Mathematics
Grade 9
15 min
Complete the addition sentence - up to two digits
Complete the addition sentence - up to two digits
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1
Introduction & Learning Objectives
Learning Objectives
Represent 'complete the addition sentence' problems as formal linear equations.
Solve for the unknown term in an addition sentence by applying algebraic properties of equality.
Identify the unknown term as the common difference (d) in a two-term arithmetic sequence.
Apply the concept of the arithmetic mean to find a missing term between two others in a sequence.
Model a simple arithmetic progression using function notation.
Generalize the solution for 'a + x = b' as a foundational rule for analyzing sequences.
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Key Concepts & Vocabulary
TermDefinitionExample
Arithmetic SequenceA sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference.The sequence 5, 12, 19, 26, ... is an arithmetic sequence with a common difference of 7.
Common Difference (d)The constant value that is added to a term in an arithmetic sequence to get the next term. In the sentence 'a + x = b', 'x' represents the common difference between terms 'a' and 'b'.In the sequence 15, 25, 35, the common difference is 10. The addition sentence is 15 + 10 = 25.
Linear EquationAn algebraic equation in which each term is either a constant or the product of a constant and a single variable. The problem '52 + __ = 99' can be written a...
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Core Formulas
Solving for the Unknown Addend (Common Difference)
If \(a + x = b\), then \(x = b - a\)
This is the algebraic transposition of the Subtraction Property of Equality. Use this to find the missing value in any one-step addition sentence, which corresponds to finding the common difference between two consecutive terms 'a' and 'b' in an arithmetic sequence.
General Term of an Arithmetic Sequence
\(a_n = a_1 + (n-1)d\)
This formula calculates the 'n-th' term (\(a_n\)) of an arithmetic sequence, given the first term (\(a_1\)), the term number (n), and the common difference (d). Completing an addition sentence is equivalent to finding 'd' when n=2.
Arithmetic Mean Formula
For consecutive terms \(a_1, a_2, a_3\), \(a_2 = \frac{a_1 + a_3}{...
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Challenging
A student is given the first term (a_1 = 25) and third term (a_3 = 75) of an arithmetic sequence. They use the rule 'x = b - a' to calculate the common difference as 75 - 25 = 50. Why is this an incorrect application of the rule for finding the common difference 'd'?
A.They should have added the numbers instead of subtracting.
B.They made a sign error; the answer should be -50.
C.The rule 'x = b - a' applies only to consecutive terms, and a_1 and a_3 are not consecutive.
D.The rule 'x = b - a' is not related to arithmetic sequences.
Challenging
The third term of an arithmetic sequence is 20 and the seventh term is -12. What is the first term of the sequence?
A.28
B.36
C.-8
D.4
Challenging
Let an arithmetic sequence be defined by its terms a_n. If a_k + x = a_{k+1} and a_{k+1} + y = a_{k+2} for any integer k, what must be the relationship between x and y?
A.x = y
B.x = -y
C.x = 2y
D.y = x + 1
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