5 Practices: Nail Polish

5 Practices

I’m continuing to dive into the use of the 5 Practices. Solving linear inequalities is the concept featured in this post. Here’s the running (short) list so far.

  1. Filling it Up! (Solving Linear Equations)
  2. Nail Polish (Solving Linear Inequalities)

Learning Goal

“To ensure that a discussion will be productive, teachers need to have clear learning goals for what they are trying to accomplish in the lesson, and they must select a task that has the potential to help students achieve those goals…The key is to specify a goal that clearly identifies what students are to know and understand about mathematics as a result of their engagement in a particular lesson.” (Smith & Stein, 2011)

My learning goal for this lesson was to create a purpose for solving inequalities, understand that a solution is the value that makes a function true, and connect student strategies to formal inequalities. I chose Nail Polish because it has been one of my students’ favorites in the past, and I wanted a problem that many of my girl students can relate to. Previously, I’ve had students ask why we don’t do more female centered problems, and I couldn’t give a good answer. I guess I’ve just created lessons based on what my interests lead me to. So, this task is intended to counter that flaw. In addition, this lesson made sense because it fell on a pep rally day at my school. The problem usually goes pretty quickly, so I figured it would fit in well with the shorter classes for the day.


“Anticipating students’ responses involves developing considered expectations about how students might mathematically interpret a problem, the array of strategies—both correct and incorrect—that they might use to tackle it, and how those strategies and interpretations might relate to the mathematical concepts, representations, procedures, and practices that the teacher would like his or her students to learn.” (Smith & Stein, 2011)

For the anticipation stage, I created the following chart based on something similar in the book, and I worked the problem out in the ways that I thought the students might approach the problem.

Possible Solutions - Nail Polish 5 Practices

Possible Mistakes - Nail Polish 5 Practices


“One way to facilitate the monitoring process is for the teacher, before beginning the lesson, to create a list of solutions that he or she anticipates that students will produce and that will help in accomplishing his or her mathematical goals for the lesson.” (Smith & Stein, 2011)

After creating the list of possible solutions in the anticipation stage, I actively looked for these strategies during the group work session in order to take note of which groups were using what strategies. I used the following chart to keep track.

Monitoring Chart - Nail Polish 5 Practices

As the classes passed by, I noticed that the great majority of groups were using the unit rate method detailed above. Surprisingly, few, if any, used a formal proportion.

In addition, I was impressed with some visuals that a couple groups used. Here is an example.

Nail Polish Visual 5 Practices

I also monitored for the possible mistakes from the anticipation stage. The most common was the conversion of 2.3 minutes. Many students thought this meant 2 minutes and 30 seconds. This lead to helpful discussions.

Finally, some groups finished ahead of others, so I had these groups analyze and discuss a commonly missed equation from a recent quiz.


“The selection of particular students and their solutions is guided by the mathematical goal for the lesson and the teacher’s assessment of how each contribution will contribute to that goal. Thus, the teacher selects certain students to present because of the mathematics in their responses.” (Smith & Stein, 2011)

Since most of the groups used the unit rate method, I had one group present this method. I also selected a group that used a table as well as any group that had a cool visual (I really like visuals for the purpose of giving the artistic students a chance to shine). Also, I selected only one student from each group to be the talker in order to prevent the most dominant student from always presenting. This method came straight from the book, and I really like it. It seems to help with student ownership because each student has to work as if he or she will be the presenter. I don’t tell the group who the talker will be until the task is near completion.

However, there is one note of caution. Be careful to give some advanced notice to who you are choosing because students with social anxiety may need some time to process before jumping in front of the class. I made this mistake this year, and I accidentally put someone in a very tough situation. Therefore, give advance notice and encouragement, and if a student has concerns, then make sure to be sensitive to his or her needs.


“Having selected particular students to present, the teacher can then make decisions regarding how to sequence the student presentations. By making purposeful choices about the order in which students’ work is shared, teachers can maximize the chances of achieving their mathematical goals for the discussion.” (Smith & Stein, 2011)

For the sequencing phase, I decided to have the most commonly used method presented first. This was one recommendation from the book because it allows the largest percentage of audience members to have investment in the presentation process. Therefore, I had one group from each class present the unit rate method.

From there, the sequence depended on the class. If there was a group that used a table, then I had that group present second. My goal was to go from concrete to abstract, and I decided a table would be next in line.

If a table wasn’t used, then I had a group with a good visual present. As mentioned earlier, I like displaying visuals whenever possible because of the artistic students and others who make connections through visuals. In addition, I’d like my students to get in the habit of making more visuals when solving problems. I believe it can help with misconceptions.


“Rather than having mathematical discussions consist of separate presentations of different ways to solve a particular problem, the goal is to have student presentations build on one another to develop powerful mathematical ideas.” (Smith & Stein, 2011)

After the presentations were complete, I lead the students through a connection handout. These handouts are the best way for me to ensure that the key concepts are driven home. The handout starts by highlighting key methods at the top. Again, I try to work from concrete to conceptual during this process. In this case, I began by showing the table and unit rate method. After that, I followed by introducing an equation. Finally, I finished by moving into “standardized test-ish” word problems.


Here are the links to all of the materials mentioned in this post and used during the lesson.

Smith, M., & Stein, M. (2011). 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA: National Council of Teachers of Mathematics

2 thoughts on “5 Practices: Nail Polish

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