I like Zip Zap Zowie because I thought the language was catchy, it lends itself to using visual representations, and it requires algebraic thinking. It’s also open-ended in that it has multiple solution pathways, which is important in my classroom. I need activities that everyone has a shot at, as the prior knowledge and ability levels of my students are all over the place. It is also a nonroutine problem that requires strategic thinking and reasoning. It has a low-entry point but a high ceiling, so students at all levels in the classroom can potentially solve this problem. It also requires that students persevere and make multiple decisions and take multiple steps to solve the problem.
Zip Zap Zowie is also the core problem in the CUNY HSE Curriculum Framework, Unit 7: Equality. Mathmemos and CollectEdNY contributor Mark Trushkowsky offers some additional background on the problem:
“Equality is a fundamental concept in algebra. It is noted through use of an equal sign, represents a relationship of equivalence, and can be conceptualized by the idea of balance. Despite the importance of the equal sign, there is tons of research that shows that students often have serious misconceptions about what the equal sign means. The research has mostly been conducted with elementary through high school students, but . . . I have found it is equally true of adult learners.
Zip Zap Zowie describes the relationship (in terms of weight) between four kinds of imaginary objects: zips, zaps, zowies, and swooshes. . . . This problem is a good way to draw out “working backwards” as a problem-solving strategy. This is the most common path students find to a solution, though it may take many students some time and trial and error before they come to that strategy.”
You can read the rest of Mark’s suggestions for teaching the problem here, beginning on page 147.
How I Solved It
I solved the problem by substituting variables for the zips, zaps, zowies, and swooshes. I let A = zip, B = zap, C = zowie, and D = swoosh. I then plugged the variables into the algebraic equations as follows:
A = 3B
2B = 5C
3C = 2D
I started at the bottom and worked backwards and plugged the known value of D into the equation in order to solve for C:
3C = 2(60)
3C = 120
3C = 120
C = 40
I then plugged the discovered value of C into the equation in order to solve for B:
2B = 5(40)
2B = 200
2B = 200
B = 100
Now that I knew the value of B I was able to plug that into the equation to solve for A, the zip we are looking for:
A = 3(100)
A = 300
So, a zip weighs 300 pounds.
Anticipating Student Thinking
The problem could be solved using a visual model, either by drawing pictorial representations or by using manipulatives. I was interesting in solving it using pattern blocks, as I use them frequently with my students when exploring algebraic thinking and fractional relationships. I grabbed the bag of pattern blocks and immediately choose the trapezoid to represent the zip and 3 triangles to represent the zaps, as there was a definite relationship there that I recognized from my prior work with pattern blocks. However, as I moved to the next equation set I quickly realized that my former understanding of the relationships between the different pattern block shapes would not work with this problem:
I had to readjust my thinking to view the pattern blocks as simply a visual representation to stand in for how many I had of each value, rather than as representing proportionately what each value is. I noticed that I processed this differently than when I used abstract symbols, as the manipulatives definitely helped to give me more of a concrete sense of what to do with the numbers in the problem. I could “see” the solution, as pictured below:
My Goal for Student Learning
I want my students to build confidence in their problem-solving abilities. Since the implementation of the TASC, I’ve heard numerous test-takers express how inadequate they felt when they opened up the mathematics test booklet on exam day. I want students to explore and practice, practice, practice a variety of problem solving strategies so when they see something that looks different than what they may have seen in class, they don’t give up before they start. I would like students to develop a routine where they start with what they know, organize that known along with the unknown, and take small steps to solve the larger puzzle.
This particular problem lays groundwork for building understanding of problem solving using algebraic thinking. On the TASC test, students may be asked to set up a problem or they might be asked which set-up is correct for a situation.
Supporting Productive Struggle
Even though the language of the problem may be fun, students may have difficulty solving the problem if they stick with zips, zaps, and zowies, as they sound so much alike. I plan to have them consult our student-created problem-solving strategy list before they start to solve the problem, and once underway, if a student seems stuck, I may refer them back to the list, which suggests using visual models, drawing a picture, or writing an equation. If they still seem stuck I might specifically ask, “Can you draw a picture to represent the situation?”
Another challenge students may face is what I experienced when I used the pattern blocks to solve. While I immediately recognized that the pattern blocks could serve as non-proportional visual representations of the problem, students may have difficulty making that transition as they have become accustomed to using them primarily to represent fractional parts of a whole, be it while studying fractions or algebraic expressions and equations.
If students finish the initial task early, I will ask students to create a real-life situation using the same equations as stated in the zip, zap, zowie problem, when D ≠ 60.
A = 3B
2B = 5C
3C = 2D
This could even be a group project that can be put on chart paper and posted on the wall. I would have some students work on the equality assessments in the HSE Curriculum Framework. And for explicit TASC connections I can have ready pages 18 & 19 in Scoreboost, Mathematics, Algebraic Reasoning, by New Readers Press.
Jackie solved the problem fairly quickly using division and addition, and mental math.
First, she deconstructed the problem by rewriting it using the equal sign to organize her information. Then, she took her known value of one swoosh and doubled it. She then used division to determine the value of a zowie.
What I find interesting is her next step. She had determined that a zowie was equal to 40 lbs and instead of multiplying the 40 x 5 to compute the value of 5 zowies, she instead used repeated addition, along with mental math to arrive at her total. Again, she used division to compute the value of one zap and finished by using mental math to compute the value of a zip. As no one in the group had drawn a pictorial representation of the situation, I asked her to explain her thinking by drawing a picture on the SMARTboard, while talking us through her solution pathway.
I like Jackie’s work because what she had written on her paper showed some organization and one of my goals in presenting this problem to my students was to make explicit the process of deconstructing a problem and organizing the information. What I like about Jackie’s pictorial representation of the problem is the way her thinking bridged to abstract representation of the situation, because instead of drawing 2 squares to represent the two zaps or 3 diamonds to present the three zowies, she used a number right next to each object to represent the number of objects. Her picture and explanation provided the perfect segue into a whole class discussion on using variables to stand in for numbers, as the pictures do in her example, as well as what it means when we see a number (coefficient) right next to a variable.
Most students talked about “working from the bottom up” and seemed to start the process similar to Jackie, but several students got lost along the way. Even those students who solved the problem found it difficult to retrace their steps and explain the process they went through, as did Jackie.
Like Jackie, Cindy began by organizing the information, and working from the bottom up. She started by taking the known value of a swoosh and doubling it, demonstrating that she understood that 2 swooshes would equal 120 lbs. However, she demonstrated a lack of understanding of the equal sign by what she did next. Instead of using division to find out what a zowie would weigh, she took the numbers that were available to her and she continued to multiply. She reasoned, if 2 swooshes is 120 lbs, that means that zowies weigh 120 lbs, so moving up to her next set of numbers she multiplied again, reasoning that 5 zowies would be 600 lbs. She then figured that 600 lbs was the weight of one zap, coming to the conclusion that the weight of one zip was 1800 lbs.
I’m interested in Cindy’s work partly because I didn’t anticipate a student making this mistake and partly because it was Cindy who made the mistake. Cindy is relatively young and remembers numerous procedures that she learned in High School, so her math skills are a bit higher than most students who enter my class. However, she many times jumps to a procedure before she thinks about what the problem means, and she has some difficulty sticking with a problem if a simple procedure doesn’t work.
As soon as Cindy jumped to the 1800 lbs conclusion, she asked me if she was correct. I asked her to explain how she got to 1800. As soon as she started to explain it to me, she immediately saw her first error that sent her down the wrong pathway and went about solving the problem by using multiplication and division where appropriate. It really underscores the task we have in helping our students to develop metacognitive skills, so they can develop the habit of asking themselves, “Does this make sense?”, instead of asking the teacher, “Is this the right answer?”
Craig got started on the right foot. He took the known weight of the swoosh and doubled it to inform him of the combined weight of the 3 zowies. He then divided the 120 by 3 which gave him the individual weight of one zowie: 40 lbs. That done, he multiplied 5 x 40 to get the weight of 5 zowies, which informed him of the total weight of 2 zaps: 200. He then divided that sum by 2 in order to find out the weight one zap: 100. Here’s where he lost his way. Instead of multiplying that 100 by 3, so as to find the weight of a zip, he instead began to divide the 100 by 3. He halted that process early on, though, I suspect because he realized that his answer wasn’t making sense. At one point during the time when students were working through this process, he asked me if the answer was 300. I asked him to explain to me how he got the 300 and he was unable to retrace his steps without working the whole process again.
I thought Craig’s work was a good example of what can happen if a student jumps right into manipulating numbers without first organizing the information. He was on the right path but lost track of where he was and when asked to defend his answer, he was unable to do so.
One thing that became clear to me as my students began to work on this problem, even though I had observed it before but not clearly defined it, is that I need to step outside my expectations of individual student understanding and learning. While many of my students exceed my expectations and surprise me with mathematical thinking dissimilar to my own that creates new learning for me, there are those few low functioning students that just don’t get it. They usually end up following along as others are solving problems and copy what they see the others doing. They may even be able to parrot a solution pathway, but I know that they haven’t really learned how to solve the problem themselves.
What I Might Change
I need to identify those students who I know will have problems deconstructing the problem, and work individually with them, perhaps away from the other students during the lesson or at a later time. Just as I need extensions of the activity for students who finish quickly, I need follow up on those students who don’t finish at all. There are two students who come to mind. One of these always jumps into verbally expressing an opinion without understanding the conditions of the problem. The other silently struggles, but both need help in separating the information into something that is manageable so that they can make sense of it.
I didn’t expect that students would have as much difficulty as they did with staying on track. They would almost get to the solution and then deviate from what was working. In analyzing what might be common to the mistakes that students were making, I tied it to a lack of understanding equality. In a hurry to solve the problem, students jumped to an operation or switched operations without giving enough thought to how that would or would not make both sides of the equation equal.
When students were stuck on this problem or had the wrong answer, asking “Where did you start?” or “What did you do first?” seemed to help get the students back on track. Those simple questions, an encouraging comment to keep moving forward, and then walking away gave students a chance to regroup. This worked even with students who had already pushed themselves away from the table and decided to give up. Strategies that seemed helpful to students were: organizing the information in a way so that they could understand equality and talking through the information that they had accumulated.
I noticed that a student who, rather than giving up on a problem he is struggling with (as was typical for him in the past), persisted and was able to prove his solution pathway verbally to the class. While he stopped short of getting up in front of the class to share his solution strategy when I invited him to do so, he did talk me through his solution pathway so I could write it on the SMARTboard as he explained how he solved the problem. I’m also hoping that our discussions that focused on organizing the information in a problem will carry over when next the students work on a non-routine problem.
I know that some students got what I wanted them to get, as was evidenced from the follow-up real-life extension activity that most students participated in as groups. Sample photos of these extensions are below, and you can download some of my reflections on their work here.
Advice for Teachers
Use this problem as an opportunity to introduce or follow up on: (1) understanding equality and how this may be presented in a word problem (2) understanding the inverse relationship between multiplication and division (3) deconstructing problems and organizing information. Give students plenty of time to work on this and opportunity to extend their understanding of equations in real-life situations.