Why Didn t My Teacher Show Me How to Do it that Way? Rich Rehberger Math Instructor Gallatin College Montana State University The Algebra in the Arithmetic Finding analogous tasks and structures in arithmetic that can be used throughout algebra 1
My Background 9 years prior experience teaching College Level Math All levels of Calculus through Differential Equations College Algebra and Precalculus Math for Liberal Arts Majors 10 years of teaching Developmental Math at Gallatin College Prealgebra Introductory/Intermediate Algebra/Algebra Gallatin College Students Below college level coursework Vast majority of demographic are traditional students Less then 0 years old Entered college directly after high school High math anxiety Low self estimation of math ability Very low placement/test scores
How do we multiply fractions? 3 5 10 7 1 we multiply across But. This explains students hatred of math.. 3
Multiply and Simplify 1 35 30 8 Oh come on it has been too weeks I can t simplify this This is too hard There is to much to remember When we multiply across we often produce large numbers, so we try better ways. Multiply and Simplify: Cancel first: 1 35 30 630 63 8 980 98 9 14 4
But this highlights misunderstanding and eventually leads to serious errors. Overgeneralization or misunderstanding of algebraic principles I can do the same thing to the top and the bottom I can cross cancel 18 8 9 14 vs. First the overgeneralization error produced by a lack of understanding of the algebraic principle. The student wrote: 18 8 9 14 But last week in my algebra class a full half of the class thought: To add we need an LCD which is good, but When she meant: 18 8 9 14 3 do the same to top and bottom 3 3 5
The Algebraic Principle of Multiplication by One is lost. The student wrote: Algebra says: 18 8 9 14 18 9 8 14 9 14 = 9 14 Understanding in algebra: To add we need an LCD 3 so multiply by one 3 3 5 3 Good Algebra: For the student who writes: I encourage: 18 8 9 14 18 9 8 14 Sameamount of writing, much better understanding! 6
Second, commonly used techniques hide algebraic principles and result in meaningless notation: The student who wrote: The individual who wrote: Explained to me that you could only cross cancel which is why she did not reduce the 6/4 Was unsure of her answer and wished she had a way to check her work. Besides, multiply across and/or canceling are really not a good ideas. 56 6 1 3 5 56 6 3 3 6 18 1 5 5 56 4 0 1 5 5 56 4 0 1 7
How do we multiply fractions in algebra? Maybe doesn t mean to multiply across. To a scholar of mathematics, the notation could be read write factors in a single fraction. not but rather To multiply we factor and remove multiplications by one! 56 6 1 3 3 1 1 3 1 1 8
Presented traditionally in arithmetic: (examples from previous slides) 6 4 15 390 195 65 39 1638 819 73 5 1 1 35 30 630 63 8 980 98 9 14 Note vertical presentation and the reduced computation Presented analogous to algebra Traditional presentation 6 4 15 39 13 3 5 3 7 3 13 5 1 9
Note consistent presentation and improved readability Presented analogous to algebra Traditional presentations 1 35 30 8 3 7 3 5 5 7 7 9 14 Try to simplify algebraically 10 1 7 15 or better yet! 15 77 14 39 55 10 10
Why didn t I get taught this before? Multiply then simplify: 10 1 7 15 70 315 70 5 315 5 14 7 63 7 Factor then multiply: 10 1 7 15 5 7 3 7 3 5 9 9 Now they yell it! Multiply and simplify: 15 77 14 39 55 10 11,550 30030 5775 5 15015 5 1155 3 385? 3003 3 1001? Factor then multiply: 15 77 14 39 55 10 3 5 7 5 11 7 11 3 13 5 5 13 11
Notice we can remove most computation and focus on what it means to multiply by one. 10 1 7 15 5 7 3 7 3 5 15 77 14 39 55 10 3 5 7 5 11 7 11 3 13 5 9 5 13 And remember the algebra that is coming! Multiply and simplify: Knowing how factors ease multiplying: 56 6 1 3 5 56 6 3 3 6 18 1 5 5 56 4 0 1 (but we don t teach factoring 4 th degree polynomials in general so would need to guess a root and try long division to reduce) 56 6 1 3 3 1 1 3 1 1 1
Algebraic properties should be used in arithmetic instruction to: Build understanding of basic algebraic principles In the preceding examples Commutative and associate laws Relationship between canceling and multiplication by one Reduce some math/number anxiety by removing large number computation (and in some cases removing computation altogether) Build on the idea that mathematics is not about numbers but about pattern recognition and connections between operations: In these examples: How are multiplication, division, and factoring related? Show the beauty of algebra that exists even in basic computation problems. How about adding fractions? Can we refocus on algebra over computation? Focus on writing factors and looking for algebraic concepts. Multiplication by factors of one (to get LCD) Removal of factors of one (canceling) 13
Can we use factors to refocus attention on algebraic principles over computation? 1 6 1 6 5 1 5 1 1 5 1 7 1 1 6 5 1 1 6 5 6 1 6 5 6 7 6 7 1 Initially this looks longer than traditional approach. 11 1 3 0 11 1 3 0 11 1 5 5 3 0 55 60 9 60 46 60 3 30 3 3 11 3 3 5 11 5 3 5 3 3 5 3 55 9 3 5 3 3 5 14
Multiple benefits to factors 11 1 3 0 Least common denominator as a list of factors is needed in algebra Finding LCD with factors can ensure LEAST common denominator Finding LCD with factors is powerful tool in arithmetic to avoid number anxiety. No need for multiples or recall of facts such as 1 times what is 60 No need for computation LCD = 3 5 Factored denominators remove all guess work and reduce steps needed in reducing fractions 11 3 3 5 11 5 3 5 3 3 5 3 55 9 3 5 46 3 5 Try to add algebraically: 1 10 1 14 15
1 10 1 14 1 5 1 7 1 7 1 5 5 7 7 5 75 5 7 6 5 7 Note: Finding LCD using factors reduces need for multiples To reduce we only need to check if 7 5 1 is divisible by,5, or 7. No other factors would cancel. Or better yet try: (and don t ever write 350) 1 14 1 50 16
Did you need to know that LCD = 350? 1 14 1 50 1 7 1 5 5 1 5 5 7 5 5 1 7 5 5 7 3 5 5 7 16 5 5 7 16 175 5 7 5 5 7 Not to mention, in algebra focus is on factors of one! 5 3 3 3 5 3 6 5 3 7 6 3 Not on same to top and bottom 5 3 3 3 5 3 10 3 17
Most problems include several factors, but the same steps: 56 3 3 1 3 1 3 1 3 1 3 1 3 3 1 3 6 1 3 Consider student difficulties: Most prealgebra students can do: 3 10 15 Many cannot even start: Why does this happen? Would the struggle remain if they learned a factor approach from the start? 18
With factors the problems are identical! 3 10 15 3 5 3 5 3 5 3 3 3 5 3 3 3 5 Many good problems would include a mix of numbers and letters But with factors the work remains unchanged. 3 5 3 10 15 3 5 3 5 3 3 9 4 10 3 5 19
So can you answer my students question? Why didn t I get taught this before? 6 4 15 39 13 3 5 3 7 3 13 5 1 1 10 1 14 1 5 1 7 1 7 1 5 5 7 7 5 75 5 7 6 5 7 0