Introducing Equations with Multiple Variables

A little over a year ago, I was privileged to attend a conference session led by Linda Levi (see her blog listed to the right). In it she showed students' work in developing place value understanding, which extended into an understanding of decimals. This understanding was further extended into experiences with algebraic expressions and equations with multiple variables. The sample student work was impressive and I was particularly fascinated by the high level of understanding in one fifth grader's work as he found multiple solutions for the variables. At the time my daughter was nearing the end of her third grade year and I knew that she was definitely not ready for this work yet. Still, I was fascinated by the question of what it would take so that she would be ready by fifth grade. What would need to happen? How does one make that "leap"? My learning journey in regards to this question is still in process, but one of the things I did was to purchase and read, Thinking Mathematically: Integrating Arithmetic and Algebra by Thomas P. Carpenter, Megan Loef Franke, and Linda Levi. I highly recommend the book. I will be mulling it over for some time to come.

I belong to a group that gets together each summer to explore aspects of Cognitively Guided Instruction. Before we begin our 2 weeks together, we articulate our goals and expectations for the experience. This year, my focus was on the development of algebraic reasoning and sense-making tied to fractions, decimals, and percents. In this blog entry, I will be describing an experience in which I attempted to use what I had learned from Linda Levi's conference session and the book Thinking Mathematically to plan and conduct an interview with an incoming fifth grader that would introduce him to algebraic expressions (including decimals!) for the very first time. His classroom teacher was my co-interviewer and she told me that decimals had been introduced during the previous year, but that it had been "rough going."

In planning the interview, I knew that I wanted to begin with ideas that were in his schema. He was a student in a school that fostered the use of Cognitively Guided Instruction (for further reading in this area, get Children's Mathematics: Cognitively Guided Instruction by Thomas Carpenter et al). Therefore, the children were accustomed to a problem-solving curriculum and they had been exposed to a lot of problems that fostered place value understanding. In addition, they were accustomed to formulating their own strategies for problem solving, based on their personal sense-making. I believe this is critical in understanding the student's success in this interview. If he had been a student in a passive learning environment, he could have quite easily simply told me that "we haven't done problems like that yet, so I don't know how." After beginning with two problems involving multiplication with groups of tens (6,10, and 13 leftovers) and a measurement division problem with groups of ten (87, 10), I left the contexts behind for a time and asked him how many tens were in 57 and 243. Thus far, none of this was a problem for him. He easily answered that 243 had 24 tens. Initially, he described a trick you could use, by covering up the three and then you would find the answer. We probed further asking him how he could convince me that it would work, using mathematical reasons, rather than tricks. He was able to do so, saying "Ten fits into 100 ten times, so it fits into 200 twenty times, and then the four tens in forty." His teacher now made an unplanned suggestion to our interview protocol, which I think was critical to his success later in the interview, because it caused him to spend considerable time "grappling" with place value concepts outside of his familiar number range. She said, "How many tens are there in 1,243?" He knew that there were 24 tens in the 243, so he just had to figure out how many tens were in 1,000. This took him some time. He was then able to tell us that it was 100, but it took him further think time (and building a base 10 representation) to be able to explain it with mathematical reasoning, saying, "There are ten hundreds in a thousand and ten tens in a hundred and 10 x 10 is 100."

After this initial opportunity to bring forth his place value understanding (this was two months after the end of the school year), we moved into fractions. He was more familiar with fractions than decimals, having worked with them longer. In addition, it allowed us to get him to think about tenths, which would translate easily to decimals. "Sam eats 1/10 of a pound of fudge a day. How many days would it take him to eat 3 1/2 pounds of fudge?" After a pause, Chris said "How many tenths are in 3 1/2. I think that's the problem." He started to model it by drawing 3 circles and a semicircle on his page, which he told us represented the pounds of fudge. He started to partition the first circle into tenths, but before he completed it, he told us that it wasn't necessary because it would obviously be 10 days, and so then 30 days for the 3 pounds of fudge and the half pound of fudge would be 5 days, since 5 is half of 10.

We switched to decimals at this point, but scaffolded this newer, more unfamilar territory by sticking with tenths. "An animal at the zoo eats .1 of a pound of food each day. If the zookeeper has 36.8 pounds of food for this animal, how many days can she feed the animal before the food runs out?" Chris spent considerable quiet think time before writing anything on his paper. Then he wrote:
10 X 30 = 300
10 X 6 = 60
360 + 8 = 368 Days

We asked him why it made sense to multiply ten. He replied that the animal eats .1 per day and it takes ten to make a whole. " So like the 6 in 36.8, it would take ten tenths to make a whole and then you would have to do that 6 times. We asked him how he knew to add 8 days. He again pointed to the .1 per day, saying. "Obviously, you would have to have 8 days to eat .8.

At this point we decided to take the plunge into equations with multiple variables. We began with a fairly easy one (we hoped), given his place value understanding. I should point out that the use of parentheses was a familiar one to this student.

(m x 10) + c = 65

I asked Chris if he had seen anything like this before. He said, "Yes, but I can't remember what m is." I told him that this was what these problems were all about; he was to suggest a number combination that would work for m and c. He did not have to think very long at all before suggesting that m be 6 and that c be 5. So we decided to try a more challenging problem for him, particularly in light of his earlier struggle with the number of tens in 1000. This was where we were really able to see the outcome of his previous cognitive processing.

6345 = m + (b x 100) + (y X 10)

You might stop at this point and try to think through for yourself what you would suggest as feasible number combinations for m, b, and y. Chris thought about this one longer than the first one. After a period of time, he wrote 3 under b, then 4 under y, and then 6005 under m. He told us in the explanation phase that he thought of m as "the leftovers after I have b and y". We told him that with these kinds of equations which have multiple variables, usually we can think of several different number combinations that would work, and challenged him to try to find another. He wowed us with his flexibility in place value understanding when he came up with this option:

6345 = 45 + (60 x 100) + (30 x 10)

His understanding that 60 hundreds would be 6000, given the struggle he experienced earlier in the interview with the number of tens in a thousand, indicated to us that the cognitive struggle he had experienced had paid huge dividends in his understanding. We asked him to try one more number combination. He then wrote:

6345 = 345 + (30 x 100) + (300 x 10)

He explained that he had decomposed (yes, this is part of the mathematical language in his school) the 6000 into a two 3000's and expressed one of them as 30 hundreds and one as 300 tens. He was able to explain the sense in this using language similar to what has been expressed earlier in this entry, which is rapidly becoming lengthy. We still had 15 minutes remaining to our scheduled one-hour interview, so we gave him the "finale"--an equation with multiple variables AND decimal numbers.

576.25 = (100 x b) + c + (.01 x p)

He had never seen one one-hundredth represented as a decimal before, so we had to tell him what it stood for. We pointed to the number 576.25 and asked him to show us how he would say this. Given that decimals were still fairly new to him, we wanted to ensure that he knew what the number was. He stated the number correctly and immediately recognized that p could be 25, "because I have 25 hundredths". He then wrote the following:

576.25 = (100 x 5) + 76 + (.01 x 25)

I was thrilled with this interview experience. I don't think it would work with all 5th graders, but I could see how children who have built strong place value understanding AND developed their sense-making skills could experience a highly scaffolded interview, which starts with ideas that clearly part of their schema, then creates cognitive conflict, and then extends their new understandings into gradually less familiar territory, could be used to help students use their own thinking and understandings to make sense of algebraic understandings.