Mathematics
Grade 9
15 min
Write addition sentences for arrays: sums to 10
Write addition sentences for arrays: sums to 10
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1
Introduction & Learning Objectives
Learning Objectives
Formalize the relationship between a visual array and its corresponding repeated addition sentences.
Apply the commutative property of addition to generate equivalent mathematical statements from a single array.
Construct all valid array representations for a given repeated addition sentence.
Analyze a given array or addition sentence to determine if it complies with a specified system constraint (sum ≤ 10).
Translate between visual (array), symbolic (repeated addition), and descriptive (m groups of n) representations of a quantity.
Use summation notation as a concise representation of a repeated addition sentence derived from an array.
How can the simple act of arranging dots in a grid reveal fundamental logical laws that govern algebra and computer scie...
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Key Concepts & Vocabulary
TermDefinitionExample
ArrayA set of elements arranged into a structured grid of `m` rows and `n` columns. In this context, the elements are identical units.A 2x3 array has 2 rows and 3 columns. It looks like this:
● ● ●
● ● ●
Addition SentenceA formal mathematical statement of equality that expresses a sum. It consists of two or more summands and a resulting sum.For a 2x3 array, a valid addition sentence is 3 + 3 = 6.
SummandA number or quantity that is added to another. In the context of arrays, a summand represents the number of elements in a single row or a single column.In the sentence 2 + 2 + 2 = 6, the number '2' is the summand.
Commutative Property of AdditionA fundamental axiom stating that the order in which numbers are added does not affect the sum. For any numbers a...
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Core Formulas
Row-Based Addition Formula
For an array with `m` rows and `n` columns, the sum `S` is given by the repeated addition of the number of elements per row (`n`), `m` times. S = \underbrace{n + n + \dots + n}_{m \text{ times}} = \sum_{i=1}^{m} n
Use this formula to generate the addition sentence by considering the array as a set of horizontal rows. The summand is the number of columns.
Column-Based Addition Formula
For an array with `m` rows and `n` columns, the sum `S` is given by the repeated addition of the number of elements per column (`m`), `n` times. S = \underbrace{m + m + \dots + m}_{n \text{ times}} = \sum_{j=1}^{n} m
Use this formula to generate the equivalent addition sentence by considering the array as a set of vertical columns. The summand is the number of rows.
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Challenging
Consider a modified system with two constraints: 1) S ≤ 10, and 2) Both dimensions m and n must be greater than 1. How many unique arrays (where m x n is considered the same as n x m) are valid in this new system?
A.3
B.4
C.5
D.6
Challenging
The sum `S` of a valid `m x n` array is a perfect square greater than 1. The row-based addition sentence is `n + ... + n = S` (`m` times). If it is known that `m=n`, which statement logically follows?
A.The sum `S` must be 4.
B.The number of terms `m` is equal to the value of the summand `n`.
C.The summand `n` must be a prime number.
D.The array must have an even number of elements.
Challenging
Let `T` be a transformation that maps the row-based addition sentence of an array to its corresponding column-based addition sentence. The existence of `T` as a valid, sum-preserving transformation for any `m x n` array is a direct consequence of which underlying principle?
A.The definition of summation notation.
B.The system constraint that S ≤ 10.
C.The commutative property of multiplication (m x n = n x m).
D.The closure property of integers under addition.
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