LBST Math - 3.1 Interpretations of Addition and Subtraction

Equality Symbol: The symbol “=” shows that whatever is on the left of the sign is exactly the same amount or value as whatever is on the right of the sign.

Operation Symbol: The symbol indicating a math operation is an operator, for example: + for addition. − for subtraction. × for multiplication. ÷ for division.

Problem Types

Word problems are often a teacher’s biggest struggle in math. Have you said things like “The kids don’t understand them.” and “They grab the numbers out and do a random operation.” We try so many different methods to try and help, and they fall short. Because often, our focus isn’t on teaching students the operations explicitly. And teaching them explicitly is what we need to do! The Common Core State Standards lay out the different addition & subtraction word problem types that students should know. By teaching them explicitly, we give students the tools they need to be successful problem solvers. We connect how the math they’re learning is relevant to their real life. We help them truly understand the mathematical operation and how and when it’s used. By focusing on the word problems and their types, we’re focusing on what makes addition and subtraction what they are.

START-CHANGE-END

While part-part-whole is the most common addition & subtraction problem type in many teacher’s minds, it’s probably not the most common in story problems. In real-world scenarios, addition and subtraction is most often demonstrated through start-change-result. In start change result scenarios, something joins or leaves the others. These are the problem types where someone got more of something, or something broke. If we’re not intentional with our word problems, we tend to default to “result unknown” problems. These problem types often have questions such as “how many are left?” and “how many are there now?”. The action has already happened.

I use the term “end” instead of “result” because “end” is the opposite of “start” and I think think students have a clearer understanding than with result. It’s also the same language I use with elapsed time problems in 3rd grade and I like to keep my language as consistent as possible.

The table below gives examples for each of the 6 start change end problem types. They’re pretty straightforward so I’m not going to explain each one in depth. Whether it’s addition or subtraction, the language indicates the location of the unknown: the subjects at the start, the subjects that changed, or the subjects at the end.

It’s important that we give students practice with unknowns in all positions. It is through these start-change-end problem types that students see that they can be asked to identify how many of something there was in the beginning. Or, what the change/action was. With start-change-end problems students get to build their understanding of unknowns being in all positions, and build their competence with addition and subtraction being inverse operations. Through unknowns in any location, students model and solve using the inverse operation.

PART-PART-WHOLE

Probably the most common addition and subtraction type in most teachers minds is part part whole. We use number bonds and bar models to model and represent part/whole relationships. We model addition as two sets of objects coming together. We introduce subtraction as separating our total number of objects into smaller parts. And it’s the foundation for future work with fractions and multiplication and division. Part-part-whole is such a critical concept for our students’ mathematical understanding. It’s important that we connect this work to our language in word problems. Many word problems can be thought of as part-part-whole scenarios (even many in the start-change-end types described next).

Total unknown...

problems are typical addition problems. There are two sets that come together. Often, these are not the same exact subject. For example, it could be red apples and green apples coming together. Or cats and dogs. When it’s the same object, it’s often, but not always, a start-change-end scenario because the change is those two sets joining. The word problem below demonstrates a total unknown that has the same subject- people.

Part unknown...

problems are subtraction. We know the total number of items, but we don’t know the number in one of the sets. Through our work with part unknown problems, we reinforce the inverse relationship between addition and subtraction. We can write the similar equation 5 + ? = 8 to connect the two operations. It’s important to work with unknowns in any position and part unknown problems are where I like to spend some work on inverse operations and fact families.

Both Parts unknown...

problems are less common. They are scenarios with multiple solutions. Students know the total number of items and they give a possible arrangement for how those items are broken up. Both parts unknown scenarios are my favorite way to introduce the unknowns to the right of the equal sign 8 = ? + ?. This helps students to know they don’t just solve from left to right and what the equals sign means.

COMPARISONS

Comparisons are the most complex word problems for students. They aren’t naturally what we think of when we think of the operations. For this reason, we need to teach them and practice them. A lot!

Difference Unknown...

problems are typically solved with subtraction. Regardless of the question being how many more or how many fewer, the question wants to know the amount between them. The word “more” can throw students off here because they want to add. You can use the addition equation to demonstrate it as presented, but ultimately subtraction is the easiest way to solve it. My favorite way to model these problems is on a number line because I can demonstrate both question types by counting forwards or backwards.

The Bigger Unknown and Smaller Unknown questions are where things get a bit more complicated. For me, as with any problem, I ask students to start by focusing on the unknown in the question. With these problems, I think it’s best to work on them together because they can be easily confused.

Bigger Unknown...

problems in the table both have the question, “How many birds does Raina have?”. Due to the table placement, we understand that Raina has more birds than Sara. But let’s take a closer look at the problems.

Sara has 3 birds. Raina has 2 more birds than Sara. How many birds does Raina have?

In this problem, it tells us that Raina has more than Sara. This is a pretty easy addition problem adding Sara’s number plus the number more- the comparison- to find Raina’s total.

Sara has 3 birds. She has 2 fewer birds than Raina. How many birds does Raina have?

This problem is not as straightforward. It tells us that Raina has more by telling us that Sara’s 3 is 2 fewer than Raina’s. When students see “fewer” they want to subtract but the bigger number is unknown. We can model that subtraction equation as the unknown – 2 gives us Sara’s 3.

Smaller Unknown...

problems both ask how many Sara has in the table since Sara has the smaller amount. Again, let’s take a closer look at the problems.

Raina has 5 birds. She has 2 more birds than Sara. How many birds does Sara have?

In this problem, the word “more” can make students feel like they need to add. But subtraction is the better operation. If Raina has 2 more than Sara, we can subtract 2 from Sara’s to Raina’s. Or, as addition, Sara’s + 2 = Raina’s 5.

Raina has 5 birds. Sara has 2 fewer than Raina. How many birds does Sara have?

This problem is more straightforward than the last. Sara has 2 less than Raina so students will naturally subtract.

For these scenarios, I find it most helpful to start with the question. The unknown. If I need to know how many birds Sara has, I keep that in mind as I go back to the word problem a second time and restate it. Raina’s 5 is 2 more than Sara’s. Sara has 2 less than Raina’s 5. By starting with the unknown I’m solving for, I can better understand what the question is asking me to do by connecting it to the known.

https://www.talesfromoutsidetheclassroom.com/addition-subtraction-word-problem-types/

Visual Models for Addition and Subtraction

You may recall that part of our problem solving strategies requires that students produce a math drawing that corresponds to the problem. Math drawings should be as simple as possible and include only those details that are relevant to solving the problem. One type of math drawing is the strip diagram, also called a tape diagram and bar model. Strip diagrams are what we will focus on in this class.

In addition to strip diagrams, there are lots of other math drawings that can be used. Which drawing type you choose can be dependent of developmental appropriateness and/or problem type.

For example:

Kindergarten students might solve problems through 10 frames, instead of bar models.

Part, Part, Whole problems might make sense using a number bond model.

Start, Change, End problems might make sense through use of a number line.

https://www.sfusd.edu/departments/mathematics-department-page/math-teaching-toolkit/math-content-support/visual-model-progressions

1.Grade 1 Unit 2 Parent Newsletter (PDF).pdf

Why Can't We Rely on Key Words Alone

In Class Activity 3D, observe how attending only to keywords in a problem might lead students to solve a problem incorrectly.

It’s important to realize that the use of keywords alone is not reliable for solving word problems. There simply isn’t any substitute for reading and understanding a word problem! For example, consider this problem: Tanya has 12 ladybugs. How many more ladybugs does she need to have 21 ladybugs altogether? A student who relies only on the keywords more and altogether might attempt to solve this problem incorrectly by adding 12 and 21 instead of subtracting 12 from 21. Problems such as this one, which is solved by the opposite operation than the one suggested by the wording of the problem, are more difficult for students than problems that can be solved with the operation that is suggested by the wording. To understand and solve problems, students must attend to the overall context as well as to words that indicate addition or subtraction.

Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 98). Pearson Education. Kindle Edition.

Information in this section corresponds to: 3.1 Interpretations of Addition and Subtraction: Beckmann, Sybilla. Mathematics for Elementary Teachers with Activities (p. 93-99). Pearson Education.

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