I'm Done, Now What? Board

My biggest challenge when I started teaching four different math levels within one class was making sure that students were not wasting time waiting for me while I was working with others. As I struggled with this at the beginning of my first year, it became clear that I needed to create a system to address this. The system needed to have differentiation built in and students had to be able to use it independently of me during class time. 

Preferring a minimalist classroom, I still had one blank bulletin board because I hadn't yet found something meaningful enough to put up on it. And so was born the "I'm Done, Now What?" board. 

I laminated sheets of colored paper to turn them into dry-erase boards and strung them together with small hooks. Every week, I choose activities that either serve as enrichment to what we are currently working on in class, review previous knowledge, or prepare students for topics to come. The levels are color-coded rather than written out to avoid stigma and make students feel more comfortable about being on their individual learning path. I put necessary materials in baskets beneath the board to make them easily accessible, and any necessary links are emailed to students at the beginning of the week. 

This system has worked well so far: when at student finishes their work, they simply move on to an activity on the board. Once the system has been explained to them and they know the color they should pick from, students are completely independent. As a result, wait time has been cut out almost completely, and students are more productive in class. It's so important to make every minute count!

Magic V

Why I like this problem: This is another problem with a lot of variation, since the arms of the V can be extended to include more numbers, and the range of numbers can be changed. 

Observations from class: To make trial and error easier and more pleasant, I printed copies of the V and put them in dry erase pockets. While some students actually tried to find all of the possible solutions, a few remembered the Chipotle Combinations problem we did the week before and applied the strategies they developed in that problem to this one to figure out how many possibilities there were. I hadn't even thought of this link myself, so I was pleasantly surprised to see the transfer of strategies. 

Follow-up questions I asked: 

1. Can you figure out a pattern to make finding all the possibilities systematic?
2. How does the first number of the range affect the solutions? 
3. Does it matter if the range starts with an even or odd number?
4. How does changing the arm length affect the number of solutions you have?

Required skills / content: Adding numbers. 

Links: PDF 

Source: https://nrich.maths.org/

Chipotle Combinations

Why I like this problem: This problem can be used as an introduction to combinations and permutations, or simply as a brain teaser. I like that students needed to think about the possible combinations in an organized manner and keep track of the meals they had accounted for to avoid counting them multiple times. Students are also easily able to connect with the problem since they all know the fast food chain. 

Observations from class: Students easily figured out that there are 24 possible variations of the major options (type of meal and "meat"), but slowed down when they reached the toppings. I could tell that they would become frustrated with the amount of options they had for the toppings, so I tried to hint to them that "only taking one topping" and "leaving out only one topping" were essentially the same thing probability-wise and would have the same number of outcomes. Some students also struggled to understand why order of toppings did not matter in this case, and how this affected their answer. 

Follow-up questions I asked: 

1. How many more possibilities do you have if you put multiple salsas or mix the meat options?
2. What do you notice about using 13 vs 1 of the topping offered? What about 12 vs 2? Etc.
3. How would the meal possibilities change if the restaurant allowed multiple servings of the same toppings?

Required skills / content: Combinations, permutations, systematic organization. 

Links: PDF 

Source: https://www.yummymath.com/

Units and Measurements Jumble

Despite the fact that students learn about units of measurements in elementary school, I find that many still do not understand their importance. Units often get forgotten at the end of problems, and some students still haven't internalized the relationship between units and the quantities they measure (capacity, length, etc). When I came across it, I immediately felt that is would be an engaging way to tackle this issue. 

Why I like this problem: This problem gives students the opportunity to practice dealing with an overwhelming amount of information. With 132 different labels to organize, students are forced to find a method to keep things manageable. In addition, it is a great review of estimation, units and conversions. 

Observations from class: I combined the three versions of the activity to make it more challenging. Students worked in teams of 3-4. It was interesting to see how students naturally assigned themselves roles. This task created a lot of opportunity for discussion and students were engaged. Different groups used various techniques to organize themselves and the labels. 

Follow-up questions I asked: 

1. How did you start organizing and matching the labels?
2. What criteria did you use to match categories with units?
3. Are there any categories that could have multiple answers? How are you accounting for these?
4. What is your reasoning for matching labels together?

Required skills / content: Units, conversions, estimation, research, organization.

Source: https://nrich.maths.org/

Can You Traverse It?

Why I like this problem: Although I chose it as a stand alone activity, it is a great introduction to graph theory for those who teach it. I liked the brain teaser feel of this task and that it is accessible to any level, even those who have no previous knowledge of  graph theory. 

Observations from class: This task really feels like a game, and ALL of my students were excited about it. Many of them had a similar game on their phones and were familiar with the concept. I had students insert the pages into page protectors and use dry erase markers to try out graphs to keep them clean in case it took several tries to traverse them. Some students got wrapped up in solving the graphs and forgot to keep track of where they started and ended, so they had to go back at the end to make those observations, which was time consuming. 

Follow-up questions I asked: 

1. How did certain characteristics affect the outcome for each graph?
2. For graphs with a center vertex, how would the graph be affected if that vertex was placed on the outside?
3. Have you counted the number of vertices / edges / edges coming out of each vertex?

Required skills / content: Following paths.

Links: PDF / Google Doc

Source: https://nrich.maths.org/

Order of Operations

Why I like this problem: This problem is a great review of order of operations and number sense. It gives younger students a chance to practice skills they have freshly learned, and older students can sharpen critical thinking skills by planning out the solution and arriving at the largest or smallest value more efficiently than trying random orders.

Observations from class: I had to clarify to students that the multiple rows were only there to give them space to try various orders and that they were not required to fill them all out if they found the answer. I also collected students' answers on the board to keep track of the current "record". Students worked to beat the value on the board and updated it when they obtained a higher value. This created healthy competition and helped students figure out how close they were to reaching the goal of the task.

Follow-up questions I asked: 

1. Can you make the value larger / smaller?
2. What strategies did you use to figure this out?
3. Are there multiple ways to obtain the same answer?
4. Are there any numbers that must be put in a certain position?
5. Are there any numbers that you know will not lead you to the answer? How do you know?

Required skills / content: Order of operations.

Links: PDF / Google Doc

Source: http://www.openmiddle.com/

The Triangle Game

Why I like this problem: It gives students the opportunity to play a game before tackling the critical thinking, which allows them to gradually immerse themselves in the problem of the week. The rules of the game are at first a bit convoluted, but get much clearer once you actually start playing.

Observations from class: At first students got hung up on the rules, so I ended up skipping over the instructions and explaining it to them on the board. I may leave them out altogether next time I use this problem. I used the following diagram to summarize the restrictions on vertex labelling:

Students quickly discovered that color-coding triangles according to which player got the point as they went along made scoring at the end much easier. In addition, some groups chose to use a third color for triangles that did not receive any points.

I found that occasionally reminding students that they could only use two letters on the sides saved us a lot of time because many played the game wrong the first few times, which caused them to have less time to dive into the real thinking.

Follow-up questions I asked: 

1. What attribute about being the first player do you think contributed to them winning all the time?
2. How could you change the game to test your hypothesis?

Required skills / content: Figuring out patterns, counting.

Links: PDF / Google Doc

Source: https://nrich.maths.org/

Number Properties Puzzle

Why I like this problem: This puzzle was a great review of basic math concepts yet challenging enough to keep students attention throughout the entire class period. I also like that it involved manipulating the labels and numbers, making it a different type of activity and engaging for tactile learners.

Observations from class: Many students did not know what a triangular number was and had to look it up. I offered them bingo chips and explained that they are numbers that can be shaped into triangles. Some used the chips to find a rule to calculate triangular numbers and others kept adding rows and recording the number of chips they had.

Follow-up questions I asked: 

1. Why did you choose to put the labels the way you did?
2. Are there any labels that must be in the same sides? What about the opposite sides?
3. Were there any numbers that must be placed at certain label intersections?
4. What knowledge did you use to place or eliminated number placements?

Required skills / content: Math vocabulary (factor, multiple, prime number, square and triangular numbers), understanding two-way tables.

Source: https://nrich.maths.org/

Pennies and Quarters

Why I like this problem: This problem is fairly straight forward and a good break from the harder ones. Students each get a different answer, which allows for class discussion afterwards.

Observations from class: I had assumed that all students would know their height and weight but they didn’t! I ended up having to ask the school nurse for a scale, and the problem unexpectedly became an exercise in measuring oneself with a measuring stick. I will come prepared for this next time I assign this problem. As students made decisions on which they would rather have, we kept a record on the board, which gave students data to formulate conclusions.

Follow-up questions I asked: 

1. What mint-year did you base yourself on for the pennies? Would your answer be different if you had chosen another year?
2. Can you make a generalization regarding which option is more advantageous based on body type?

Required skills / content: Understanding height and weight, measuring height, the value of a penny and a quarter, multiplication.

Links: PDF / Google Doc

Source: http://www.wouldyourathermath.com/

Route to Infinity

Why I like this problem: It is a great review of coordinate plane attributes. Many avenues can be taken to solve the problem, and the questions of increasing difficulty make it ideal for all levels of thinking.

Observations from class: Seeing the animation on the Nrich maths website was very helpful to students. Without it, some of them thought that the path remained in the 6X6 square pictured in the poster. Seeing the animated version showed students that the area covered was not restricted, and extended to coordinates (1,7) and (7,1) and so on. A lot of students were intimidated by the last task of finding the 1000th coordinate and did not attempt it. They enjoyed trying to figure out the patterns of the points that will be reached.

Follow-up questions I asked: 

1. What patterns do you notice?
2. Can you find a system that will allow you to predict the nth point that will be crossed?

Required skills / content: Understanding of how coordinate planes work.

Links: PNG / Google Doc

Source: https://nrich.maths.org/

Hercules' Hand

Why I like this problem: It is very open to interpretation and students can bring in a lot of outside knowledge. It ties together many different subjects and can be incorporated in mythology, History, architecture and art lessons. This problem is also great for developing research skills.  

Observations from class: Once students figured out that using the internet for more information would be a good strategy (and was allowed), I had a hard time controlling their tendency to look up the answer. When I assign it again, I will only have a few computers available (as opposed to a Chromebook per student) to ensure that students only use technology for relevant research and to make it easier to keep tabs on the pages they are using for research.

Follow-up questions I asked: 

1. What assumptions did you make? 
2. What background knowledge did you use?
3. Why did you decide to use this method?

Required skills / content: Research, measuring, proportions.

Links: PDF / Google Doc

Source: Christian Courtemanche

Area Fractions

Why I like this problem: This problem can be solved in many different ways, which makes it accessible to all types of learners. 

Observations from class: My class found at least 7 different ways to solve this. Some used algebra, others took out scissors and still some others decided to measure and calculate the area to approximate the fraction. 

Follow-up questions I asked: 

1. Are there any other ways you could prove that your answer is correct?
2. For those who know algebra, can you prove it algebraically?
3. How can you explain your answer to a visual person?

Required skills / content: Basic understanding of fractions and geometry.

Links: PDF / Google Doc

Source: http://donsteward.blogspot.com/

Dice in a Corner

Why I like this problem: This it the perfect example of a low-floor and high ceiling problem. Anyone who can count to 18 can access this problem, but through questioning, teachers can bring it to a very high level. The manipulation part of the task is great for tactile learners, and student who struggle with focus or sitting still.

Observations from class: Students must have access to dice for this one, and I found it helpful to give them a strip of card stock which they folded into a corner. Some students wanted to stop after finding one solution, and asking questions really helped them dig in and think more deeply.

Follow-up questions I asked: 

1. What were the touching values on your dice? Can you find a solution with different sides touching?
2. Can you find any patterns with the solutions?
3. What strategy are you using to figure out your solutions?
4. How will using a different number of dice affect your solutions?
5. How many solutions can you find with 4 dice? With 5 dice?

Required skills / content: Counting to 18.

Links: PDF / Google Doc

Source: https://nrich.maths.org/

Four Factors

Why I like this problem: This problem is a great review of basic, sometimes long-forgotten concepts. Although the knowledge required dates back to elementary grades, students can develop strategies to make the solution more efficient. It also requires some thinking outside the box and perseverance for the last two questions. I provided students with the 100 grid below to help the visual learners. 

Observations from class: I was surprised to see that students struggled with the concept of factors and multiples despite being in high school. They often confused the two and had to remind each other of the definitions several times throughout the class period. I gave students a 100 grid to help them organize their thoughts and work with number patterns.

Follow-up questions I asked: 

1. What number facts can you use to make the process of elimination more efficient?
2. How can you make the process of finding the numbers systematic?

Required skills / content: Knowledge of factors and multiples.

Links: PDF / Google Doc

Source: http://donsteward.blogspot.com/

Snowmen Buttons

Why I like this problem: It is easy and gives students a break from the more complex problems, yet it leaves room for interpretation due to it’s wording. When I was asked if all the buttons needed to be used up, I acknowledged: "it doesn't say...." Students are able to practice data organization, and explain their reasoning in various ways.

Observations from class: Students found this one too easy and some got bored when asked to find ALL the possibilities. Not sure that I will use it again for older students, but it would work well for younger grades.

Follow-up questions I asked: 

1. How can you organize the data to make it systematic?
2. How can you ensure that you do not miss any configurations?

Required skills / content: Counting, organizing data.

Links: PDF / Google Doc

Source: Christian Courtemanche

Area Mazes

Why I like this problem: There several different levels allowing students to choose the ones they are comfortable working on, and working their way up in difficulty. The problems seem straight forward at first, until students realize that they need to think outside the box to obtain certain missing measurements.

Observations from class: Students had a difficult time initially figuring out how to get the missing measures, but did well once they were able to get past this struggle. 

Follow-up questions I asked: 

1. Is the route you used the most efficient one? Could you have gotten to the answer in less steps?
2. Do you need to use all of the rectangles to get to the answer?

Required skills / content: Figuring out patterns, counting.

Links: PDF / Google Doc

Source: http://donsteward.blogspot.com/