Formative Assessment 84

Assessment Problem: 

If you have “n + 5”
• Can “n” stand for “4”?
• Can “n” stand for “37”?
• Can “n” stand for “3r + 2”? 

Common Responses: 

Can “n” stand for “4”?
Student: It could because letters are also used as variables and so n could be any number.
Can “n” stand for “4”?
Yes (56%, 77%, 87%)*
Can “n” stand for 37?
Yes (30%, 67%, 81%)*
Can “n” stand for 3r + 2?
Yes (26%, 30%, 47%)*
*Percent of 373  6th, 7th, and 8th grade students (respectively) in Weinberg, et. al’s (2004) study with that answer.

Mathematical Issues: 

One of the first cognitive obstacles for students to overcome when learning algebra is the idea that letters mean different things in different contexts. Students use their prior experience to make sense of variables and generalize their thinking. However, depending on the situation, the representation can confuse them or help them. They know, for instance, that p6 means page six. The “p” represents a word. “3a” could mean the first section of item number 3. “7m” can mean 7 meters, 7 times the variable “m”, or an abbreviation for 7 monsters. In algebra, “m” all by itself could mean 1 meter, an unknown number of meters, or have nothing to do with meters at all. Therefore, in an expression such as 10x – 1, students might think that it parallels “10m – 1” or “ten meters minus one is nine meters”. Students have a difficult time knowing when letters mean words and when they represent single numbers or when they represent multiple numbers or a range of numbers. Research identifies this stru

Domain: 
Variables & Expressions
Theme: 
Representation
Common Core Standards: 
6.EE Expressions and Equations
Research References: 
MacGregor, M.E. (1986). A Fresh Look at Fruit Salad Algebra. [Journal Article]. The Austrailian Mathematics Teacher, 42(3), 9-11. MacGregor, M.; Stacey, K. (1993). What is X? [Journal Article]. Australian Association of Mathematics Teachers, 49(4), 28-30. MacGregor, M.; Stacey, K. (1993). What is X? [Journal Article]. Australian Association of Mathematics Teachers, 49(4), 28-30.