1. Transforming Geometric Instruction Part 2
Understanding the Instructional Shifts in the Iowa Core Mathematics Standards Grades 6-8

2. Session Outcomes
In order to effectively implement Iowa Core Geometry Standards, teachers will:
become familiar with the content shifts in middle school geometry (not all geometry content represents a shift).
experience those shifts through activity based learning.
identify connections between content and the Standards for Mathematical Practice.
understand the importance of the van Hiele levels of geometric learning.

3. Where Are We Going? Middle School
Solve problems involving area, surface area,
and volume (6th)
Draw, construct, and describe geometrical figures and describe the relationships between them (7th)
Solve problems involving angle measure, area, surface area, and volume (7th)
Understand congruence and similarity using physical models, transparencies, or geometry software (8th)
Understand and apply the Pythagorean Theorem (8th)
Solve problems involving cylinders, cones, and spheres (8th)

4. Why Transformational Geometry?
Take a few moments with your table group and discuss why it is critical for students to develop spatial sense.
Collect groups ideas on a shared poster

5. Graphic Arts Architecture Engineering Drafting Construction
Math Matters pg. 248
Topics like reflections and rotations are important to many career fields.

6. Why is Spatial Sense Important to Mathematics?
These and similar topics are formal aspects of geometry, statistics, and algebra. Students must be able to identify, analyze, and transform shapes and figures in order to make sense of data and solve problems.
Students are constantly presented with visual images that they must be able to interpret (Ex. 3-D represented as 2-D and different orientations).
Many researchers believe that when data and information are represented spatially that students are better able to generalize and remember the underlying mathematical concepts.
Research suggests that students who make sense of visual images are better problem solvers.
Spatial + analytic thinking = flexible thinking and multiple approaches
“Math Matters” pg. 248

7. Some students have a natural spatial sense; others are easily confused by spatial information. Nevertheless, with proper experiences ALL students’ spatial sense can be improved.
- Ben-Chaim, Lappan, and Houang (1988)

8. Essential Vocabulary - Translations
The shift of a figure on the plane to a new position without changing its orientation.

9. Essential Vocabulary - Reflection
A figure in a transformation is flipped or reflected across a specific reflection line. The figure’s orientation changes.

10. Essential Vocabulary - Rotation
In a rotation, a figure is turned any number of degrees about a rotation a point. This point can either be within or outside of the figure.

11. Essential Vocabulary - Dilation
A dilation is a non-rigid transformation that produces similar figures. A figure can be dilated to make a larger shape (enlargement) or a smaller shape (reduction).

12. Flip and Slide http://calculationnation.nctm.org/
Have participant sign in as a guest or create an account. The goal of this activity is for participants to explore and experiment with transformations.

13. 8.G.1 –
Verify experimentally the properties of rotations, reflections, and translations:
Lines are taken to lines, and line segments to line segments of the same length
Angles are taken to angles of the same measure
Parallel lines are taken to parallel lines

14. adapted from Illustrative Mathematics
Congruent Segments
adapted from Illustrative Mathematics
8.G.2 Congruent Segments-Teacher Pages
8.G.2 Congruent Segments-Student Page

15. adapted from Illustrative Mathematics
Congruent Rectangles
adapted from Illustrative Mathematics
Show that the rectangles are congruent by finding a translation followed by a rotation which maps one of the rectangles to the other.
Explain why the congruence of the two rectangles can not be shown by translating Rectangle 1 to Rectangle 2.
Can the congruence of the two rectangles be shown with a single reflection? Explain.
8.G.2 Congruent Rectangles – Teacher Pages
8.G.2 Congruent Rectangles – Student Page

16. adapted from Illustrative Mathematics
Congruent Triangles
adapted from Illustrative Mathematics
8.G.2 Congruent Triangles- Teacher pages
8.G.2 Congruent Triangles – Student Page

17. Mystery Transformations
How can L1 be moved onto L2 with as few transformations as possible?
L 2
8.G.2 Mystery Transformations
L 1

18. 8.G.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.

19. Transformations and Coordinates
8.G.3 Transformations and Coordinates Lesson Plan

20. Launch Dilation Rotation Reflection Translation
Explain to an elbow partner each of the following transformations:
Dilation
Translation

21. Reflection: a transformation creating a mirror image of a figure on the opposite side of a line. The line is the line of symmetry between the original figure and its image.
Rotation: a transformation in which a figure is turned a given angle and direction around a point (the center of rotation)
Translation: a transformation that slides a figure a given distance in a given direction.
Dilation: a transformation that shrinks or enlarges a figure by a scale factor from a given point.
Definition from activity 8.G.1
Reflection: A figure in a transformation is flipped or reflected across a specific reflection line. The figure’s orientation changes.
Rotation: In a rotation, a figure is turned any number of degrees about a rotation a point. This point can either be within or outside of the figure.
Translation: The shift of a figure on the plane to a new position without changing its orientation.
Dilation: A dilation is a non-rigid transformation that produces similar figures. A figure can be dilated to make a larger shape (enlargement) or a smaller shape (reduction).

22. Explore: With your Partner…
…work through the student handout, “Transformations and Coordinates”. At the end of each type of transformation, check in with the teacher with your results.
With Geogebra:
8.G.3 Transformations and Coordinates Recording Sheet (geogebra)
8.G.3 Transformations and Coordinates Student Handout (geogebra)
8.G.3 Transformations and Coordinates Recording Sheet (geogebra) answer key
With paper/pencil
8.G.3 Transformations and Coordinates Recording Sheet (pencil)
8.G.3 Transformations and Coordinates Student Handout (pencil)
8.G.3 Transformations and Coordinates Recording Sheet (pencil) answer key

23. Summarize
In performing reflections over the axes and the lines y = x or y = -x, what did you observe about the coordinates of the triangle’s vertices?
If you were to draw a segment between the point A and its image, what would you observe about that segment with the reflection line?
You performed rotations in a CW direction. How do those compare with rotations done CCW?
Can you explain what would happen to your triangle’s vertices if you translated it 15 units left and 10 units up?
If your triangle’s image landed on (-9, 6) after a dilation with scale factor of 3 and center at the origin, what would have been the original point?

24. Iowa Core
8.G.3: Describe the effect of dilations, translations, rotations, and reflections on two- dimensional figures using coordinates.

25. Transformations and Coordinates
Which Standards for Mathematical Practice are evident when doing this activity? Use your template to record any notes.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriately tools strategically.
Attend to precision.
Look for and make use structure.
Look for and express regularity in repeated reasoning.
Refer to SMP template

26. Similar Figures and Transformations
8.G.4 Similar figures and transformations Lesson Plan

27. Launch
Work through the sheet, “Thinking about Transformations”. Be prepared to discuss your answers.
8.G.4 Similar figures and transformations Launch activity
8.G.4 Similar figures and transformations Launch activity answer key

28. Explore, Part 1
Two similar figures have congruent angles and sides that are proportional. If a two-dimensional figure is similar to another, then there should be a way to obtain the second figure from the first by using a single dilation or a sequence of transformations (rotation, reflection, translation, dilation). To explore this statement, with a partner, work through Part 1 of “Similar Figures and Transformations” student handout.
8.G.4 Similar figures and transformations (geogebra)
8.G.4 Similar figures and transformations (pencil)
8.G.4 Similar figures and transformations (geogebra) answer key

29. Summarize, Part 1
Were all the triangle pairs similar? If not, how did you know they were not?
Can you describe the transformations that you did to determine your answer? Is there another way that you could have done the problem?
How can/did knowing about “orientation” help you choose a transformation to use?
Could you create a set of triangles that another partnership could check?

30. Explore, Part 2
In Part 2 of this activity, you know that the given triangles are similar. But what transformations can make D ABC map onto D DEF?

31. Summarize, Part 2
What series of transformations mapped the first onto the second in each problem?
Was there a different order you could have used?
Did you determine if the triangles had the same orientation before you chose any transformations?
Could you do these problems without having coordinate grids?

32. Iowa Core
8.G.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.

33. Angles Everywhere
8.G.5 Angles Everywhere Lesson Plan

34. Launch
On your own, write down at least three angle relationships found in the diagram to the left. Share your findings with an elbow partner and compare what your conclusions

35. How do you define each of the following?
Vertical angles Adjacent angles Straight Angle Supplementary Angles

36. Explore New Relationships
Complete the activities in the student handout. There will be times when you will need to check with your teacher before you can proceed.
8.G.5 Angles Everywhere handout
8.G.5 Angles Everywhere handout (answer key)

37. Summarize
What conclusion did you make about the sum of the angles of any triangle?
Exterior Angles of a Triangle:
What did you discover about the sum of angles E, X, and T?
How does knowing the sum of each interior and exterior angle pair help you find the sum of the exterior angles in a different way?
How does knowing the sum of each interior and exterior angle pair help you determine that an exterior angle is equal to the sum of the remote interior angles?

38. Summarize Angles formed by a transversal and two parallel lines:
What did you determine about the:
corresponding angles?
alternate interior angles?
alternate exterior angles?
If you know the measure of one of the eight angles formed by the transversal, can you determine the other seven without measuring them? Explain.
Similar Triangles:
What must you know, at a minimum, to tell if two triangles are similar?

39. Iowa Core
8.G.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.

40. Where Are We Going? Middle School
Solve problems involving area, surface area,
and volume (6th)
Draw, construct, and describe geometrical figures and describe the relationships between them (7th)
Solve problems involving angle measure, area, surface area, and volume (7th)
Understand congruence and similarity using physical models, transparencies, or geometry software (8th)
Understand and apply the Pythagorean Theorem (8th)
Solve problems involving cylinders, cones, and spheres (8th)

42. LAUNCH What kind of triangle is this?
The two sides that form the right angle are called the legs of the right triangle.
The side opposite the right angle is called the hypotenuse.
What are the areas of the squares on the legs?
What is the area of the square on the hypotenuse?

43. EXPLORE Draw a right triangle with the given lengths on dot paper.
Draw a square on each side of the triangle.
Find the areas of the squares. Be prepared to explain how you partitioned the squares to determine the areas. Record your results in the table.
8.G.6 - Proof of the Pythagorean Theorem
8.G.6 Cm Grid Paper

44. SUMMARIZE
For each triangle look for a relationship among the areas of the three squares. Make a conjecture about the areas of squares drawn on the sides of any right triangle. How can we find the length of the hypotenuse of each right triangle? Extend Your Thinking: On dot paper, find two points that are 32 units apart. Label the points X and Y. Explain how you know the distance between the points is 32 units.

45. A Proof Without Words I’m speechless….

46. Snowball Problem (adapted from Big Ideas Math, 2014)
You and a friend stand back-to-back. You run 20 feet forward and then 15 feet to your right. At the same time, your friend runs 16 feet forward, then 12 feet to her right. She stops and hits you with a snowball.
How far does your friend throw the snowball?

47. Agree or Disagree?
Hermione has a shipping box that measures 6 inches long, 8 inches wide, and 12 inches tall. She says that the longest wand she can ship to Hogwarts is 12 inches. Do you agree or disagree? What is the longest wand she can fit in her shipping box?

48. Is That Right? (adapted from Math Matters, 2006)
Use cm grid paper to cut strips of paper in the following centimeter lengths:
6 , 6 , 7 , 8 , 8 , 9 , 10 , 11 , 12 , 13, 14
Use one 6 cm and one 8 cm strip to form the legs of a right triangle.
Use each of the remaining segment lengths in conjunction with the 6 cm and 8 cm segments to form triangles. Not all the strips will make a triangle unless you adjust the angle between the 6 cm and 8 cm legs.
In the table, name the type of triangle each combination forms – acute, right, or obtuse.
Substitute the values into the Pythagorean Theorem and record your results in the table.
8.G.6 – 7 “Is That Right?” Participant Handout

49. Is That Right? (adapted from Math Matters, 2006)
Discuss in your group:
What is the relationship between the sum of the squares of the two legs and the type of triangle?
What is the relationship between the lengths of the sides of the triangles and the angles opposite each of those sides?

50. Is That Right?
Which Standards for Mathematical Practice are evident when solving this problem? Use your template to record any notes.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriately tools strategically.
Attend to precision.
Look for and make use structure.
Look for and express regularity in repeated reasoning.
Refer to SMP template

51. Making the Play!
You design a football play in which a player runs down the field, makes a 90˚ turn, and runs to the corner of the end zone. Your friend runs the play as shown.
Did your friend make a 90˚ turn? Use the Pythagorean Theorem to prove your answer.
If your friend made a turn at (20,10), would it be a 90˚ turn?
Turn
8.G.6 – 7 “Making the Play” Participant Handout

52. 8.G.6 – Explain a proof of the Pythagorean Theorem and its converse.
8.G.7 – Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Of Special Note to Participants (see language highlighted in red above):
Important to point out that we can use a right triangle to prove the Pythagorean Theorem, but we can also use the Pythgorean Theorem to prove a triangle is right (the converse). Problems like “Is it Right?” and “Making the Play!” support the understanding about the converse of the Pythagorean Theorem.
It is also stated in the standards that students apply the Pythagorean Theorem to objects in three dimensions. Problems like the “Agree or Disagree – Will Hermione’s Wand Fit?” Problem is an example of this.

53. What is this Figure? Adapted from “A rectangle in the coordinate plane”, Illustrative Mathematics
8.G.8 What is this Figure Lesson Plan

54. Launch
Find the missing side of right ∆ ABC and explain your thinking. Side AB is 8 units and side AC is 5 units.
8.G.8 What is this Figure Launch Activity

55. Launch
Use this segment as the hypotenuse of a right triangle. Draw in the other sides to show a triangle. How can you use your new triangle sides and the Pythagorean theorem to find the length of the segment?

56. Explore
Students will work through the student handout, “What is this Figure?”. There are two “Check with your teacher before you proceed” in the student handout. Be prepared to share your thinking about your work.
8.G.8 What is this Figure Student Activity
8.G.8 What is this Figure Student Activity (answer key)

57. Summarize
How can you determine the length of any line segment in the coordinate plane?
Was there more than one way to create right triangles in order to determine segment lengths? Explain.
What did you discover about the diagonals of any rectangle by doing this activity? (Extension)

58. Iowa Core
8.G.8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

59. Where Are We Going? Middle School
Solve problems involving area, surface area,
and volume (6th)
Draw, construct, and describe geometrical figures and describe the relationships between them (7th)
Solve problems involving angle measure, area, surface area, and volume (7th)
Understand congruence and similarity using physical models, transparencies, or geometry software (8th)
Understand and apply the Pythagorean Theorem (8th)
Solve problems involving cylinders, cones, and spheres (8th)

60. Understanding and Applying Volume Formulas
8.G.9 Understanding and Applying Volume Formulas Lesson Plan

61. Launch
Solve each of the following problems individually: Determine the volume of a rectangular prism with Base area of 6 cm2 and a height of 7 cm. Determine the area of a circle with radius of 3 cm.

62. Compare your work with an elbow partner and discuss the following:
What is the label you had for each problem? Why?
What does “area of the base” mean in the first problem? What is the shape of the base?
Would you use the same formula for a triangular prism? Why or why not?
How is the area of a circle formula different from the circumference of a circle?

63. Explore “deriving the formulas….”
Volume of a cylinder: Volume of a cone: Volume of a sphere:

64. Summarize the formulas
Cylinder: V = π r2h Cone: V = π r2h
Sphere: V = π r3
How are the formulas for volume of a prism and volume of a cylinder alike? Different?
What is the relationship between the volume of a cone and the volume of a cylinder?
If we had explored the volume of a pyramid, what other volume formula might it relate to? Why?
What is one “real world” application of the volume of a sphere? How could this formula help you in find the volume of a hemisphere?

65. “Glasses” Adapted from “8.G Glasses”, Illustrative Mathematics
8.G.9 Glasses Problem Student Activity
8.G.9 illustrative Mathematics 112 Glasses (answer key)
Apply the volume formulas you have learned by doing “Glasses” with a partner. Be prepared to discuss your results.

66. Summarize - “Glasses”
What relationship did you have to use to find the height of glass 3? What number did you determine for the height? Did any glass have the shape of a sphere? Did you use the formula for the volume of a sphere for any glass? Explain. Question 5 seems like you can just find the volume of the hemisphere to get the answer. Is that correct?

67. Iowa Core
8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real- world and mathematical problems.

68. Glasses
Which Standards for Mathematical Practice are evident when solving this problem? Use your template to record any notes.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriately tools strategically.
Attend to precision.
Look for and make use structure.
Look for and express regularity in repeated reasoning.
Refer to SMP template