Topic G closes Module 3 with questions such as, “What number is 10 less than 402?” and “What number is 100 more than 98?” As students have been counting up and down throughout the module, these three lessons should flow nicely out of their work thus far and provide a valuable transition to the addition and subtraction of the coming module where more and less will be re-interpreted as addition and subtraction of one, ten, and a hundred (2.NBT.8). The language component of this segment is essential, too. Students need to be encouraged to use their words to make statements such as, “452 is 10 less than 462 and 100 less than 562.” This allows for a greater understanding of comparison word problems (2.0A.1) wherein the language of more and less is a constant presence.
In Module 4, Topic A, students build upon their understanding of the base ten system and their prior knowledge of place value strategies presented in Module 3. In Lesson 1, students relate 10 more and 10 less and 1 more and 1 less to addition and subtraction. They recognize that they must add and subtract like units and that the digit in the tens place changes when adding and subtracting 10, just as the digit in the ones place changes when adding or subtracting 1. Students see numbers in terms of place value units; 54 – 10 is 5 tens 4 ones minus 1 ten. They learn to record the addition and subtraction of multiples of 10 using arrow notation. In Lesson 2, students apply place value understanding to add and subtract multiples of 10 before counting on by tens. For example, when adding 20 to 43, they may count 53, 63. Students also develop flexibility in using related addition problems. For example, to solve 92 – 60, one student might think 9 tens – 6 tens is 3 tens, plus 2 is 32, while another starts at 60, adds on 3 tens, and then 2 ones to reach 92, so 32.
In Lesson 17 of Module 4, Topic D, students extend the base ten understanding developed in Topic A to numbers within 200. Having worked with manipulatives to compose 10 ones as 1 ten, students relate this to composing 10 tens as 1 hundred. For example, students might solve 50 + 80 by thinking 5 ones + 8 ones = 13 ones, so 5 tens + 8 tens = 13 tens = 130. They use place value language to explain when they make a new hundred. They also relate 100 more from Module 3 to + 100 and mentally add 100 to given numbers. In Lesson 18, students use number disks on a place value chart to represent additions with the composition of 1 ten and 1 hundred. They use place value language to explain when they make a new ten and a new hundred, as well as where to show each new unit on the place value chart. In Lesson 19, students relate manipulatives to a written method, recording compositions as new groups below in vertical form. As they did in Topic B, students use place value language to express the action as they physically make 1 hundred with 10 tens disks and 1 ten with 10 ones disks. Working in partners, one student records each change in the written method step by step as the other partner moves the manipulatives. In Lessons 20 and 21, students move from concrete to pictorial as they use math drawings to represent compositions of 1 ten and 1 hundred. Some students may need the continued support of place value drawings with labeled disks, while others use the chip model. In both cases, students relate their drawings to a written method, recording each change they make to their model on the numerical representation. They use place value language to explain these changes. Lesson 22 focuses on adding up to four two-digit addends with totals within 200. Students now have multiple strategies for composing and decomposing numbers, and they use properties of operations (i.e., the associative property) to add numbers in an order that is easiest to compute. For example, when solving 24 + 36 + 55, when adding the ones, a student may make a ten first with 4 and 6. Another student may decompose the 6 to make 3 fives (by adding 1 to the 4).