Building Geometric Understanding

Hands-on Exploration in Geometric MeasurementGrades 3-5

WALT:

We are learning to:– Understand the concepts of area and volume as

they are sequenced in the CCSS for 3-5th grades and incorporate the Math Practice Standards in our teaching

– Describe relationships between perimeter and area

– Describe relationships between surface area and volume

Success Criteria:

We know we are successful when we can describe how explorations in geometric measurement meet the criteria of the CCSS in both content and practice standards.

Effective Classroom Practices

Manipulatives Cooperative groups Goal setting - WALT Effective questioning Student thinking explained Connections to prior knowledge Multiple exposures

CCSS Practice Standards

#2 Reason abstractly and quantitatively #3 Construct viable arguments and critique

the reasoning of others #4 Model with mathematics #5 Use appropriate tools strategically #6 Attend to precision

CCSS Content Standards

Grade 3– Geometric measurement: Understand concepts of area

and relate area to multiplication and to addition– Geometric measurement: recognize perimeter as an

attribute of plane figures and distinguish between linear and area measures

Grade 4– 4.MD.3 Apply the area and perimeter formulas for

rectangles in real world and mathematical problems Grade 5

– Geometric measurement: understand concepts of volume and relate volume to multiplication and addition

Battista

Powerful mathematics learning can occur in problem-centered inquiry-based teaching

To develop powerful mathematical thinking, instruction must carefully guide and support students’ personal construction of concepts and ways of reasoning while the students intentionally try to make sense of situations.

Pay careful attention to classroom talk

Battista, M.T. (1999). Fifth graders’ enumeration of cubes in 3D arrays: Conceptual progress in an inquiry-based classroom. Journal for Research in Mathematics Education, 30, 417-448. In Lessons Learned from Research, NCTM, 2002, pg. 75-83. In Adding It Up, National Research Council, 2001, p 284-288.

Why Examine Perimeter and Area Relationships?

Woodward and Byrd (1983) found that almost two-thirds of 8th graders studied believed that rectangles with the same perimeter occupy the same area.*

This is a 3rd grade content piece in the new CCSS.

*Stone, Michael E. (1994). Teaching relationships between area and perimeter using geometer’s sketchpad. Mathematics Teacher, Nov. 590-594.

Erika was wondering how to arrange 20 pieces of fencing to make a rectangular dog run.

Table Task 1: Build a rectangle with 20 toothpicks (fencing

pieces) Sketch, label dimensions and find area. Display all rectangles on chart paper. Label which arrangement has the largest

area and which has the smallest. Post

Wait a minute…

We have the same number of toothpicks for the perimeter but different areas. How can this happen?

Discuss with your table group how students in 3rd – 5th may respond to the above question.

Area

Although students can recall standard formulas for areas and perimeters, other aspects of area measure remain problematic.

Rectangular area is treated as simply multiplying length times width; research suggests many elementary students do not see this product as a measurement.

A Research Companion to Principles and Standards for School Mathematics. Reston: NCTM, 2003,p.185

Erika has 20 square pieces of sod (grass) for the dog run. Which rectangular arrangement of sod would take the most fencing? The least fencing?

Table Task 2: Build a rectangle with 20 tiles Sketch, label and find the perimeter Display all rectangles on chart paper Label which requires the most fencing and

which requires the least fencing Post

Wait a minute….

We always have the same number of tiles but the number for our perimeter changes. How can this happen?

Discuss with your table group how students in 3rd – 5th may respond to the above question.

From Perimeter to Area to Volume

As students progress in their understanding of geometric measurement, underlying concepts build upon one another.

Fourth grade focuses on angle measurement but perimeter and area should be reinforced.

Fifth grade introduces the measurement of volume.

Why explore understanding of volume?

In one study, Lehrer and Schauble found that fifth graders who had a wide range of experience with representations of volume and its measurement typically organized space into three-dimensional arrays.*

Three dimensional thinking is vital in the fields of engineering and science

Lehrer and Schauble.( 2000). Inventing data structures for representational purposes: Elementary grade students’ classification models. Mathematical Thinking and Learning, 2, pg.49-72. In Adding It Up, Helping Children Learn Mathematics, National Research Council. Washington, DC: National Academy Press, 2001.

Patrick Thompson, Vanderbilt University

Students in a 5th grade teaching experiment on area and volume alerted us to the distinction between understanding a formula numerically and understanding it quantitatively.

Assessment Item

What is the volume of this box?

17 in2

6 in

Thompson, Patrick W. and Saldanha, Luis. Fraction and multiplicative reasoning. In A Research Companion to Principles and Standards for School Mathematics, NCTM, 2003.

Student Interview A

Discussion about how to find volume of the figure: Student: “There’s not enough information”Interviewer: “What information do you need?”

“I need to know how long the other sides are.”“What would you do if you knew those numbers?”“Multiply them.”“Any idea what you would get when you multiply them?”“No, it would depend on the numbers.”“Does 17 have anything to do with these numbers?”“No, it’s just the area of that face.”

Student Interview B

Discussion about how to find volume of the figure:

Student: “Somebody’s already done part of it for us.”

Interviewer: “What do you mean?”

“All we have to do now is multiply 17 and 6.”

“Some children think that you have to know the other two dimensions before you can answer this

question. Do you need to know them?”

“No, not really.”

“What would you do if you knew them?”

“I’d just multiply them.”

“What would you get when you multiplied them?”

“17”

Difficult for students: 3D

Students have considerable difficulty determining # of cubes in 3-D rectangular buildings

Students told to first predict, then check with cubes, then reflect and refine mental models

Student Reflection: discrepancies between predicted and actual number of cubes

Battista, M.T. (1999). Fifth graders’ enumeration of cubes in 3D arrays: Conceptual progress in an inquiry-based classroom. Journal for Research in Mathematics Education, 30, 417-448. In Lessons Learned from Research, NCTM, 2002, pg. 75-83. In Adding It Up, National Research Council, 2001, p 284-288.

How do you find the volume and surface area of a cube?

Table Task 3 Build cubes with various side lengths Sketch, label dimensions and find volume Use “Examining Cubes Record Sheet” to

gather information

Examining Cubes

Look at the Examining Cubes record sheet Look for patterns. What is the relationship

between surface area and volume in a cube? How might students be led to discover how to

generalize finding Volume, Area of each face and Total Surface Area for a cube with side length n?

Discuss with your table group

Kelly wants to wrap 20 golf balls, each in a cube-shaped box, together in one larger box. Which arrangement will use the least wrapping paper?

Build a box with 20 cubes Sketch each box, label dimensions, find area

of each face and the total surface area Display all boxes on chart paper Label which arrangement has the largest

surface area and which has the smallest. Post

Wait a minute…

How can the boxes have the same volume of 20 cubes and have different surface areas?

Discuss with your table group how students in 5th grade may respond to the above question.

CCSS Practice Standards

Reread these practice standards and answer: How do the exercises and the discussion questions help students experience the richness of these Practice Standards?

#2 Reason abstractly and quantitatively #3 Construct viable arguments and critique the

reasoning of others #4 Model with mathematics #5 Use appropriate tools strategically #6 Attend to precision

WALT:

We are learning to:– Understand the concepts of area and volume as

they are sequenced in the CCSS for 3-5th grades and incorporate the Math Practice Standards in our teaching

– Describe relationships between perimeter and area

– Describe relationships between surface area and volume

Success Criteria:

We know we are successful when we can describe how explorations in geometric measurement meet the criteria of the CCSS in both content and practice standards.