In this section, students solve Compare, Difference Unknown story problems, reinforcing their understanding of the relationship between addition and subtraction.

Like Put Together/Take Apart problems, Compare problems are about a relationship, rather than an action that is easy to connect to adding to or taking from. Another challenge with Compare problems is that one of the quantities in the relationship (the difference) is not there physically. To top it off, the language of the problems can be more complex and ambiguous.

For these reasons, students begin this section by revisiting the “are there enough?” problems they solved in kindergarten to practice comparing quantities using the language “more” and “fewer.” Students then solve “how many more?” and “how many fewer?” problems in contexts that elicit matching strategies. Students are asked to identify which quantity has “more” or “fewer” and “how many more” and “how many fewer”.

There are 9 dry erase boards at the table.
There are 6 markers.
Are there more dry erase boards or more markers?
How many more?

Matching strategies help students visualize the relationships involved. Students make sense of the bigger amount and the smaller amount. They understand the difference to be the answer to “how many more?” or “how many fewer?”

For example, “How many more cubes does Clare have than Andre?”

In this case, students may count the cubes in Clare’s tower that are unmatched. Some may add cubes physically to Andre’s tower to make it the same as Clare’s. Others may remove the matched cubes (4) and count the remaining cubes in Clare’s tower.

Students relate the ways to find the difference in Compare problems to the unknown addend problems they solved in the previous section. After the teacher annotates, students analyze both addition ( ) and subtraction ( ) equations that can be used to represent the same problem. As students compare different ways to identify the difference, they deepen their understanding of subtraction as an unknown addend problem.

In this section, students revisit Add To and Take From story problems. They share different ways to act these stories out and strategies for adding and subtracting within 10.

Students also work with Add To, Change Unknown story problems for the first time. As students represent and solve these problems, they notice that what they need to do to solve this problem type is very different from the Add To, Result Unknown problems they solved before. For example:

Students notice that they need to figure out how much to add to 6 to get 8 and may think, “6 and something is 8.” This naturally elicits counting strategies, such as using objects to keep track of counting on from 6 to 8.

In this section, it is important to highlight the ways students make sense of the problems and demonstrate their understanding of addition and subtraction with objects and drawings. Teachers connect this thinking to expressions and equations (for example, ), but students are not expected to do this on their own until the end of the unit.

In this section, students solve Put Together/Take Apart, Addend Unknown problems, developing their understanding of the relationship between addition and subtraction.

Put Together/Take Apart problems involve part-part-whole relationships rather than actions or a change over time. To provide visual support for making sense of the relationships between the total and the two parts, students play the center game Shake and Spill. It’s also used to help generate strategies when students are introduced to problems where one addend is unknown.

There are 9 counters total.
How many counters are under the cup?

The lack of an action makes it more likely that students may think of this problem type in different ways. Some students will see it as finding how many to add to the known part, while others may think of the problem as removing the known part from the total to find the unknown part.

Students continue to highlight the ways they make sense of the problems with objects and drawings. Teachers annotate student thinking with equations (for example, or ) and invite them to make sense of the equations in context.

These discussions help students understand subtraction as an unknown-addend problem and deepen their understanding of the relationship between addition and subtraction as more than just thinking of them as opposites.

In this section, students bring together the work of the unit to solve and compare a variety of story problem types and represent them with equations, including equations with a symbol for the unknown.

Students begin by matching equations to both familiar problem types and the new types introduced in the unit. Students deepen their understanding of the ways they can use the relationship between addition and subtraction to solve problems as they consider how subtraction could be used to solve an Add To, Change Unknown problem.

Students practice solving and comparing a variety of problems and writing equations to represent their thinking. As students share and compare the equations they write, they also have opportunities to deepen their understanding of the meaning of the equal sign.

Students also demonstrate their progress toward addition and subtraction within 10. Although this is a year-long goal, some students may move from representing quantities concretely to trying counting strategies and using known facts during this unit.