Copy of Area and Volume
Published on May 25, 2016
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PRESENTATION OUTLINE
Area and Volume
Wendy Curtner and Dusty Burcham
Participants will make foldable to capture their thinking.
Photo by Arian Zwegers
That's Me!
I have more steps on my fitbit than I've had all summer break
I am a classroom teacher
I want to slide at the Ron Clark Academy!
I am an administrator
I am an instructional coach or curriculum coordinator
I bought one of Ron Clark's books or took a selfie with him
I am a classroom teacher
I want to slide at the Ron Clark Academy!
I am an administrator
I am an instructional coach or curriculum coordinator
I bought one of Ron Clark's books or took a selfie with him
Photo by evilpeacock
What is area?
Find your sole mate and tell them your definition of area.
Show Terminology Inventory Probe
Show Terminology Inventory Probe
Photo by Theen ...
What is area?
- Area is length times width
- Area is any formula that starts with A =
- Area is the measure of the surface
- Area is the two-dimensional space inside a region
- Area equals base times height
Students who say area is length times width tend to be more focused on a formula.
A = b x h is a more "unifying" idea because it can be generalized to all parallelograms (not just rectangles) and is useful for developing the area formulas for triangles and trapezoids. Furthermore, the same approach can be extended to three dimensions - volumes of pyramids, prisms, cylinders, and cones. (Van de Walle Teaching Student-Centered Mathematics p. 312)
Students can know a formula, but not understand it!
A = b x h is a more "unifying" idea because it can be generalized to all parallelograms (not just rectangles) and is useful for developing the area formulas for triangles and trapezoids. Furthermore, the same approach can be extended to three dimensions - volumes of pyramids, prisms, cylinders, and cones. (Van de Walle Teaching Student-Centered Mathematics p. 312)
Students can know a formula, but not understand it!
Photo by Theen ...
Common Misconception: Failure to conceptualize the meaning of height and base in geometric figures.
Students confuse a slanted side with height. Any side figure can be called a base, if you slide the figure into a room on the selected base, the height would be the height of the shortest door it could pass through without tipping.....the perpendicular distance to the base.
Do students connect length to the idea of base? Some people call base: length or width.
How will you surface those misconceptions or underdeveloped concepts? Are you using your IFD?
Do students connect length to the idea of base? Some people call base: length or width.
How will you surface those misconceptions or underdeveloped concepts? Are you using your IFD?
How do we develop formulas conceptually?
"When students develop formulas, they gain conceptual understanding of the ideas and relationships involved and there's less chance they will confuse the formulas later or forget them altogether."
Grades 5-8 Reference Materials handout
Have students go outside and use side walk chalk to draw shapes, write formulas, solve problems - they don't mind showing their work!
Grades 5-8 Reference Materials handout
Have students go outside and use side walk chalk to draw shapes, write formulas, solve problems - they don't mind showing their work!
Conceptual Development of Formulas
How is a parallelogram like a rectangle? How can it be changed into a rectangle?
Can you find a parallelogram that is related to your triangle? Might need to nudge by giving students 2 identical copies of the same triangle. Can you find more than 1 possible parallelogram? Will the areas of these parallelograms be equivalent?
Can you find a parallelogram that is related to your trapezoid? Might need to nudge by giving students 2 identical copies of the same trapezoid. Can you find more than 1 possible parallelogram? Will the areas of these parallelograms be equivalent?
Find the area of the trapezoid by using only the formulas for area of rectangles, area of parallelograms, and area of triangles.
Can you find a parallelogram that is related to your triangle? Might need to nudge by giving students 2 identical copies of the same triangle. Can you find more than 1 possible parallelogram? Will the areas of these parallelograms be equivalent?
Can you find a parallelogram that is related to your trapezoid? Might need to nudge by giving students 2 identical copies of the same trapezoid. Can you find more than 1 possible parallelogram? Will the areas of these parallelograms be equivalent?
Find the area of the trapezoid by using only the formulas for area of rectangles, area of parallelograms, and area of triangles.
Nets
Great interactive where students can change the length, width, and height to see the surface area of a rectangular prism
https://www.geogebra.org/m/713727
https://www.geogebra.org/m/713727
Should students learn special formulas for Surface Area?
Surface Area = area of surfaces! Are you introducing surface area using nets or interactives?
manipulatives with slant height strings from ETA
manipulatives with slant height strings from ETA
Untitled Slide
Common error: students confuse the meaning of height and base in their use of formulas!
V = Bh
height of base = h
height of 3d figure = h
b = base of shape
B = area of the base shape
Use color to connect the pictorial representation to the variables. For example: B = green, b = blue and h (height of base) = yellow; h (height of 3d figure) = pink
V = Bh
height of base = h
height of 3d figure = h
b = base of shape
B = area of the base shape
Use color to connect the pictorial representation to the variables. For example: B = green, b = blue and h (height of base) = yellow; h (height of 3d figure) = pink
Untitled Slide
A probe is a great way to uncover student thinking!
Uncovering Student Thinking in Mathematics Grades 6-12
30 Formative Assessment Probes for the Secondary Classroom
Uncovering Student Thinking in Mathematics Grades 6-12
30 Formative Assessment Probes for the Secondary Classroom
Popcorn Picker
Untitled Slide
Cone vs. Cylinder
Volume & Surface Area
Understanding the relationship between volume and surface area:
volume doesn't dictate surface area
cubelike prisms have less surface area than long, narrow prisms with the same volume
Fixed Volume Lesson -
Van de Walle
need cm cubes
graph paper
Blackline Master 37
volume doesn't dictate surface area
cubelike prisms have less surface area than long, narrow prisms with the same volume
Fixed Volume Lesson -
Van de Walle
need cm cubes
graph paper
Blackline Master 37
Untitled Slide
Great visual and explanation of why P is in the SA formula - http://www.virtualnerd.com/geometry/surface-area-volume-solid/prisms-cylind...
This video is long, but 4min 45sec - 7min 3sec is where she makes the connection.
This video is long, but 4min 45sec - 7min 3sec is where she makes the connection.
"The connectedness of mathematical ideas can hardly be better illustrated than with the connections of all of these formulas to the single concept of base times height."
John Van de Walle
Photo by peg
https://goo.gl/io2AVN
Photo by vernhart
