Transforming Geometric Instruction

Part 2

Understanding the Instructional Shifts in the Iowa Core Mathematics

Standards Grades 6-8

Session OutcomesIn 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.

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)

Why Transformational Geometry?Take a few moments with your table group and discuss why it is critical for students to develop spatial sense.

Graphic Arts

DraftingEngineering

Architecture

Construction

Why is Spatial Sense Important to Mathematics?

1. 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.

2. Students are constantly presented with visual images that they must be able to interpret (Ex. 3-D represented as 2-D and different orientations).

3. Many researchers believe that when data and information are represented spatially that students are better able to generalize and remember the underlying mathematical concepts.

4. 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

- Ben-Chaim, Lappan, and Houang (1988)

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.

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

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

Essential Vocabulary - RotationIn 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.

Essential Vocabulary - DilationA 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).

Flip and Slidehttp://calculationnation.nctm.org/

8.G.1 –Verify experimentally the properties of rotations, reflections, and translations:a. Lines are taken to lines, and line

segments to line segments of the same length

b. Angles are taken to angles of the same measure

c. Parallel lines are taken to parallel lines

Congruent Segmentsadapted from Illustrative Mathematics

Congruent Rectanglesa. Show that the rectangles are

congruent by finding a translation followed by a rotation which maps one of the rectangles to the other.

b. Explain why the congruence of the two rectangles can not be shown by translating Rectangle 1 to Rectangle 2.

c. Can the congruence of the two rectangles be shown with a single reflection? Explain.

adapted from Illustrative Mathematics

Congruent Trianglesadapted from Illustrative Mathematics

Mystery Transformations

L 1

L 2

How can L1 be moved onto L2

with as few transformations

as possible?

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.

Transformations and Coordinates

Launch

Explain to an elbow partner each of the following transformations:

Reflection Rotation

Translation Dilation

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.

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.

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?

Iowa Core

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

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.

Similar Figures and Transformati

ons

Launch

Work through the sheet,

“Thinking about Transformations”.

Be prepared to discuss your

answers.

Explore, Part 1Two 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.

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?

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?

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?

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.

Angles Everywhe

re

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

How do you define each of the following?

Vertical angles

Adjacent angles

Straight Angle

Supplementary Angles

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.

SummarizeWhat 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?

SummarizeAngles 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?

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.

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)

LAUNCHWhat 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?

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.

SUMMARIZEFor 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 units apart. Label the points X and Y. Explain how you know the distance between the points is units.

A Proof Without Words

http://illuminations.nctm.org/Activity.aspx?id=6376

I’m speechless….

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?

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?

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.

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?

Is That Right?(adapted from Math Matters, 2006)

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.

Making the Play!

Turn

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?

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.

What is this Figure?

Adapted from “A rectangle in the coordinate plane”, Illustrative Mathematics

Launch

Find the missing side of right ∆ ABC and explain your

thinking.

Side AB is 8 units and

side AC is 5 units.

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?

ExploreStudents 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.

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)

Iowa Core

8.G.8. Apply the Pythagorean Theorem to

find the distance between two points in a coordinate

system.

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)

Understanding and Applying Volume Formulas

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.

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?

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

https://www.youtube.com/watch?v=jP4P50lA-SE

Volume of a cone:

https://www.youtube.com/watch?v=xwPiA0COi8k

https://www.youtube.com/watch?v=0ZACAU4SGyM

Volume of a sphere:

https://www.youtube.com/watch?v=YNutS8eIhEs

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?

“Glasses”Adapted from “8.G Glasses”, Illustrative Mathematics

Apply the volume formulas you have learned by doing “Glasses” with a partner. Be prepared to

discuss your results.

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?

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.

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.