Ohio Resource Center
Lessons
Square Circles
Discipline
Mathematics
3, 4
Professional Commentary

This lesson features two creative twists on the standard lesson of having students measure several circles to discover that the ratio of the circumference to the diameter seems always to be a little more than 3.  This lesson starts with squares, so students can first identify a simpler constant ratio (4) of perimeter to length of a side, before moving to the more difficult case of the circle. The second good idea is to measure with a variety of units, so students can more readily see that the ratio of the measurements remains constant, not only across different sizes of figures, but even for the same figure with different measurements. An activity sheet, overheads, discussion questions, lesson extensions, suggestions for assessment, and prompts for teacher reflection are included. (sw)

Common Core State Standards for Mathematics
Measurement and Data
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
3.MD.D.8
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
Operations and Algebraic Thinking
Generate and analyze patterns.
4.OA.C.5
Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
Measurement and Data
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
4.MD.A.3
Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
Ohio Mathematics Academic Content Standards (2001)
Measurement Standard
Benchmarks (5–7)
C.
Identify appropriate tools and apply appropriate techniques for measuring angles, perimeter or circumference and area of triangles, quadrilaterals, circles and composite shapes, and surface area and volume of prisms and cylinders.
2.
Use strategies to develop formulas for finding circumference and area of circles, and to determine the area of sectors; e.g., 1/2 circle, 2/3 circle, 1/3 circle, 1/4 circle.
3.
Estimate perimeter or circumference and area for circles, triangles and quadrilaterals, and surface area and volume for prisms and cylinders by: a. estimating lengths using string or links, areas using tiles or grid, and volumes using cubes; and b. measuring attributes (diameter, side lengths, or heights) and using established formulas for circles, triangles, rectangles, parallelograms and rectangular prisms.
Principles and Standards for School Mathematics
Measurement Standard
Understand measurable attributes of objects and the units, systems, and processes of measurement
Expectations (6–8)
understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.
Apply appropriate techniques, tools, and formulas to determine measurements.
Expectations (6–8)
understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.
select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision;
develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes;