This problem encourages higher order thinking and the use of problem-solving skills. It introduces algebraic concepts at their most elementary level, and gives the students the opportunity to use manipulatives in order to solve. Although this problem is a pasture problem and relates more to students living in rural or more agricultural areas, the context of the problem could be altered to suit a different audience or demographic and thus make this mathematical concept more relevant to students.
Topic: Algebra
Grade: 4-6
Expected Heuristics: Act it out, draw a picture, use logic
Problem: Pasture
There are cows in each pasture. The pastures are connected by bridges. The number on each bridge tells the total number of cows in the two pastures it connects. How many cows are in each pasture?
Answer: Purple—4, Orange—3, Yellow—3
Discussion: Using manipulatives to “act it out” would be the most effective way to solve this problem, although drawing a representation would work as well, or one who does not need the use of visual aid may be able to simply look at the picture and see that if there are 4 cows in the purple pasture, and three in both the yellow and orange pastures, that equates to seven cows between both purple and yellow, and purple and orange, and six cows between yellow and orange.
Source: I snagged this problem from Session 4 of the Math Conference, Using What We Already Know to Solve Multi-Step Equations presented by Alfreda Jernigan.
(1990). Pastures 3. Algebra thinking: First experiences. Creative Publications.
Problem Teaching Critiqu/Reflection:
I taught this problem to a group of four 4th grade students at Waller Mill Elementary School. Students were really confused at first, while I thought the directions were fairly straightforward, they needed some additional guidance and modeling. The numbers on the bridges they weren’t sure what to do with even though it seemed to be pretty evident from the directions I gave, but the students were trying to subtract, add, or multiply by the bridge numbers even though the directions say “The number on each bridge tells the total number of cows in the two pastures it connects.” I still had to explain that more, I’m not sure how I could have been more clear in the directions. I had colored dienes unit cubes for them to use, 3 of the four students used the cubes, to help them solve the problem. The other student preferred to use paper and pencil and reasoned it out using logic. I have learned that the next time I will have to do a separate but similar problem in order to model for them first so they can better visualize what the directions mean. Additionally, I’ve learned that fourth graders are capable of learning and practicing algebraic concepts in their elementary forms! Also, I don’t observe these students working with manipulatives in math; they were extremely excited to get to work with dienes materials for the problem solving work here. I realized just how much these kids love math and how much better they can perform in math given the chance to work with manipulatives.