Previously, students learned that there are 10 hundredths in 1 tenth and explored tenths and hundredths in fraction notation. In this section, they learn to represent and reason about tenths and hundredths in decimal notation.

Students relate to the notation 0.1 and to 0.01. They learn to read 0.1 as “one tenth” and 0.01 as “one hundredth,” the same way these numbers are read when written in fraction notation. To see the connections between the fraction notation, decimal notation, and the word name, students reason with unit squares (representing 1) divided into hundredths.

The structure of the unit square grid helps to illustrate the equivalence of and . It also allows students to see that 0.10 is equivalent to 0.1 and to generalize this understanding to other equivalent tenths and hundredths (for example,  and ). 

In this section, decimals less than 1 are expressed with a leading zero. Consider explaining to students the zero is sometimes omitted and this doesn’t impact the value of the decimal.

Later in the section, students use benchmarks such as 0.5 and the relationship between tenths and hundredths to locate and label decimals on a number line. They compare and order decimals based on size and write comparison statements using the symbols >, <, and =. 

In IM Grade 3, students used various representations and strategies to add and subtract within 1,000, including strategies that rely on place value. In this section, they build on those strategies while also learning about the standard algorithm for addition and subtraction. They begin working toward the end-of-grade expectation of fluency with addition and subtraction within 1,000,000.

As in earlier grades, students attend to the relationship between addition and subtraction, and find sums and differences by composing and decomposing numbers. They compare an algorithm that uses expanded form and the standard algorithm, and observe the role of place value in both algorithms.

In IM Grade 3, students compared, ordered, and rounded numbers within 1,000. In this section, they extend that work to include numbers within 1,000,000.

Students begin by placing multi-digit numbers on a number line with increasing levels of precision and then making comparisons. In comparing numbers, including those that are missing digits, they make use of structure to determine the size of numbers and the significance of the value of the digits (MP7).
 

Is it possible to fill in the two blanks with the same digit to make:

less than ?

less than ?

Previously, students rounded numbers to the nearest multiple of 10 or 100. In this section, students round numbers within 1,000,000 to the nearest multiples of 10, 100, 1,000, 10,000, and 100,000. When a number is exactly halfway between two consecutive multiples of 1,000, 10,000, or 100,000, they round up. This follows the convention used in IM Grade 3 when rounding to the nearest multiple of 10 or 100.

Students apply their understanding of place value and rounding to solve contextual problems. They also engage in aspects of mathematical modeling as they consider the implications of rounding large numbers in different situations (MP4).

In this section, students make sense of whole numbers up to the hundred-thousands place, learn to read and write them, and deepen their understanding of place value.

Students begin by using base-ten blocks and diagrams to represent and reason about multi-digit numbers. They quickly see the limits of using base-ten blocks to represent large numbers when the smallest cube represents 1. For example, this collection represents 1,325. If the smallest block has a value of 10 or ten times as much, the same collection would represent 13,250. The reasoning here prepares students to think about place-value relationships.
 

As students analyze and draw base-ten diagrams and write multi-digit numbers in expanded form, they observe structure and begin to understand the value of the digit in each position (MP7). They see the “ten times” relationship between the value of a digit in one place and that of the same digit in a place to its right. For example, , so the 3 in 347,000 has a value ten times that of the 3 in 34,700.

Students also see this “ten times” relationship as they locate numbers on a number line. If the endpoints of a number line are each ten times those on another number line, points that are in the same position on the two number lines are related by a factor of 10 as well.

Students use these observations to compare, order, and round numbers in the next section.