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The Cube Challenge - Discovering Three-Dimensional Congruence
The Cube Challenge - Discovering Three-Dimensional Congruence
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Content
Content
Expectations:
Expectations:
E1.3 identify congruent lengths, angles, and faces of three-dimensional objects by mentally and physically matching them, and determine if the objects are congruent
Key Concepts
Congruence is a relationship between three-dimensional objects that have the same shape and the same size. Congruent three-dimensional shapes match every face exactly, in the exact same position.
Checking for congruence is closely related to measurement. Side lengths and angles can be directly compared by matching them, one against the other. They can also be measured.
Two objects that are not congruent can still have specific elements that are congruent. For example, two objects might have a face that is congruent (i.e., the face is the same size and shape), but if the other faces are different in any way (e.g., the faces have different angles or side lengths), then the two objects are not congruent. Likewise, even if all faces are congruent but they are in a different arrangement, the two objects would not be congruent because they would not be the exact same shape.
The skill of visualizing congruent objects – mentally manipulating and matching objects to predict congruence – can be developed through hands-on experience.
Learning Goal
Learning Goal
We want students to understand...
We want students to understand...
Spatial visualization
3D congruence
Composing and decomposing 3D objects
Success Criteria
Success Criteria
I can use information to visualize building a 3D structure.
I can build and compare 3D structures.
I can decide if a 3D structure is congruent to another by matching them in my mind or by turning, rotating and flipping them.
I can solve problems without giving up!
Pedagogy
Pedagogy
Number Talk Resources
Number Talk Resources
Number Talks to Support B1.1 read, represent, compose, and decompose whole numbers up to and including 1000, using a variety of tools and strategies, and describe various ways they are used in everyday life
Fosnot: Minilessons for Extending Addition and Subtraction
Page 11 - 17 (Around the Circle)
Page 43 -45 (Partial Sums, Splitting, Expanded Notation
Page 43 -45 (Partial Sums, Splitting, Expanded Notation
Sherry Parrish - Number Talks
Page 198 - 200 (Breaking Each Number into Its Place Value)
Minds on
Minds on
“Build it In Your Mind” (5-10 minutes) - Supporting Spatial Visualization
(Adapted from “Taking Shape” - Page 160)
Tell students they are going to use only their imaginations to build a 3D cube structure. Explain that you are going to read the instructions and their task is to build the structure in their minds and then tell you about it. Invite them to close their eyes if this might help them to picture what you are saying. Read the building instructions slow, repeating as necessary, so that all children can follow:
Imagine that you have three red cubes. Take the three red cubes and snap them together to make a tower. Imagine the tower standing up like a building.
Now imagine you have one blue cube. Take the blue cube and snap it on the top of the red tower.
You should now have a red tower with a blue cube on the top of it.
This final sentence describes the structure explicitly for students. Providing this much information will be helpful for the first one or two challenges you present, as it allows children to be successful as they gain experience with the activity. After the first few challenges, you can be less explicit in describing the structure.
Once you have read the instructions, present students with an image of four cube structures, only one of which is the one you described. You can build the actual structures ahead of time and present those, or show photos of the structures.
Have students identify which structure, if any, mathes what they built in their minds.
Invite students to share their visualization process and any challenges that arose or any strategies that help them through the process.
Repeat with increasingly more difficult challenges (limit to 2 or 3).
Key Questions:
How were you able to remember the shape of the structure you built?
What sorts of strategies did you use to keep track of all the information?
Did you close your eyes to help “picture” the structure? Do you think this strategy helps? Why?
Did you use your hands to act out building the structure? Do you think this strategy helps? Why?
How did you know it wasn’t this one [pointing to the other options]?
Were there any parts of the structure that were easy to picture in your mind? Were there any parts of building the structure that were more difficult to imagine?
SEL connection: What strategies did you use to help you persevere when the structure got more difficult?
Action!
Action!
Materials Required: 1 Bag of snap cubes for each student
Discovering Three-Dimensional Congruence
(Adapted from “Taking Shape”: The Cube Challenge - Discovering Three-Dimensional Congruence, Page 149)
Hand out individual bags of cubes to each student. Tell students to take 3 cubes from their bag and make an object with those cubes. Ensure that they understand that each cube must line up with the adjacent cubes so all the edges are flush - cubes should not rest on one another in a twisted fashion. When students have each produced one object, invite children to place their objects in front of them.
Challenge students to take out 3 more cubes and make a second different object. Direct children to leave the second object beside the first.
Ask students whether they are able to create any other objects with 3 cubes that are different from their first two. At this point, some children might come to realize that it is not possible to build a third unique object with 3 cubes. Others might build a third and fourth object and insist that they are different because of how they can be oriented.
Invite students to hold up their various objects. As a class, discuss how some can be seen as the same (congruent) and some can be seen as different (incongruent). Eventually, children will come to see that only two unique objects exist and that while some objects may look different, when they are flipped or rotated, we can see that they are identical in structure.
Introduce children to the next challenge by inviting them to see how many different structures they can make using 4 cubes. Before children begin to build, ask them to estimate how many different objects they think they can make using only 4 cubes. Provide students with lots of time to construct their objects. Students might flip, rotate, and overlay one on top of the other to test or prove whether two structures are the same or unique.
Key Questions:
How do you know you have all the combinations?
Are these two objects the same or different? How do you know?
How do you know that these two objects are different, even though they might look the same?
Is there anything you can do to this one to make it look and go the same way as this one?
Is there a way of building the objects so that they can’t lie flat on the ground and they have cubes sticking out another direction?
Once children have had time to make at least 5 different structures, have students share their structures with the group. Challenge the class the think and talk about whether two people have the same or different structures, and have children work together to determine whehter all of the possible combinations have been made. Encourage students to view and compare structures from different perspectives (turning, rotating, and flipping). Explore the idea that 3D maintains congruence (sameness), despite differences in appearance when viewed from different perspectives or orientations.
The next challenge is to construct unique structures using 5 cubes. This time, the maximum number of unique combinations is 29. Follow the same lesson sequence as above. As they did in earlier challenges, have the students estimate how many different configurations they think are possible and do not tell them the number of possible solutions at the outset.
Have students place their structures in front of them.
Consolidation Ideas
Consolidation Ideas
As a class, work together to eliminate any duplicate structures so that only unique 3D objects remain in the centre of the circle. Because they are on display, the students are encouraged to make visual comparisons. They can see whether the objects are the same or different and engage in spatial visualization and mental rotation.
Use prompts to promote class discussion:
Are these two objects the same or different? How do you know?
How do you know that these two pibects are different, even though they might look the same?
Is there anything you can do to make it look and go the same way as this one? How could you describe one of your structures to someone else?
How could you describe the easiest way to build your structure to someone else?
Was your estimate accurate?
Do the shapes of any of the structures remind you of anything?
How might you describe one of your structures to someone else?
Have students share their experiences with the task.
SEL connections:
What strategies did you use when it became difficult or frustrating to find unique structures?
Did you encourage others around you who were having difficulty?
How did you persevere when you felt like you were stuck?
How did your mistakes help you in today’s tasks?
How did you react when you made a mistake?
What happened to your thinking when you saw someone else try something that didn’t work?
How has your thinking changed about mistakes?
Independent Task / Assessment Opportunities
Independent Task / Assessment Opportunities
Asking children to compare structures and determine whether they are congruent provides ample opportunities to listen to and observe student understanding of 3D congruence. While some children might use gestures and the actual objects to demonstrate congruence, other children might rely more heavily on a verbal explanation.
KnowledgeHook: Grade 3 Congruent 3D shapes
SEL Self Assessments and Teacher Rubric
Extension Activities:
Extension Activities:
Super Source: “One of A Kind” (Adapted from Grade ¾ Super Source: Snap Cubes - Page 47)
“Plans and Structures” (Adapted from Grade ¾ Super Source: Snap Cubes - Page 54)
Technology
Technology
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