Reasoning about Multiplication and Division and Place Value, Part 2

Task

Place a decimal on the right side of the equal sign to make the equation true. Explain your reasoning for each.

1. $3.58\times 1.25 = 044750$
2. $26.97 \div 6.2 = 04350$

IM Commentary

Standard 6.NS.3 calls for students to fluently compute with decimals. A companion of fluency is the extension of the students’ existing number sense to decimals. It is insufficient to merely teach procedures about “where to move the decimal.” Rather, the focus of instruction and student work should be on operations and number sense.

The tasks in this set are not examples of tasks asking students to compute using the standard algorithms for multiplication and division because most people know what those kinds of problems look like. Instead, these tasks show what kinds of reasoning and estimation strategies students need to develop in order to support their algorithmic computations.

Solution

1. $3.58\times 1.25 = 4.475$. We are multiplying a number between $3$ and $4$ by a number a little more than $1$. More specifically, we can appeal to the meaning of multiplication and ask, “How many $3.58$’s do we have?” A little more than one of them. Thus, the product must be a number around $4$.

   We can also say that the product must be greater than $3 \times 1 = 3$ and less than $4 \times 2 = 8$. Assuming the digits shown are correct, the only place one could put the decimal that would result in a value between 3 and 8 would be 4.475.

2. $26.97 \div 6.2 = 4.35$ We are dividing a number around $27$ by a number a little more than $6$. More specifically, we can appeal to the meaning of division and ask, “How many $6.2$’s go into $26.97$?” Since $4$ sixes go into $24$, and $5$ sixes go into $30$. Thus, it is reasonable for the quotient to be a number around $4.5$.