Connecting Two- and Three-Dimensional Figures Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success Core Math July 25, 2016 8:30 – 10:15

Learning Intention We are learning to help children make connections between two- and three-dimensional figures. We will be successful when we can… Describe the development of 3-D ideas in the CCSSM. Construct a net for a 3-dimensional figure, and reconstruct a 3-dimensional figure from a net.

Build and Describe What makes this challenging?

What do you notice? What are you wondering? What are we noticing? It appears to be a cube because 3 faces all look the same. The front face is a square. The other two faces we see are squares. What are we wondering? Do the back faces have the same shape? Is this a solid cube? Are there any missing cubes? Is there another representation of this shape that can help us? How can we be sure? Is there another representation of this shape that can help us make sense of what we’re wondering?

How might this representation help you think about the figure? What would students need to understand to produce a representation like this?

Find the Block Connecting 2D and 3D Use the wooden block that matches the “footprints”. What are young children learning about the relationship between 3D figures and 2D representations?

Keeping it Simple What are young children learning about the relationship between 3D figures and 2D representations? Students see a 3-D figure as a whole. Students see the attributes of the 3-D figures. Students see the relationships between the attributes.

CCSSM Standards Geometry: Solve real-world and mathematical problems involving area, surface area, and volume. 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

PRR: 2-D and 3-D in the CCSSM Study the Critical Focus Areas in the CCSSM for the following grade levels. Kindergarten  p. 9 #2 Grade 1  p. 13 #4 Grade 2  p. 17 #4 Grade 3  p. 21 #4 Talk about the development of 3-D understanding from Kindergarten through Grade 3. What do you notice? What is the message the authors of the CCSSM are sending?

What is a “Net” (for a 3-Dimensional Figure)? Answer this question (silently!) on your own. Add a drawing to help clarify your explanation. Compare your answer with those of others at your table. Come to consensus on an answer to report out to the whole group. Thinking back on our quick image of the cube. A net is a 2-dimensional figure that can be folded into a 3-dimensional object.

Essential Understandings of Geometry Classification of Shapes Features or properties of geometric shapes can be analyzed and described to define and refine classification schemes with growing precision. Spatial Visualization Spatial relationships and spatial structuring involves developing, attending to, and learning how to work with imagery, as well as to specify locations. Geometry is the branch of mathematics that addresses spatial sense and geometric reasoning. Take out and study. What do you notice? Try to make sense in these. Talk in groups. Did any of this come up in your posters? Transformations Transformation involves working with geometric phenomena in ways that build on spatial intuition by explaining what does and does not change when moving and altering the objects and the space that they occupy.

Nets for a cube Think about the attributes of a cube. What would a net look like for a cube? Use ¾ inch grid paper. Work with a partner to draw/build as many nets as you can for a cube. How do you know you have found all possible nets? “Cube” means closed cube, so every net should have exactly 6 squares. The intent of the second question is to have participants question whether two nets that are rotations or reflections of each other are truly different. The last question is challenging, and we may not reach a complete answer (so maybe we shouldn’t ask it?) but gets to the necessary structure of a net for a cube—or, at first, the structure of a 2-D shape formed by 6 squares joined along a common side. What did you have to understand about cubes to make the net?

How Many Different Nets for A Cube? Date: 11/15/2001 at 07:15:47 From: Ashuk Braham Subject: Cubes My teacher says that there are 11 combinations to make a cube without reversing them, but I can only find 6. Please tell me how many there really are, as I'm puzzled. Thank you.

Gr. 6 – 7 Task Folding A Cube Task: Part 1 Complete the task. Take turns sharing your explanations with your table group. Where do you anticipate students struggling with this task? What would you expect for students explanations to include?

Gr. 6 – 7 Task Toblerone Candy Bar Box Complete the task. Take turns sharing your net. Where do you anticipate students struggling with this task? What would you expect for students explanations to include?

Create nets for rectangular prisms Work as a trio. Build nets for rectangular prisms with the following dimensions. Make 2 of each. Cut them out, fold, and check. 2 x 3 x 4 1 x 3 x 4 ½ x 3 x 4 What discussions did you have as you worked to make the nets?

Learning Intention We are learning to help children make connections between two- and three-dimensional figures. We will be successful when we can… Describe the development of 3-dimensional ideas in the CCSSM. Construct a net for a 3-dimensional figure, and reconstruct a 3-dimensional figure from a net.

Core Mathematics Partnership Project Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.