The Orange Juice Problem

   Contents

 Task

In this lesson, students use proportional reasoning to determine a) which of the four orange juice mixtures contains the highest and the lowest percent of orange concentrate and b) the orange juice recipe for each of the four mixtures if 120 cups are needed.  


 GPS Addressed (click [+] to expand)

    [-] M6N1. Students will understand the meaning of the four arithmetic operations as related to positive rational numbers and will use these concepts to solve problems.
     

    c. Determine the greatest common factor (GCF) and the least common multiple (LCM) for a set of numbers.
    f. Use fractions, decimals, and percents interchangeably.
    g. Solve problems involving fractions, decimals, and percents.

      M6A1. Students will understand the concept of ratio and use it to represent quantitative relationships.
    [-] M6A2. Students will consider relationships between varying quantities.
     

    b. Use manipulatives or draw pictures to solve problems involving proportional relationships.
    c. Use proportions (a/b=c/d) to describe relationships and solve problems, including percent problems.
    g. Use proportional reasoning (a/b=c/d and y=kx) to solve problems.

    [-] M6P1. Students will solve problems (using appropriate technology).
     

    a. Build new mathematical knowledge through problem solving.
    b. Solve problems that arise in mathematics and in other contexts.
    c. Apply and adapt a variety of appropriate strategies to solve problems.
    d. Monitor and reflect on the process of mathematical problem solving.

    [-] M6P2. Students will reason and evaluate mathematical arguments.
     

    a. Recognize reasoning and prof as fundamental aspects of mathematics.
    b. Make and investigate mathematical conjectures.
    c. Develop and evaluate mathematical arguments and proofs.
    d. Select and use various types of reasoning and methods of proof.

    [-] M6P3. Students will communicate mathematically.
     

    a. Organize and consolidate their mathematical thinking through communication.
    b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
    c. Analyze and evaluate the mathematical thinking and strategies of others.
    d. Use the language of mathematics to express mathematical ideas precisely.

    [-] M6P4. Students will make connections among mathematical ideas and to other disciplines.
     

    a. Recognize and use connections among mathematical ideas.
    b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
    c. Recognize and apply mathematics in contexts outside of mathematics.

    [-] M6P5. Students will represent mathematics in multiple ways.
     

    a. Create and use representations to organize, record, and communicate mathematical ideas.
    b. Select, apply, and translate among mathematical representations to solve problems.


 Video
The Orange Juice Problem: Day 1
Exploring Proportional Reasoning (duration: 22:48)

Use these questions to guide your thinking about some of the important teacher ideas in the lesson featured in the video clip.

 

  1. What kinds of questions does the teacher ask to make sure the students understand the task?
  2. How did the teacher determine the students level of understanding at the end of the day's lesson?
  3. How can the study of ratios help students solve the problems?

 

 
The Orange Juice Problem: Day 2
Analyzing Proportional Reasoning - Part A
(duration: 20:38)
  1. How does the teacher help students dispel their mathematical misconceptions?
  2. How effective are the teacher's questioning techniques in cultivating student mathematical thinking?
  3. How does the teacher encourage students to engage in discourse while shaping their mathematical thoughts?
  4. Does the classroom discourse between students stimulate their understanding of the content?
  5. What could the teacher do differently to support the students' understanding of ratios in this lesson?

 
Ratio and Proportion - A Mini Lesson
(duration: 12:04)
  1. How does the teacher encourage students to justify their mathematical arguments?
  2. How does the teacher encourage students to analyze each other's mathematical thinking during the discussions?
  3. How does the teacher use questions to direct students to think mathematically?
  4. How effective is the uses of mathematical notations in helping students understand "part" and "whole" relationships?

The Orange Juice Problem: Day 3
Analyzing Proportional Reasoning - Part B
(duration: 20:56)

Understanding the students were not grasping the mathematics behind the problem, Mrs. Powell spent the beginning of the third day reviewing to help clarify the concepts.

 

  1. How does the teacher’s use of questioning help clarify students’ misunderstandings?
  2. How do students draw upon the concept of part-to-part and part-to-whole, learned in a previous lesson, to help them arrive at the correct fraction for each mixture?
  3. How does the teacher utilize the students’ memory of the taste of each mixture to help guide their proportional reasoning? 
  4. Identify the different of roles the teacher assumes during the class period. 

The Orange Juice Problem: Day 3
Analyzing Proportional Reasoning - Part C
(duration: 9:11)

 
  1. How does the teacher’s use of questioning help students in solving the problem?
  2. Why do you suppose some students have difficulties applying fundamental mathematical knowledge in other contexts? 

The Orange Juice Problem: Day 4
Clarifying and Building on Student Understanding
(duration: 23:16)

  1. Notice how the teacher helps the students work through the problems without telling them.
  2. How does the teacher use the students' misunderstandings to clarify their thinking?
  3. How does the teacher have the students work with multiple representations for the same problem?

 Classroom Materials / Lesson Outlines