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Geometrical Applications of Fractions in Elementary Education: Problems and Proofs - Prof., Assignments of History of Education

An assignment for a university course on mathematics for elementary teachers, which includes various problems related to finding rectangles with specific area-perimeter ratios, covering fraction circles with other fractions, and representing fractions as sums of unit fractions. Students are required to provide mathematical proofs using algebra and equations.

Typology: Assignments

Pre 2010

Uploaded on 08/09/2009

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GRED 505 (Topics in mathematics for elementary teachers)
Assignment 5: Geometrical applications of operations on fractions
Brain Teaser. On an electronic (computational) geoboard, find all rectangles
with area numerically equal to perimeter.
Alternative Problem 1. In how many ways can one cover completely (i.e.,
without gaps or overlaps) fraction circle 1/2 using two other fraction circles?
Represent your finding both through fractions and fraction circles. Provide a
mathematical proof (formal demonstration) of your finding. Use algebra (i.e.,
equations) to show the relationship between this problem and Brain Teaser.
Alternative Problem 2. In how many ways can one cover completely (i.e.,
without gaps or overlaps) fraction circle 1/3 using two other fraction circles?
Represent your finding both through fractions and fraction circles. Provide a
mathematical proof (formal demonstration) of your finding.
Alternative Problem 3. In how many ways can one cover completely (i.e.,
without gaps or overlaps) fraction circle 1/4 using two other fraction circles?
Represent your finding both through fractions and fraction circles. Provide a
mathematical proof (formal demonstration) of your finding.
Task 1. The sum of two adjacent sides of a rectangle is called the semi-perimeter.
On a geoboard, find all rectangles with area and semi-perimeter being numerically
in the ratio of a) 3:1; b) 4 :1. How are Task 1 and the last two problems related?
Use algebra (i.e., equations) to show the relationship between: (i) case a) and
Alternative Problem 2; (ii) case b) and Alternative Problem 3.
Instructor Sergei Abramovich
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GRED 505 (Topics in mathematics for elementary teachers)

Assignment 5: Geometrical applications of operations on fractions

Brain Teaser. On an electronic (computational) geoboard, find all rectangles with area numerically equal to perimeter. Alternative Problem 1. In how many ways can one cover completely (i.e., without gaps or overlaps) fraction circle 1/2 using two other fraction circles? Represent your finding both through fractions and fraction circles. Provide a mathematical proof (formal demonstration) of your finding. Use algebra (i.e., equations) to show the relationship between this problem and Brain Teaser. Alternative Problem 2. In how many ways can one cover completely (i.e., without gaps or overlaps) fraction circle 1/3 using two other fraction circles? Represent your finding both through fractions and fraction circles. Provide a mathematical proof (formal demonstration) of your finding. Alternative Problem 3. In how many ways can one cover completely (i.e., without gaps or overlaps) fraction circle 1/4 using two other fraction circles? Represent your finding both through fractions and fraction circles. Provide a mathematical proof (formal demonstration) of your finding. Task 1. The sum of two adjacent sides of a rectangle is called the semi-perimeter. On a geoboard, find all rectangles with area and semi-perimeter being numerically in the ratio of a) 3:1; b) 4 :1. How are Task 1 and the last two problems related? Use algebra (i.e., equations) to show the relationship between: (i) case a) and Alternative Problem 2; (ii) case b) and Alternative Problem 3. Instructor Sergei Abramovich

2 Task 2. Represent 1/2 as a sum of three different unit fractions in as many ways as possible both numerically and geometrically. Show in detail how the numerical representations were found. Alternative Problem 4. How many right rectangular prisms of different integer sides and volume numerically equal to surface area are there? What are the dimensions of the prisms? What are the volumes of the prisms? Show your work. Explain the formula used to find volume in terms of counting unit cubes. Use algebra (i.e., equations) to show how this problem and Task 2 are related? Reflection Task. Answer the following questions as a way of reflecting on this assignment.

  1. What did you learn about the relationship between experiential and theoretical knowledge in mathematics from this assignment?
  2. How did the use of electronic manipulatives (fraction circles) help you articulate mathematical proof?
  3. What is your understanding of the meaning of mathematical proof and why is the proof important?
  4. Describe your previous experience with mathematical proofs. Instructor Sergei Abramovich