Student Work from Comparing & Scaling, Problem 1.2: Making Juice

Problem 1.2: Mixing Juice from the Student Book Analysis of Student Work Student Work


The “Orange Juice” Problem in the Grade 7 Unit: Comparing and Scaling has become a classic CMP problem. Iterations of the problem have been in all three versions of CMP. While working with the problem, students generate many interesting strategies for determining the “most orangey” tasting juice from four different recipes.

In CMP3 this is not students first encounter with ratios and rates. The development of ratio and rates began in grade 6, Comparing Bits and Pieces and Variables and Patterns, and continues in grade 7 with Stretching and Shrinking and then Comparing and Scaling.

The student work comes from a Grade 7 CMP 3 teacher during the school years, 2012 and 2013. The teacher also taught CMP1 and CMP2 and was a CMP3 Field Teacher. With a change in the location of the problem in the unit for CMP3, it was not known if the problem would still generate a rich set of student strategies.The teacher commented that the student thinking in this set is similar to the thinking of CMP1 and CMP2 students. The problem continues to generate a variety of student strategies and discussions.

The student work from this problem has been used in many presentations and several articles about teaching and learning proportional reasoning concepts.


The CMP3 Problem 1.2: Mixing Juice occurs early in the unit and provides insights into students’ knowledge about rates and ratios. This group of 22 pieces of student work shows multiple strategies utilized by students to solve the problem. Students use ratio and rates, both part-to-part and part-to-whole, and a variety of number representations including fractions, decimals, and percents.

Three Artifacts are provided:

  • A copy of Problem 1.2 from the student book and the Focus Question from the Teacher Guide
  • Pictures of the chart papers students produced in class showing their strategies
  • A document with analysis of the student strategies


This student work can serve as a guide for planning, teaching, assessing, and reflecting on the mathematics of this Problem. The work can be downloaded for teachers to examine during planning time, for use in a collaborative meeting, or for use in professional development activities.

Questions to Consider

Planning for the Lesson
  • What strategies do you anticipate students using to accomplish the task?
  • How do these pieces of student work reflect the mathematical goals of the problem? Could these students answer the Focus Question?
  • How does the student impact how you might consider teaching the lesson?
Formative Assessment
  • What mathematical strengths do the students have? Weaknesses?
  • What types of questions might you ask to help these students clarify or extend their thinking?
  • If your students produce work like this, what impact will it have on how you would teach the next few problems?
Planning for the Summary
  • Are there 22 different strategies?
  • What opportunities could be provided for students to compare and contrast the strategies?
  • How might you orchestrate the Summarize of the lesson if your students produce work like these samples?
Teacher Reflections
  • Have the students demonstrated sufficient understanding of the Focus Question posed for this problem?
  • How will this set of student work affect your planning for the next lesson?

This student work has been used in a Professional Development activity as an example of The 5 Practices for Orchestrating Mathematics Discussions[1]. After learning about the 5 Practices by reading the article Orchestrating Discussions[2], teachers solved the Comparing and Scaling Problem 1.2: Mixing Juice to anticipate student strategies. Then using about ten examples of student work, the teachers worked in pairs or groups of three to simulate monitoring, selecting, sequencing, and connecting the work. Discussions were held after each practice was discussed to learn from the decisions of others.


[1] Smith M., Stein M. K. (2011). Five practices for orchestrating productive mathematical discourse. Reston, VA: National Council of Teachers of Mathematics.
[2] Smith M., Hughes E., Engle R., Stein M.K. (2009). Orchestrating Discussions. Mathematics Teaching in the Middle School, 14(9), 548-556.