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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.
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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.