Download Unit 8 - Geometry - Congruence Similarity

7th/8th Grade Mathematics Curriculum Guide
2016 – 2017
Unit 8: Congruence & Similarity- Use Calculators
Time Frame: Quarter 4 – about 20 days
Connections to Previous Learning:
In previous grades, students made scale drawings of geometric figures and solved problems involving angle measure, surface area, and volume.
Focus of this Unit:
Geometric sense allows students to comprehend space and shape. Students analyze the characteristics and relationships of shapes and structures, engage in
logical reasoning, and use tools and techniques to determine measurement. Students learn that geometry and measurement are useful in representing and
solving problems in the real world as well as in mathematics.
Connections to Subsequent Learning:
At the high school level, students will apply transformations to numbers, functional representations, and data. They will experiment with transformations in the
plane where they will do rigid motion transformations, transformations in terms of a function, and transformations such as stretches that lead to shapes that are
not similar or congruent.
Mathematical Practices
1. Make Sense of Problems and Persevere in Solving Them.
2. Reason Abstractly and Quantitatively.
3. Construct Viable Arguments and Critique the Reasoning of Others.
4. Model with Mathematics.
5. Use Appropriate Tools Strategically.
6. Attend to Precision.
7. Look for and Make Use of Structure.
8. Look for and Express Regularity in Repeated Reasoning.
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7th/8th Grade Mathematics Curriculum Guide
2016 – 2017
Stage 1 Desired Results
Transfer Goals
Students will be able to independently use their learning to…
 Determine how seemingly dissimilar objects in real life might be related.
Meaning Goals
UNDERSTANDINGS
Students will understand that…
 Plane figures can be measured, translated, reflected,
rotated, and dilated.
 Congruence of a geometric figure is preserved under
rotations, reflections and translations.
 Similarity of a geometric figure is preserved under
rotations, reflections, translations and dilations.
 A two-dimensional figure is congruent to another if the
second can be obtained from the first by a sequence of
rotations, reflections and translations.
 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.
 If the scale factor of a dilation is greater than 1, the image
resulting from the dilation is an enlargement, and if the
scale factor is less than 1, the image is a reduction.
 Lines intersect to form measurable, identifiable and
predictable angles.
 When parallel lines are cut by a transversal, corresponding
angles, alternate interior angles, alternate exterior angles,
and vertical angles are congruent.
 Polygons have unique and specific characteristics that can
be measured and predicted.
Unit 8
ESSENTIAL QUESTIONS
 What are translations, rotations, reflections, and dilations?
 How is the figure changing? What geometric properties remain the same?
 What are transformations and what effect do they have on an object?
 How can transformations be used to determine congruency or similarity?
 What angle relationships are formed by a transversal?
 What relationships are present in polygons?
 How are points, rays, and lines the building blocks of geometry?
 How can special angle relationships be used to find missing angles in a figure?
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7th/8th Grade Mathematics Curriculum Guide
2016 – 2017
Acquisition Goals
Students will know…
 Properties of rotations, reflections and translations (8.G.1)
 Angle-angle criterion for similarity of triangles.
 Exterior angle and angle sum of triangles. (8.G.5)
 Angles created when parallel lines are cut by a transversal.
(8.G.5)
Students will be skilled at…
 Describe a series of transformations that exhibits congruence between two congruent
figures. (8.G.2)
 Describe transformations (dilations, translations, rotations, and reflections) with words and
with coordinates. (8.G.3)
 Describe a series of transformations that exhibits similarity between two similar figures.
(8.G.4)
 Justify congruence or similarity of figures using a series of transformations. (8.G.2 and
8.G.4)
 Identify pairs of angles to find the measure of missing angles.
 Find the measures of angles using transversals, the sum of angles in a triangle, the exterior
angles of triangles. (8.G.5)
 Determine if triangles are similar using the angle-angle criterion. (8.G.5)
 Will be able to determine unknown angle measures of polygons.
Calculators
8.G.1
8.G.1a
8.G.1b
8.G.1c
8.G.2
8.G.3
8.G.4
8.G.5
Materials Needed for Unit
Holt Course 3
Holt Course 3 Common Core Curriculum Companion
Unit 8
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Additional Materials if needed:
Engage NY Grade 8 Module 3
Clover Park School District 6/3/16
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7th/8th Grade Mathematics Curriculum Guide
Unit 8
Clover Park School District 6/3/16
2016 – 2017
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7th/8th Grade Mathematics Curriculum Guide
2016 – 2017
Stage 1 Established Goals: Common Core State Standards for Mathematics
8.G.A Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations:
8.G.A.1a Lines are taken to lines, and line segments to line segments of the same length.
8.G.A.1b Angles are taken to angles of the same measure.
8.G.A.1c Parallel lines are taken to parallel lines.
8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and
translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
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.
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. For example, arrange three copies of the same triangle so that the sum of the three angles
appears to form a line, and give an argument in terms of transversals why this is so.
 Major Clusters  Supporting Clusters  Additional Clusters
Vocabulary
≊ (7)- “congruent to” Having the same size and shape.
A’ (8)– “A prime” the prime symbol is used to show the result of a transformation.
Adjacent Angles (7)-Angles in the same plane that have a common vertex and a common side.
Alternate Interior Angles (8)-A pair of angles on the inner sides of two lines cut by a transversal that are on opposite sides of the transversal.
Angle (2)-A figure formed by two rays with a common endpoint called the vertex.
Angle-Angle Criterion (8)-If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar.
Center of Rotation (8)-The point about which a figure is rotated.
Clockwise (4)-A circular movement to the right in the direction shown.
Complementary Angles (7)-Two angles whose measures add to 90 ˚
Congruent (4)-Having the same size and shape.
Corresponding Angles (5)- Matching angles of two or more polygons.
Counterclockwise (4)-A circular movement to the left in the direction shown.
Deductive Reasoning (?)-Method of reasoning from general to particular.
Dilation (8)-A transformation that enlarges or reduces a figure.
Exterior Angle (8)-Angles on the outer sides of two lines cut by a transversal.
Image (8) –A figure resulting from a transformation.
Interior Angle (8)-Angles on the inner sides of two lines cut by a transversal.
Line of Reflection (8)-A lines that a figure is flipped across to create a mirror image of the original figure.
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Line Segment (3)-A part of a line between two endpoints.
Line (2)- A straight path that extends without end in opposite directions.
Parallel Lines (3) -Lines in a plane that do not intersect.
Point (3)-An exact location in space.
Polygon (2)-A closed plane figure formed by three or more line segments that intersect only at their endpoints (Vertices).
Pre-Image (8)-The original figure prior to a transformation.
Reflection (8)- A transformation of a figure that flips the figure across a line.
Rigid (8)- Not moving, where the angles cannot be changed.
Rotation (8)-A transformation in which a figure is turned around a point.
Scale Factor (8)-The constant that is multiplied by the length of each side of a figure that produces an image that is the same shape as the original figure, but a
different size.
Similar (8)-Figures with the same shape but not necessarily the same size are similar.
Supplementary Angles (7)- Two angles whose measures have a sum of 180 ˚.
Transformation (8)-A change in the size or position of a figure.
Translation (8)- A movement (slide) of a figure along a straight line.
Transversal (8)-A line that intersects two or more lines.
Triangle (K)- A three-sided polygon.
Vertical Angles (7)-A pair of opposite congruent angles formed by intersecting lines.
8.G.A.5
Students use exploration and deductive reasoning to determine relationships that exist between the following: a) angle sums and exterior angle sums of triangles,
b) angles created when parallel lines are cut by a transversal, and c) the angle-angle criterion for similarity of triangle.
Students construct various triangles and find the measures of the interior and exterior angles. Students make conjectures about the relationship between the
measure of an exterior angle and the other two angles of a triangle. (the measure of an exterior angle of a triangle is equal to the sum of the measures of the
other two interior angles) and the sum of the exterior angles (360°). Using these relationships, students use deductive reasoning to find the measure of missing
angles.
Students construct parallel lines and a transversal to examine the relationships between the created angles. Students recognize vertical angles, adjacent angles
and supplementary angles from 7th grade and build on these relationships to identify other pairs of congruent angles. Using these relationships, students use
deductive reasoning to find the measure of missing angles.
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7th/8th Grade Mathematics Curriculum Guide
2016 – 2017
Example 1:
You are building a bench for a picnic table. The top of the bench will be parallel to the ground. If m∠ 1 = 148°, find m∠ 2 and m∠ 3. Explain your answer.
Solution:
Angle 1 and angle 2 are alternate interior angles, giving angle 2 a measure of 148°. Angle 2 and angle 3 are
supplementary. Angle 3 will have a measure of 32° so the m∠ 2 + m∠ 3 = 180°
Example 2:
Show that m∠3 + m∠ 4 + m∠ 5 = 180° if line l and m are parallel lines and t1 and t2 are transversals.
Students can informally conclude that the sum of the angles in a triangle is 180º (the angle-sum theorem) by applying their understanding of lines and alternate
interior angles.
Example 3:
̅̅̅̅ . Prove that the sum of the angles of a triangle is 180°.
In the figure on the right Line X is parallel to Line𝑌𝑍
Solution:
Angle a is 35° because it alternates with the angle inside the triangle that measures 35°. Angle c is 80° because it
alternates with the angle inside the triangle that measures 80°. Because lines have a measure of 180°, and angles a + b + c form a straight line, then angle b must
be 65 ° ⤏ 180 – (35 + 80) = 65. Therefore, the sum of the angles of the triangle is 35° + 65 ° + 80 °.
Example 4:
What is the measure of angle 5 if the measure of angle 2 is 45° and the measure of angle 3 is 60°?
Solution:
Angles 2 and 4 are alternate interior angles; therefore the measure of angle 4 is also 45°. The measure
angles 3 and 4 add to 105° the angle 5 must be equal to 75°
Unit 8
Clover Park School District 6/3/16
of angles 3, 4 and 5 must add to 180°. If
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7th/8th Grade Mathematics Curriculum Guide
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Students construct various triangles having line segments of different lengths but with two corresponding congruent angles. Comparing ratios of sides will
produce a constant scale factor, meaning the triangles are similar. Students solve problems with similar triangles.
8.G.A.1
Vocab - image, reflection, rotation, translation, transformations, line segments, pre-image, rigid
Students use compasses, protractors and rulers or technology to explore figures created from translations, reflections and rotations. Characteristics of figures,
such as lengths of line segments, angle measures and parallel lines, are explored before the transformation (pre-image) and after the transformation (image).
Students understand that these transformations produce images of exactly the same size and shape as the pre-image and are known as rigid transformations.
8.G.A.2
Vocab - clockwise, counterclockwise, congruent, ≊, A’
This standard is the students’ introduction to congruency. Congruent figures have the same shape and size. Translations, reflections and rotations are examples of
rigid transformations. A rigid transformation is one in which the pre-image and the image both have exactly the same size and shape since the measures of the
corresponding angles and corresponding line segments remain equal (are congruent).
Students examine two figures to determine congruency by identifying the rigid transformation(s) that produced the figures. Students recognize the symbol for
congruency (􂄳) and write statements of congruency.
Example 1:
Is Figure A congruent to Figure A’? Explain how you know.
Solution:
These figures are congruent since A’ was produced by translating each vertex of Figure A 3 to the right and 1 down
Example 2:
Describe the sequence of transformations that results in the transformation of Figure A to Figure A’.
Solution:
Figure A’ was produced by a 90° clockwise rotation around the origin.
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7th/8th Grade Mathematics Curriculum Guide
2016 – 2017
8.G.A.4
Vocab – similarity
Similar figures and similarity are first introduced in the 8th grade. Students understand similar figures have congruent angles and sides that are proportional.
Similar figures are produced from dilations. Students describe the sequence that would produce similar figures, including the scale factors. Students understand
that a scale factor greater than one will produce an enlargement in the figure, while a scale factor less than one will produce a reduction in size.
Example1:
Is Figure A similar to Figure A’? Explain how you know.
Solution:
Dilated with a scale factor of then reflected across the x-axis, making Figures A and A’ similar.
Students need to be able to identify that triangles are similar or congruent based on given information.
Example 2:
Describe the sequence of transformations that results in the transformation of Figure A to Figure A’.
Solution:
90° clockwise rotation, translate 4 right & 2 up, dilation of ½.. In this case, the scale factor of the
dilation can be found by using the horizontal distances on the triangle (image = 2 units; pre-image = 4 units)
8.G.A.3
Vocab - dilation, center of rotation, line of reflection
Students identify resulting coordinates from translations, reflections, and rotations (90°, 180° and 270° both clockwise and counterclockwise), recognizing the
relationship between the coordinates and the transformation.
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Translations
Translations move the object so that every point of the object moves in the same direction as well as the same distance. In a
translation, the translated object is congruent to its pre-image. Triangle ABC has been translated 7 units to the right and 3 units up. To
get from A (1,5) to A’ (8,8), move A 7 units to the right (from x = 1 to x = 8) and 3 units up (from y = 5 to y = 8). Points B and C also move
in the same direction (7 units to the right and 3 units up), resulting in the same changes to each coordinate.
Reflections
A reflection is the “flipping” of an object over a line, known as the “line of reflection”. In the 8th grade, the line of reflection will be the
x-axis and the y-axis. Students recognize that when an object is reflected across the y-axis, the reflected x-coordinate is the opposite
of the pre-image x-coordinate (see figure on right).
Likewise, a reflection across the x-axis would change a pre-image coordinate (3, -8) to the image coordinate of (3, 8) -- note that the
reflected y-coordinate is opposite of the pre-image y-coordinate.
Rotations
A rotation is a transformation performed by “spinning” the figure around a fixed point known as the center of rotation. The figure may be
rotated clockwise or counterclockwise up to 360º (at 8th grade, rotations will be around the origin and a multiple of 90º). In a rotation, the
rotated object is congruent to its pre-image.
Consider when triangle DEF is 180° clockwise about the origin. The coordinate of triangle DEF are D(2,5), E(2,1), and F(8,1). When rotated
180° about the origin, the new coordinates are D’(-2,-5), E’(-2,-1) and F’(-8,-1). In this case, each coordinate is the opposite of its pre-image
(see figure on right).
Dilations
A dilation is a non-rigid transformation that moves each point along a ray which starts from a fixed center, and multiplies distances from this
center by a common scale factor. Dilations enlarge (scale factors greater than one) or reduce (scale factors less than one) the size of a figure
by the scale factor. In 8th grade, dilations will be from the origin. The dilated figure is similar to its pre-image.
The coordinates of A are (2, 6); A’ (1, 3). The coordinates of B are (6, 4) and B’ are
(3, 2). The coordinates of C are (4, 0) and C’ are (2, 0). Each of the image coordinates is the value of the pre-image coordinates indicating a
scale factor of ½
The scale factor would also be evident in the length of the line segments using the
ratio: image length pre-image length
Unit 8
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7th/8th Grade Mathematics Curriculum Guide
2016 – 2017
Students recognize the relationship between the coordinates of the pre-image, the image and the scale factor for a dilation from the origin. Using the
coordinates, students are able to identify the scale factor (image/pre-image).
Students identify the transformation based on given coordinates. For example, the pre-image coordinates of a triangle are A(4, 5), B(3, 7), and C(5, 7). The image
coordinates are A(-4, 5), B(-3, 7), and C(-5, 7). What transformation occurred?
Stage 2 - Evidence
Evaluative Criteria/Assessment Level Descriptors (ALDs):
8.G.A (SBAC Target G)
Level 4 students should be able to describe a sequence that exhibits the similarity between two shapes and understand that the angle measures are unchanged.
Level 3 students should be able to understand and describe the impact of a transformation on a figure and its component parts with or without coordinates. They
should be able to use or describe a sequence of transformations to determine or exhibit the congruence of two figures. They should also be able to construct
rotations and dilations of figures in a coordinate plane.
Level 2 students should be able to construct reflections and translations of figures in a coordinate plane and identify dilations and the results of dilations on
figures.
Level 1 students should be able to identify reflections, rotations, and translations and the result of these rigid motions on figures.
SBA Released
Claim 1 Item Specs
See Sample Assessments for Unit 8.
Stage 3 – Learning Plan Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
LEARNING ACTIVITIES: **Days may change depending on any tasks or assessing you choose to do.
NOTES:
8.G.A.5
8.G.A.5
Day 1: Points, Lines, Planes, and Angles
IXL
N.1, N.2, N.3, N.6, N.7, N.9, N.14
 Holt Course 3: 7-1
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7th/8th Grade Mathematics Curriculum Guide
2016 – 2017
Stage 3 – Learning Plan Sample
Day 2: Parallel and perpendicular Lines
 Holt Course 3: 7-2
Suggested Performance Tasks
 Lunch Lines
 Window Pain
Day 3: Angles in Triangles
 Holt Course 3: 7-3
Day 4: Classifying Polygons
 Holt Course 3: 7-4
Day 5: Coordinate Geometry
 Holt Course 3: 7-5
8.G.A.1
Day 6: Hands-On Lab: Explore Congruence
 Holt Course 3 7-6A
8.G.A.1
IXL
Q.1, Q.2, Q.4, Q.6
8.G.A.2
Day 7: Congruence
 Holt Course 3 Lesson 7-6 Congruence
8.G.A.2
IXL
N.10, N.12, N.13
Day 8: Transformations
 Holt Course 3 Lesson 7-7 Transformations
Day 9: Similarity & Congruence Transformations
 Holt Course 3 Common Core Curriculum Companion Lesson 7-7A EXTRAS
Day 10: Hands on Lab
 Holt Course 3 Lesson 7-7
Day 11: Identifying Combined Transformations
 Holt Course 3 Common Core Curriculum Companion Lesson 7-7B EXTRAS
8.G.A.4
IXL
8.G.A.4
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7th/8th Grade Mathematics Curriculum Guide
2016 – 2017
Stage 3 – Learning Plan Sample
Day 12: Similar Figures
 Holt Course 3 Lesson 5-5
N.10, N.11
Additional Materials
Engage NY Grade 8 Module 3 Topic B Lessons 8, 9,
10
Suggested Performance Task
 Aaron’s Design – 8.G.3 & 8.G.4
 Similar Triangles
8.G.A.3
Day 13: Hands-On Lab: Explore Dilations
 Holt Course 3 Lesson 5-6
8.G.A.3
IXL
Q.3, Q.5, Q.7, Q.8, Q.9
Day 14: Dilations
 Holt Course 3 Lesson 5-6
Performance Tasks
 Dilations in the Coordinate Plane 7.G.3
 Playing with Dilations 7.G.3
 Coordinating Reflections, Translations, and
Rotations
 Changing Shapes
Common Assessment
Daily Lesson Plan
Learning Target:
Opening Activity:
Activities:
 Whole Group:
 Small Group/Guided/Collaborative/Independent:
 Whole Group:
Checking for Understanding (before, during and after):
Assessments:
Additional Materials
Engage NY Grade 8 Module 3 Topic A
https://www.engageny.org/resource/grade-8mathematics-module-3-topic-overview
TI Activities
Dilations
Reflections
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