THE IMPORTANCE OF
NON-STANDARD UNITS IN AREA
MEASUREMENT
F U N DA G O N U L AT E S A N D A S H L E Y TAG L AU E R
M I C H I G A N STAT E U N I V E R S I T Y
M AT H I N A C T I O N 2 0 1 2
STRENGTHENING TOMORROW’S EDUCATION IN MEASUREMENT (STEM)
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
CRAZY CAKES
SAMPLE STUDENT SOLUTIONS
UNIT AND UNITIZING A REGION
What does a unit do?
What is the distinction between standard units and nonstandard units?
Why working with non-standard units might help understanding
area better?
CCSSM: AREA AND AREA UNITS
Recognize area as an attribute of plane figures and
understand concepts of area measurement.
a) A square with side length 1 unit, called “a unit square,” is
said to have “one square unit” of area, and can be used to
measure area.
b) A plane figure which can be covered without gaps or
overlaps by n unit squares is said to have an area of n square
units.
PROBLEMS TO EXPLORE WORKSHEET
5 problems into explore
While exploring problems consider:
What is my unit of measure?
What does unit do in this problem?
How does this problem support conceptualization of area?
PROBLEM 1 TO EXPLORE
Find the area of the entire tangram:
PROBLEM 2 TO EXPLORE SCALING UP
Use Rectangle, Trapezoid build similar figures with scale factors of 2
and 3. Then calculate the area of each enlargement in terms of the
original polygon, and record your results in the table provided.
PROBLEM 3 TO EXPLORE
PROBLEM 4 TO EXPLORE
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? (NAEP,
2009, Grade 8)
PROBLEM 4 TO EXPLORE
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? (NAEP,
2009, Grade 8)
PROBLEM 5 TO EXPLORE
REFLECTIVE DISCUSSION
In Problems we explored:
Comparing /creating areas by using a practical unit or an available unit (P1,P2,CC)
Area is quantification of a region enclosed in a boundary (as an alternative to seeing
area just as count of unit squares)
Dynamic view of area in addition to static view (P2)
Area changes as the region change – area increases as we enlarge the boundary
Changes in boundary-scaling up by k – change in area k2
There might be variety in unit selection (or in provided units) depending on purpose (P1,
P2,P3, P4, P5)
Any objects that provide a 2D measure –beans, cut out foot, …
Any 2D shape –triangles, rectangles, trapezoid, …
Squares and Unit squares
REFLECTIVE DISCUSSION
In Problems we explored:
Measuring area by actually covering or structuring the actual or a representative
region in contrast to counting number of squares provided in already structured
regions (P4)
REFLECTIVE DISCUSSION
In Problems we explored:
Communication of our measurement (P1, P3)
Stating our unit of measure at the end of our measurement
Describing our unit of measure as best as we can
REFLECTIVE DISCUSSION
Encourage use of proper names of units in stating area measurement
Area is …. smallest triangle/blue rectangles/(5 by 5) squares/ in. squares
Area is …..
Allow students to voice confusion instead of moving ahead
Why we had different numbers for the same area
What is the most appropriate unit of measure
Why we do prefer square units over other units
Why area goes by k2 when you increase each side by k
What is the difference between practical and standard
Provide visuals to allow others to see
Units move us from seeing area as quality of 2D regions to measurable quantities
SUGGESTION
• Improve lesson we already have
• EM Grade 2 : Lesson Find the area of each shape by using pattern blocks.
• Compare and contrast area measurement by using different units
•
Bring what is missing as problems to explore
•
Allow violations
We want to thank the National Science
Foundation for funding this work
We want to thank you for coming!
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For more information starting you can go to: http://www.msu.edu/~stemproj
If you have any questions please e-mail us at: stemproj@msu.edu