# Conceptual Development in Area Measurement

Motion and Measurement Nic Gilbertson & Funda Gonulates Strengthening Tomorrows Education in Measurement (STEM) Project Michigan State University Math-in-Action Conference 2012 Grand Valley State University STEM What: Research analyzing the capacity of three written elementary curricula to support robust learning of spatial

measurement (length, area, volume) Professional Development work with pre-service teachers on measurement lesson study working with facilitators throughout Michigan who are in turn working with teachers in their areas STEM 2012- Math-in-Action 2 STEM Three Curricula The three carefully chosen

curricula are: Scott ForesmanAddison Wesley Mathematics UCSMPs Everyday Mathematics Saxon Math STEM 2012- Math-in-Action 3 STEM Our Analysis In our analysis we are looking at every lesson,

problem, and activity of teach curricula for two important aspects: Knowledge elements - Spatial measurement knowledge (conceptual, procedural, conventional) Textual elements - The ways in which this knowledge is expressed (statements, demonstrations, worked examples, questions, problems, games) STEM 2012- Math-in-Action 4

Some overarching questions How might dynamic (as opposed to static) ways of measuring help develop measurement concepts? How might measurement and motion help students better understand formulas? STEM 2012- Math-in-Action 5 Pentagon Task

1) Find the area 2) Draw a second pentagon with the same area STEM 2012- Math-in-Action 6 Pentagon Task With a partner, 1) Share your solutions 2) What might students have difficulties with in

solving this problem? 3) What would students need to know in order to solve this problem? STEM 2012- Math-in-Action 7 Pentagon Task NAEP results Scoring Rubric: Incorrect (both parts a & b incorrect)

Partial (either a or b correct) Correct (both a & b correct) Omitted Off Task STEM 2012- Math-in-Action 8 Pentagon Task NAEP results Scoring Rubric: Incorrect (both parts a & b incorrect)48%

Partial (either a or b correct)47% Correct (both a & b correct)2% Omitted2% Off Task(rounded to 0%) STEM 2012- Math-in-Action 9 Pentagon Task NAEP results Grade 8 2005 Scoring Rubric: Incorrect (both parts a & b incorrect)48%

Partial (either a or b correct)47% Correct (both a & b correct)2% Omitted2% Off Task(rounded to 0%) STEM 2012- Math-in-Action 10 Pentagon Task NAEP results Some possible explanations Constraints limit possible solutions

No straightforward formula Dynamic situation STEM 2012- Math-in-Action 11 Task 2 Tables Task STEM 2012- Math-in-Action 12

Task 2 Tables Task Part 1) Using the isosceles right triangles in your group, find the area of the top of your table. Part 2) Describe a formula (symbols, words, pictures) for how you can determine the area units any rectangular table top. Once you are finished let one of us know STEM 2012- Math-in-Action 13

Task 2 Tables Task Part 3) Using the right triangles in your group, find the area of the top of your table. Part 4) Describe a formula (symbols, words, pictures) for how you can determine the area for any rectangular table top. STEM 2012- Math-in-Action 14 Task 2 Tables Task

STEM 2012- Math-in-Action 15 Task 2 Tables Task Whats the area unit - squares, triangles, rectangles? Can we use base x height if the length units are different? STEM 2012- Math-in-Action 16

Task 2 Tables Task Whats the area unit - squares, triangles, rectangles? Can we use base x height if the length units are different? STEM 2012- Math-in-Action 17 Task 2 Tables Task Whats the area unit - squares, triangles, rectangles? Can we use base x height if the length units are different?

STEM 2012- Math-in-Action 18 Task 2 Tables Task Whats the area unit - squares, triangles, rectangles? Can we use base x height if the length units are different? STEM 2012- Math-in-Action 19

Task 2 Tables Task Whats the area unit - squares, triangles, rectangles? Can we use base x height if the length units are different? STEM 2012- Math-in-Action 20 Task 2 Tables Task AN ALTERNATIVE TASK What would this lesson have looked like had it

been taught this way? STEM 2012- Math-in-Action 21 Task 2 Tables Task BIG IDEAS

Different shapes have many different measures Defining 1 unit is important The area unit affects the formula because of how the space is structured The formula depends on the unit and attribute Structuring space is not simple for many students The formula is a description of a general situation, not just a procedure to calculate something. STEM 2012- Math-in-Action

22 Some other NAEP items (Grade 8) Megan drew a rectangle that has an area of 24 square centimeters. Which of the following could be the dimensions of her rectangle?

2 cm by 12 cm 3 cm by 9 cm 4 cm by 20 cm 6 cm by 6 cm 12 cm by 12 cm STEM 2012- Math-in-Action 23 Some other NAEP items (Grade 8)

How many square tiles, 5 inches on a side, does it take to cover a rectangular area that is 50 inches wide and 100 inches long? Open Response STEM 2012- Math-in-Action 24 Some other NAEP items (Grade 8) Make a drawing in the space below to show how the four triangles shown above could fit together without

overlapping to make a rectangle that is not a square. Show the dimensions of the rectangle on your drawing. What is the area of this rectangle? STEM 2012- Math-in-Action 25 Dynamic Simulations Some Video Resources Circumference of Circle http://www.rkm.com.au/ANIMATIONS/animation-pi.html Area of Circle http://www.rkm.com.au/ANIMATIONS/animation-Circle-Area-Derivation.html

Area of Circle 2 http://curvebank.calstatela.edu/circle2/circle2.htm Area of Circle 3 http://www.youtube.com/watch?v=06VSISqNmoI Surface Area of Sphere http://www.rkm.com.au/ANIMATIONS/animation-Sphere-Surface-Area-Derivation.html STEM 2012- Math-in-Action 26 Dynamic Simulations Some Applet Resources

STEMs Simulations https://www.msu.edu/~stemproj/simulations.html National Library of Virtual Manipulatives http://nlvm.usu.edu/ NCTM Illuminations http://illuminations.nctm.org/

Freudenthal Institute http://www.fi.uu.nl/wisweb/en/ STEM 2012- Math-in-Action 27 Thank you! We want to thank the National Science Foundation for funding this work

We want to thank you for coming! Please take a few minute to fill out our evaluation. For more information starting you can go to: http://www.msu.edu/~stemproj If you have any questions please e-mail us at: [email protected] STEM 2012- Math-in-Action 28

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