Overview of Investigation 4
This Investigation has four Problems. The first Problem reviews student understanding of dilations and their properties that have been developed earlier in Stretching and Shrinking from Grade 7. The second Problem parallels the earlier work showing how constructing combinations of dilations and flips/turns/slides can be used to prove the similarity of figures. The third Problem develops the key similarity result for triangles—that any two triangles with two pairs of congruent corresponding angles must be similar. The fourth Problem applies notions of similarity to problems about finding lengths of inaccessible objects.
Goals of Investigation 4
Congruence and Similarity
Understand congruence and similarity and explore necessary and sufficient conditions for establishing congruent and similar shapes
Understand congruence and similarity and explore necessary and sufficient conditions for establishing congruent and similar shapes
- Recognize that two figures are congruent if one is derived from the other one by a sequence of reflection, rotation, and/or translation transformations
- Recognize that two figures are similar if one can be obtained from the other by a sequence of reflections, rotations, translations, and/or dilations
- Use transformations to describe a sequence that exhibits the congruence between figures
- Use transformations to explore minimum measurement conditions for establishing congruence of triangles
- Use transformations to explore minimum measurement conditions for establishing similarity of triangles
- Relate properties of angles formed by parallel lines and transversals, and the angle sum in any triangle, to properties of transformations
- Use properties of congruent and similar triangles to solve problems about shapes and measurements
Common Core Content Standards
8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equationy = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. Problem 1 and 3
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Problem 1 and 3
8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. Problems 1, 2, 3, and 4
8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Problem 3
Facilitating the Mathematical Practices
Students in Connected Mathematics classrooms display evidence of multiple Common Core Standards for Mathematical Practice every day. Here are just a few examples of when you might observe students demonstrating the Standards for Mathematical Practice during this Investigation.
Practice 1: Make sense of problems and persevere in solving them.
Students are engaged every day in solving problems and, over time, learn to persevere in solving them. To be effective, the problems embody critical concepts and skills and have the potential to engage students in making sense of mathematics. Students build understanding by reflecting, connecting, and communicating. These student-centered problem situations engage students in articulating the “knowns” in a problem situation and determining a logical solution pathway. The student-student and student-teacher dialogues help students not only to make sense of the problems, but also to persevere in finding appropriate strategies to solve them. The suggested questions in the Teacher Guides provide the metacognitive scaffolding to help students monitor and refine their problem-solving strategies.
Practice 2: Reason abstractly and quantitatively.
In the Problems of this Investigation, students use rigid and nonrigid transformations to discover that, for triangles, they only need to know that two corresponding angles are congruent to determine similarity. This minimum requirement is practical for determining vertical heights and distances that are unreachable assuming that the angle the vertical object makes with the ground is 90°.
Practice 3: Construct viable arguments and critique the reasoning of others.
In Problem 3, students evaluate the arguments of other students about whether or not two triangles are similar. To do this, they nest the smaller triangle in the larger triangle using one corresponding angle and observe that the third sides that do not form the angle are parallel. Then, students use the results from Problem 3.5 about parallel lines cut by a transversal to deduce the equality of corresponding angles in the triangles. They also use the sum of the interior angles of a triangle. They reason that two angle measures are enough to determine similarity, since the third angle can be found by subtracting the sum of the two known angles from 180°.
Practice 4: Model with mathematics.
In Problem 4, students use the scale factor between two similar triangles to determine vertical heights that they are unable to measure. They use a mirror with the fact that angles of reflection are congruent and relate the corresponding parts of similar triangles in their diagram of the situation. With the angle of reflection and the assumption that the object they cannot measure forms a 90° angle with the ground, they know that two triangles are similar. The smaller triangle is used in this exploration as a model of the larger triangle. For example, they use height from the ground to their eyes when they are able to see the reflection of the top of the object that they want to measure in the mirror. They also use the distances from the mirror to the object and to their position given a pair of corresponding sides to use to calculate the scale factor. With the scale factor, they can use multiplication to estimate the height that they are unable to measure.
Practice 5: Use appropriate tools strategically.
In Problems 1–3, students use their rulers and angle rulers or protractors to verify measurements of corresponding parts of triangles. They learn that they only need to measure two corresponding angles and verify congruence to determine similarity. If they decide to use measurements of sides, they learn that they need to verify that the scale factor is constant for all pairs of corresponding sides. In Problem 4, students use a mirror and meter stick to find the appropriate position to take measurements to estimate the height of an object they cannot reach using properties of similarity.
Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of the Investigation.
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. Problem 1 and 3
8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. Problems 1, 2, 3, and 4
8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Problem 3
Facilitating the Mathematical Practices
Students in Connected Mathematics classrooms display evidence of multiple Common Core Standards for Mathematical Practice every day. Here are just a few examples of when you might observe students demonstrating the Standards for Mathematical Practice during this Investigation.
Practice 1: Make sense of problems and persevere in solving them.
Students are engaged every day in solving problems and, over time, learn to persevere in solving them. To be effective, the problems embody critical concepts and skills and have the potential to engage students in making sense of mathematics. Students build understanding by reflecting, connecting, and communicating. These student-centered problem situations engage students in articulating the “knowns” in a problem situation and determining a logical solution pathway. The student-student and student-teacher dialogues help students not only to make sense of the problems, but also to persevere in finding appropriate strategies to solve them. The suggested questions in the Teacher Guides provide the metacognitive scaffolding to help students monitor and refine their problem-solving strategies.
Practice 2: Reason abstractly and quantitatively.
In the Problems of this Investigation, students use rigid and nonrigid transformations to discover that, for triangles, they only need to know that two corresponding angles are congruent to determine similarity. This minimum requirement is practical for determining vertical heights and distances that are unreachable assuming that the angle the vertical object makes with the ground is 90°.
Practice 3: Construct viable arguments and critique the reasoning of others.
In Problem 3, students evaluate the arguments of other students about whether or not two triangles are similar. To do this, they nest the smaller triangle in the larger triangle using one corresponding angle and observe that the third sides that do not form the angle are parallel. Then, students use the results from Problem 3.5 about parallel lines cut by a transversal to deduce the equality of corresponding angles in the triangles. They also use the sum of the interior angles of a triangle. They reason that two angle measures are enough to determine similarity, since the third angle can be found by subtracting the sum of the two known angles from 180°.
Practice 4: Model with mathematics.
In Problem 4, students use the scale factor between two similar triangles to determine vertical heights that they are unable to measure. They use a mirror with the fact that angles of reflection are congruent and relate the corresponding parts of similar triangles in their diagram of the situation. With the angle of reflection and the assumption that the object they cannot measure forms a 90° angle with the ground, they know that two triangles are similar. The smaller triangle is used in this exploration as a model of the larger triangle. For example, they use height from the ground to their eyes when they are able to see the reflection of the top of the object that they want to measure in the mirror. They also use the distances from the mirror to the object and to their position given a pair of corresponding sides to use to calculate the scale factor. With the scale factor, they can use multiplication to estimate the height that they are unable to measure.
Practice 5: Use appropriate tools strategically.
In Problems 1–3, students use their rulers and angle rulers or protractors to verify measurements of corresponding parts of triangles. They learn that they only need to measure two corresponding angles and verify congruence to determine similarity. If they decide to use measurements of sides, they learn that they need to verify that the scale factor is constant for all pairs of corresponding sides. In Problem 4, students use a mirror and meter stick to find the appropriate position to take measurements to estimate the height of an object they cannot reach using properties of similarity.
Students identify and record their personal experiences with the Standards for Mathematical Practice during the Mathematical Reflections at the end of the Investigation.