Learning from Comparison in Algebra

Transcription

1 Learning from Comparison in Algebra The Harvard communit has made this article openl available. Please share how this access benefits ou. Your stor matters. Citation Accessed Citable Link Terms of Use Star, J.R., Pollack, C., Durkin, K., Rittle-Johnson, B., Lnch, K., Newton, K., & Gogolen, C. (in press). Learning from comparison in algebra. Contemporar Educational Pscholog. April 6, 20 :7:5 PM EDT This article was downloaded from Harvard Universit's DASH repositor, and is made available under the terms and conditions applicable to Open Access Polic Articles, as set forth at (Article begins on net page)

2 Which is correct? Ale and Morgan were asked to solve Which is correct? Ale and Morgan were asked to solve Ale s combine like terms wa Morgan s combine like terms wa Ale s combine like terms wa Morgan s combine like terms wa I first combined like terms on the left side of the equation. Then I subtracted both sides b 60. both sides b 75 to get the answer. First I subtracted 5 on either side; 60 5 is 5. both sides b 5 to get the answer. I first combined like terms on the left side of the equation. Then I subtracted both sides b 60. both sides b 75 to get the answer. Like terms contain the same variable or group of variables raised to the same power. In He Morgan, what did we learn from comparing these right and wrong was? order for two or more terms to be like terms, their coefficients can be different, but the terms need to have the same variables raised to the same powers. Unlike terms cannot be combined b addition or subtraction. First I subtracted 5 on either side; 60 5 is 5. both sides b 5 to get the answer. * How did Ale solve the equation? * How did Morgan solve the equation? * Wh did Ale combine the terms on the left as a first step? * Wh did Morgan subtract 5 as a first step? * Which wa is correct, Ale's or Morgan's wa? How do ou know? * Can ou state a general rule about combining like terms that describes what ou have learned from comparing Ale's and Morgan's was of solving this tpe of problem?.2.2 * How did Ale solve the equation? * How did Morgan solve the equation? * Wh did Ale combine the terms on the left as a first step? * Wh did Morgan subtract 5 as a first step? * Which wa is correct, Ale's or Morgan's wa? How do ou know? * Can ou state a general rule about combining like terms that describes what ou have learned from comparing Ale's and Morgan's was of solving this tpe of problem?.2.2

3 Which is better? Ale and Morgan were asked to simplif 6 Which is better? Ale and Morgan were asked to simplif 6 Ale s rewrite the eponent wa Morgan s use perfect squares to rewrite the eponent wa Ale s rewrite the eponent wa Morgan s use perfect squares to rewrite the eponent wa fractional eponent as times /. (6 ) (6 ) fractional eponent as / times. fractional eponent as times /. He Morgan, what did we learn from comparing (6 these ) two (6 was? ) fractional eponent as / times. I epanded the I got 096. Then I applied the eponent. (6 6 6) (096) (2) I simplified the Since 2 is 6, I know that 6 / is 2. This is m answer. I epanded the I got 096. Then I applied the eponent. (6 6 6) When deciding which factors to use to rewrite the fractional eponent, be on the lookout for perfect squares. (096) (2) I simplified the This is m answer. This is m answer. This is m answer. * How did Ale simplif the epression? * How did Morgan simplif the epression? * What are some similarities and differences between Ale s and Morgan s was? * Which strateg do ou think is more efficient for this problem? Wh? 9..2 * How did Ale simplif the epression? * How did Morgan simplif the epression? * What are some similarities and differences between Ale s and Morgan s was? * Which strateg do ou think is more efficient for this problem? Wh? 9..2

4 Wh does it work? Ale and Morgan were asked to simplif the epression (5 + 2) + (5 + 2) Wh does it work? Ale and Morgan were asked to simplif the epression (5 + 2) + (5 + 2) Ale s wa Morgan s wa Ale s wa Morgan s wa (5 + 2) + (5 + 2) (5 + 2) + (5 + 2) (5 + 2) + (5 + 2) (5 + 2) + (5 + 2) First I epanded the epression using the distributive propert ( + )(5 + 2) First I factored the epression. First I epanded the epression using the distributive propert He Ale, (5 what + 2)(5 did + we 2) learn from comparing these two was? First I factored the epression. the epression Then I epanded the epression. the epression Then I epanded the epression the epression. Like epressions enclosed b grouping smbols, such as parentheses, can be combined as like terms are combined. * How did Ale simplif the epression? How did Morgan simplif the epression? * What are some similarities and differences between Ale's and Morgan's was? * Is Morgan s wa OK to do? Wh or wh not? * How did Ale simplif the epression? How did Morgan simplif the epression? * What are some similarities and differences between Ale's and Morgan's was? * Is Morgan s wa OK to do? Wh or wh not?

5 How do the differ? Ale was asked to graph the equation = 2, and Morgan was asked to graph the equation. = 2 How do the differ? Ale was asked to graph the equation = 2, and Morgan was asked to graph the equation. = 2 Ale s graph = 2 wa! Morgan s graph = -2 wa Ale s graph = 2 wa Morgan s graph = -2 wa = 2 = 2 = 2 = 2 = m + b form. I graphed the - counted up 2, right and down 2, left to plot other = m + b = = m + b = = m + b form. I graphed the - counted down 2, right and up 2, left to plot other = m + b form. I graphed the - counted up 2, right and down 2, left to plot other = m + b = = m + b In the slope-intercept = form of a line ( = m + b), the coefficient of, which is m, indicates the slope. Changing the sign of m changes the slope, or the steepness, of the line. When a line has a positive slope, its height increases from left to right. When a line has a negative slope, its height decreases from left to right. = m + b form. I graphed the - counted down 2, left and up 2, right to plot other * How did Ale graph the line given b his equation? How did Morgan graph the line given b her equation? * Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? * What are some similarities and differences between Ale s and Morgan s problems? * What are some similarities and differences between Ale s and Morgan s graphs? * How does changing the sign of m affect the graph of a line?.5.5 * How did Ale graph the line given b his equation? How did Morgan graph the line given b her equation? * Can ou think of another wa that Ale and Morgan could have used to find the graphs of their lines? * What are some similarities and differences between Ale s and Morgan s problems? * What are some similarities and differences between Ale s and Morgan s graphs? * How does changing the sign of m affect the graph of a line?.5.5