Math Routines
Primary objectives of the activity:
- numeracy and arithmetic practice
- working flexibly with numbers
Secondary objectives of the activity:
- perseverance in challenging tasks ("productive struggle")
- collaborative problem solving
Sample work
Student Testimonials
- “It helped me to understand different ways of solving math problems.”
- “testing my patience and opening my mind to think outside of the box!”
- "It helped me challenge my brain."
- “It showed me to see math in a different way.”
Tips and Tricks
The keys to Four 4s are perseverance and collaboration. What may seem like an impossible task quickly becomes manageable when students start to see any possible solutions (or non-solutions). Stress that sharing strategies is a good thing; we learn from each other and can collectively evaluate shared solutions.
So for perseverance, continuously remind students that making mistakes is a good thing as we will learn what works (and what doesn’t) and be able to refine our strategy through that iterative process. So encourage students to try anything; start modeling examples and non-examples on the board; and give them time and space to really dig into the project without fear of being graded.
For collaboration, encourage students to not only share their solutions for each value but to question each other; does a shared solution work? Why or why not? And as students find success with a few of the easier values, they will gain confidence and excitement as they strive towards finding expressions for all 20 values.
How to implement
- Face to Face
- Allow students to work in groups and encourage them to share possible solutions. At the start, give students the task and give them a few minutes to just puzzle; ask probing questions like “What are the rules of this puzzle? What is a possible solution for any value?” After a solution is presented, place it on the board and ask students to evaluate it → “Does this work? Why or why not?” Be ready for students to share solutions that equal the final value but don’t follow the rules (i.e. they may use more than four 4s). These scenarios are great teachable moments since we can facilitate student-led discussions around figuring out what works and how to defend our answers. If calculators are available, encourage students to use them to test the computations (so they can focus more on creative solutions/problem solving and not just computational practice).
- Blended
- This puzzle can work very well in a blended setting where students have access to the internet/devices. Free apps like PhotoMath can be used to test solutions. And most relevantly, a well-structured Google Doc/Sheet can enable students to share solutions in real time while the teacher can use colors to help students visualize what works (and what doesn’t) as they try to figure out why. (sample doc | sample doc (for similar game, “Five 5s”).
And again, the teacher’s role here is to motivate, share tips, and help students persevere (while adding some useful feedback along the way). The first few minutes are always confusing for students, so be patient as they build confidence in sharing ideas and working together in small groups. - Distance
The biggest challenge of this puzzle in a distance setting is that it feels overwhelming (and nearly impossible at first). So students that are working alone at home have a tendency to give up before they see prospective solutions and build confidence along the way. In our distance program, we use 3 strategies to combat the perseverance challenge: - teacher “think-alouds” are videos recorded by the teacher to help students understand how to get started with a task. In a face to face setting, early confusing can be quickly clarified through live discussion; when it’s online, students need a resource like a short video of the teacher helping them understand what’s being asked and how to begin
- coaching into productive stretch. one great aspect of this activity is that it is a “low floor, high ceiling” task, which means it is “easy to start and hard to finish.” That means high level students can find almost infinite ways to challenge themselves and make the task harder and harder. Low level students can quit at any point they feel “overstretched”. So if students turn in an assignment with only a few solutions, that’s okay! They can pause and return to the puzzle for a few minutes everyday.
- google as our “classmates”. on campus, students can bounce ideas off each other and work collectively through the struggle. That’s not always possible online. So in this case, we encourage students to use Google. Students will often think this is “cheating” as there are tons of solutions to this puzzle freely available with a quick search. So our role here is to help students understand the goal is not “completion” (i.e. finish all 20 as quickly as you can and turn it in) but rather understanding. So if a student is stuck, we want them to Google and find possible solutions. And when they do, the key is to spend time evaluating it (i.e. does it work? does it make sense to you? only use it if and when you’ve made sense of it yourself).