2010-2011 Workshop Series for YISD Teachers of 6 th Grade Math Conceptual Understanding & Mathematical Thinking

1. 2010-2011 Workshop Series for YISD Teachers of 6th Grade Math Conceptual Understanding & Mathematical ThinkingWorkshop 2 (Dec 9) Kien Lim Dept. of Mathematical Sciences, UTEP

2. Goals: • Strengthen our mathematical knowledge for teaching areas and perimeters • Get a sense of what it takes to teach in a manner to foster conceptual understanding and mathematical thinking • Increase our commitment to help our students develop mathematical thinking

3. Plan of Activities: • Discuss TAKS items on areas and perimeters • Solve the “Which Artwork is Larger” problem • Discuss key ideas about measurements • Consider other problems for classroom use

4. 2009 7.9A #39 Is this problem easy or difficult for our students? Why?

5. 2010 Released Item Objective 6, SE 6.12A #2 Is this problem easy or difficult for our students? Why? Which equation could be used to determine A, the area of Joshua’s house only?

6. What Do You Think? • Do our students often get confused between area and perimeter? If so, why? • Do our students really understand the meaning of area and the meaning of perimeter? • How can we help students understand the difference between areas and perimeters conceptually?

7. “Which Artwork is Larger” Activity You have two pieces of artwork, which piece do you think is larger? Why? Solving Time Q1

8. Strategies from Participants • Overlap & shade common areas & compare leftovers (2) • Split into rectangles, add their areas (3) • Cover space with 1 inch tiles (2) • Place 1 inch tiles along the sides and used them to gauge number of tiles per row (1) • Compute area of large rectangle and subtract the missing region (1) • Measure dimensions of figures with a ruler (1)

9. Key Ideas About Measurement • What are we comparing? Largeness • What do we mean by “largeness”? Amount of material needed to “cover” it or to “surround” it • Can we be more precise? Area or perimeter • How do we quantify area? perimeter?

10. “Which Artwork is Larger” Activity Solving Time If you want to cover each figure with shiny stickers (1 inch by 1 inch), which figure do you think needs more stickers? Q3

11. “Which Artwork is Larger” Activity 8 2 2 3 3 3 3 3 3 1 4 1 1 1 5 9 Solving Time 4 8 8 8 If you want to surround each artwork with an expensive golden string around its border, which figure do you think needs a longer piece of string? Q2

12. Thinking Questions from Students • Why do we use square inches (or square meters) to measure area? Why not inches (or meters)? How would you respond?

13. Thinking Questions from Students • Why do we use square inches (or square meters) to measure area? Why not inches (or meters)? 1 in. 4 in. Consider the area of a 3 in. by 4 in. rectangle. Area = Length x Width = 3 in. x 4 in. 1 in2 1 in2 1 in2 1 in2 1 in2 1 in2 1 in2 1 in2 1 in2 1 in2 1 in2 1 in2 3 in. = 12 in2 • What does 12 in2 really mean? 12 units of 1 in2 1 in2 1 in. It is important that students “see” 12 in2 as … iterating the base-unit of 1 in2twelve times.

14. Key Ideas About Measurement • First, help students have a sense of the attribute that we are measuring (concrete experience is critical) • Then use pictorial representation to help students see the BASE-UNIT. • Make sure students understand the value of a measurable attribute consists of a number and its unit of measurement.

15. Key Ideas About Measurement An Object An Attribute of the Object Non-Measurable Attribute Measurable Attribute(Quantity) Qualitative Description Value of the Attribute Number Unit of Measurement

16. Thinking Questions from Students • We can multiply 3 in. by 4 in. to get 12 in2. Can we multiply 3 in. by 4 cm to get 12 units of something? 1 cm 4 cm How would you respond? Area = Length x Width 3 in. = 3 in x 4 cm = 3 x 1 in x 4 x 1 cm = 3 x 4 x 1 in x 1 cm = 12 x 1 in-cm = 12 in-cm 1 in. 1 in-cm 1 in-cm 1 in-cm 1 in-cm 1 in-cm 1 in-cm 1 in-cm 1 in-cm 1 in-cm 1 in-cm 1 in-cm 1 in-cm 1 in-cm Is it correct to say 1 inch2 is the same as 1 inch-inch?

17. Create Your Own Artwork • You have a piece of golden string with 28 inches to form the border of an artwork to be filled with shiny stickers. • Create a single-piece artwork (connected) that contains the least number of stickers. How do you know you there won’t be another artwork that can contain fewer or same number stickers than yours? • Create an artwork that contains the most number of stickers. How do you know you there won’t be another artwork that can contain more stickers than yours? Solving Time Q4

18. Create Your Own Artwork You have a rectangle that has a length of 3/4 foot and a width of 2/3 foot to be bordered by a piece of golden string and be filled with shiny stickers. • How long is the golden string? • How many stickers do you need? • What is the perimeter (measured in feet) of the rectangle? • What is the area (measured in feet2) of the rectangle? • Susan was surprised to find that he area is only ½ foot2 because she expected the area to be larger than 3/4 and larger than 2/3 since multiplying two numbers makes the product larger. Can you help Susan understand why the area is only ½ foot2? Solving Time Q5

19. Create Your Own Artwork You have a rectangle that has a length of 3/4 foot and a width of 2/3 foot to be bordered by a piece of golden string and be filled with shiny stickers. Area = Length x Width = 3/4 ft x 2/3 ft 2/3 foot = 6/12 ft2 = 1/2 ft2 Area = Length x Width 3/4 foot Q5

20. Create Your Own Artwork You have a rectangle that has a length of 3/4 foot and a width of 2/3 foot to be bordered by a piece of golden string and be filled with shiny stickers. 2/3 foot = 8 inches Area = Length x Width = 3/4 ft x 2/3 ft = 6/12 ft2 = 1/2 ft2 Area = Length x Width 3/4 foot = 9 inches = 9 in. x 8 in. = 72 in2 Is 1/2 ft2 equal to 72 in2? 1 ft2 = 1 ft x 1 ft = 12 in x 12 in = 144 in2 Q5

21. Jorge’s Square vs. Melinda’s Square Jorge has a square. The length of each side is 1 foot. What is the area (measured in sq. foot) of the rectangle? What is the area (measured in sq. inch) of the rectangle? How many stickers can fit Jorge’s square? If we frame Jorge’s square with a golden string, how long is the string? Melinda has a square. The length of each side is ½ foot. How many of Melinda’s squares are needed to cover Jorge’s square? How many stickers can fit Melinda’s square? If we frame Melinda’s square with a golden string, how long is the string? Q6 & Q7

22. Classroom Use: • Are these problems appropriate for use in your classroom? • How can we tailor these problems to benefit your students? • What learning objectives do you hope to achieve from using these problems?

23. What have we learned? • Different ways to find the area and perimeter. • Make it concrete. • Differentiate the area and perimeter. • Relate inches and square-inches. • Be precise, largeness in terms of area or perimeter. • Quantify area. • Let the students think, explore, etc. • We can have in-cm, worker-days, inch-inch for units • Area of a circle > area of a square (same perimeter)

24. Thank You