Problem Based Learning in Math and the 2018 Question

Fanny Sosenke, Middle Grades Math Teacher

Every year I tell my students that I have a challenge for them. I tell them that my past students have been able to use the digits of the current year (in this case 2, 0, 1 and 8) to generate expressions equivalent to the numbers 1 through 100.

The “2018” problem is a good example of how problem based learning (PBL) happens in 6th grade math. In a PBL based math classroom, the focus is on unique problems that teach content and skills in a student-centered manner. The problem is what motivates students to learn mathematics by encouraging them to be curious and inquisitive.

The main goal of “2018” is to look at order of operations and decimal operations. However, this problem always turns into an activity in which students, hungry to be able to complete the challenge, push themselves to learn concepts that are not part of our regular curriculum.

2018 PBL Day 1

Results at the end of the first day

When the activity starts, students concentrate on making sure the equations follow order of operations and are mathematically correct. For example, when a student offers a solution to “2018” as 2 – 1 x 8 + 0 = 8, we have a whole group discussion on whether that answer is correct or not. Students realize that the first student wasn’t following order of operations and suggest that he or she insert parenthesis to make the equation true. When we follow order of operations, we have to multiply 1 x 8 first, so your answer would be -6 instead of 8. The correct equation to get 8 is (2 – 1) x 8 + 0 = 8.

After students work on “2018” for a while they always run out of answers. When that happens I tell them that there are way to get more answers but they need to learn math that is not taught in 6th grade. That means I’ve “caught” them! They often say: “What do you mean math that is not 6th grade math?” And that’s when students realize that solving problems sometimes means having to get out of their comfort zones and stretch their minds. I mention to them that there is something called factorials and that factorials might help them get more answers. I purposefully do this at the end of class. There is always one student in each class who comes back the next day and asks me, “Can I share what factorials are with the class?” And this opens many more options to solving “2018.”

2018 Day 2 student working at whiteboard

A students works at the whiteboard on day two of the challenge

When they get stuck again my question to the class is, “Do you know what happens when you divide a number by 0.1?” We have a whole group discussion about how to divide numbers by powers of 10 (10, 100, 0.1, 0.01). Students explore this by changing 0.1 to a fraction and investigating dividing decimals by small numbers. Once they realize that dividing by 0.1 makes a number 10 times bigger I tell students that for 2018 they are allowed to use the “1” as a “.1” Now students start getting 80 by dividing 8 by 0.1!

The expression sheet at the end of the second day

At the end of day two, spaces still remain

The final stretch involves a third suggestion. Students now have about half of the answers to “2018” and they are stuck again.  My new question is, “Have you heard of trigonometry? This is something you will learn in 10th grade.” Again, there are always one or two students who come the next day and tell the class, “Did you know that tan-1(1) = 45?”

Through this problem, students are presented with a problem that has an entry level for all students but that also stretches them to grow. I asked David what made him be so eager to find the answers to “2018.” He said: “I liked how this problem had many different answers. I like that I didn’t know all the math needed to solve it, and I liked a challenge that might not be solved.” By persisting and pushing themselves, students go further than they would with a teacher telling them the answers.

100 expressions filled out in 2018 problem sheet

After stretching beyond 6th grade math, students provide all 100 expressions!

Are you ready for 2019? What about for the Jewish year of 5,779?


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