Mathematics Content Courses for Elementary Teachers
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1 Mathematics Content Courses for Elementary Teachers Sybilla Beckmann Department of Mathematics University of Georgia Massachusetts, March 2008 Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 1 / 38
2 Mathematics content courses for elementary teachers at UGA for teachers who will be certified to teach grades PreK 5 5 specialized math courses in addition to core math : 3 content, 2 methods content courses: an arithmetic course a course in geometry and measurement a course in algebra, number theory, and a little probability and statistics our 5 courses form a mathematics endorsement approved by the Professional Service Commission in Georgia separate but similar courses for prospective middle grades teachers, certified grades 4 8. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 2 / 38
3 Mathematics content courses for elementary teachers at UGA for teachers who will be certified to teach grades PreK 5 5 specialized math courses in addition to core math : 3 content, 2 methods content courses: an arithmetic course a course in geometry and measurement a course in algebra, number theory, and a little probability and statistics our 5 courses form a mathematics endorsement approved by the Professional Service Commission in Georgia separate but similar courses for prospective middle grades teachers, certified grades 4 8. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 2 / 38
4 Mathematics content courses for elementary teachers at UGA for teachers who will be certified to teach grades PreK 5 5 specialized math courses in addition to core math : 3 content, 2 methods content courses: an arithmetic course a course in geometry and measurement a course in algebra, number theory, and a little probability and statistics our 5 courses form a mathematics endorsement approved by the Professional Service Commission in Georgia separate but similar courses for prospective middle grades teachers, certified grades 4 8. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 2 / 38
5 Mathematics content courses for elementary teachers at UGA for teachers who will be certified to teach grades PreK 5 5 specialized math courses in addition to core math : 3 content, 2 methods content courses: an arithmetic course a course in geometry and measurement a course in algebra, number theory, and a little probability and statistics our 5 courses form a mathematics endorsement approved by the Professional Service Commission in Georgia separate but similar courses for prospective middle grades teachers, certified grades 4 8. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 2 / 38
6 Development of mathematics content courses for elementary teachers at UGA Development began when the Board of Regents of the University System of Georgia mandated a minor in mathematics, consisting of 5 upper division courses for prospective elementary teachers. All 3 content courses have been in place since spring During early development there were collaborative projects in which Arts & Sciences faculty and Education faculty (elementary education, mathematics education, special education) worked together and considered aspects of teacher preparation. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 3 / 38
7 Development of mathematics content courses for elementary teachers at UGA Development began when the Board of Regents of the University System of Georgia mandated a minor in mathematics, consisting of 5 upper division courses for prospective elementary teachers. All 3 content courses have been in place since spring During early development there were collaborative projects in which Arts & Sciences faculty and Education faculty (elementary education, mathematics education, special education) worked together and considered aspects of teacher preparation. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 3 / 38
8 Development of mathematics content courses for elementary teachers at UGA Development began when the Board of Regents of the University System of Georgia mandated a minor in mathematics, consisting of 5 upper division courses for prospective elementary teachers. All 3 content courses have been in place since spring During early development there were collaborative projects in which Arts & Sciences faculty and Education faculty (elementary education, mathematics education, special education) worked together and considered aspects of teacher preparation. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 3 / 38
9 Mathematics content courses should make sense to stakeholders with various perspectives We should work together to design and revise mathematics content courses for prospective elementary teachers that are recognized as worthwhile, informative, and helpful by key stakeholders: by the prospective teachers who take them by the professors who teach them by other professors who are in charge of preparing teachers (elementary education faculty, mathematics education faculty, special education faculty) by practicing teachers by other stakeholders in business, science, engineering,... Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 4 / 38
10 Mathematics content courses should make sense to stakeholders with various perspectives We should work together to design and revise mathematics content courses for prospective elementary teachers that are recognized as worthwhile, informative, and helpful by key stakeholders: by the prospective teachers who take them by the professors who teach them by other professors who are in charge of preparing teachers (elementary education faculty, mathematics education faculty, special education faculty) by practicing teachers by other stakeholders in business, science, engineering,... Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 4 / 38
11 Mathematics content courses should make sense to stakeholders with various perspectives We should work together to design and revise mathematics content courses for prospective elementary teachers that are recognized as worthwhile, informative, and helpful by key stakeholders: by the prospective teachers who take them by the professors who teach them by other professors who are in charge of preparing teachers (elementary education faculty, mathematics education faculty, special education faculty) by practicing teachers by other stakeholders in business, science, engineering,... Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 4 / 38
12 Mathematics content courses should make sense to stakeholders with various perspectives We should work together to design and revise mathematics content courses for prospective elementary teachers that are recognized as worthwhile, informative, and helpful by key stakeholders: by the prospective teachers who take them by the professors who teach them by other professors who are in charge of preparing teachers (elementary education faculty, mathematics education faculty, special education faculty) by practicing teachers by other stakeholders in business, science, engineering,... Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 4 / 38
13 Mathematics content courses should make sense to stakeholders with various perspectives We should work together to design and revise mathematics content courses for prospective elementary teachers that are recognized as worthwhile, informative, and helpful by key stakeholders: by the prospective teachers who take them by the professors who teach them by other professors who are in charge of preparing teachers (elementary education faculty, mathematics education faculty, special education faculty) by practicing teachers by other stakeholders in business, science, engineering,... Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 4 / 38
14 Mathematics content courses should make sense to stakeholders with various perspectives We should work together to design and revise mathematics content courses for prospective elementary teachers that are recognized as worthwhile, informative, and helpful by key stakeholders: by the prospective teachers who take them by the professors who teach them by other professors who are in charge of preparing teachers (elementary education faculty, mathematics education faculty, special education faculty) by practicing teachers by other stakeholders in business, science, engineering,... Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 4 / 38
15 A focus on explaining why A focus on explaining why has been found to be productive: why standard procedures and formulas are valid why nonstandard methods can also be valid why other seemingly plausible ways of reasoning are not correct Explanations: multiple explanations when helpful draw explicitly on fundamental principles and concepts of elementary mathematics Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 5 / 38
16 A focus on explaining why A focus on explaining why has been found to be productive: why standard procedures and formulas are valid why nonstandard methods can also be valid why other seemingly plausible ways of reasoning are not correct Explanations: multiple explanations when helpful draw explicitly on fundamental principles and concepts of elementary mathematics Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 5 / 38
17 A focus on explaining why A focus on explaining why has been found to be productive: why standard procedures and formulas are valid why nonstandard methods can also be valid why other seemingly plausible ways of reasoning are not correct Explanations: multiple explanations when helpful draw explicitly on fundamental principles and concepts of elementary mathematics Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 5 / 38
18 A focus on explaining why A focus on explaining why has been found to be productive: why standard procedures and formulas are valid why nonstandard methods can also be valid why other seemingly plausible ways of reasoning are not correct Explanations: multiple explanations when helpful draw explicitly on fundamental principles and concepts of elementary mathematics Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 5 / 38
19 A focus on explaining why A focus on explaining why has been found to be productive: why standard procedures and formulas are valid why nonstandard methods can also be valid why other seemingly plausible ways of reasoning are not correct Explanations: multiple explanations when helpful draw explicitly on fundamental principles and concepts of elementary mathematics Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 5 / 38
20 Why focus on explaining why? Assertions should have reasons From NCTM Principles and Standards for School Mathematics (PSSM), 2000: From children s earliest experiences with mathematics, it is important to help them understand that assertions should always have reasons. Questions such as Why do you think it is true? and Does anyone think the answer is different, and why do you think so? help students see that statements need to be supported or refuted by evidence. (chapter 3) Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 6 / 38
21 Why focus on explaining why? Reasoning is essential to understanding math From PSSM: Being able to reason is essential to understanding mathematics. By developing ideas, exploring phenomena, justifying results, and using mathematical conjectures in all content areas and with different expectations of sophistication at all grade levels, students should see and expect that mathematics makes sense. (chapter 3) Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 7 / 38
22 Why focus on explaining why? Development of a rationale Teaching Mathematics in Seven Countries: Results From the TIMSS 1999 Video Study Percentage of 8th grade mathematics lessons in sub-sample that contained the development of a rationale Australia 25% Switzerland 25% Hong Kong SAR 20% Czech Republic 10% Netherlands 10% United States 0% Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 8 / 38
23 Why focus on explaining why? What are the advantages? The focus on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid: is true to the discipline of mathematics is a means for thinking about mathematical ideas is a means for considering ideas from multiple perspectives and for using multiple representations is a means for connecting ideas is a means for focusing on fundamental concepts, principles, and ideas, such as: the meanings of the operations the definition of fraction place value properties of arithmetic additivity of area and volume decomposition as a generally useful technique Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 9 / 38
24 Why focus on explaining why? What are the advantages? The focus on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid: is true to the discipline of mathematics is a means for thinking about mathematical ideas is a means for considering ideas from multiple perspectives and for using multiple representations is a means for connecting ideas is a means for focusing on fundamental concepts, principles, and ideas, such as: the meanings of the operations the definition of fraction place value properties of arithmetic additivity of area and volume decomposition as a generally useful technique Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 9 / 38
25 Why focus on explaining why? What are the advantages? The focus on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid: is true to the discipline of mathematics is a means for thinking about mathematical ideas is a means for considering ideas from multiple perspectives and for using multiple representations is a means for connecting ideas is a means for focusing on fundamental concepts, principles, and ideas, such as: the meanings of the operations the definition of fraction place value properties of arithmetic additivity of area and volume decomposition as a generally useful technique Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 9 / 38
26 Why focus on explaining why? What are the advantages? The focus on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid: is true to the discipline of mathematics is a means for thinking about mathematical ideas is a means for considering ideas from multiple perspectives and for using multiple representations is a means for connecting ideas is a means for focusing on fundamental concepts, principles, and ideas, such as: the meanings of the operations the definition of fraction place value properties of arithmetic additivity of area and volume decomposition as a generally useful technique Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 9 / 38
27 Why focus on explaining why? What are the advantages? The focus on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid: is true to the discipline of mathematics is a means for thinking about mathematical ideas is a means for considering ideas from multiple perspectives and for using multiple representations is a means for connecting ideas is a means for focusing on fundamental concepts, principles, and ideas, such as: the meanings of the operations the definition of fraction place value properties of arithmetic additivity of area and volume decomposition as a generally useful technique Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 9 / 38
28 Why focus on explaining why? What are the advantages? The focus on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid: is true to the discipline of mathematics is a means for thinking about mathematical ideas is a means for considering ideas from multiple perspectives and for using multiple representations is a means for connecting ideas is a means for focusing on fundamental concepts, principles, and ideas, such as: the meanings of the operations the definition of fraction place value properties of arithmetic additivity of area and volume decomposition as a generally useful technique Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 9 / 38
29 Why focus on explaining why? What are the advantages? Advantages of focusing on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid, why nonstandard methods can also be valid, and why other seemingly plausible ways of reasoning are not correct: expecting to explain why is part of a mathematical habit of mind is do-able and productive we can expect every teacher to be able to explain why standard solution methods and formulas are valid prospective teachers believe they should know explanations prospective teachers often don t know that procedures and formulas can be explained provides a way to work on mathematical reasoning within the topics that should be the focus of elementary math instruction has the potential to travel into the school classroom Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 10 / 38
30 Why focus on explaining why? What are the advantages? Advantages of focusing on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid, why nonstandard methods can also be valid, and why other seemingly plausible ways of reasoning are not correct: expecting to explain why is part of a mathematical habit of mind is do-able and productive we can expect every teacher to be able to explain why standard solution methods and formulas are valid prospective teachers believe they should know explanations prospective teachers often don t know that procedures and formulas can be explained provides a way to work on mathematical reasoning within the topics that should be the focus of elementary math instruction has the potential to travel into the school classroom Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 10 / 38
31 Why focus on explaining why? What are the advantages? Advantages of focusing on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid, why nonstandard methods can also be valid, and why other seemingly plausible ways of reasoning are not correct: expecting to explain why is part of a mathematical habit of mind is do-able and productive we can expect every teacher to be able to explain why standard solution methods and formulas are valid prospective teachers believe they should know explanations prospective teachers often don t know that procedures and formulas can be explained provides a way to work on mathematical reasoning within the topics that should be the focus of elementary math instruction has the potential to travel into the school classroom Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 10 / 38
32 Why focus on explaining why? What are the advantages? Advantages of focusing on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid, why nonstandard methods can also be valid, and why other seemingly plausible ways of reasoning are not correct: expecting to explain why is part of a mathematical habit of mind is do-able and productive we can expect every teacher to be able to explain why standard solution methods and formulas are valid prospective teachers believe they should know explanations prospective teachers often don t know that procedures and formulas can be explained provides a way to work on mathematical reasoning within the topics that should be the focus of elementary math instruction has the potential to travel into the school classroom Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 10 / 38
33 Why focus on explaining why? What are the advantages? Advantages of focusing on explaining why common procedures, formulas, and solution methods of elementary mathematics are valid, why nonstandard methods can also be valid, and why other seemingly plausible ways of reasoning are not correct: expecting to explain why is part of a mathematical habit of mind is do-able and productive we can expect every teacher to be able to explain why standard solution methods and formulas are valid prospective teachers believe they should know explanations prospective teachers often don t know that procedures and formulas can be explained provides a way to work on mathematical reasoning within the topics that should be the focus of elementary math instruction has the potential to travel into the school classroom Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 10 / 38
34 The multiplication algorithm Key ingredients in understanding and explaining it 1 Definition of multiplication: A B means the total in A groups of B m 4 m 4 m Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 11 / 38
35 The multiplication algorithm Key ingredients in understanding and explaining it 2. Properties of arithmetic, what they are and why they should be true, especially the distributive property 4 7 = (5+2) = Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 12 / 38
36 The multiplication algorithm Key ingredients in understanding and explaining it 3. Basic multiplication facts knowing relationships among them is key to scaffolding student learning for automaticity 6 7 = = 2 (3 7) 6 7 = Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 13 / 38
37 The multiplication algorithm Key ingredients in understanding and explaining it 4. Why multiplication by 10 and powers of 10 shifts digits to the left 5. Decomposing: The algorithm decomposes in terms of place value and applies the distributive property multiple times in order to reduce to basic multiplication facts Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 14 / 38
38 The multiplication algorithm Key ingredients in understanding and explaining it 4. Why multiplication by 10 and powers of 10 shifts digits to the left 5. Decomposing: The algorithm decomposes in terms of place value and applies the distributive property multiple times in order to reduce to basic multiplication facts Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 14 / 38
39 The multiplication algorithm Key ingredients in understanding and explaining it 4. Why multiplication by 10 and powers of 10 shifts digits to the left 5. Decomposing: The algorithm decomposes in terms of place value and applies the distributive property multiple times in order to reduce to basic multiplication facts Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 14 / 38
40 The Multiplication Algorithm the partial products algorithm is a step toward the condensed standard algorithm Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 15 / 38
41 Calculating by decomposing informally, with benchmark fractions and percentages Mentally calculate 95% of 80,000 by calculating 1 10 of 80,000, calculating half of that result, and then taking this last amount away from 80,000. Use the picture and the percent diagram to help you explain, record, and clarify your thinking. 100% 80,000 10% 5% 95% Percent diagram : Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 16 / 38
42 Calculating by decomposing more formally, to show the algebra in mental math For each arithmetic problem below, find ways to use properties of arithmetic to make the problem easy to do mentally. Describe your method in words, and write equations that correspond to your method. Write your equations in the form: original = some expression =. = some expression Try to find several different ways to solve this problem mentally % $44 Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 17 / 38
43 Area formulas Key ideas in explaining area formulas 1 The additivity principle and the moving principle (invariance of area under isometries) which allow us to decompose shapes and move pieces around 2 Areas of rectangles Aside: note that items 1 and 2 are linked to explaining properties of arithmetic (distributivity especially!) and the multiplication and division algorithms! 3 Reduce to a case handled previously Used throughout mathematics! Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 18 / 38
44 Area formulas Key ideas in explaining area formulas 1 The additivity principle and the moving principle (invariance of area under isometries) which allow us to decompose shapes and move pieces around 2 Areas of rectangles Aside: note that items 1 and 2 are linked to explaining properties of arithmetic (distributivity especially!) and the multiplication and division algorithms! 3 Reduce to a case handled previously Used throughout mathematics! Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 18 / 38
45 Area formulas Key ideas in explaining area formulas 1 The additivity principle and the moving principle (invariance of area under isometries) which allow us to decompose shapes and move pieces around 2 Areas of rectangles Aside: note that items 1 and 2 are linked to explaining properties of arithmetic (distributivity especially!) and the multiplication and division algorithms! 3 Reduce to a case handled previously Used throughout mathematics! Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 18 / 38
46 Area formulas Key ideas in explaining area formulas 1 The additivity principle and the moving principle (invariance of area under isometries) which allow us to decompose shapes and move pieces around 2 Areas of rectangles Aside: note that items 1 and 2 are linked to explaining properties of arithmetic (distributivity especially!) and the multiplication and division algorithms! 3 Reduce to a case handled previously Used throughout mathematics! Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 18 / 38
47 Area formulas Key ideas in explaining area formulas 1 The additivity principle and the moving principle (invariance of area under isometries) which allow us to decompose shapes and move pieces around 2 Areas of rectangles Aside: note that items 1 and 2 are linked to explaining properties of arithmetic (distributivity especially!) and the multiplication and division algorithms! 3 Reduce to a case handled previously Used throughout mathematics! Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 18 / 38
48 Steps to explaining why the area formula for triangles is valid 1 Find areas of shapes on graph paper 1 cm initial primitive methods involve moving small pieces Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 19 / 38
49 Steps to explaining why the area formula for triangles is valid 1 Find areas of shapes on graph paper 1 cm initial primitive methods involve moving small pieces Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 19 / 38
50 Steps to explaining why the area formula for triangles is valid 2. Any one of the three sides of a triangle can be the base! Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 20 / 38
51 Steps to explaining why the area formula for triangles is valid 3. Explain why the formula is valid for right triangles develop multiple explanations corresponding to different ways of writing the formula: 1 2 (b h) (1 2 b) h b (1 2 h) Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 21 / 38
52 Explaining why the area formula for triangles is valid one method, corresponding to ( 1 2 b) h One method: h b Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 22 / 38
53 Explaining why the area formula for triangles is valid one method, corresponding to ( 1 2 b) h One method: h b Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 23 / 38
54 Explaining why the area formula for triangles is valid one method, corresponding to ( 1 2 b) h One method: h b h b 2 Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 24 / 38
55 Explaining why the area formula for triangles is valid another method, corresponding to 1 (b h) 2 Another method: h b Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 25 / 38
56 Explaining why the area formula for triangles is valid another method, corresponding to 1 (b h) 2 Another method: h h b b Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 26 / 38
57 Explaining why the area formula for triangles is valid another method, corresponding to 1 (b h) 2 Another method: h h b b h b Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 27 / 38
58 Explaining why the area formula for triangles is valid 4. Explain the formula in the case where the height is inside the triangle. h 5. Explain the formula in the oblique case. b Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 28 / 38
59 Explaining why the area formula for triangles is valid 4. Explain the formula in the case where the height is inside the triangle. h 5. Explain the formula in the oblique case. b Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 28 / 38
60 Explaining why the area formula for triangles is valid analyze an incorrect argument in the oblique case What is wrong with the following reasoning that claims to show that the area of the triangle ABC is 1 2 (b h) square units? Draw a rectangle around the triangle ABC, as shown. The area of this rectangle is b h square units. The line AC cuts the rectangle in half, so the area of the triangle ABC is half of b h square units in other words, 1 2 (b h) square units. F C h A b B E Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 29 / 38
61 Explaining why the area formula for triangles is valid give a correct argument in the oblique case What is a valid way to explain why the triangle has area 1 2 (b h) square units for the given choice of b and h? C h A b B E view the oblique triangle as a difference of right triangles Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 30 / 38
62 Explaining why the area formula for triangles is valid give a correct argument in the oblique case What is a valid way to explain why the triangle has area 1 2 (b h) square units for the given choice of b and h? C h A b B E view the oblique triangle as a difference of right triangles Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 31 / 38
63 Area problem solving The techniques used in explaining the formula are also useful in solving problems Determine the area of the shaded triangle in the next figure in two different ways. Explain your reasoning in each case. 7 units 9 units 5 units 15 units Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 32 / 38
64 Area problem solving The techniques used in explaining the formula are also useful in solving problems Determine the areas of the shaded shapes in the next pair of figures. The figure on the left consists of a 3-unit by 3-unit square and a 5-unit by 5-unit square. Explain your reasoning in each case. 3 units 14 units 5 units Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 33 / 38
65 Explaining why seemingly plausible reasoning is not correct a fraction comparison example Claire says that because 4 9 > > 3 and 9 > 8 Discuss whether or not Claire s reasoning is correct. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 34 / 38
66 Why seemingly plausible reasoning is not correct a fraction subtraction example Denise says that = 1 3 and gives the reasoning indicated in the figure to support her answer. Is Denise right? If not, what is wrong with her reasoning and how could you help her understand her mistake and fix it? Don t just explain how to solve the problem correctly; explain where Denise s reasoning is flawed. First I showed 2/3. Then when you take away half of that you have 1/3 left. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 35 / 38
67 Understanding why a non-standard method is valid Dividing without using a calculator or long division Zane is working on the division problem Zane writes: left = 16 R 5 Explain why Zane s strategy makes sense. It may help you to work with a story problem for Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 36 / 38
68 Understanding the standard method of long division Use standard long division to calculate 39 4 (to the hundredths place). Explain how to interpret each step in the process in terms of dividing $39 equally among 4 people. Include a discussion on how to interpret the bringing down steps. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 37 / 38
69 Fraction division Using a story problem to explain why we multiply by the reciprocal Write a how many in one group? story problem for and use your story problem to explain why it makes sense to solve by inverting and multiplying, in other words, by multiplying 8 by 3 2. Sybilla Beckmann (University of Georgia) Mathematics for Elementary Teachers 38 / 38
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