of the activity. Follow up by asking them whether it is easier to measure the height of a flagpole, for example, or if it is easier to measure the length of the flagpole's shadow. Ask for instances of other objects that people may want to determine the heights of, but can't directly measure.</p><p>Remind students that the two triangles are similar according to the AA postulate, and the two angles that are congruent in each triangle are the right angle at the ground and the angle formed by sunlight with the top of the object (or the ground). By demonstrating that the two triangles are similar, we may utilize the scale factor to determine unknown lengths.</p><p><strong>Activity 3: Creating Similar-Triangle Problems</strong></p><p>Have students create a similar-triangle problem for a classmate to solve. Their challenges should include a triangle with known height and shadow of a single object (they can use a triangle from the previous activity) as well as the length of a particular object's shadow. Let students exchange and solve problems. Students should identify and clearly explain the scale factor for each problem as they solve them.</p><p>Distribute the Similarity worksheet (M-G-4-3_Similarity Worksheet and KEY) as a summarizing task, and go over the proportionality relationships between corresponding sides and angles.</p><p><strong>Extension:</strong></p><p>According to the Side Splitting Theorem, when a line parallel to one of the triangle's sides intersects the other two sides, the two sides are divided into proportional segments.<br>In Δ<i>GHJ</i>, line <i>KL</i> is parallel to base <i>GJ</i>. What is the relationship between Δ<i>KLH</i> and Δ<i>GHJ</i>, and how is that relationship proved?</p><figure class="image"><img style="aspect-ratio:483/289;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_26.png" width="483" height="289"></figure><p>(Answer: Because \(\overline{KL}\) is parallel to \(\overline{GJ}\), the angles of the transversals <i>HG</i> and <i>HJ</i> are congruent. The two triangles are similar according to the AA postulate. Prove that Δ<i>UVW</i> is not congruent to Δ<i>XYZ</i>?)</p>