In this section, students find the value of unknown addends using methods that are based on place value and are introduced to base-ten blocks. Students continue to rely on the relationship between addition and subtraction to solve problems involving differences.

Students begin by solving Compare story problems. They use any methods and tools that make sense to them—including diagrams and connecting cubes—to find differences of two-digit numbers.

Lin and Clare used cubes to make trains.
What do you notice? What do you wonder?

Students then analyze the structure of base-ten blocks and use them to find unknown addends (MP7). Unlike connecting cubes, base-ten blocks cannot be pulled apart, which helps emphasize the structure of two-digit numbers in base ten.

To reason about an unknown addend, students may add tens and ones to the known addend until they reach the value of the sum. They may also start with the total amount and subtract tens from tens and ones from ones to reach the known addend. The numbers encountered here do not require students to decompose a ten when they subtract by place value.

In this section, students subtract one- and two-digit numbers from two-digit numbers within 100. To reason about differences of two numbers, students use methods based on place value, base-ten blocks and diagrams, and properties of operations. The numbers here require students to decompose a ten when subtracting by place.

Students also make sense of different representations of subtraction by place, including those that show their peers’ reasoning. For example, to find the value of , students might use base-ten blocks or drawings to represent tens and ones. In this case, they might decompose 1 ten from 63 and exchange it for 10 ones, making 5 tens and 13 ones. From here, some students may first take away 8 ones, then 1 ten. Others may take away 1 ten, then 8 ones. 

When students discuss different approaches and explain why they result in the same value, they deepen their understanding of the properties of operations and place value.

The reasoning here builds a foundation for students to understand the standard algorithm for subtraction, but students should not be encouraged to use the notation for standard algorithm at this point. Allow them to build conceptual understanding by reasoning with base-ten blocks and drawings and articulating their thinking.

This section allows students to apply their knowledge to solve story problems that involve addition and subtraction within 100. The story problems include all types—Add To, Take From, Put Together/Take Apart, and Compare—and have unknowns in all positions.

Previously, students worked with diagrams that represent Compare problems. Throughout this section, students also make sense of diagrams that could represent Put Together/Take Apart story problems. 

Clare and Han playing a game with seeds.
Clare has 54 seeds on her side of the board.
Han has 16 seeds on his side.
How many seeds are on the board in all?

Which diagram matches this story? Explain your match to your partner.

As students relate quantities in context and diagrams that represent them, they practice reasoning quantitatively and abstractly (MP2). For Add To/Take From story problems, the best representations show the actions of the story in the correct order. However, it is not incorrect to show them in a different order or to represent an Add To story problem with a subtraction equation or vice versa. For this reason, students should be encouraged to explain why they think a diagram or equation matches a story problem.

Throughout the section, students are invited to interpret and solve problems in the ways that make sense to them (MP1). Math tools, such as connecting cubes and base-ten blocks, should be made available to encourage methods based on place value and the properties of operations to solve the problems.