From Whole Numbers to Rational Numbers

1986 ◽  
pp. 42-60
Author(s):  
Saunders Mac Lane
Keyword(s):  
Rational Numbers ◽  
Whole Numbers

A number pencil

1967 ◽  
Vol 14 (7) ◽  
pp. 557-559
Author(s):  
David M. Clarkson
Keyword(s):  
Elementary Teachers ◽  
Sixth Grade ◽  
Rational Numbers ◽  
Class Discussion ◽  
Irrational Numbers ◽  
Whole Numbers ◽  
Number Lines

So much use is being made of number lines these days that it may not occur to elementary teachers to represent numbers in other ways. There are, in fact, many ways to picture whole numbers geometrically as arrays of squares or triangles or other shapes. Often, important insights into, for example, oddness and evenness can be gained by such representations. The following account of a sixth-grade class discussion of fractions shows how a “number pencil” can be constructed to represent all the positive rational numbers, and, by a similar method, also the negative rationals. An extension of this could even be made to obtain a number pencil picturing certain irrational numbers.

scholarly journals Order and Equivalence of Rational Numbers: A Clinical Teaching Experiment

1984 ◽  
Vol 15 (5) ◽  
pp. 323-341 ◽  
Author(s):  
Merlyn J. Behr ◽  
Ipke Wachsmuth ◽  
Thomas R. Post ◽  
Richard Lesh
Keyword(s):  
Clinical Teaching ◽  
Fourth Grade ◽  
Rational Numbers ◽  
Teaching Experiment ◽  
Previous Knowledge ◽  
Whole Numbers ◽  
Fourth Grade Students

Fourth-grade students' understanding of the order and equivalence of rational numbers was investigated in 11 interviews with each of 12 children during an 18-week teaching experiment. Six children were instructed individually and as a group at each of two sites. The instruction relied heavily on the use of manipulative aids. Children's explanations of their responses to interview tasks were used to identify strategies for comparing fraction pairs of three types: same numerators, same denominators, and different numerators and denominators. After extensive instruction, most children were successful but some continued to demonstrate inadequate understanding. Previous knowledge relating to whole numbers sometimes interfered with learning about rational numbers.

Teaching one of the differences between rational numbers and whole numbers

1971 ◽  
Vol 18 (5) ◽  
pp. 317-320
Author(s):  
Robert W. Prielipp
Keyword(s):  
Elementary School ◽  
School Children ◽  
Rational Number ◽  
Elementary School Children ◽  
Rational Numbers ◽  
Whole Numbers

What are some ways in which rational number (fractions) differ from whole numbers? How can we make these differences evident to elementary school children? We begin by looking at two ways in which rational numbers and whole numbers differ, and then we consider in depth a procedure that can be followed to teach one of these differences.

scholarly journals Do we have a sense for irrational numbers?

2017 ◽  
Vol 2 (3) ◽  
pp. 170-189 ◽  
Author(s):  
Andreas Obersteiner ◽  
Veronika Hofreiter
Keyword(s):  
Correct Response ◽  
Response Times ◽  
Number Sense ◽  
Irrational Number ◽  
Rational Numbers ◽  
Cube Root ◽  
Irrational Numbers ◽  
Whole Numbers ◽  
Symbolic Notation ◽  
Whole Number

Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers’ whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers.

Ratio-based perceptual foundations for rational numbers, and perhaps whole numbers, too?

2021 ◽  
Vol 44 ◽  
Author(s):  
Edward M. Hubbard ◽  
Percival G. Matthews
Keyword(s):  
Numerical Cognition ◽  
Processing System ◽  
Number System ◽  
Rational Numbers ◽  
Approximate Number System ◽  
Access Points ◽  
Whole Numbers ◽  
Two Systems ◽  
Systems View

Abstract Clarke and Beck suggest that the ratio processing system (RPS) may be a component of the approximate number system (ANS), which they suggest represents rational numbers. We argue that available evidence is inconsistent with their account and advocate for a two-systems view. This implies that there may be many access points for numerical cognition – and that privileging the ANS may be a mistake.

Ten Often Ignored Applications of Rational–Number Concepts

1984 ◽  
Vol 31 (6) ◽  
pp. 48-50
Author(s):  
Zalman Usiskin ◽  
Max S. Bell
Keyword(s):  
Rational Number ◽  
Rational Numbers ◽  
Number Concepts ◽  
Whole Numbers

When we speak of applications of rational numbers, we refer to those applications that involve more than just whole numbers or integers. Most textbooks outline such applications.

Teaching Rational Numbers—Junior High School

1984 ◽  
Vol 31 (6) ◽  
pp. 43-46
Author(s):  
Fernand J. Prevost
Keyword(s):  
High School ◽  
Junior High School ◽  
Junior High ◽  
Rational Numbers ◽  
Whole Numbers ◽  
The Given

Whole numbers are used to count sets, report measures, and order a set. You can report the sum and difference of two counts, report the product of twice a measure, and divide two whole numbers to determine how many subsets of a given size are contained in a larger set or find how many objects will be in each subset if you divide the given set into a specified number of parts.

New Publications

1974 ◽  
Vol 67 (2) ◽  
pp. 152-155
Keyword(s):  
Word Problems ◽  
Rational Numbers ◽  
Natural Numbers ◽  
Irrational Numbers ◽  
Whole Numbers

The text provides a refresh on these topics: natural numbers, whole numbers, integers, rational numbers, decimals, and irrational numbers. There are two final chapters on geometry and selected applications. A good many “word” problems are included, along with lots of drill exercises. Exposition is rather brief. The treatment of topics is elementary throughout.— Skeen.

Understanding Multiplication as One Operation

2002 ◽  
Vol 7 (8) ◽  
pp. 466-471
Author(s):  
Diane Azim
Keyword(s):  
Rational Numbers ◽  
Negative Numbers ◽  
Simple Process ◽  
Whole Numbers

Some of the wonder of numbers becomes apparent when we use numbers to perform calculations. Understanding multiplication with positive rational numbers is not a simple process. It requires reconceptualizing the meanings of multiplication with whole numbers to include numbers that are less than 1 or are mixed numbers. (Numbers in this article are all non-negative numbers.) To learn how multiplication works with fractions is to experience a whole new meaning for multiplication.

Sneaking in Basic Drill with Sneaky Squares

1982 ◽  
Vol 30 (4) ◽  
pp. 32-34
Author(s):  
Joseph E. Kuczkowski
Keyword(s):  
Rational Numbers ◽  
Magic Square ◽  
Whole Numbers ◽  
Grade Levels ◽  
Scientific American

“Sneaky Squares” are versions of a type of magic square introduced in an article by Martin Gardner many years ago (Scientific American, January 1957, pp. 138–42). His main example was that of a square using whole numbers under addition. As Gardner indicated, it is possible to construct such squares using addition or multiplication and to involve whole numbers, integers, rational numbers, or decimals. With a little variation, subtraction and division can also be involved. Thus Sneaky Squares can be conveniently used for a variety of basic drill work at all grade levels.

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