Measuring Circles Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success Core Math July 26, 2016 10:30 – 2:00 PM

Learning Intention We are learning to help children understand the geometry and measurement of circles We will be successful when we can Provide two definitions of π; Explain why these two definitions agree; Explain the proportional relationships that hold between the radius, diameter, circumference, and area of a circle.

Measuring Round Objects Measure the diameter and the circumference of the selection of round objects at your table. Record your answers on the white board at the front of the room. What pattern(s) do you see in the class data?

What is π? Answer this question (silently!) on your own Compare your answer with those of others at your table. Come to consensus on an answer to report out to the whole group.

CCSSM Circle Measurement Standards Geometric measurement: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Relating to the Standards Which part(s) of 7.G.4 are addressed by the measurement activity you have just completed? Goal: have participants realize that, by defining π as the ratio of the diameter of a circle to its circumference, they “know” the formula for the circumference of a circle—either as C = πD, or C = 2πr.

Warm-up How many square feet in a square yard? Goal: have participants gain some preliminary understanding that area scales as the square of the linear dimension. (1 yard is 3 times as long as a foot, so 1 square yard is 3 x 3 = 9 times as large as 1 square foot.) Reinforce the idea immediately after, by asking “How many square inches in a square foot?” and “How many square centimeters in a square meter?”

Areas of Round Objects Choose one of your round objects—one that will fit easily on a sheet of grid paper. Draw around your object, to get a circle on the grid paper. Find the diameter and the (approximate) area of your circle, and record your results at the front of the class. What pattern(s) do you see in the class data? Goal: have participants realize that, by defining π as the ratio of the diameter of a circle to its circumference, they “know” the formula for the circumference of a circle—either as C = πD, or C = 2πr.

Areas of Round Objects Our data seems to show that the area of a circle is proportional to its radius (or, equivalently, its diameter). Explain why this result is reasonable. Goal: have participants realize that, by defining π as the ratio of the diameter of a circle to its circumference, they “know” the formula for the circumference of a circle—either as C = πD, or C = 2πr.

Estimating the Area of a Circle What can you conclude from this picture about the area of a circle of radius r? Explain why this next picture shows that the area of the circle must be between 2π2 and 4π2.

Relating Circumference and Area We now know (or believe): The circumference of a circle is proportional to its radius (in fact, C = 2πr); The area of a circle is proportional to the square of the radius. What is the constant of proportionality in this second relationship? That circumference is proportional to radius (or diameter), and that area is proportional to the square of the radius, should be intuitively obvious”; the interesting mathematics is that the two proportionality constants are so closely related.

Relating Circumference and Area Cut a paper plate along two of its diameters, to obtain 4 equal wedges Re-arrange the wedges to form an (approximate!) parallelogram Repeat with other plates, cutting them into 8 and 16 equal wedges.

Relating Circumference and Area What are the length and height of your approximate parallelograms (in terms of the circumference and radius of the original circle)? What is the area of your approximate parallelograms (in terms of the area of the original circle)? What do you conclude about the constant of proportionality between the area of a circle and its circumference?

Relating Circumference and Area Where do you see the circumference and radius of the circle in the approximate parallelogram? Picture credit: Wikipedia

CCSSM Circle Measurement Standards Geometric measurement: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Misconceptions What are some misconceptions students have as they begin the study of circle measurement? What instructional strategies are effective in helping students to overcome those misconceptions?

Recall: A Classic Problem Old McDonald had a farm, and on that farm he wanted to build a sheep pen in the shape of a rectangle. He wanted to make the pen as large as possible, but he only had 100 feet of fencing wire. What dimensions should he choose for the pen? What was the largest rectangular pen Old McDonald could build? Would a circular pen be larger or smaller than the largest rectangular pen? (Reason and estimate first, then calculate.)

Volumes of Cylinders and Cones We learned yesterday how to calculate the volume of a prism and a pyramid So … how can we find the volume of a cylinder? Of a cone?

CCSSM Volume Measurement Standards Geometry: Solve real-world and mathematical problems involving volume of cylinders, cones and spheres. 8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Note the exact parallel with 3.MD.5 and 3.MD.6 for area!

Learning Intention We are learning to help children understand the geometry and measurement of circles We will be successful when we can Provide two definitions of π; Explain why these two definitions agree; Explain the proportional relationships that hold between the radius, diameter, circumference, and area of a circle.

Desmos: Polygraph Navigate to student.desmos.com (you do not need to sign in). Play one of the following games using the class codes noted: Polygraph: Basic Quadrilaterals weug Polygraph: Advanced Quadrilaterals 6yas Polygraph: Hexagons kvz8

Core Mathematics Partnership Project Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.