Algebraic Relations: Rational Numbers

Contents

Theme
Common Core State Standards
Introduction
Symbolic Representation
Interviews with students
Mathematical Issues
Teaching Strategies
References
Your Experience

Theme

Foundational Knowledge

Key Words

Fractions, Decimals, Operation with rational numbers

Common Core State Standards

6.EE.2b

Introduction

Students might need review on operations with fractions even as they move to algebra. Students need to review the meaning of fractions and use of common denominators during algebra courses. Students who are not clear on operations with fractions will struggle even more with operations on rational expressions.

Symbolic Representation

Problem Response
1) Find the sum of 5/12 + 3/8:

Incorrect solution: adding both numerator and denominator to get: 8/20

To help students understand why they need to find a common denominator teachers can use manipulatives.

2) Find the sum a/b + a/c:

Incorrect solution: a/(b + c)

To help students overcome this misconception, the teacher could ask students to pick different numbers for a, b, and c and see that only for a = 0 their incorrect answer a/(b+c) would be correct.

3) In an election candidate A got 1/3 of the votes, candidate B got 9/20 of the votes and candidate C got 2/15 of the votes. What fraction of the votes did candidate D get if those are the only four candidates?

Such an addition and subtraction problem is difficult for students, since the fractions do not have the same denominators.

Correct Solution: 1/3+9/20+2/15=20/60+27/60+8/60=55/60, so candidate D go the remaining 5/60=1/12 of the votes.

4) If n gets very large, then 1/n

a) gets close to 1 b) gets close to 0 c) gets very large

This question helps students to understand fractions on number line while also helping to improve their inductive reasoning skills.

5) 12/13 - 3/7 is about

a) 1 b) 1/2 c) 0 d) don’t know

These informal estimating activities are valuable precursors to a more formal treatment of fraction operations. Students should be able to see "(about 1)-(about 1/2)" and reason that the difference is about 1/2.
6) Find 18/0:

Division by 0 is an abstract notion that needs to be connected to the complete rational number concept. Since the solution is undefined, there is no process for determining the outcome. However, the results can be logically developed using a pattern.

18/3 = 6 because 3(6) = 18.

18/2 = 9 because 2(9) = 18.

18/1 = 18 because 1(18) = 18.

18/0 = A would then mean that 0(A) = 18, but it doesn't! Since we can't make this work, 18/0 is undefined.

Interviews with students

(I: Interviewer, S: Student)

I: Is this correct: a/b + a/c = a/(b+c)
S:

Incorrect answer: Yes, there is a common numerator.

Correct answer: No you need common denominator.

I:  Find numbers for a, b, and c such that a/b + a/c = a/(b+c)
S:

Incorrect: all numbers will work OR there is no number that will work.

Correct answer: a=0 and b, c can be anything except zero. (There are also non-real complex solutions: b=c(-1 +/- i sqrt(3))/2.)

Mathematical Issues

Even capable high school students often complain that they cannot operate with fractions. Only 46% of high school seniors are successful with grasping decimals, percentages, fractions and simple algebra. Understanding of rational numbers and fluency with operations on them are important prerequisites for algebra and predictors of success in algebra.

In order to be fluent with rational numbers, students need to understand what they mean and what each operation means. They need to do the operations with understanding. One of the reasons for difficulties with fractions is that students are often encouraged to memorize algorithms without developing an understanding of them. Developing the ratio meaning of fraction is important for students to develop proportional reasoning.

The development of fractions and operations on them should be an extension of the same ideas for whole numbers, and students should be given enough time to develop their own understanding with the teacher’s guidance.

Teaching Strategies

Children should be familiar with whole number concept and meaning of operations with whole numbers before being introduced to fractions. The informal treatment of fractions should start with manipulation of concrete objects and the use of pictorial representations such as unit rectangles and number lines. Fraction notation must be developed, but formal fraction operations using teacher-taught algorithms should be postponed. Learning the subject of fractions should revolve around informal strategies for solving problems involving fractions. The objective at this level is to build a broad base of experience that will be the foundation for a progressively more formal approach to learning fractions.

Grade Band Suggestions:

Grades 1-4:

● Students should be given time to develop whole number concepts and operations informally with many references to concrete examples.

● Students should also informally practice fraction concepts, but it should be limited to naturally occurring situations such as money or sharing.

Grade 5:

● Students in this grade level should be given opportunities to extend their concept of whole numbers.

● Students should have experience with fractions using partitioning ideas in problem solving.

● Students should be given time to develop fraction operations informally through the manipulation of concrete objects and pictorial representations.

● The notation of fractions should be developed formally, but developing fraction operations formally (especially teacher taught algorithms) should be postponed.

● Providing concrete examples and informal understanding and reasoning of fraction operations will provide a firm base for students' formal understanding of fraction operations.

Grades 6-7:

● Fraction operations should be developed as a generalization of whole number operations and students should be given the chance to construct their own algorithms.

● The development of the formal definitions of fraction operations should progress through students’ experiences and algorithms and also prepare students for abstraction necessary in algebra.

Some of the manipulatives that could be used to develop the meaning of fraction and operations with fractions are:

Cuisenaire Rods, pattern blocks, fraction bars (these can be found online through National Library of Manipulatives).

References

Brown, G., & Quinn, R. J. (2006). Algebra Students' Difficulty with Fractions: An Error Analysis. Australian Mathematics Teacher, 62(4), 28-40.

Brown, G., & Quinn, R. J. (2007). Investigating the Relationship between Fraction Proficiency and Success in Algebra. Australian Mathematics Teacher, 63(4), 8-15.

Brown, G., & Quinn, R. J. (2007). Fraction Proficiency and Success in Algebra: What Does Research Say? Austrailian Mathematics Teacher, 63(3), 23-30.

Rossi, P. S. (2008). An Uncommon Approach to a Common Algebraic Error. PRIMUS, 18(6), 554-558.

Your Experience

  • What have you learned about your students’ thinking regarding this topic?
  • What have you learned that is an effective teaching strategy to help students understand this topic?
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