All students - place homework (worksheet)in Green folder and get out binders and turn to next available page for writing and get yourselves going today: RM - pick up graph paper and tracking paper RR - have students write today’s date on Their note paper Facilitators - have all students turn to Lesson 3.1.5 (page 139), students should read to get ready for the lesosn. TM - only 3 minutes to get ready this morning. 20 Oct 2017 Agenda Homework turn in

Lesson 3.1.5 Using transformations to create polygons Lesson 3.1.6 Symmetry

Homework 3.1.5 What shapes can I create with triangles? Using Transformations to Create Polygons

In Lesson 3.1.4, you practiced reflecting, rotating, and translating polygons. Since these were rigid transformations, the image always had all the same angle measures and side lengths as the original. In this lesson, you will combine the image with the original to make new, larger polygons from four basic “building-block” polygons. As you create new shapes, consider what information the transformation gives you about the resulting new shape. As you create new shapes, consider what information the transformation gives you about the resulting new shape.

3-51. THE SHAPE FACTORY The Shape Factory, an innovative new company, has decided to branch out to include new shapes. As product developers, your team is responsible for finding exciting new shapes to offer your customers. The current company catalog is shown at right. As you create new shapes, consider what information the transformation gives you about the resulting new shape.

Since your boss is concerned about production costs, you want to avoid buying new machines and instead want to reprogram your current machines. As you create new shapes, consider what information the transformation gives you about the resulting new shape.

The factory machines cannot only make all the shapes shown in the catalog, but are also able to rotate or reflect a shape exactly one time. For example, if the half-equilateral triangle is rotated 180º about the midpoint (the point in the middle) of its longest side, as shown, the result is a rectangle. On white board or scratch paper, how many shapes you can make? For each triangle in the original catalog, determine which new shapes are created when it is rotated or reflected exactly one time so that the image shares an entire side with the original triangle. Be sure to make as many new shapes as possible. Use tracing paper or any other reflection tool to help.

Rectangle (by rotation of half equilateral triangle) On graph paper or white paper - what shapes can you make with a rotation? 5mins How many did you get? On graph paper or white paper - what shapes can you make with a reflection? How many did you get? 3.1.6 What shapes have symmetry? Symmetry

You have encountered symmetry several times in this chapter. So far you have seen how symmetry arises from some transformations. Now we will look at this relationship between transformations and symmetry further, and this will deepen your understanding of symmetry. By the end of this lesson, you should be able to answer these questions: 3.1.6 What shapes have symmetry? Symmetry What is symmetry?

How can we determine whether or not a polygon has symmetry?

What types of symmetry can a shape have? 3-61 Reflection Symmetry

A figure has reflection symmetry if a reflection carries it onto itself. See an example of a quilt design that has reflection symmetry at right.

Obtain the Lesson 3.1.5 Resource Page. It is important to keep this page for the remainder of the course. You may also want to refer to the Math Notes box about polygons in Lesson 3.1.2. 3-61 Reflection Symmetry a. Which of the shapes on the resource page have reflection symmetry? For each figure on the resource page, draw all the possible lines of symmetry. If you are not sure if a figure has reflection symmetry, use tracing paper or a reflective tool to explore. b. Which types of triangles have reflection symmetry? c. Which types of quadrilaterals have reflection symmetry? d. Which shapes on your resource page have more than three lines of symmetry? 3-61 Reflection Symmetry a. Which of the shapes on the resource page have reflection symmetry? For each figure on the resource page, draw all the possible lines of symmetry. If you are not sure if a figure has reflection symmetry, use tracing paper or a reflective tool to explore. b. Which types of triangles have reflection symmetry? c. Which types of quadrilaterals have reflection symmetry? d. Which shapes on your resource page have more than three lines of symmetry? 3-61 Reflection symmetry a) regular hexagon, kite, rectangle, equilateral triangle, square, isosceles right triangle, circle, rhombus, isosceles triangle, isosceles trapezoid, regular pentagon - all have reflection symmetry. 3-61 Reflection symmetry

b) triangles with reflection symmetry: equilateral triangle, isosceles right triangle, isosceles triangle 3-61 c) Which types of quadrilaterals have reflection symmetry? square, rectangle, rhombus, kite, And isosceles trapezoid 3-61 d) Which shapes on your resource page have more than three lines of symmetry? square, regular pentagon, regular hexagon, and circle 3-62 Rotation Symmetry a) Is this shape a polygon? b) Can this figure be rotated onto itself? Trace the shape on tracing paper and test your conclusion. If it is possible where is the point of rotation? 3-62 a) Is this shape a polygon? Yes b) and it can be rotated onto itself (see desmos demonstration) using the center of the shape as the center of rotation. Rotate 120 degrees or any multiple of it. (360 divided by 3 is 120) 3-62 c) Jessica claims that she can rotate all polygons in such a way that they will not change. How does she do it? 3-62 c) Jessica claims that she can rotate all polygons in such a way that they will not change. How does she do it? Rotate 360 degrees! (All the way around until the shape is back where it started.) 3-62 d) Since all polygons can be rotated 360° without change, that is not a very special quality. However, the polygon in part (a) above was special because it could be rotated less than 360° and still remain unchanged. A polygon with this quality is said to have rotation symmetry.

But what shapes have rotation symmetry? Examine the shapes on your Lesson 3.1.5 Resource page and identify those that have rotation symmetry. 3-62 d) Since all polygons can be rotated 360° without change, that is not a very special quality. However, the polygon in part (a) above was special because it could be rotated less than 360° and still remain unchanged. A polygon with this quality is said to have rotation symmetry.

Regular hexagon, rectangle, equilateral triangle, square, circle, rhombus, parallelogram, and regular pentagon all have rotation symmetry. 3-62 e) Which shapes on the resource page have 90o rotation symmetry? (That is which can be rotated about a point 90o and remain unchanged?) 3-62 e) Circle and square have 90 degree rotation symmetry (can be rotated about a point 90° and remain unchanged). 3-63 Translation symmetry

a. Is there a polygon that can be translated so that its end result is exactly the same as the original object? If so, draw an example and explain why it has translation symmetry. NO. (not moving at all isn’t a translation)

b. Is there any kind of shape or figure that has translation symmetry?

3-63 Translation symmetry

a. Is there a polygon that can be translated so that its end result is exactly the same as the original object? If so, draw an example and explain why it has translation

symmetry. NO.

b. Is there any kind of shape or figure that has translation symmetry? All lines can be translated so that the end result is exactly the same as the original. Also, a tessellation that extends infinitely in all directions can be translated so that the end result is exactly the same as the original.

3-64 a. The decagon has 10 lines of symmetry which pass through opposite vertices and the midpoints of opposite sides of the polygon. b. The polygon can be rotated about its center (which can be found by finding the point of intersection of the diagonals connecting opposite vertices). The angle of rotation can be 36° or any multiple thereof (notice, 360 divided by 10 is 36). 3-64 c) There is no translation that moves a regular decagon onto itself, so it does not have translation symmetry. As soon as I slide the shape it is not in the same place anymore. 3-65 Connections with Algebra During this lesson, you have focused on the types of symmetry that can exist in geometric objects. But what about shapes that are created on graphs? What types of graphs have symmetry? a. Examine the graphs below. Decide which have reflection symmetry, rotation symmetry, translation symmetry, or a combination of these. Draw each one into your notes and answer the questions. 3-65 Symmetries?

Translation Rotation 180 degrees 3-65 Symmetries?

Reflection across y-axis 3-65 Symmetries?

Reflections Translation rotation 3-65 Symmetries?

Reflections rotation 3-65 Symmetries?

Translation rotation 3-65 Symmetries?

Reflection 3-65 Symmetries? b) If the y-axis is a line of symmetry of a graph, then its function is referred to as even. Which of the graphs in part (a) are even functions? 2) and 6) 3-65 Symmetries? c) If the graph has rotation symmetry about the origin (0, 0), its function is called odd. Which of the graphs in part (a) are odd functions? 3 and 4 3-66 Learning Log: Symmetry

Reflect on what you have learned during this lesson. In your Learning Log, answer the questions posed at the beginning of this lesson, reprinted below. When helpful, give examples and draw a diagram. Title this entry “Symmetry” and include today’s date. What is symmetry? How can we determine whether or not a polygon has symmetry? What types of symmetry can a shape have? Homework - Read Math Notes: Parallel and Perpendicular lines - make notes, include diagrams Lesson 3.1.5 Review and Preview 3-55, 3-56, 3-59, 3-60

Lesson 3.1.6 Review and Preview 3-67 through 3-69 and 3-71

Finish Initial Learning Log in Aleks is not finished yet. Will be “grading” that for completeness tomorrow.