Precision and Structure as Tools to Build Understanding in Algebra. Rich Rehberger Math Instructor Gallatin College Montana State University

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1 Precision and Structure as Tools to Build Understanding in Algebra Rich Rehberger Math Instructor Gallatin College Montana State University

2 What do I mean by Precision? Think of some common phrases overheard in a typical classroom: Two negatives is a positive Absolute values make things positive Get x by itself Let x be the amount These statements are a kind of overgeneralization error and are the result of a lack of precise language.

4 Good vocab leads to good understanding Two negatives is a positive A negative times a negative is a positive A negative added to a negative is a negative Get x by itself Divide by the coefficient Collect like terms

5 Counter examples address overgeneralizations Two negatives are a positive = = = 16 Absolute values make things positive Absolute value is the distance from zero 0 = 0 x = x if x 0 x if x < 0

6 Example of Precision Commonly Missing in Early Algebra Translating a phase in English into an Algebraic Expression

7 Translate to an algebraic expression. x more than y y + x or x + y x less than y y x My question for you is why do we accept x + y?

8 What do you mean by translate? Translate 10 less than y y 1 9 y 5 5 y 7 sin π cos 180

9 Current culture works against precision

10 Ask a more precise questions (and expand student learning) Translate directly into mathematical notation and then apply the commutative property. (2pts) English Direct Translation (mindful of indicated order) Commutative Property Applied to Direct Translation y more than x x + y y + x y less than x x y y + x

11 Consider changing tables

12 Add a column of words that determine order. Order Words than to from by then (They are all prepositions in English.)

13 Give examples that demonstrate order and separate commutativity from translation English Translates Directly to Commutative Property applied to 5 more than subtract subtracted from x add y x + y y + x x added to y y + x x + y a divided by b a 1 b b a 4 divide x 4 1 x x 4

14 Example in Algebra where Precision is Critical Percent and Change Percent * Base = Amount vs. (Percent Increase)(Base) = (Amount Increase)

15 In 2006, there were 50 thousand pharmacy aides in the United States. This number is expected to drop to 45 thousand by What is the percent of decrease? Common tools used are often imprecisely presented. P B = A or P 100 = A B Both commonly spoken as Percent, Base, and Amount Do we expect students to correctly identify the amounts in this problem? There are three: 50k, 45k, and 5k (only one is A). Do we expect students to find the percent? If so it is 90% but answer to the question asked is 10%. This problem is about change. How can change be measured? 90% 10% or 5k all represent change.

16 Two ways to measure change. Multiplicatively (includes division) A Inc = P Inc Base Additively (includes subtraction) A Inc = A Final Base A Inc Base = P Inc Base + A Inc = A Final All problems involving a change have three amounts and two percents: Original Amount (Base), Amount Change, Final Amount, Percent, Percent Change

17 A case for precision. (This time with the change as a decrease.) P P Dec P B = A P Dec B = A Dec B A Dec = A Final

18 In 2006, there were 50 thousand pharmacy aides in the United States. This number is expected to drop to 45 thousand by What is the percent of decrease? Since we know P P Dec then we need A Dec = B A Final and P Dec B = A Dec Precise Identifier Knowns & Unknowns B 50 A Final 45 A Dec 5 Since we know P P Dec then we need A Dec = P Dec B and B A Dec = A Final Precise Identifier Knowns & Unknowns B 50 A Final 45 A Dec x (50) P Dec x P Dec x x 50 = 5 50 x (50) = 45

19 With three amounts and two percents we need precision. Precise Identifier Result In 2006, there were 50 thousand pharmacy aides in the United States. This number is expected to drop to 45 thousand by What is the percent of decrease? B 50 A Final 45 A Dec 5 P Dec 10% P 90%

20 Back to order words and their precise meaning. (Look for the prepositions.) Hint: the order word is to In 2006, there were 50 thousand pharmacy aides in the United States. This number is expected to drop to 45 thousand by What is the percent of decrease? Source: Occupational Outlook Handbook taken from, Introductory Algebra 11 th ed., Bittinger Hint: the order word is by In 2006, there were 50 thousand pharmacy aides in the United States. This number is expected to drop by 45 thousand by What is the percent of decrease? I changed a word so perhaps I should have changed pharmacy aides to newspaper employees

21 See how the preposition changes the order in the chart and why more precise definitions of the amounts are needed: In 2006, there were 50 thousand pharmacy aides in the United States. This number is expected to drop to 45 thousand by What is the percent of decrease? English Math Problem Original Amount B 50 Final Amount A Final 45 Amount Change A Dec 5 Percent Change P Dec x In 2006, there were 50 thousand pharmacy aides in the United States. This number is expected to drop by 45 thousand by What is the percent of decrease? English Math Problem Original Amount B 50 Final Amount A Final 5 Amount Change A Dec 45 Percent Change P Dec x

22 Joe went out to dinner and spent $48. This included a 20% tip. What was his bill before the tip? Since we know P P Dec then we need A Tip = P Tip B and B + A Tip = A Final Since we know P P Dec then we need B + A Tip = A Final and P Tip B = A Tip Precise Identifier Knowns & Unknowns Precise Identifier Knowns & Unknowns B x B x A Final 48 A Final 48 A Tip 0.2 (x) A Tip 48 x P Tip 0.2 (x) + (0.2x) = (48) P Tip x = (48 x)

23 Examples of Structure in Algebra Distributing and Factoring

24 Also need to identify and present idea of structure in algebra Seemingly unrelated to the percent problems just posed but look for the analogous tasks that lead to a better understanding of structure in the equations below. 9x 6 = 12 9 x = x 8 6 = x 8 6 = 12 In calculus this is analogous to u-substitution 9u 6 = 12

25 Review Precise Language Divide by b Divide both sides by b

26 Solve for c: a = bc + d Common approach (subtract first) a = bc + d a d = bc a d b = bc b Less common, AND incorrect, approach (divide first) a = bc + d a b = bc b + d a b = c + d a d b = c a b d = c

27 Result: Common to discourage alternative techniques don t divide first most students who do make an error if you subtract first you will do better Students are denied experience with structure Done correctly, both techniques use exactly the same number of steps. Showing both examples shows importance of the need for precision divide both sides by the coefficient of the variable

28 Solve for c: a = bc + d Correct approach (subtract first) a = bc + d Correct approach (divide first) a = bc + d a d = bc a b = bc + d b a d b a d b = bc b = c a b = bc b + d b a b d b = c

29 Solve c = k a+b for b order of operations approach structure approach a + b c = k a + b a + b a + b c = k a + b a + b ac + bc = k bc = k ac a + b = k c b = k ac c b = k c a

30 Review Precise Language

31 Why do we teach FOIL? When we mean distribute? 3x 4 x + 5 = 3x x x + 5 = 3x x 4x 20 = 3x x 20

32 Do you teach LIOF when you factor? You can t! If you remove structure you remove understanding! Distribute x + 3 x 5 = x 2 2x 15 Factor

33 What is structure of distributing? a b + c = ab + ac a + b c + d = ac + ad + bc + bd a + b c + d + e = ac + ad + ae + bc + bd + be Inherent in the distributive property are the concepts of terms and factors The words are often incorrectly used interchangeably an imprecise classroom Term refers to addition Factor refers to multiplication

34 So factoring must also be a consideration of terms and factors: Not signs x 2 4x + 21 = x x is already incorrect. Not guess and check Distributing isn t, why should factoring be? Reverse the order of operations and the distributive property. Think right to left and outside first inside last. Not PEMDAS reversed to SADMEP Recall PEMDAS SDNKM Not mathematical magic

35 Precise definitions of operations on signed numbers: Collecting Like Terms If two terms have the same sign: 1. Find the sum of the absolute values. 2. The result has the sign of the larger absolute value. If two terms have different signs: 1. Find the difference of the absolute values. 2. The result has the sign of the larger absolute value. Multiplying Factors If two factors have the same sign 1. Find the product of the absolute values 2. The result is positive If two terms have different signs 1. Find the product of the absolute values 2. The result is negative

36 Consider: Distribute: x 7 x + 4 = x 2 + 4x 7x 28 Working left to right we distribute the x then the 7 creating four distinct terms before simplification Factor: x 2 3x 28 Thinking right to left: 1. begin with 28 (came from factors) NOTE: not then the negative sign (how do signs work with factors) 3. then the 3x (came from terms composed of factors) NOTE: not -3x 4. then the negative sign (how do signs work with terms)

37 x 2 3x List the factors of c = Consider then the sign of c. Positive sign would have meant two same signs and a sum of factor absolute values, but Negative sign means two different signs so find difference of factor absolute values. 3. Look for b Created from difference of factors with different signs. 4. Consider the sign of b What combination of signs creates desired result? 2. Difference Factors b = 3 so this polynomial factors 4. 3x = +4x 7x Answer: (x + 4)(x 7)

38 x 2 11x List the factors of c = Consider then the sign of c. Negative sign would mean two different signs so a difference of factor absolute values, but Positive sign means two same signs so find sum of factor absolute values. 3. Look for b Created from sum of factors with different signs. 4. Consider the sign of b What combination of signs creates desired result? 2. Sum Factors b = 11 so this polynomial factors 4. 11x = 3x + ( 8x) Answer: (x 3)(x 8)

39 Keys to success: 1. Never find sum and difference of factors. Be sure table in second step has only one label chosen from sign of c. 2. Use examples where b can be found with sum and difference to show that precise vocab and understanding of structure are important. 3. Use structure of distributive property and read factoring questions in the reverse order.

40 1. Never find sum and difference of factors. Have a precise examples of products and sums using absolute value. Use non-factorable trinomials to highlight point. Factor: x 2 4x Sum Factors Students using guess and check find a factorization they think works. 3. b 22, 10 so this polynomial does not factor. Since sign of c is positive we never consider the difference 3 7 = 4

41 2. Use examples where b can be found with sum and difference to show that precise vocab and understanding of structure are important. 2. Sum Factors Factor: x x Difference Factors b = x = 4x + 6x x x + 24 = (x + 4)(x + 6) 3. b = x = 2x + 12x x x + 24 (x 2)(x + 12)

42 3. Use structure of reverse distribute and read in the reverse order (if a = 1). x 2 5x 6 Factors of six whose difference is negative five since +1 6 = 5 then x 2 5x 6 = (x + 1)(x 6) x 2 5x + 6 Factors of six whose sum is negative five since = 5 then x 2 5x + 6 = (x 2)(x 3)

43 3. Use structure of reverse distribute and read in the reverse order (if a = 1). x 2 + 5x 6 Factors of six whose difference is five since +6 1 = 5 then x 2 + 5x 6 = (x + 6)(x 1) x 2 + 5x + 6 Factors of six whose sum is five since = 5 then x 2 + 5x + 6 = (x + 2)(x + 3)

44 1. and 3. together: x 2 + 7x 6 Factors of six whose difference is seven since we can sum to seven, but not subtract, this must be prime.

45 Some good examples to use Primes (constructed from imprecise understanding of signs) x 2 3x + 28 Sum AND difference result in b. (but only one is correct) x x 24 x 2 7x 6 x 2 8x + 20

46 Can be expanded to ax 2 + bx + c Working outside to inside, factor: 15x 2 19x 10 Factors of ax 2 Factors of c 1x 15x x 5x 2 5 Diference of factors right to left Difference of factors swapped 10 15x 1 1x = 149x 15x x = 5x 10 5x 1 3x = 47x 3x x = 25x 5 15x 2 1x = 63x 15x 2 5 1x = 25x 5 5x 2 3x = 19x Consider sign of b: 19x = 2 3x 5 5x 3x 5 (5x + 2)

47 Ideas from: Teaching Strategies to Improve Algebra Learning Rose Mary Zbiek and Matthew R. Larson The Mathematics Teacher, Vol. 108, No. 9 (May 2015), pp Institute of Education Sciences [IES] practice guide, Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students (Star et al. 2015)

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