Beyond Base Ten

A Mathematics Unit for High-Ability Learners in Grades 3-6

Dana T. Johnson

• Publisher: Routledge
• Year: 2021
• ISBN:
  9781000490725
• Edition: 1
• Topic: Didattica
• Subtopic: Didattica della matematica

Looking for a way to challenge your gifted students in math class? Look no further! Beyond Base Ten investigates the concept of place value and the representation of numbers by using place value and non-place-value systems. Number bases other than Base Ten are featured, especially through historical contexts of early civilizations that developed number systems different from the one we use today.Place value is a fundamental and powerful concept that is the foundation for the number system used by all cultures. Typical curriculum materials address this concept in a rote method. This unit goes beyond this and encourages students to analyze the structure of our number system and other systems; examine the historical foundations of place value systems (Babylonian and Mayan) and non-place-value systems (Roman and Greek) over thousands of years in different civilizations; analyze why Base Ten is the surviving number system; and investigate applications of other number bases in areas such as computers and electricity. Beyond Base Ten is perfect for any student who loves mathematics! Beyond Base Ten was developed by the Center for Gifted Education at The College of William and Mary.Grades 3-6

Part I: Introduction

DOI: 10.4324/9781003233282-1

Introduction to the Unit

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DOI: 10.4324/9781003233282-2

Unit Introduction: Beyond Base Ten investigates the concept of place value and the representation of numbers by using place value and non-place-value systems. Number bases other than Base Ten will be featured, especially through historical contexts of early civilizations that developed number systems that are different from the one we use today.

Unit Rationale: Place value is a fundamental and powerful concept that is the foundation for the number system used by all cultures today. Typical curriculum materials address this concept in a rote method. This unit goes beyond this by:

• analyzing the structure of our number system and other systems;
• examining the historical foundations of place value systems (Babylonian and Mayan) and non-place-value systems (Roman and Greek) over thousands of years in different civilizations;
• analyzing why Base Ten is the surviving number system; and
• presenting applications of other number bases in areas such as computers and electricity.

Differentiation for Gifted Learners: This unit gives students a much broader and deeper experience with place value using number bases other than Base Ten. This is challenging because it requires a deep understanding of the mathematical concept of place value, rather than just a familiarity with one system or the memorization of the values of the places in Base Ten without an understanding of the structure of the system. Reasons to use this unit with gifted learners include the following:

• The regular school curriculum generally does not probe place value and numeral concepts other than Base Ten. This unit provides those additional experiences.
• Number operations done in a base other than Base Ten require a much deeper understanding of the algorithms and require serious thinking.
• The study of number systems other than Base Ten is analogous to understanding the structure of one's native language by studying a foreign language.
• The unit also includes some enriching experiences in applications of number bases, problem solving, and history of mathematics. These are appropriate for all students but are especially well suited for students who are mathematically adept because they are more likely to take advanced coursework.
• Roman numerals are not included in National Council of Teachers of Mathematics (NCTM) standards, but they are useful for students to know because they are still used in numbering prefaces and chapters, on watch faces, to give dates on old films, and the like. Remember that NCTM standards are not a ceiling; gifted students can and should go beyond what is stated as curriculum for all students.
• Higher level questions are included throughout the unit.
• Writing is required beyond just giving numerical answers to questions and this requires more cognition. These activities will reinforce language arts standards.
• The task demands are more strenuous than in other curriculum materials. Students sometimes are asked to complete tasks with less teacher support than would be given in a typical math class.
• There is a large amount of mathematical content included in this unit that is covered in an accelerated time frame.
• Examination of number systems from different historical cultures requires analysis, a higher level thinking skill.
• Substantive extensions are suggested for students who want to investigate a concept further. These often require independent work and there is room for creativity in completing them.
• Much of the work in this unit is inquiry-based. Although this approach may benefit all students, inquiry lessons are a good approach to unleash the thinking abilities of gifted students.
• Appropriate problem-solving challenges are posed for mathematically gifted students.

Even a full treatment of all lesson extensions might not be sufficiently challenging for most highly gifted students. However, these students will benefit from this unit if they experience it at an earlier age than most other students.

Links to NCTM Standards: Place value is addressed in the Number and Operations Standard in grades K-5. However, this unit also is very appropriate for middle school students due to the depth and complexity of the way the concepts are addressed. Standards include the following for grades 3-5, "Students should understand the place-value structure of the Base Ten number system and be able to represent and compare whole numbers and decimals." This unit not only will address Base Ten, but will require students to understand representations of the same number in different bases and perform operations within other number bases.

Suggested Grade Level Range: 3-6

Prerequisite Knowledge: Students should have the following background knowledge prior to beginning this unit:

• Students should be familiar with Base Ten. Lesson 2 refreshes the basic principles, but it should not be the first time students have talked about place value.
• Students should be proficient in whole number operations of addition, subtraction, multiplication, and division.
• Students should understand the partial products method of multiplication. For example:

Length of Lessons: These lessons can be customized to an individual classroom by selecting or omitting various activities.

Timing: This unit can be completed as a 1-2-week experience or spread throughout the semester. It may be used in a pull-out or enrichment class. Not all lessons need to be included in the unit, and the order of lessons may be determined by the teacher. Table 1 gives suggestions for selecting lessons for various classroom needs.

Table 1
Suggested Lessons for Various Classroom Needs

Minimum Treatment of This Unit	Intermediate Treatment of This Unit	Maximum Treatment of This Unit
• Younger students • Limited time frame • Students who need more support	• Upper elementary students • Fewer than seven class periods available	• Middle school students • Upper elementary grades with at least eight class periods available • Students who are very capable
Lessons 2, 4, 6, and 8. Be selective about activities within the lessons. Adapt questions as needed.	Lessons 2, 4, 6, and 8. Select either Lesson 5 or 7. Do only the easier extensions.	All lessons and some extensions. Lesson 2 may be summarized as a quick introduction.
Align the pre- and postassessment questions to match what will be covered in the lessons selected.	Align the pre- and postassessment questions to match what will be covered in the lessons selected.	Do all assessment questions.

Extensions: Suggestions for extension activities are included within lessons and at the end of the unit. Some can be completed either individually or in groups. You might also keep a piece of poster paper hanging in your classroom and encourage students to generate questions regarding material they want to know more about; individuals or groups can be asked to find out the answers to these questions and report back as additional extensions. The extensions often require students to function somewhat independently. However, you may choose to assign extensions to less able students by writing a more scaffolded version of the task.

Assessment

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DOI: 10.4324/9781003233282-3

Each lesson has suggested assessments but teachers will find many more ways to determine student understanding.

Journals: If students maintain a math journal, they can be asked to solve a single problem in their journal and explain their reasoning. A good technique to create an audience for student writing is to suggest that they write a postcard to a friend who has asked for help in solving the problem.

Preassessment: This is not a readiness test; however, it assumes that students know about Base Ten concepts and operations. It is intended to give you a baseline indicator of what students know before they start the unit. Typically, they should not do well on the preassessment. If some students do perform well on the preassessment, you should use the lesson and unit extensions to extend their learning.

Postassessment: This is included at the end of the unit, it is parallel in structure to the preassessment. If you administer both instruments, you will be able to tell if students learned the concepts as a result of participating in this unit.

Unit Glossary

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DOI: 10.4324/9781003233282-4

Binary number system: Base Two.

Decimal number system: Base Ten.

Decimal point:The symbol that separates whole numbers from fractional parts in a BaseTen numeral. In the United States, a period is used to designate the decimal point, but in other countries, it might be a comma or another symbol.

Digit: One of 10 Hindu-Arabic numerals: 0,1,2,3,4,5,6, 7,8, and 9.

Googol: name given to the number resulting from taking 10to the 100th power.

Logarithm: The power to which a base must be raised to get a certain number. If a^x= y, then the logarithm of y in base a is x. We can write log_a y = x.

Number: The idea of how many of something exist.

Numeral: The written representation for a number of things.

Radix: The base of a system of numbers, such as two in the binary system and ten in the decimal system.

Radix point: The location in a number that separates the integral part from the fractional part. For example, in the decimal system, it is the decimal point.

Sexagesimal system: Base Sixty.

Vigesimal number system: Base Twenty.

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Contents
• Part I: Introduction
• Part II: Lesson Plans
• Part III: Unit Extensions
• Common Core State Standards Alignment