Mathematics
Grade 9
15 min
Add three one-digit numbers
Add three one-digit numbers
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Analyze addition as a binary operation within a defined set of integers.
Formally apply the Commutative and Associative Properties of Addition to prove the equivalence of expressions.
Decompose the process of adding three numbers into a sequence of logical steps based on mathematical axioms.
Differentiate between the roles of grouping (Associative Property) and ordering (Commutative Property) in simplifying expressions.
Evaluate the property of closure for the set of one-digit numbers under the operation of addition.
Construct logical arguments to justify why the sum of three one-digit numbers is constant regardless of the order of operations.
You've known that 2+3+4 = 9 for years, but have you ever considered *why* 4+2+3 gives the exact same result?...
2
Key Concepts & Vocabulary
TermDefinitionExample
Binary OperationAn operation that takes two input elements from a set and produces a single output element. Addition is a binary operation because it combines two numbers (e.g., a + b).In the expression 5 + 8 + 2, we perform the binary operation of addition twice. First, (5 + 8) = 13, and then (13 + 2) = 15.
SetA well-defined collection of distinct objects. In this lesson, our primary set is the set of one-digit integers, D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.The numbers 3, 7, and 9 are all elements of the set D of one-digit integers.
Commutative Property of AdditionA property of a binary operation stating that changing the order of the operands does not change the result.For any numbers a and b, a + b = b + a. For instance, 7 + 4 = 11 and 4 + 7 = 11.
Associative Proper...
3
Core Formulas
The Commutative Property of Addition
\forall a, b \in \mathbb{Z}, a + b = b + a
This rule states that for any integers 'a' and 'b', the order in which they are added does not matter. It allows us to reorder terms in a sum freely.
The Associative Property of Addition
\forall a, b, c \in \mathbb{Z}, (a + b) + c = a + (b + c)
This rule states that for any integers 'a', 'b', and 'c', the way we group them with parentheses for addition does not change the final sum. It allows us to regroup terms, which is the logical basis for why we can add a list of numbers without specifying which pair to add first.
4 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
Let a new ternary operation '∇' be defined on one-digit numbers as a ∇ b ∇ c = (a + b) * c. Is this operation associative, i.e., is (a ∇ b ∇ c) ∇ d ∇ e equivalent to a ∇ b ∇ (c ∇ d ∇ e)?
A.Yes, because the underlying operations of addition and multiplication are associative.
B.No, a counterexample is (1 ∇ 2 ∇ 3) ∇ 4 ∇ 5 = 45, while 1 ∇ 2 ∇ (3 ∇ 4 ∇ 5) = 1 ∇ 2 ∇ 35 = 107.
C.Yes, because the order of operations (PEMDAS) ensures a consistent result.
D.No, because the operation is not commutative.
Challenging
Imagine a mathematical system where the Commutative Property (a+b=b+a) holds, but the Associative Property ((a+b)+c=a+(b+c)) does NOT. In this system, what is the most significant consequence for the expression 4 + 9 + 5?
A.The expression becomes ambiguous; its value depends on which pair of numbers is added first.
B.The expression's value would be different from 9 + 4 + 5.
C.The expression would be undefined because the axioms of arithmetic are incomplete.
D.The property of closure would be violated for the set of integers.
Challenging
Consider the set E of all even one-digit numbers: E = {0, 2, 4, 6, 8}. Is E closed under the operation of 'add three numbers' from the set?
A.No, because 8 + 6 + 4 = 18, and 18 is not in E.
B.Yes, because the sum of any three even numbers is always an even number.
C.No, because the set does not contain an odd number to make the operation complete.
D.Yes, because all possible sums (e.g., 0+2+4=6, 2+4+6=12) are even.
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free