This string was developed by our colleague Bill Jacob, University of California, Santa Barbara, who is, among other things, an algebraist. The post is written by our colleague Monica Mendoza, University of California, Santa Barbara who leads a summer algebra institute for teachers as part of her work at The Center for Mathematical Inquiry with Bill.
This early algebra string emphasizes landmark numbers and the relationship between addition, multiplication and division.
The ability to read an entire expression for meaning before computing is an essential algebraic skill. Students will likely notice this as the string progresses and hopefully make use of it as a helpful strategy. For example, in the problem (54 + 54 + 54 + 54) ÷ 4 there is no need to add the 54’s and then divide by 4 (an arithmetic strategy). Instead we hope students notice the structure of the problem — that there are four 54’s and they are divided by 4 suggests an “undoing strategy” (an algebraic strategy) rather than sheer computation.
The string also encourages students to “look inside” a number to see how it is composed and how it is related to other numbers at play. In the fourth problem the 68 is equivalent to two 34’s and therefore there are three 34’s within the parentheses. This idea is a beautiful illustration of what it means to “look for and make use of structure” (Standards for Mathematical Practice, CCSS-M). Recall that this practice standard encourages students to:
look closely to discern a pattern or structure…… step back for an overview and shift perspective….. see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.
Enjoy the string. Model students’ thinking on an open number line — to underscore the idea of equivalence.
34 + 34 + 34 =
334 + 334 + 334 =
(54 + 54 + 54 + 54) ÷ 4 =
(34 + 68) ÷ 3 =
(150 + 27) ÷ 3 =
Check out Bill and Cathy’s book Young Mathematicians at Work: Constructing Algebra if you are eager to see more. Or find out more about the The Center for Mathematical Inquiry