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## 10 Practical Tips for Making Fractions Come Alive and Make Sense

Using a Journal Article as a Professional Development Experience
Number and Operations

Title:            10 Practical Tips for Making Fractions Come Alive and Make Sense
Author:         Doug M. Clarke, Anne Roche, and Anne Mitchell
Journal:        Mathematics Teaching in the Middle School
Issue:           March 2008, Volume 13, Issue 7, pp. 373 – 380

Rationale for Use

The opening sentence in this article says, “Fractions are difficult to teach and learn, but they should not be viewed as a lost cause”.  The authors then proceed to make ten suggestions, based on research and classroom practice that make fractions make sense for middle school students.
Principles and Standards for School Mathematics (PSSM) NCTM, 2000, suggests that students in the middle grades should deepen their understanding of fractions, decimals, percents, and integers.  This article gives practical suggestions how this might happen.

Procedures

1. Working in groups, present the following task: Represent ¾ (words, drawings) in as many ways as you can.  (Note that answers could be written on acetate, scanned to PowerPoint, or drawn on chart paper to facilitate sharing with the large group.)
• Participants share with the group their different representations.  Discussion could center on different interpretations of fractions implied by each.
• Read the section in the article, p. 373-374, “What Makes Fractions so Difficult to Teach and Learn”.  Discuss whether all five interpretations of fractions identified in this section were included in the representations presented.  Were there some interpretations included but not listed in the article?  Discuss.
• Take each representation made by Darcy in Fig 1, p. 374, and classify according to fraction interpretation.
• Suggest participants give the Darcy assignment to their own students and bring student work to the next session for discussion.  Discuss what was learned about students’ understanding of fractions.
2. Assign one each of Tips 1, 2, 3, 7 from the article to groups (arranged according to number of participants in the group).  Each group should read the tip assigned and report back to the large group, addressing the following questions:
• What was the main suggestion made by the authors to make teaching/learning fractions easier?
3. Related to Tip 4 - Prepare the Color in Fractions worksheet (p. 380) for each participant.  Have participants play the game with a partner. When finished discuss the following in the large group:
• Fractional concepts and skills encountered while playing the game;
• Strategies used to be a winner; and
• Would you use this with your class?  Why or why not?
4. Participants read Tip 5, and do the model tasks in Figs 3 and 4.  Participants should share experiences they have had with the model tasks involving partitioning with unequal parts, and perceptual distracters, as described in Tip 5. Suggest participants design similar tasks and assign to their students. Bring student work to the follow-up session and share experiences of what was learned about students’ understanding of fractions.
5. Participants do the task in Figure 5. Answers are found mentally and a pencil is only used to record answers. Share answers and strategies with the whole group.
• As in #4, suggest using task in Figure 5 with students. When students have discussed strategies for finding solutions assign a similar sheet of fraction comparisons. Bring student work and record of discussion to the next session.
• The authors, p. 376, Tip 6, say that when students share their strategies for comparing fractions during class, other student s may be convinced to use the two strategies of “benchmarking’ and “residual thinking” to tackle relative size and ordering of fraction problems. Discuss if this was so from experiences of doing the activity described above with students.
6. Prepare the set of fraction cards as suggested in Tip 8.  Participants do the activity as suggested.
• Discuss strategies used to order themselves.
7. Prepare the cards and board for the Construct a Sum activity in Figure 7.
• Suggest participants use the activity in a one-to-one interview with a select number of students.  Based on each student’s thinking, actions, and dialogue while doing the task, write an evaluation of what was learned about each student’s attempt to make sense of fractions and each student’s understanding of improper fractions.
8. When all or most of the activities above are completed, ask teachers to discuss if the article has, as suggested by the authors in the Conclusion, p. 378, given ideas and suggestions for making fractions come alive and make sense in the middle years. What leaves you wondering? What else would you like to know more about in regard to the teaching and learning of fractions?

Next Steps/ Extensions

• On page 374 of the article cited above, the authors say, “Curriculum documents sometimes give the sense that the ultimate goal of teaching fractions is that the students will be able to carry out the four operations with them. We believe that students need to be given time to understand what fractions are about (rather than moving too quickly to computation)…”
• In Canada students first learn operations with fractions in Grades 7 and 8.  The earlier grades (grades 2 to 6), are used developing the meaning of fractional numbers and developing fractional sense.
Discuss the advantages and disadvantages of delaying learning operations with fractions until grades 7 and 8.  Do you think it would be easier for your students to learn fractional concepts and operations?
• The Number and Operation Standard for Grades 6-8 in Principles and Standards for School Mathematics (NCTM, 2000), p. 215 states: “In the middle grades, students should become facile in working with fractions, decimals and percents. Teachers can help students deepen their understanding of rational numbers … that call for flexible thinking”.  The problem below is suggested as an example of such a problem.

a. Work each problem above.  Discuss strategies used to find solutions.
b. Would you use these problems with your students?  Why, or why not?
c. As noted in the excerpt above, taken from PSSM, problems that require flexible thinking deepen students’ understanding of rational numbers.  Discuss why the three problems above are examples of using flexible thinking.
d. Make up another problem, involving rational numbers that may require students to use flexible thinking.
• The problem below is taken from Addenda Series, Grades 5-8, Developing Number Sense (NCTM). The problem is intended to help teachers gain a sense of students’ understanding of the effects of certain operations on fractions.
e. Give each participant a copy of the number line, as drawn in the problem. Present the questions to the group, discuss answers explaining and justifying the reasoning used.
f. Encourage participants to use the activity with their students.  Share the experiences with students at a follow-up session. Discussion at this session can center around what was learned about students’ fractional operation sense.

Connections to Other NCTM Publications