Understanding measurable attributes is key to developing deep understanding of area formulas.
View Progressions in Measurement
See the standards throughout the grades: http://www.corestandards.org/Math/Content/MD/
M1.a. Describe several measurable attributes of a single object. Include various quantities such as time, temperature, money, and angle measure.
M1.b. Compare the lengths of two objects indirectly by using a third object.
M2.a. Know basic unit conversions such as 1 yard = 3 feet, 1 foot = 12 inches, 10 mm = 1 cm, 100 cm = 1 meter, 1000 meters = 1 kilometer
M2.b. Choose appropriate measurement tools and use the tools to take measurements; understand that the same length (size) can have many different measurements (depending on the units used).
M3.a. Derive the formula for area of rectangles, parallelograms, triangles, and trapezoids and justify via partitioning arguments.
M3.c. Investigate whether the area of a parallelogram is determined by the lengths of its sides. Realize that all parallelograms and triangles with a certain height and base have the same area.
M3.d. Explore the distinction and relationship between perimeter and area, such as by fixing a perimeter and finding the range of areas possible or by fixing an area and finding the range of perimeters possible.
G2.g. Informally derive the formulas for circumference (by measuring diameter and circumference of circles) and area of a circle (by decomposing into a parallelogram-like shape).
Derive Pythagorean Theorem
G1.c. Given a set of coordinates as vertices, plot the points and identify what shape is represented by referencing lengths of segments, parallel and perpendicular properties of segments (be as specific as possible). Include examples where the sides of the shape are not parallel to the axes, and where the Pythagorean Theorem is used to determine the length.
When students derive the formulas, they are much more likely to remember them and apply them correctly.
Estimation of measures and the development of benchmarks support students to reason about tools for measuring, units of measurement, and conversions.
Slides for Chapter 11 in Mathematics for Elementary Teachers with Activities, 5th edition by Sybilla Beckmann.
Slides for Chapter 12 in Mathematics for Elementary Teachers with Activities, 5th edition by Sybilla Beckmann.
M1.a Blog post about measurable attributes.
M3.a Very short, wordless videos to demonstrate derivation of formula for area of polygons and circles
M3.a Online activity has draggable figures to help show how the formulas for area are derived MathOpenRef.com
M3.a Online activity to derive the area of a trapezoid Area of Trapezoid by Tap Into Teen Minds
G2.g Overview of a lesson on deriving the area of a circle Area of Circles by Fawn Nguyen
Blackline master of a circle cut into twelfths which can be used to informally derive the formula for area of a circle
Proof without words for the Pythagorean Theorem.