Two Arrays, One Product: Justifying a Decomposition
During a multiplication unit, Ms. Alvarez writes 6 × 8 on the board and asks her third graders to find the product using a strategy they can explain out loud. Devon builds six rows of eight tiles and says the answer is 48 because he already knows that fact by heart. Priya builds six rows of five tiles and, separately, six rows of three tiles, then adds 30 + 18 to get 48. Marcus builds eight rows of six and also reaches 48, writing 8 × 6 on his paper. Later the class must compute 6 × 9. Devon says arrays cannot help him this time because 6 × 9 is not a fact he has memorized. Priya says she can split the 9 into 5 and 4 instead. Before the lesson ends, the teacher asks the class to decide whether Priya's split really accounts for all 48 tiles in the original array or whether it quietly adds extra tiles, and to defend that decision by describing the array in words.
1. Priya claims her two smaller arrays together contain exactly the same 48 tiles as Devon's single array. Is she correct? Justify using the structure of the rows, and write an equation that records the split.
Read the worked answer
Priya is correct. Each of Devon's six rows holds 8 tiles, and she cuts every row into a group of 5 and a group of 3; since 5 + 3 = 8, no tile is created and none is lost.
Equation: 6 × 8 = 6 × (5 + 3) = (6 × 5) + (6 × 3) = 30 + 18 = 48.
In array language, the two smaller rectangles have the same height of 6 rows, so their areas add to the area of the original 6-by-8 rectangle. The key idea is that the split happens inside each row, not between rows.
2. Marcus wrote 8 × 6 and got 48 for the same array that Devon described as 6 × 8. Explain why both expressions describe the same collection of tiles, and what that shows about which factor may be split.
Read the worked answer
A 6-by-8 array turned a quarter turn becomes an 8-by-6 array made of the very same tiles, so the two products must be equal: 6 × 8 = 8 × 6 = 48.
Because the two factors can be read as rows and columns interchangeably, a student may decompose either one. Priya split the 8 (into 5 + 3); Marcus could instead split the 6 (into 3 + 3) and write 8 × (3 + 3) = 24 + 24 = 48. Both are valid because both describe cutting the same rectangle along one direction.
3. Devon says arrays are useless for 6 × 9 because he has not memorized it. Respond to him using Priya's idea: give at least one decomposition, compute the product, and state the general rule he should carry forward.
Read the worked answer
Devon does not need the fact memorized, because any row of 9 tiles can be cut into parts he does know.
Split 9 into 5 + 4: 6 × 9 = 6 × (5 + 4) = 30 + 24 = 54.
A second split works too: 9 = 10 − 1, so 6 × 9 = 60 − 6 = 54.
General rule: decomposing one factor into parts and multiplying each part by the other factor keeps the total, because the pieces reassemble into the original array. Arrays are a tool for reasoning, not just a way to display facts already memorized.