As a teacher/mathematician, I hope that Open Middle Problems may improve problem solving strategies and bring excitement and challenge to all the students in my classroom. Our current topic of study was multiplying fractions by whole numbers.  I used boxes to create a whole number and a fraction. I wanted to see what students could do with the open ended problem and what it would show about their understanding of the concept without a word problem context.  I displayed it as an Engage activity at the beginning of our lesson:

Multiplying a whole number by a fraction Use digits 1 through 9 to complete the boxes below & create the greatest product possible. You may only use each digit one time.

Multiplying whole numbers and fractions

All of the students were able to attempt the problem at some level. Many of the students used friendly numbers such as 9 x 1/2.  As I walked around the room, I noticed that a few students had thought about using a denominator of 1 to create a larger product. I asked several students to share their answers. One of the students shared 3/1 x 9.  The lightbulbs went on with other students as he shared how a numerator over 1 is equivalent to a whole number. When I asked him if he could find a larger product by using different digits, he was able to do so.  One student who had discovered 1/8 x 9 shared his answer with the class.  I asked the class what the product would be if the denominator and whole number were switched and it led to a short class discussion of the commutative property. The problem allowed different solutions and sharing of math understanding in a fun and trusting way.

The next week, I used a few Open Middle problems as challenge/extension problems at the end of a review of prior concepts.  I am trying to find additional ways to differentiate so that the fast finishers and above level mathematicians in my class are challenged. Using it as an extension provided both benefits and a downside. The open middle problems provided enough challenge for the students who needed it but using it at that particular time did not give them time to find out if they had the best solution or not. One student wanted desperately to know if he had the “right” answer.  I discovered that open middle problems can provide the challenge and excitement that some of my students need but adequate time needs to be built in to discuss and share out their solutions.

This week we are concentrating on number patterns with multiplicative/additive patterns and graphing these realtionships on coordinate grids.  I am planning to use these problems as either an Engage or Enrichment/Extentsion with my class:

  1. Triathalon Training

Ms. Smith is training for a Triathalon.  Each day that she trains, she plan to bike twice as long as she runs. Her goal is to stay under 30 miles each training day. Which ordered pairs will fit this goal?

Running Distance Biking Distance

2.  Write an equation that would fit this graph:

Open Middle Graph

3. Equations:

Write four equations whose solutions is y = 5.  If you graph each of these equations, would the graps be similar or different?

4. Create a possible Input and Output table for y = ⅔ of x. What is the smallest digit you can use for x?