Teaching fractions and geometry according to the CCSSM

Transcription

1 Teaching fractions and geometry according to the CCSSM Selden, NY April 25, 2015 H. Wu

2 Part 1. Definition and the addition of fractions. Part 2. Congruence and similarity (geometry of grade 8 and high school).

3 Recall the basic mathematical requirements: (1) Every concept is clearly defined. (2) Every statement is precise about what is true and what is not true. (3) Every statement is supported by reasoning. (4) Mathematics is presented as a coherent story. (5) A purpose is given to each skill and concept.

4 Part 1. Definition of a fraction We first introduce the number line. On a horizontal line, let two points be singled out. Identify the point to the left with 0 and the point to the right with 1. This segment, denoted by [0,1] is called the unit segment and 1 is called the unit. 0 1

5 Now mark off equidistant points to the right of 1 as on a ruler, as shown, and identify the successive points with 2, 3, 4, This line, with a sequence of equidistant points on the right identified with the whole numbers, is called the number line.

6 Intuitive discussion. Suppose the whole is taken to be the length of the segment [0,1] on the number line (or any of [1,2], [2,3], etc.). Divide [0,1] into three equal parts (= three segments of equal length). The part adjoining 0 is a third. Denote its right endpoint by

7 Fix the distance between 0 and 1 3. Marking off equidistant points to the right of 1 3 as we would with whole numbers, we obtain a sequence of points, denoted by 2 3, 3 3, 4 3, etc etc The segment [0, 1 3 ] may as well be identified with its right endpoint, 1 3. Similarly, the segment [0, 3 2 ] may as well be identified with its right endpoint, 2 3, etc.

8 Each of these parts-of-a-whole (in the context of thirds) is now replaced by a point on the number line. Two-thirds can be replaced by the 2nd point to the right of 0, denoted by 2 3. Seven-thirds can be replaced by the point that is the 7th point to the right of 0, denoted by 7 3. m-thirds can be replaced by the mth point to the right of 0, denoted by m 3.

9 The sequence of thirds should remind you of the sequence of whole numbers. The only difference: for the whole number sequence, we start with 0 and 1, but for the sequence of thirds, we start with 0 and 1 3. The sequence of thirds is thus entirely analogous to the sequence of whole numbers.

10 This is one of several features that exhibit the parallel between the study of whole numbers and that of fractions. This puts whole numbers and fractions on the same footing. Contrary to common misconceptions, fractions are not essentially different from whole numbers.

11 Fractions with denominator equal to 5 are similarly placed on the number line: 8 5 is the 8th point to the right of 0 in the sequence of fifths. And so on

12 We also agree to identify n 0 with 0 for any nonzero whole number n. In this way, all fractions are unambiguously placed on the number line. Intuitively, we have identified parts-of-a-whole with points on the number line.

13 A fraction such as 5 8 is therefore not three things: 8, and 5, and the action of taking 8 out of a division of the unit segment into 5 equal parts. Rather, it is one thing: a certain point on the number line. Every part of the symbol 8 5 is needed to locate the position of the fraction on the number line: the 8th point to the right of 0 in the sequence of 5ths. The importance of recognizing a fraction m n a single object cannot be overemphasized. as

14 Formal definition of a fraction: A fraction is one of the points on the number line as described above. What does this definition mean? It means: Any time we explain something about fractions, there is no need to guess what a fraction is (is it a piece of pizza or is it part of a square?). Everything we want to say about a fraction can be and has to be realized on the number line.

15 Part 1. (cont.) Equivalent fractions We are going to convince students that: 1 3 = = 2 6, 5 3 = = 10 6, 2 5 = = 6 15, etc.

16 The general statement is: Theorem on equivalent fractions. Given any fractions k l and a nonzero whole number c, then: k l = c k c l i.e., the two fractions k l and c c k l on the number line. are the same point

17 Let us explain why 7 3 = Here is the common explanation from TSM: 7 3 = = = = 14 6

18 Here are two possible reactions by students: (1) So = Fractions are so simple! I can now add fractions! =

19 Here are two possible reactions by students: (1) So = Fractions are so simple! I can now add fractions! = (2) I am supposed to know that = ? I have just learned that 4 4 and 3 2 are pieces of pizzas, and now I am supposed to multiply two pieces of pizza? I give up.

20 Let us try again: In order to show that 7 3 = 14 6, we must show that the 7th point to the right of 0 in the sequence of thirds is also the 14th point to the right of 0 in the sequence of sixths. Moral: With a clear-cut definition of a fraction, there will be no ambiguity about what must be proved.

21 We divide each of the thirds into 2 equal parts, getting sixths (2 3 = 6): Clearly the 7th point to the right of 0 in the sequence of thirds is also the 14th point to the right of 0 in the sequence of sixths.

22 Another example: prove 5 6 = We must show that the 5th point to the right of 0 in the sequence of sixths is also the 15th point to the right of 0 in the sequence of eighteenths.

23 We divide each of the sixths into 3 equal parts, getting eighteenths (3 6 = 18): We see that the 5th point to the right of 0 in the sequence of sixths is also the 15th point to the right of 0 in the sequence of eighteenths.

24 Observe that the reasoning for each of the equalities 7 3 = 14 6 and 5 6 = is the same. So this reasoning proves the Theorem in general. Application: Given 2 7 and 4 5, we can rewrite them as two fractions with the same denominator: and

25 Generalization: Given two fractions m n and k l, the Theorem says we can always rewrite them as two fractions with equal denominators, e.g., lm ln and kn ln This is the FFFP (Fundamental Fact of Fraction pairs): Any two fractions may be regarded as two fractions with the same denominator.

26 Part 1. (concluded) Addition of fractions First, how do we add whole numbers? The sum of 4 and 3 is the length of the concatenation of a segment of length 4 and a segment of length 3. (Concatenation: joining an endpoint of one segment to an endpoint of the other and putting them on a straight line, as shown.) }{{} 4 }{{} 3

27 Now, because whole numbers are also fractions, the meaning of should not be different from the addition of whole numbers. We define the sum to be the length of the concatenation of one segment of length 4 5 segment of length 3 5 : and a second } {{ } 4 5 } {{ } 3 5

28 In terms of segments of length 1 5, is just the length of the concatenation of 4 such segments and 3 such segments, and is therefore exactly such segments, i.e., =

29 The same reasoning allows us to add any two fractions with the same denominator: k n + m n = k + m n Observe: the meaning of is not different from that of The addition of fractions with the same denominator is not different from the addition of whole numbers.

30 Next, something more complicated: does it mean? What We define this sum in exactly the same way: it is the length of the concatenation of one segment of length 4 7 and another segment of length 2 5 : } {{ } 4 7 } {{ } 2 5 Therefore, by definition, is the total length of 4 of the 7 1 s and 2 of the 1 5 s.

31 This looks forbidding, because the addition of becomes something like adding 4 feet and 2 meters. But FFFP tells us that there is never any need to face two fractions with different denominators: = = 34 35

32 In general, we define the addition of k l and m n in exactly the same way: k l + m n is the length of the concatenation of one segment of length k l and another of length m n : }{{} k l }{{} m n By FFFP, k l + m n = kn ln + lm ln = kn + lm ln

33 Using LCD: = (4 1) + (3 5) 24 = Without using LCD: = = Same answer (of course).

34 By the definition of concatenation, addition of fractions is commutative and associative, e.g., k l + m n = m n + k l This is something you cannot easily prove using LCD. LCD is a distraction. It should be brought up only as a specialized skill, not as part of the definition of the addition of fractions.

35 The continuity from whole numbers to fractions is of critical importance in the learning of fractions. The continuity lightens the cognitive load: students have less to learn. It also enhances their incentive to learn: they see that what they learned about whole numbers is still valid and has an immediate payoff.

36 Looking back: (1) Every concept is clearly defined. (2) Every statement is precise about what is true and what is not true. (3) Every statement is supported by reasoning. (4) Mathematics is presented as a coherent story. (5) A purpose is given to each skill and concept.

37 Part 2. Congruence and similarity. The most striking aspect of the school geometry curriculum is the discontinuity from middle school to high school. First students are told in K 8 that congruence is same size and same shape, and that similarity is same shape but not necessarily the same size. Middle school students also study rotations, reflections, and translations for artistic reasons: they learn about the beauty of symmetries. It is fun.

38 We have already seen that the definition of similarity as same shape and not necessarily the same size is fraudulent as mathematics. The definition of congruence as same size and same shape is no better: Can we use same size and same shape to prove theorems?

39 Perhaps for these reasons, all that is forgotten in high school. High school geometry is taught using axioms. As is well-known, axiomatic geometry is a radical departure from the rest of the school mathematics curriculum. Whereas up to this point not much reasoning is given, students are suddenly thrust into an arena where everything, no matter how trivial or obvious, has to be proved.

40 Congruence and similarity are now tied down to triangles and polygons, and only triangles and polygons. They are only discussed in terms of equal angles and equal (or proportional) sides. Such restrictions facilitate the discussion of congruent and similar rectilinear figures. Congruence and similarity between curvy figures like parabolas? Nowhere to be found.

41 In other words, the way we teach students about the two cornerstones of school geometry congruence and similarity is to teach them first as metaphors, and then as abstractions unrelated to the metaphors. This can hardly be Exhibit A of good teaching. Not surprisingly, the rampant nonlearning in geometry classes became a scandal.

42 In the 1990 s, textbooks began to appear that taught geometry by hands-on activities alone, with no proofs. Are these the only viable alternatives: teach axiomatic geometry with proofs but no understanding and no connection to the rest of the school curriculum, or teach geometry with no proofs? The CCSSM offer a third alternative.

43 CCSSM s approach is roughly the following. (i) Introduce informally, using manipulatives such as transparencies translations, reflections, rotations, and dilations in grade 8. (ii) Use this informal knowledge to define the concepts of congruence and similarity in general, and then explore elementary facts about congruent and similar triangles, such as why SAS, ASA are true for congruent triangles and why AA is true for similar triangles.

44 (iii) Give precise definitions of translations, reflections, rotations, and dilations in high school and, retracing the steps in grade 8, use them to define congruence and similarity for all geometric figures. (iv) Prove the basic congruence criteria for triangles (ASA, SAS, SSS, HL) and the basic similarity criteria for triangles (AA, SAS, SSS). Use these as the foundation for developing Euclidean geometry.

45 This approach at least avoids making school geometry impossible to teach from the beginning. There is another important consideration: to rescue the teaching of linear equations in two variables ax + by = c

46 TSM never gives any reason why the graph of a linear equation (in two variables) is a line. This is because TSM never gives the correct definition of the slope of a line. Consequently, TSM forces students to learn the geometry of linear equations by rote. This is one of the main reasons why students have trouble learning algebra (V. Postelnicu and C. Greenes, NCSM Newsletter, Winter ).

47 In TSM, the definition of the slope of a (nonvertical) line L in the coordinate plane is the following: P = (p 1, p 2 ) and Q = (q 1, q 2 ) be distinct points on L. Then, the slope of L is the ratio: p 2 q 2 p 1 q 1. Q Is anything wrong with that? P R O L let

48 Yes, because: If A = (a 1, a 2 ) and B = (b 1, b 2 ) are two other points on L, then the slope of L would be: a 2 b 2 a 1 b 1. Q P R B O A C L So which of these ratios should be the slope of L: p 2 q 2 p 1 q 1 or a 2 b 2 a 1 b 1?

49 This question must be answered if slope is to be a general property of the line L and not of the two specific chosen points on L. It turns out that p 2 q 2 p 1 q 1 = a 2 b 2 a 1 b 1. The fact that p 2 q 2 p 1 q 1 = a 2 b 2 a 1 b 1 proof of similar triangles: is true requires the ABC P QR.

50 This is exactly why eighth graders, who begin the study of linear equations in two variables, need some familiarity with similar triangles in order to learn a correct definition of slope. It will be difficult for these eighth graders, not to say impossible, to solve problems related to slope without the explicit knowledge that slope can be computed by choosing any two points that suit one s purpose.

51 CCSSM s departure from the standard school curriculum is most pronounced in the two areas of fractions and geometry. I hope I have given you some idea of the reasons for and benefits of this departure from the standard school curriculum. At least I hope you appreciate why the departure is absolutely necessary.

52 References: Understanding Numbers in Elementary School Mathematics, American Mathematical Society, Also go to wu/ for: Teaching Fractions According to the Common Core Standards Teaching Geometry According to the Common Core Standards Teaching Geometry in Grade 8 and High School According to the Common Core Standards