In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
Understand congruence and similarity using physical models, transparencies, or geometry software.
Verify experimentally the properties of rotations, reflections, and translations:
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Properties
Angles [b]
Lines [c]
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5 - Verify (Properties of rotations)
5 - Verify (Properties of reflections)
5 - Verify (Properties of translations)
2 - Understand (lines taken to lines of same length [a])
2 - Understand (line segments taken to line segments of same length [a])
2 - Understand (angles taken to angles of same measure [b])
2 - Understand (parallel lines taken to parallel lines [c])
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5 - I can verify experimentally the properties of rotations.
5 - I can verify experimentally the properties of reflections.
5 - I can verify experimentally the properties of translations.
2 - I can understand that lines are taken to lines of same length [a].
2 - I can understand that line segments are taken to line segments of same length [a].
2 - I can understand that angles are taken to angles of same measure [b].
2 - I can understand that parallel are lines taken to parallel lines [c].
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A. The properties of rotations, reflections and translations can be verified experimentally.
B. Rotations, reflections, and translations all preserve figure congruence.
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A.1 What is a rotation?
A.2 What are the properties of rotations?
A.3 What is a reflection?
A.4 What are the properties of reflections?
A.5 What is a translation?
A.6 What are the properties of translations?
A.7 How can one experimentally verify the properties of rotations?
A.8 How can one experimentally verify the properties of reflections?
A.9 How can one experimentally verify the properties of translations?
B.1 What effect does rotation, reflection, or translation have on the figure being transformed?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Figures
Sequence
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1 - Understand (Congruence between two figures)
2 - Describe (Sequence of rotations, reflections and translations)
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1 - I 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 or translations.
2 - I can describe the sequence of rotations, reflections and translations that exhibits congruence between two figures.
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A. Two different two dimensional figures are congruent only if the second figure can be obtained from the first figure through a sequence of rotations, reflections and translations.
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A.1 What makes two figures congruent?
A.2 What is a rotation?
A.3 What is a reflection?
A.4 What is a translations?
A.5 What is a sequence?
A.6 What is a two dimensional figure?
A.7 How does one find the sequence of rotations, reflections or translations that exhibits congruence between two figures?
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Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Effects
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2 - Describe (Effect of dilations)
2 - Describe (Effect of translations)
2 - Describe (Effect of reflections)
2 - Describe (Effect of rotations)
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2 - I can describe the effect of dilations on two dimensional figures using coordinates.
2 - I can describe the effect of translations on two dimensional figures using coordinates.
2 - I can describe the effect of reflections on two dimensional figures using coordinates.
2 - I can describe the effect of rotations on two dimensional figures using coordinates.
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A. The effect of dilations, translations, rotations and reflections on a two dimensional figure can be described by using coordinates.
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A.1 What does the term effect mean?
A.2 What is a dilation?
A.3 What is a translation?
A.4 What is a rotation?
A.5 What is a reflection?
A.6 What is a two dimensional figure?
A.7 What is a coordinate?
A.8 How does one describe the effect of dilations, translations, rotations, and reflections of two dimensional figures using coordinates?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Figures
Sequence
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1 - Understand (Similarity between two figures)
2 - Describe (Sequence of rotations, reflections, translations and dilations)
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1 - I 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, dilations and translations.
2 - I can describe the sequence of rotations, reflections, translations and dilations that exhibits similarity between two figures.
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A. A two dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, dilations and translations.
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A.1 What makes two figures similar?
A.2 What is a rotation?
A.3 What is a reflection?
A.4 What is a translation?
A.5 What is a dilation?
A.6 What is a sequence?
A.7 What is a two dimensional figure?
A.8 How does one find the sequence of rotations, reflections or translations that exhibits congruence between two figures?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Arguments
Facts of triangles
Facts of parallel lines/transversal
Similarity
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3 - Use (Informal arguments)
3 - Establish (Facts about angle sum in triangles)
3 - Establish (Facts about exterior angles of triangles)
3 - Establish (Facts about angles created when parallel lines are cut by a transversal)
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3 - I can use informal arguments to establish facts.
3 - I can establish facts about the angle sum in triangles.
3 - I can establish facts about the exterior angles of triangles.
3 - I can establish facts about the angles created when parallel lines are cut by a transversal.
3 - I can establish facts about the angle/angle criteria for similarity of a triangle.
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A. Informal arguments can be used to establish facts about the angle sum and exterior angles of triangles, about the angles created when parallel lines are cut by a transversal, and about the angle/angle criterion for similarity of triangles.
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A.1 What is an informal argument?
A.2 What facts can be established about the sum of the angles in a triangle?
A.3 What facts can be established about the exterior angles of triangles?
A.4 What are parallel lines?
A.5 What is a transversal?
A.6 What facts can be established about the angles formed when parallel lines are cut by a transversal?
A.7 What facts can be established about the angle/angle criteria for triangle similarity?
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Explain a proof of the Pythagorean Theorem and its converse.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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2 - Explain (proof of Pythagorean Theorem)
2 - Explain (proof of the converse)
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2 - I can explain a proof of Pythagorean Theorem.
2 - I can explain a proof of the converse of the Pythagorean Theorem.
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A. The Pythagorean Theorem is related to other formulas in math.
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A. What is the Pythagorean Theorem?
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Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Theorem
Triangles
Problems
Objects
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3 - Apply (Pythagorean theorem)
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3 - I can apply the Pythagorean theorem to determine unknown side lengths in right triangles.
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A. Real world and mathematical problems can be solved by using the Pythagorean theorem to determine unknown side lengths in right triangles.
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A.1 What is the Pythagorean theorem?
A.2 In what types of triangles can the Pythagorean theorem be used?
A.3 What is a right triangle?
A.4 How does one use the Pythagorean theorem to determine unknown side lengths in right triangles?
A.5 What are some real world applications of the Pythagorean theorem?
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Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Theorem
Distance
System
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3 - Apply (Pythagorean theorem)
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3 - I can apply the Pythagorean theorem to find the distance between two points in a coordinate system.
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A. The distance between two points in a coordinate system can be found by applying the pythagorean theorem.
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A.1 What is a coordinate system?
A.2 How does one apply the Pythagorean theorem to find the distance between two points in a coordinate system?
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Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Formulas
Problems
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1 - Know (Formula for volume of a cone)
1 - Know (Formula for volume of a cylinder)
1 - Know (Formula for volume of a sphere)
3 - Solve (Real world problems)
3 - Solve (Mathematical problems)
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1 - I know the formula for the volume of a cone.
1 - I know the formula for the volume of a cylinder.
1 - I know the formula for the volume of a sphere.
3 - I can solve real world problems involving the volumes of cones, cylinders and spheres.
3 - I can solve mathematical problems involving the volumes of cones, cylinders and spheres.
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A. Some real world and mathematical problems can be solved by finding the volumes of cones, cylinders and spheres.
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A.1 What is the formula for the volume of a cone?
A.2 What is the formula for the volume of a cylinder?
A.3 What is the formula for the volume of a sphere?
A.4 What are some real world applications for the volumes of cones, cylinders and spheres?
A.5 How does one solve real world problems involving the volume of cones, cylinders and spheres?
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