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!
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?
"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!
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.
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
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
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
"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