In this section, students deepen their understanding of the base-ten structure by representing two-digit numbers with different amounts of tens and ones. They also extend their comparison work by comparing numbers expressed in different ways.

Students begin by making a number with towers of 10 and singles, using more than 9 single cubes. This prompts students to consider how to represent numbers in different ways. For example, students compose a ten from 10 ones or decompose a ten into 10 ones.

3 tens 2 ones

2 tens 12 ones

Students use the symbols , , or to compare numbers that are represented with different compositions of tens and ones.

This reasoning prepares students to use their understanding of place value and the properties of operations as methods to add within 100.

Throughout the section, observe students for the look-fors on the Sections D Checkpoint Assessment. Or use the list given at the end of the section.

In an earlier unit, students used a 10-frame to represent a unit of ten. In this section, they make sense of other representations of units of ten—towers of 10 connecting cubes, base-ten drawings, words, and numbers—to build their understanding of the base-ten system.

Students count collections with a multiple of 10 objects in each. As they represent the quantity in each collection, they see that counting by one and counting by ten yields the same number.

Students learn that each name and numeral used to skip-count by 10 represents an amount of tens, so 30 represents 3 tens, 40 represents 4 tens, and so on. This understanding helps students add and subtract multiples of tens.

4 tens and 1 ten is 5 tens.

5 tens take away 2 tens is 3 tens.

The focus is on connecting written numbers to their word names and the amounts of tens they represent. Terms such as “two-digit number,” “digits,” “multiples,” “tens place,” and “ones place” are not used. “Multiple of 10” is used in teacher-facing text, but isn’t a term that students use until grade 3. Students should be encouraged to use any language that makes sense to them.

Throughout the section, observe students for the look-fors on the Sections A Checkpoint Assessment. Or use the list given at the end of the section.

In this section, students learn that the two digits in a two-digit number represent amounts of tens and ones. They count collections of objects that don’t contain a multiple of 10 and express the quantity in a way that makes sense to them. Their understanding of teen numbers helps them see the collections in terms of tens and ones.

Students continue to use familiar representations, such as connecting cubes and base-ten drawings, to make sense of the digits in two-digit numbers. Some base-ten drawings use long rectangles to represent the tens and smaller squares to represent the ones. Students also begin to use words such as “___ tens ___ ones.”

Students then interpret addition expressions to show the value of each digit in two-digit numbers. These expressions are shown in expanded form and with the value of the ones before the tens .

Later in the section, students use their understanding of two-digit numbers to add multiples of ten to any two-digit number and mentally find 10 more or 10 less than any number.

Throughout the section, observe students for the look-fors on the Sections B Checkpoint Assessment. Or use the list given at the end of the section.

In this section, students use their understanding of the base-ten structure to compare and order numbers to 99. Students notice that if a two-digit number has more tens it will be greater than a number with fewer tens, no matter how many ones there are. This is true because we know the value of the digit in the ones place will always be less than 1 ten because the place value system only allows up to 9 ones. Students generalize this understanding to compare numbers based on the digits.

Students are introduced to the and symbols in this section. Students gain familiarity by reading and interpreting comparison statements with symbols before they use them to write true statements. They have opportunities to work with the symbols throughout the section.

17 is less than 35.

35 is greater than 17.

The lesson activities intentionally use mathematical language to support students in recalling how to read or write the symbols. For example, initially students are encouraged to notice that the side of the symbol with the greater amount of space between the top and the bottom segments faces the greater number. Avoid using non-mathematical or imaginative language that may distract from the focus of the unit and delay fluency with reading and writing the symbols.

Throughout the section, observe students for the look-fors on the Sections C Checkpoint Assessment. Or use the list given at the end of the section.