Ant’s Picnic Imagine that you and the other members of your group are a team of picnic basket ants, and you have just reached a picnic basket filled with supplies and food. Suppose that you are small, but you walk on long, narrow feet that are the size and shape of paper clips!

Ant’s Picnic We’ve given you a set of picnic items. Check out the items by stepping around the items’ edges with your unusual feet. Record your work on the sheet. Give participants ~5 minutes to complete roughly 3 items or as many as they can do in that time.

By participating in the Ants’ Picnic, have you been exploring a math idea called perimeter. Individually, jot down what you think perimeter is on the backside of your sheet. Compare your answer with those of others at your table. Come to consensus on an answer to report out to the whole group.

Measuring Perimeter Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success Core Math July 27, 2016 8:00 – 10:30 AM

Misconceptions What are some misconceptions students have as they begin the study of perimeter? Confuse the attributes of area and perimeter Misapply the formula Don’t see the relationship between the numbers and the sides

Learning Intention We are learning to help children understand the concept of perimeter. We will be successful when we can Explain measurement of perimeter as an example of the general measurement process; Distinguish between perimeter and area; Find perimeters of simple geometric shapes; Explain the extent to which the moving and combining principles apply to perimeter.

CCSSM Perimeter Measurement Standards Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 3.MD.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.

Difficulties with Perimeter What is the perimeter of this figure? 30 ft 10 ft Eduardo is raising goats. He has to protect the tree in the middle of his garden from them nibbling. He needs to put a fence around the garden to keep in the goats and to protect the tree. What is the perimeter of his garden? 30 ft 10 ft 10 ft 30 ft 10 ft 30 ft

Difficulties with Perimeter Eduardo is raising goats. He has to protect the tree in the middle of his garden from them nibbling. He needs to put a fence around the garden to keep in the goats and to protect the tree. How many feet of fence will he need? Eduardo is raising goats. He digs a square watering hole in the middle of his garden for them to drink from. He needs to put a fence around the garden to keep in the goats. How many feet of fence will he need?

What is the Perimeter of This Figure?

CCSSM Perimeter Measurement Standards Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. 3.MD.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.

Difficulties with Perimeter What are the perimeters of the three hexagon trains below? What unit did you choose to measure these perimeters? Do the moving and combining principles work for perimeter?

Properties of Area “Moving principle”: the area of a shape is not changed under a rigid motion. (Congruent shapes have equal areas.) “Combining principle”: the total area of two (or more) non-overlapping shapes is equal to the sum of their areas.

Properties of Perimeter Do the Moving and Combining principles work for perimeter? Moving property does; combining property does not. (Participants should give examples to show why not.)

Grades 3-5 Student Work: Planning a Garden Sam is planning a garden. He wants to build a fence around the garden and he needs to buy mulch to cover the garden to control weeds. It will have the shape shown below in the picture. You can see what he has measured so far. Be sure to include the units in your answer and to show your work. What is the perimeter of the garden? What is the area of the garden? How much fence does Sam have to buy?

Area and Perimeter Use unifix cubes or tiles to make several rectangles with area 16 square units. (Make as many differently-shaped rectangles as you can.) Record each of your rectangles on grid paper. What are the perimeters of your rectangles? Is 12 sq. units the best number here? It intentionally does not allow for squares—is that what we want in this activity?

Area and Perimeter What is the largest possible perimeter for a rectangle with area 16 square units? What is the smallest possible perimeter? This question worked nicely, because 16 is a square number! What is the smallest possible perimeter for a rectangle with area 12 square units?

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? In your groups, solve this problem. First discuss: what does “as large as possible” mean in the context of this problem?

A Slightly Less Classic Problem Old McDonald realizes he might be able to build a bigger sheep pen if he builds one side along the wall of his large barn (so that he doesn’t need fence wire on that side). What dimensions should he choose for his pen if he does this? How much extra area does the largest pen have with this strategy? Optional: we will do this version of the problem only if time.

What is “Surface Area”? 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.

Surface Area of a Rectangular Prism What is the volume of a 2 x 3 x 4 rectangular prism? (How do you know?) What is the surface area of this prism? Note: Participants could build the 2 x 4 x 5 prisms with unifix cubes—or draw nets.

Properties of Surface Area Do the Moving and Combining principles work for surface area? Discuss: what similarities do you see between the attributes of perimeter and surface area? How do they compare to length, area and volume? Moving property does; combining property does not. (Participants should give examples to show why not.)

Volume and Surface Area Use unifix cubes to make several rectangular prisms with volume 32 square units. (Make as many differently-shaped prisms as you can.) Record (sketch) each of your prisms. What are the surface areas of your prisms?

Learning Intention We are learning to help children understand the concept of perimeter. We will be successful when we can Explain measurement of perimeter as an example of the general measurement process; Distinguish between perimeter and area; Find perimeters of simple geometric shapes; Explain the extent to which the moving and combining principles apply to perimeter.

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.