2010-2011 Workshop Series for 2010-2011 Workshop Series for YISD Teachers of 6YISD Teachers of 6thth Grade Math Grade Math

Conceptual Understanding Conceptual Understanding & Mathematical Thinking& Mathematical Thinking

Workshop 2 (Dec 9)Workshop 2 (Dec 9)

Kien LimKien LimDept. of Mathematical Sciences, UTEPDept. of Mathematical Sciences, UTEP

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

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

2009 7.9A #39

Is this problem easy or difficult for our students?

Why?

2010 Released Item Objective 6, SE 6.12A #2

Which equation could be used to determine A, the area of Joshua’s house only?

Is this problem easy or difficult for our students?

Why?

• 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?

What Do You Think?

“Which Artwork is Larger” Activity

You have two pieces of artwork, which piece do you think is larger?

Why?Q1

Solving Time

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)

• What are we comparing?

Key Ideas About Measurement

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?

“Which Artwork is Larger” Activity

If you want to cover each figure with shiny stickers (1 inch by 1 inch), which figure do you think needs more stickers?

Q3

Solving Time

“Which Artwork is Larger” Activity

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

1

3

8

9

8

1

13

4

4

51

88

3333 22

Solving Time

• Why do we use square inches (or square meters) to measure area? Why not inches (or meters)?

Thinking Questions from Students

How would you respond?

• Why do we use square inches (or square meters) to measure area? Why not inches (or meters)?

Thinking Questions from Students

Consider the area of a 3 in. by 4 in. rectangle.3 in.

4 in.• What does 12 in2 really mean?

Area = Length x Width= 3 in. x 4 in.= 12 in2

12 units of 1 in2 1 in.

1 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 1 in2

It is important that students “see” 12 in2 asiterating the base-unit of 1 in2 twelve times.

…

• 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.

Key Ideas About Measurement

An Object

An Attribute of the Object

Measurable Attribute(Quantity)

Non-Measurable Attribute

Value of the Attribute Qualitative Description

Number Unit of Measurement

Key Ideas About Measurement

Thinking Questions from Students

How would you respond?

• We can multiply 3 in. by 4 in. to get 12 in2.

3 in.

4 cm

1 cm1 in.

Area = Length x Width= 3 in x 4 cm

= 12 in-cm1 in-cm

1 in-cm1 in-cm1 in-cm1 in-cm1 in-cm1 in-cm1 in-cm1 in-cm1 in-cm1 in-cm1 in-cm1 in-cm

= 3 x 1 in x 4 x 1 cm = 3 x 4 x 1 in x 1 cm = 12 x 1 in-cm

Is it correct to say 1 inch2 is the same as 1 inch-inch?

Can we multiply 3 in. by 4 cm to get 12 units of something?

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.

a.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?

b.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?

Q4

Solving Time

Create Your Own ArtworkYou 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.

Q5

Solving Timea. How long is the golden string?b. How many stickers do you need?c. What is the perimeter (measured in feet) of the

rectangle?d. What is the area (measured in feet2) of the rectangle?e. 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?

Create Your Own ArtworkYou 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.

Q53/4 foot

Area = Length x Width

Area = Length x Width= 3/4 ft x 2/3 ft= 6/12 ft2

= 1/2 ft2

Create Your Own ArtworkYou 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.

Q53/4 foot = 9 inches

Area = Length x Width= 9 in. x 8 in.= 72 in2

Area = Length x Width= 3/4 ft x 2/3 ft= 6/12 ft2

= 1/2 ft2

Is 1/2 ft2 equal to 72 in2?

1 ft2 = 1 ft x 1 ft = 12 in x 12 in = 144 in2

Jorge’s Square vs. Melinda’s Square

Jorge has a square. The length of each side is 1 foot.a.What is the area (measured in sq. foot) of the rectangle?b.What is the area (measured in sq. inch) of the rectangle?c.How many stickers can fit Jorge’s square?d.If we frame Jorge’s square with a golden string, how long is the string?

Q6 & Q7

Melinda has a square. The length of each side is ½ foot.a.How many of Melinda’s squares are needed to cover Jorge’s square? b.How many stickers can fit Melinda’s square?c.If we frame Melinda’s square with a golden string, how long is the string?

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?

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)

Thank You