Thinking Like a Mathematician

Lessons That Develop Habits of Mind and Thinking Skills for Young Mathematicians in Grade 3

Mary-Lyons Walk Hanks, Jennifer K. Lampert, Katherine Plum

• Publisher: Routledge
• Year: 2021
• ISBN:
  9781000489910
• Edition: 1
• Topic: Education
• Subtopic: Education General

Thinking Like a Mathematician focuses on high-interest, career-related topics in the elementary curriculum related to mathematics. Students will explore interdisciplinary content, foster creativity, and develop higher order thinking skills with activities aligned to relevant content area standards. Students will engage in exploration activities, complete mathematical challenges, and then apply what they have learned by making real-world connections. Thinking Like a Mathematician reflects key emphases of curricula from the Center for Gifted Education at William & Mary, including the development of process skills in various content areas and the enhancement of discipline-specific thinking and habits of mind through hands-on activities. Grade 3

UNIT 1 SYSTEMS OF COUNTING

DOI: 10.4324/9781003239079-1

RATIONALE

This unit shows students different ways of counting. Students will learn that the base-10 numbering system is not always used. The binary system will be introduced, and students will learn that computers use the binary system. This will solidify students’ understanding of place value and how the number system works. This unit will also introduce basic computer concepts to students.

PLAN

In Lesson 1.1, students will explore the base-10 numbering system. Students will also explore other base numbering systems. In Lesson 1.2, students will apply different numbering systems to write binary code. In Lesson 1.3, students will discover different ways of counting around the world as they develop and present a project.

LESSON 1.1 EXPLORE DIFFERENT COUNTING SYSTEMS

DOI: 10.4324/9781003239079-2

RESOURCES AND MATERIALS

• Lesson 1.1 Base-4 Counting
• Three stackable cups with the numerals 0, 1,2, and 3 written around each cup lip (per group; see Figure 1 for reference)
• Student math notebooks

ESTIMATED TIME

40-45 minutes

OBJECTIVES

In this lesson, students will:

• discover a pattern from a base-4 numbering system,
• understand the differences between numbering systems, and
• compare and contrast the base-10 numbering system with a base-4 numbering system.

PRIOR KNOWLEDGE

Students will need a basic understanding of a base-10 number system. They should also understand place value and how to add and subtract three-digit numbers.

INSTRUCTIONAL SEQUENCE

Anticipatory Set (5-10 minutes): Have students discuss the following in pairs: Explain how our number system works. When you count, is there a pattern? What is the pattern? What would happen if there were no numerals after the numeral 3? How would that kind of counting work?

Activity (30 minutes):

1. Divide students into groups of 2-3. Each group should receive three cups. Distribute Lesson 1.1 Base-4 Counting.
2. Have students stack the cups and turn them sideways so the numerals are showing (see Figure 1). Students should start with the number 000. Explain that (moving from left to right) cup 1 represents the hundreds digit, cup 2 represents the tens digit, and cup 3 represents the ones digit. Students will be turning the cups to find a pattern and recording their numbers on Lesson 1.1 Base-4 Counting.
3. Have students turn the ones digit (cup 3) to 1 and record the number shown: 001. They should then turn the ones digit again, and write the corresponding numbers until they reach the number 003.
4. Once students reach the number 003, have them turn the ones digit back to 0,and then turn the tens digit (cup 2) to 1. They should write down the number: 010.
5. Students should continue the pattern until they have reached 333.
6. Explain that this number system is called a base-4 number system (consisting of four numerals: 0,1,2,3).
7. Lead a whole-group discussion about patterns of numbers.

Figure 1. Stackable cups for base-4 counting.

Exit Ticket (5 minutes): Have students respond to the following in their math notebooks: How would our numbering system change if there were no numeral zero? What if there were only the numerals 0 and 1?

EXTENSION ACTIVITY

Have students add and subtract the following numbers in base 4, using carrying and borrowing rules. Students may also create their own addition and substraction problems.

• 110 + 011
• 223 + 121
• 323 - 122
• 301 -223

ASSESSMENT OBSERVATIONS

• Students should discuss patterns in the base-4 numbering system.
• Students should be using the correct counting process with their cups: After students reach 003, they should reset cup 3 to the digit 0 and turn cup 2 to the digit 1.
• Students should discuss place value similarities and numeral differences.

NAME:________________________________________________DATE:________

LESSON 1.1
Base-4 Counting

Directions: Fill in the chart with the base-4 digits as you count the numbers. Then answerthe questions on the next page.

NAME:________________________________________________DATE:________

1. Write the patterns you see in the new number system.
2. How is this counting system similarto the way we count numbers? Explain.
3. How is this counting system different from the way we count numbers? Explain.
4. What happens when you get to the highest possible ones digit?
5. What happens when you get to the highest possible tens digit?

LESSON 1.2 APPLY BINARY COUNTING

DOI: 10.4324/9781003239079-3

RESOURCES AND MATERIALS

• Lesson 1.2 Robot Maze
• Lesson 1.2 Writing in Binary Code
• Student math notebooks
• Coins (one per student)

ESTIMATED TIME

60-65 minutes

OBJECTIVES

In this lesson, students will:

• create a written block code to guide a robot through a maze,
• learn how binary code is formed, and
• crea...

Table of contents

• Cover Page
• Half Title Page
• Title Page
• Copyright Page
• Table of Contents Page
• Introduction
• Introductory Lesson: What Is a Mathematician?
• Unit 1: Systems of Counting
• Unit 2: Order of Operations
• Unit 3: Patterns
• Unit 4: Time and Measurement
• Unit 5: Graphing, Data, and Charts in Algebra
• Unit 6: Geometry
• Unit 7: Data Analysis and Statistics
• Final Project: Designing a Garden
• Answer Key
• About the Authors