3: Building Number Sense- Understanding Whole Number Operations and Their Properties
- Page ID
- 181690
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Numbers are everywhere! From counting seats on a bus to sharing churros at a snack stand (Morgan and Grandpa, see story below :) ), whole number operations—addition, subtraction, multiplication, and division—help us make sense of the world around us. In this chapter, we will explore these essential operations using models and strategies that highlight their fundamental properties. We will see how numbers relate through commutativity, associativity, and distributive properties, and how inequalities and exponents fit into the bigger picture of mathematical thinking. By understanding these concepts deeply, we can develop strong mental computation skills that will serve us in various real-life situations.
As future elementary teachers, mastering whole number operations is more than just learning procedures—it’s about fostering number sense in young learners. Through this chapter, we will uncover how whole number properties help solve everyday problems and why they are crucial for effective math instruction.
Chapter Learning Objectives
- SLO 3.1 Students will demonstrate a comprehensive understanding of whole number operations, including addition, subtraction, multiplication, and division, by modeling each operation with set and measurement models and analyzing their properties.
- SLO 3.2 Students will apply strategies for mental computation in addition, subtraction, and multiplication across different bases and demonstrate mastery of basic facts using commutativity, associativity, and distributive properties.
- SLO 3.3 Students will analyze and apply the properties of whole numbers in relation to inequality, including transitivity, addition, and multiplication properties of "less than," as well as use addition to establish comparative relationships.
- SLO 3.4 Students will demonstrate an understanding of whole number exponents by defining exponents through repeated multiplication and proving fundamental exponent rules.
- SLO 3.5 Students will evaluate numerical expressions involving whole numbers using the conventional order of operations (including parentheses, exponents, multiplication, division, addition, and subtraction), to arrive at a unique and correct solution.
Essential Questions
- How do the properties and operations of whole numbers help us make sense of and solve real-world problems?
- How can understanding whole number concepts and strategies support effective math instruction for young learners?
Learning That Transfers
Students will analyze and apply whole number operations—addition, subtraction, multiplication, and division—to real-world contexts by developing mathematical models that demonstrate problem-solving strategies for everyday situations and elementary-level teaching. They will evaluate and adapt these models to support young learners’ conceptual understanding and problem-solving skills.
Number Sense Groove
Morgan and Grandpa’s Mathematical Adventure
One bright morning, as Morgan arose, Grandpa said, “Let’s go where the number wind blows! Numbers are grand, they’re everywhere seen, Let’s find their magic in places between!”
They hopped on a bus—oh what a delight! Numbers on seats, left and right. “See here, my dear,” Grandpa said with a grin, “Addition is easy, let’s begin!”
Morgan looked, eyes open wide, “Two kids sit here, and three more beside. If we add them together, it’s easy to do, Two plus three, why look, it’s just five—woohoo!”
Down at the park, the swings went high, Morgan saw seagulls swoop through the sky. “Grandpa, if eight birds fly off in a dash, And three fly back, what’s left in a flash?”
Grandpa chuckled, “That’s subtraction, my dear! Take eight, remove three, and five remain here!” Morgan beamed, “Oh, it’s quite clear! Numbers are fun, let’s find more near!”
They stopped for a snack at a sweet little stand, Grandpa held three churros in his hand. “If each one holds two bites just right, How many bites will we have in sight?”
Morgan thought, “Three times two, Means six sweet bites—one for me, one for you!” Grandpa clapped, “That’s multiplication, you see, Numbers at work, so brilliantly!”
Then at the pond, a puzzle was found, Ten ducks lined up, splashing around. “They split into groups, two per a row, How many rows do they make in a show?”
Morgan counted, “One, two, three, four, five, Dividing by two keeps math alive! Ten ducks split makes five neat lines, Division, Grandpa, is full of designs!”
The sun dipped low, the sky turned gold, Morgan felt proud, so brave and bold. “Numbers are everywhere, big and small, And now I see, I can use them all!”
- 3.1: Addition and Subtraction
- This page explains the foundational concepts of addition and subtraction, emphasizing their interrelationship and the importance of properties like commutative and associative for mathematical fluency. It highlights how children learn these operations through practical experiences and the role of educators in nurturing a positive attitude toward math.
- 3.2: Developing Multiplicative and Divisional Thinking
- This page explains foundational concepts in multiplication and division for children, highlighting informal strategies before formal procedures. It covers key properties (commutativity, associativity, distributivity) through hands-on learning and various problem-solving approaches. It also discusses the distributive property and order of operations while introducing models of division, including repeated subtraction and partitioning.
- 3.3: Ordering Whole Numbers and Exponents
- This page emphasizes the importance of intuitive understanding of number comparison before formalizing inequalities. It covers concepts like subsets, properties of inequalities, and connections between multiplication and addition. Additionally, it details rules of exponents, simplification techniques, the definition of zero as an exponent, and the order of operations (PEMDAS) to clarify expression evaluation. Together, these concepts lay the foundation for advanced mathematical reasoning.
- 3.4: Counting Collections- Uncovering Number Sense and Mathematical Thinking Project
- This page outlines an assignment designed to improve mathematical teaching skills by exploring counting strategies across different age groups. Participants will observe and document counting methods, identify misconceptions, and engage in discussions. Activities include gathering counting objects and presenting final analyses in video or written form, emphasizing interaction and teaching implications. Peer review is encouraged to foster collaborative learning.
- 3.5: Chapter 3 Resources
- This page offers a bibliography of resources on mathematical education, featuring works by prominent authors like Jo Boaler and Carl Boyer. It includes insights on mathematical mindsets, teaching strategies, and the historical context of math. The focus is on innovative and equitable teaching practices for elementary mathematics, aiming to improve educators' understanding and approaches in early education.