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#WithMathICan Understand Complex Shapes!
1 Review + Hook
To get students' cognitive juices flowing, have them work in small collaborative groups to create a mind map of everything they know about shapes in Coggle. You might encourage them to include branches such as "properties," "types," or "measurements" - but try to stay out of the way as much as possible. After about ten minutes, have each group share their mind map with the other groups in class and compare content. Point out to your students that (most likely) no one group had exactly the same map as any other group - we have to "zoom out" and look at others' ideas to get the whole picture. This is an important component of growth mindset!
Using SketchUp, show students how shapes can transform from 2-dimensional to 3-dimensional by using the "pull" tool. Transform a square into a cube, a triangle into a triangular prism, a circle into a cylinder, etc. As you model this, ask students what they notice about the measurements of the shape - what parts do they have in the 2-D shape (length and width) vs. the 3-D shape (adds height)? This should lead you to the concept of volume and how it relates to area, and the relationship between the area of a 2-D shape and the surface area of a 3-D shape.
Once students are clear on the concepts of volume and surface area, go over the formulas for volume and surface area in different 3-D shapes. Your students might find it helpful to create notecards (digital or paper) for their formulas.
If you have extra time or students who need an extra challenge, you might first see if your students can figure out what the formulas are, using what they already know about 2-D and 3-D shapes. Even if your students original guesses are incorrect, encourage them to explain their process and see where they went off track. Breaking down the problem and being flexible is how we get to solutions that work!
In SketchUp, your students can build complex 3D figures with specific measurements. The video tutorials linked here will show students how to build a doghouse shape. You might ask your students what other buildings or objects they can think of which have this shape (a house, a milk carton, etc.). Emphasize that students can build any composite figure they'd like (bonus points for creating something that looks like a real everyday object!).
The SketchUp software can be a bit overwhelming to some students, because there are so many options and tools to use. Reassure your students that it's OK to stick with a few tools they feel most comfortable with, and praise those students that take a risk with different tools. Make sure that students use precise measurements when making their figures (telling SketchUp exactly how long, deep, and tall they'd like their figures to be).
Last, have students create an "answer key" on a piece of paper or an index card that shows the smaller pieces that make up their shape (e.g. a triangular pyramid and a cube, for the doghouse), as well as the calculations for surface area and volume of their shape.
Next, have your students share their SketchUp models with their classmates. You might do this by having students rotate around to other computers, or by sharing the SketchUp files to a cloud service like Dropbox or Google Drive. For each of their classmates' composite figures, have your students identify the smaller pieces that make up the shape, then calculate the surface area and volume (this is best done on paper).
Your class can create an "answer key" by snapping a photo of their own index cards and posting them to a shared bulletin board like Padlet. Students can compare their calculations to the answers, and make corrections or ask for assistance as needed. Encourage your students to work together to identify and fix mistakes if they notice that an answer does not match the correct solution. You might designate a few "expert" students that can assist classmates if they get stuck.
To wrap up what the activity, do a live Q&A with your students in a backchannel tool like TodaysMeet. This can be a great "exit ticket" activity to check for deep understanding in many math activities! Here are some ideas for questions you might ask:
- How are area and volume related?
- What does the word "composite" mean?
- Give some examples of composite shapes we see in everyday life.
- Why is it important to be able to figure out the surface area of something?
- Why is it important to be able to figure out the volume of something?
- If you ever get stuck solving a problem about volume or surface area, what could you think about to get your brain going?
Key Standards Supported
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.