On first thought one might be hard pressed to connect the ideas of rational numbers and geometry, and yet a strong case for a connection is made by the authors of Developing Essential Understanding of Geometry for Teaching Mathematics in Grades 6-8 (Sinclair, Pimm, Skelin, & Zhiek, 2012). The book authors put forth that there is a plethora of connections between geometry and ratio. They state, “much of geometry involves the idea of ratio – how one thing compares to another.” Often learners of geometry are asked to compare the ratio of side lengths to diagonals and conjecture about relationships and congruence based on ratios. Geometry, at its core, deals with measurement, properties and relationships. Before students can delve deeply into these concepts it is paramount that they have a conceptual and fluent understanding of rational numbers. With this in mind, and as part of the rational numbers unit that precedes my geometry unit, I posed the following multi part task/lesson to my students.
- Use strips of paper 8 ½ inches long. Each strip represents 1 whole. Fold the strips to show halves, thirds, fourths, fifths, sixths, eighths, ninths, tenths, and twelfths. Mark the folds so you can see them easily, as shown below.
- As you fold your strips, think about the strategies you use to make them, and about the relationships you notice.
On the surface, this task may not seem to demand higher order thinking or be considered a high cognitive demand task. According to Stein and Smith (Stein & Smith, 1998),
Tasks that ask students to perform a memorized procedure in a routine manner lead to one type of opportunity for student thinking; tasks that require students to think conceptually and that stimulate students to make connections lead to a different set of opportunities for student thinking. High cognitive demand tasks involve making connections, analyzing information, and drawing conclusions. High-level tasks require students to think abstractly and make connections to mathematical concepts. These tasks can use procedures, but in a way that builds connections to mathematical meanings and understandings.
Therefore, when this task was posed to middle school students with a wide array of disabilities the rigor begins to skyrocket as it was anticipated that they would make connections to their previous study of common factors and common multiples when creating unit fractions. It is predicted that this foundational task will lead to connections in creating equivalent fractions and ratio relationships.
When launching the task, I only provided the above directions and an endless supply of strips of colored paper. I told students that they could work with their shoulder partners as well.
I anticipated that students would be able to create a 1/2 strip and would discover pretty quickly if they folded that in half they could create a 1/4 strip, followed by a 1/8 strip. I also anticipated they would believe they would create a 1/12 strip by folding the 1/8 strip in half. Not all students made this error, but most did.
When students are failing to move forward, or when an error is quite common, we stop working and have a whole group discussion. We utilize these questions when we are problem solving and discussing in our class.
We noticed the following and decided to put the information in a table. When the information was in the table the students noticed that each time we folded a fraction strip in half the denominator doubled, and that the denominator was a multiple of the previous denominators.
|When Folded in half created
We wondered then how we could create the remaining fraction strips of thirds, fifths, sixths, ninths tenths and twelfths. The students conjectured that if they folded a third strip in half that they would make a sixth strip and if they folded a fifths strip in half they could make a tenth strip . They wondered what they would fold in half to make a ninth strip, but when they thought of half of nine they concluded that there would be no strip that would have unit fractions that were half of nine. They decided that this would be their next big challenge and set to work to discover a solution.
Students struggled for quite some time in their efforts to create the third and fifth strips. They tried several ways of folding the strips that depicted the desired unit fraction, but wrestled with creating equal units. It was fun to hear them struggle, encourage one another and also critique one another. In anticipation of struggles I had these questions on hand to promote the thought processes of the working groups:
- What fraction strips can you make by partitioning halves strips?
- Which strips can you NOT make by partitioning halves?
- What fraction strips, besides halves, could you refold to make twelfths?
- Which strips can you NOT make by partitioning halves?
- Can you see a relationship between repeated partitioning (folding) and equivalent fractions?
- What happens when you fold fifths and then fold the fifths in half? What happens when you fold others in half?
When students are working in groups or independently and ask me for affirmation about their work or a correct answer, I almost always ask them to convince a friend or ask a friend before they ask me for help. This practice encourages students to collaborate with one another and work to communicate their ideas effectively. This also aids in the development of self-reliance skills and confidence. During this task, when I noticed groups becoming too frustrated I asked them the above questions as well as what strategies and methods they used to create their halves and fourths. Students could explain the process of lining up edges so that the units were equivalent. I then asked if they could use the same strategy for the other unit fractions. These questions and support served to move the collective thinking along and allowed reentry to the task.
This task took two full class periods to complete. Students grumbled and struggled, but encouraged each other to persevere. In one of my seventh-grade classes, a student declared, “Mrs. Naegele, you are evil! This is making my brain hurt!” He quickly followed that statement with a “just kidding, I know you are letting us struggle so our brains will grow.” Eventually the students were able to discover what they called a “Z” fold that allowed them to create equivalent unit pieces for the 1/3 and 1/5 strips . Once this was shared among the group the fraction strips began to come together quickly.
When students are working I utilize the collection sheet below both as a way to conduct formative assessment, and as a tool for deciding how to scaffold follow-up discussions.
The follow-up discussion entailed some of the ideas shared about numerators doubling when unit fractions were folded in half. In addition to using the Notice and Wonder Routine, I also regularly ask my students what relationships and patterns that they notice. Students were able to notice that the fraction strips that have equivalent parts are those that have common factors and multiples.
The following day I introduced my students to Annie Forrest’s Fraction Strip Creation Lesson and challenged them to create a stop motion animation movie that would teach someone how to create a 1/12 fraction strip. I introduced them to the Stop Motion app and showed this tutorial video about creating stop motion movies using the Stop Motion Studio app available free for Apple and Android. Students were instructed to create a plan and write a script before they began filming.
I chose the 1/12 strip because there are multiple ways students could choose to create the strip, and I chose the stop motion project because my students often have difficulty explaining in writing their ideas and strategies.
I started this task and lesson during a four-day week. I felt that two days for folding the strips and two days with the long weekend to finish if needed, would be adequate time for the stop motion project. Little did I know that my district would have a bomb threat that week and that Oklahoma teachers would stage a walkout. So, following 2.5 weeks of lost instruction, we returned to class and students resumed their stop motion project.
With such a long break, it was as if we had not done the initial paper-folding task. Students were allowed to use their fraction strips, but they had forgotten all about their discoveries during the paper folding task. Instead of going back and doing that activity again I encouraged students to talk in their production groups and explore their fraction kits using our routine of notice and wonder and our other regularly used problem solving questions. It was as if they were discovering the information all over again. This experience affirmed that students need multiple experiences and interactions with a concept to truly know and learn it.
For the stop motion piece of this activity, I anticipated that most students would create a third strip, fold that in half to create a sixth strip, and then fold that in half to create a 1/12 strip. I also knew that students could opt to create a 1/3 strip and then fold that in half to create a sixth and then half that to create a twelfth strip. I also knew that students could create a fourth strip, fold that into thirds to achieve their goal. I was pleased that all three of these strategies were used. I did have some students that created a 1/2 and took that all the way to a 1/16 thinking they were creating the correct unit strip as they had done in the initial exploration of this task. When this occurred, I asked them how they could create a relationship with 2 and 12. If they were not able to grapple with that and come up with an idea, I asked, what numbers are we exploring that you know that have a relationship with 12? Some students were able to think about the factor pairs of twelve and consider the pair of two and six. They then rationalized that to create a 1/6 from a 1/2 that they would need to use the “Z”fold to fold three equal parts to create a sixth strip. Once at this point they could half that to create a 1/12 strip. Some groups did not make this connection and chose to use the three and four factor pair and start with a 1/3 strip.
To date, students have finished their stop motion videos and submitted their scripts and plans. An overwhelming majority of scripts and plans mirrored the movies, but there were a few that the script and plan did not depict the movie submitted. Before students uploaded their videos to http://www.youtube.com they had to show their movie to another group and the other group had to create a 1/12 strip following the movie tutorial. All of the movies did teach someone how to create a 1/12 strip if the viewer was really watching! The plan going forward, is for students to present their movies and entertain questions and critiques from the other students when state testing is over. I will begin this follow-up by showing them my movie and letting them critique me! I am looking forward to doing this kind of activity again in the future, and so are my students!
Sinclair, N., Pimm, D., Skelin, M., & Zhiek, R. M. (2012). Connections: lookingback and ahead in learning. In N. Sinclair, D. Pimm, M. Skelin, & R. M. Zhiek, Developing essential understanding of geometgry for teaching mathematics in grades 6-8 (pp. 69-79). Reston: National Council of Teachers of Mathematics, Inc.
Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection: from research to practice. Mathematics teaching in the middle school, 268-275.
The MNmMath Nerds had wonderful couple of days watching and critiquing our 1/12 fraction strip stop motion videos! Following each video we spent time noticing and wondering about each production. We utilize the Notice Wonder Routine almost daily making this was a wonderful way to facilitate conversations in this situation . Students are familiar with the process and eagerly contributed their ideas. We critique one another’s thinking regularly as well, but we have never critiqued a final production before. These critiques were much more personal, therefore we spent some time talking about how to provide and accept productive feedback.
Following each notice and wonder the film producers answered questions from the audience. This was an awesome learning experience for all. To close this exploration we held a whole group discussion about what we learned about math and fractions from our experiences. We noticed that every video utilized methods that created fractions that had denominators that are factors of 12. This was a wonderful connection to our previous Prime Time Unit that focused on factors, multiples, greatest common factors and least common multiples. Students also noticed patterns and relationships between division and multiplication as related to fraction denominators.
This was a wonderful investigation! All mathematicians decided we MUST do more explorations like this in the future!