In the second resource of our three part series on addition and subtraction situations, we will take a closer look at the compare situation. In our first resource of the series, we introduce the four problem situations and describe what students might understand about addition and subtraction from problem types learned in Kindergarten. Now, we take a look at how they build on their understanding of the operations in Grade 1 through learning the compare situation.
We will begin this resource with an example compare problem to familiarize ourselves with the story situation. Then, looking at this problem, we discuss how students build on their understanding of subtraction through solving compare problems where the difference between two quantities is unknown. We end on discussing compare situations with different unknown quantities — specifically, when might be a good time to teach this in relation to other problem types and what challenges students may face while learning it.
An Example Compare Problem
Take a look at the compare problem below, taken from Mathematics International by Tokyo Shoseki. Then, try exercise 1 with your team.
(Tokyo Shoseki, G1, p. 58)
In this problem, as with all compare situations, two quantities are being compared. These quantities are commonly referred to as Smaller and Larger, and the amount between them as Difference. In this problem, we have a Larger quantity of 8 (people on red team), a Smaller quantity of 5 (people on white team), and we solved for our unknown Difference quantity — discovering that it is 3.
Compare Problems with an Unknown Difference
Compare problems are typically first introduced with the Difference unknown, since it is a more natural context. It makes sense to ask, given two quantities, how much more is the larger? In contrast, it is much less common in real life to know one amount (i.e. the Larger or Smaller quantity) and how much more or less it is than another amount (the Difference), but not the other amount itself. Since solving compare problems with an unknown difference is the typical entry point for the situation, students must first extend their definition of subtraction to encompass this context.
Drawing on their prior understanding of subtraction from take-away situations (finding what’s left over when an amount is removed from a larger amount) and take-apart situations (finding the missing part when you know the whole and the other part), how might students understand the textbook problem above? Young children may be able to represent the situation using manipulatives to make matches of elements in the two groups, as is shown in the picture below. They even may be able to answer the question with the correct answer, “there are 3 more people on the red team”. However, many students have difficulty connecting this situation to the equation 8 – 5 = 3 because students are not used to using subtraction when comparing between two groups.
(Tokyo Shoseki, G1, p. 59)
So, how can we help students connect this new situation to what they already understand subtraction to be? An initial thought may be to demonstrate how the problem can be modeled to represent 8 – 5 = 3, as is done in the figure above. However, many children are likely to interpret this as showing that 5 yellow blocks and 5 white blocks are being removed, or 10 blocks altogether.
One way to help students is to reframe the situation. Of the 8 members of the red team, how many can be partnered with members of the white team? How many are left over? When the question is rephrased this way, “5” is not the number of people on the white team, but rather it is the number of people on the red team who can be partnered with people on the white team. Therefore, we are not removing 5 partner pairs (or 10 team members total) as the blocks might suggest, we are removing the 5 red team members who can be partnered.
Reframing the problem to focus more on the larger quantity, rather than the comparison between two quantities, makes it is easier for students to relate this problem type to the ones they’ve used subtraction in previously. If we “take away” the red team members who can be partnered with white team members, students may think back to take-from problems where they solved for an unknown result quantity. We can also think of the problem as a take-apart problem by asking, “if there are 8 total red team members and 5 can be partnered, how many cannot be partnered”? Making explicit these connections to students would likely cause confusion, but using partnering to reframe this problem type can help remind students of previous situations they’ve subtracted in. Then, students can reason that they can subtract in this situation, too.
Compare problems with Unknown Smaller or Larger Quantities
Solving compare problems when the difference is unknown may be a good place to start with the compare situation, but students must eventually also learn how to solve compare problems with either of the other quantities (the smaller or larger quantity) unknown. In fact, according to the CCSS, students need to be able to solve for all missing quantities in all problem situations by the end of grade 1.
The remainder of this resource will focus on these remaining types of compare problems. First, we discuss when students should learn this, in relationship to other problem types. (Specifically, we discuss advantages to varying the unknown quantity in compare situations before doing so in add-to and take-from situations). Then, we discuss challenges students may face when beginning to solve compare problems with either the larger or smaller quantity unknown.
Where should these problem types fit into the progression of student understanding?
As we mentioned previously, the CCSS calls for students to learn to solve for all quantities as unknown in all of the situations. Below, we list out all the problem types within each situation, and provide example problems.
From Kindergarten through when compare problems are introduced in Grade 1, a typical elementary school curriculum will have covered the problem types in green. In part one of this series, we discussed how the add-to, take-from, and put-together/take-apart problem types inform students’ initial understanding of addition and subtraction in Kindergarten. Earlier on in this resource, we discussed how student understanding of subtraction is extended by solving compare problems with an unknown Difference (typically in grade 1).
Out of the remaining problem types, we recommend teaching how to solve for unknown Smaller and Larger quantities in compare problems before solving for unknown Start and Change quantities in add-to and take-from situations. Because compare situations are static, all three quantities (Larger, Smaller, and Difference) are easily represented simultaneously. In contrast, add-to and take-from situations involve a sequence of events, making it difficult to show all quantities at the same time. Consequently, solving for unknown Start or Change quantities is challenging, even though solving for an unknown Result is quite intuitive. Let’s see how these problem types compare in the exercise below.
You may have noticed that problem (b), which requires you to solve for an unknown Change quantity, is much more difficult to model than problem (a), which asks you to solve for the unknown Result quantity. Problem (b) is difficult to model because you must keep track of how many more you are adding while at the same time ensuring that you end up with a Result of 9. On the other hand, when you finish modeling the situation in problem (a), you also end up with your solution — the Result quantity. Now let’s experience modeling a compare situation, where all quantities can be represented simultaneously, by trying the following exercise.
You may have found it easier to keep track of all the quantities in this problem than it was in problem (b) in the previous exercise because the quantities are all present at the same time. For this reason, we suggest having students experience solving for unknown Smaller and Larger quantities in compare situations before solving for the unknown Change and Start quantities in add-to and take-from problem situations.
Challenges in problems with unknown larger or smaller quantities
As students tackle these new problem types, one challenge students may face is identifying which quantity they have been given — Smaller or Larger — and which one they are solving for. This requires students to firmly understand the story situation because words like “more” or “less” (fewer) are relative, and thus must be interpreted in context. For example, in the problem, “There are 6 dogs. There are 3 more cats than dogs. How many cats are there?”, students need to understand that since there are 3 more cats than dogs, the number of cats must be greater than the number of dogs. Thus, “6” is the smaller quantity and they are solving for an unknown larger quantity (the number of cats). If we had described the same relationship between cats and dogs in a different way, by saying there are 3 less dogs than cats, students should be able to come to the same conclusion.
Many students have been taught previously to search for key words in a story problem. If the emphasis is put on this strategy, instead of understanding the relationship between the quantities, students may struggle with certain word choices. Try the following exercise to experience this.
You may have found problem (a), which uses the word “fewer”, easier to understand using the key word approach because “fewer” often causes us to assume we will be subtracting — thus, it better matches the operation used to solve. Problem (b), on the other hand, may have been more difficult because “more” does not cause us to assume we need to subtract. (To see a teaching through problem-solving lesson where 2nd grade students grapple with the wording in problem (b), click on this link). While the key word strategy is unreliable for determining how to solve a problem, there is nothing wrong with recognizes phrases like “more than” or “less than” to identify a problem as a compare situation. Once this is known, though, students should read the story closely to determine the relationship between the quantities.
After reading this resource, it may be helpful to discuss the following questions as a team.
- What ideas about teaching-learning compare problems discussed in this essay strike you as important in your setting?
- How does your curriculum handle compare problems? How does it situate the topic within the other problem situation types? What about it do you like and/or notice that has opportunity for improvement?
- What experiences have your students had that currently inform their understandings of compare problems?
- What makes it difficult for students about compare problems?
- What do you want to learn more about?
This concludes part 2 of our series on addition and subtraction. In our third and final part, we discuss how students’ understanding of addition and subtraction is deepened when they learn about the remaining problem types: add-to and take-from with Start and Change quantities unknown.
See the Resource tiles below for expanded discussions on a variety of topics related to addition and subtraction.