Measurement 4.3
Developing Formulas
Developing Formulas
TASK: Exploring Spatial Relationships
TASK: Exploring Spatial Relationships
Watch
Watch
Your assignment, should you choose to accept it...
Your assignment, should you choose to accept it...
Visualize
Visualize
Take a minute and look at the 11 shapes below. Visualize and predict what relationships you think exist between and among these specific shapes (e.g., which shapes have the same area, same lengths, etc.). Also consider which shapes have no relationships and "do not belong".
Take a minute and look at the 11 shapes below. Visualize and predict what relationships you think exist between and among these specific shapes (e.g., which shapes have the same area, same lengths, etc.). Also consider which shapes have no relationships and "do not belong".
Jot down your predictions so you don't forget.
Jot down your predictions so you don't forget.
Verbalize
Verbalize
Share your predictions and rationale with a partner. Take note of the spatial language you are using to describe the relationships.
Share your predictions and rationale with a partner. Take note of the spatial language you are using to describe the relationships.
Verify
Verify
Download and print the "Rectangles, Parallelograms and Triangles" set of shapes for every 1-2 people. All participants will need scissors. Tape could also be helpful.
Download and print the "Rectangles, Parallelograms and Triangles" set of shapes for every 1-2 people. All participants will need scissors. Tape could also be helpful.
Cut out the shapes and test your predictions. Prove that those relationships you suspect to be true actually are. Check out those shapes you considered to have no relationships. Look for new relationships.
Cut out the shapes and test your predictions. Prove that those relationships you suspect to be true actually are. Check out those shapes you considered to have no relationships. Look for new relationships.
Want to Extend the Learning?
Want to Extend the Learning?
Download and print Set 2 to extend your exploration into spatial relationships that include trapezoids.
Download and print Set 2 to extend your exploration into spatial relationships that include trapezoids.
Discuss and Share
Discuss and Share
So, what did you find? Take photos of your "visual proofs", post them to our Collaborative Notebook and describe the relationships you see.
So, what did you find? Take photos of your "visual proofs", post them to our Collaborative Notebook and describe the relationships you see.
Reflect on the role that spatial reasoning, proportional reasoning and even algebraic reasoning play in this measurement task. Consider how a task like this could be used to help students construct indirect measurements (formulas) for these quadrilaterals. Share any insights you've had from this task or how you might strengthen it further.
Reflect on the role that spatial reasoning, proportional reasoning and even algebraic reasoning play in this measurement task. Consider how a task like this could be used to help students construct indirect measurements (formulas) for these quadrilaterals. Share any insights you've had from this task or how you might strengthen it further.
CONSOLIDATE and SHARE
CONSOLIDATE and SHARE
Spatial reasoning is at the heart of measurement. We hope that this has been evident throughout our entire online time together. What follows are three videos that try to pull things together and show some of the ways we can use spatial reasoning to build powerful formulas that underpin the indirect measurement of area and volume.
Spatial reasoning is at the heart of measurement. We hope that this has been evident throughout our entire online time together. What follows are three videos that try to pull things together and show some of the ways we can use spatial reasoning to build powerful formulas that underpin the indirect measurement of area and volume.
Big Idea 1: Measurement formulas flow out of spatial relationships between shapes and solids.
Big Idea 1: Measurement formulas flow out of spatial relationships between shapes and solids.
Discuss with your group how your exploration of shapes from the task above aligns with these observations.
Discuss with your group how your exploration of shapes from the task above aligns with these observations.
Big Idea 2: Base x Height is the underlying structure behind all array-based shapes and solids.
Big Idea 2: Base x Height is the underlying structure behind all array-based shapes and solids.
The move from spatial relationships to algebraic relationships is elegant and beautiful. Take time to explore and unpack each of these formulas to ensure that each one makes sense.
The move from spatial relationships to algebraic relationships is elegant and beautiful. Take time to explore and unpack each of these formulas to ensure that each one makes sense.
Share your own pedagogical practices around the development of area formulas and talk about what you've found to work and what ideas you'd like to incorporate.
Share your own pedagogical practices around the development of area formulas and talk about what you've found to work and what ideas you'd like to incorporate.
Big Idea 3: When we consider ideas 1 and 2 together, we can move seamlessly from area to volume, from triangles and quadrilaterals to prisms, cylinders and beyond.
Big Idea 3: When we consider ideas 1 and 2 together, we can move seamlessly from area to volume, from triangles and quadrilaterals to prisms, cylinders and beyond.
There's lots to consider in this video, both from a measurement and a geometry perspective. What are your thoughts? What do you need to think about further? What questions or insights does this provoke?
There's lots to consider in this video, both from a measurement and a geometry perspective. What are your thoughts? What do you need to think about further? What questions or insights does this provoke?
Share
Share
Any Aha moments or inspirations? Any questions or musings? Any further ideas you want to share?
Any Aha moments or inspirations? Any questions or musings? Any further ideas you want to share?
You know the drill. Post them to our Collaborative Notebook for others to consider and enjoy.
You know the drill. Post them to our Collaborative Notebook for others to consider and enjoy.
Looking for More?!?
Looking for More?!?
Do yourself a favour and check out these GREAT online visual representations created by Ontario educators!
Do yourself a favour and check out these GREAT online visual representations created by Ontario educators!
- Measurement Math Vines | Mike Jacobs, Chad Richard and Dan Allen, Durham Catholic DSB. I especially like the area of a trapezoid visualization! But they will all make you think and take you into areas we don't consider here... and in ways that you might not have considered!
- TapIntoTeenMinds | Kyle Pearce, Lambton Kent DSB. Three-part lessons and videos to help visualize and understand the formulas for the volume of Cones and Spheres, Triangular Prisms, and much more!
Are there any other Ontario educators you've appreciated who have done some cool spatial representations of measurement? Let us know through email.
Are there any other Ontario educators you've appreciated who have done some cool spatial representations of measurement? Let us know through email.
