The final section invites students to use multiplication and division of whole numbers to estimate large quantities and solve real-world and mathematical problems.

Students encounter area and volume problems in the context of geography—the area of U.S. states—and everyday consumption—the volume of milk consumed, the area of plastic waste in the Pacific Ocean, and the volume of recyclable plastic shipped abroad for processing.

New Mexico is about 596 kilometers long and 552 kilometers wide. Which is larger, the garbage patch or New Mexico?

The section ends with an additional opportunity for mathematical modeling. Students estimate and calculate the weight of food waste produced in the United States per year, using an average per-person amount. They also estimate and reflect on the amount of their own food waste.

In grade 4, students found whole-number quotients and remainders, with up to four-digit dividends and one-digit divisors, using strategies based on place value and partial quotients. In grade 5, they extend this work to include quotients involving two-digit divisors.

Students begin with an exploration that relates division of large numbers to a real-world context, a world-record event for making the largest Peruvian folk dance. They use strategies based on place value and the relationship between multiplication and division to determine how many groups of dancers there were at the record-breaking event. Then they analyze and use different ways to decompose a dividend.

For instance, here are two ways to divide 448 by 16:

Divide. four hundred forty eight divided by 16, 11 rows. First row: 28. Second row: 3. Third row: 5. Fourth row: 20. Fifth row: 16, long division symbol with four hundred forty eight inside. Sixth row: minus three hundred twenty. In parentheses, 20 times 16. Horizontal line. Seventh row: one hundred twenty eight. Eighth row: minus 80. In parentheses, 5 times 16. Horizontal line. Ninth row: 48. Tenth row: minus 48. In parentheses, 3 times 16. Horizontal line. Eleventh row: 0.

Students see that some decompositions may be more helpful than others for finding whole-number quotients. They use this insight to make sense of partial-quotients algorithms that are more complex. Throughout the section, students take a closer look at division problems that do not have whole-number quotients and interpret their remainders in the context of the problems.

Note that use of the standard algorithm for division is not an expectation in grade 5, but students can begin to develop the conceptual understanding needed to do so. The partial-quotients algorithms seen here are based on place value, which will allow students to make sense of the logic of the standard algorithm they’ll learn in grade 6.

This section introduces the standard algorithm for multiplication, extending students’ earlier work on multiplication. In grade 4, students used diagrams and partial-products algorithms to find the product of a number up to four digits and a one-digit number, and the product of 2 two-digit numbers. They attended to the role of place value along the way.

Students revisit these strategies and representations here, but work with factors with more digits than encountered in grade 4. They make connections between the partial products in diagrams and previous algorithms to the numbers in the standard algorithm. They also learn the notation for recording new place-value units that result from finding partial products.

When using the standard algorithm to multiply a three-digit number by a two-digit number, students account for the place value of the digits being multiplied, as they had done before.

For example, the 3 in 23 represents 3 ones, so is 369.

The 2 in 23, however, represents 2 tens, so the partial product is or 2,460, instead of or 246.

The partial products 369 and 2,460 can be seen in a diagram as well.

multiply. one hundred twenty three times 23. 5 rows. First row: one hundred twenty three. Second row: multiplication symbol, 23. Horizontal line. Third row: three hundred sixty nine. Fourth row: plus two thousand four hundred sixty. Horizontal line. Fifth row: two thousand eight hundred twenty nine

Once students have practiced recording products this way, they learn to multiply factors that require composing new units, such as .

multiply. two hundred sixty four times 38. 5 rows. First row: two hundred sixty four. Second row: multiplication symbol, 38. Horizontal line. Third row: two thousand one hundred twelve. Fourth row: plus seven thousand nine hundred twenty. Horizontal line. Fifth row: ten thousand thirty two