Solving Problems with the Unknown in any position

Beginning as early as 1st grade, students should have experience solving addition and subtraction problems with the unknown in any position. Traditionally, we have tended to focus on result unknown problems, such as 3 + 2 = ⬜. But students also need to be able to solve problems such as 3 + ⬜ = 5. So exactly how do we go about building that understanding?

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number bonds

Flexibility with numbers begins in Kindergarten when students learn all the combinations for the numbers through 10. These combinations are often referred to as number bonds. Here you see a common way to show number bond relationships. In this case, you see all the ways to make 5.

You may notice that number bonds look a lot like fact families, and they are similar. The difference is how we approach building an understanding of the relationship between the numbers. When we taught fact families, it was typically done as a rote skill. Students knew that they needed to have two addition equations and two subtraction equations using the same numbers, yet they didn’t have an understanding of what the equations represented. We knew that because we would often see unreasonable results, like 2 – 5 = 3.

Students in Kindergarten need to first work with number bonds in a totally concrete way. You can find lots of different games, many with free downloads, in this post.

We can use number bond cards to help students understand the part/whole relationship of the numbers that make up a number bond. We want to start out with result unknown cards and provide students the concrete support of counters.

Here we see that they start out by putting teddy bear counters on the two parts that are known—3 bears on one part and 2 on the other. Next, students move the bears representing the two parts to the unknown whole section, finding that the whole is 5.

After lots of practice with result unknown, students can move to working with the cards with the unknown as one of the parts. Keep in mind that this won’t happen at the same time for all children. Differentiation is critical. We begin by placing the 5 bears on the whole. Next, we move 2 of the bears to the part we know. Finally the remaining 3 bears are moved to the unknown part. Be sure to provide plenty of guided practice during small group instruction before asking students to work with part unknown cards independently.

3 reads protocol

Using word problems makes abstract concepts more concrete because they put the numbers in a familiar context. However, we need to make sure that we help students develop reading comprehension skills to allow them to understand what the numbers represent in the context of the story. Enter a powerful strategy called the 3 Read Protocol.

The Three Reads Protocol, not surprisingly, involves reading a word problem three times, with each read having a different purpose.

But here are some things that might surprise you.

The problem in Read 1 has NO numbers and NO question. When you take out the numbers, students have to focus on the words. This helps them learn to make mental pictures of what’s taking place, which helps them understand what math to do.

Read 2 provides the numbers, but still does not have a question. Now the focus shifts to the numbers and what they represent in the story.

Finally, in Read 3, students come up with questions that could complete the word problem.

While this sample problem is a result unknown problem (but could also be a comparison problem, right?), you would gradually introduce stories that have the unknown in other positions.

For more information on 3 Reads and how to incorporate it into your instructional routine, check out this post.

Part/whole models

Another tool that can be used to help students understand that the unknown can be in any position is a part/whole diagram. To illustrate, let’s revisit the word problem from the last section, but now let’s make at a part unknown problem.

Now let’s listen in on what it would sound like to incorporate part/whole thinking.

TEACHER: [displays the word problem and a blank part/whole diagram] Let’s read this problem together and decide how each number fits into our part/whole diagram. First we’ll read the whole problem and talk about what’s happening in this story. Then we’ll read each sentence and add the numbers to our diagram. [teacher and students read the story]

TEACHER: Who is this story about? [Juliet and her grandmother] What’s happening in the story? [Juliet is saving money for a video game. She already has some money. Her grandmother gives her money for her birthday.]

TEACHER: Okay, let’s go back to the first sentence: Juliet has saved $15 for a video game. Is the $15 she had already saved the whole, her total money, or is it part of her money. [part of her money] What part is it? [the part she had already saved] Great! Let’s add that to our part/whole diagram and label it money she had. Sound good?

TEACHER: Next sentence: Her grandmother gave her some [teacher shrugs her shoulders when she says some] money for her birthday. Huh, do we know how much her grandmother gave her? [no] So that is our unknown in this problem! Is the money her grandmother gave her part of her money or all of the money, the total? [part] Can we label that part money her grandmother gave her? If we don’t have a number for the money her grandmother gave her, what should we put in that part of our diagram? [a question mark]

TEACHER: Next sentence: Now she has $33. Is that all of her money, the whole, or one of the parts? [all of her money] Why don’t we label the whole All of her money and add the $33 to our diagram.

TEACHER: It seem like we’re getting really close to solving this problem! Let’s read the question and make sure it matches where we placed the unknown in our diagram: How much money did her grandmother give her? Does that match our diagram? [yes] Yes, because we have our question mark in the part labeled money her grandmother gave her.

TEACHER: Great work! Now work with your partner to solve the problem.

Let me add an important note—I used a missing part problem for this example. Keep in mind that your students would have been using the part/whole diagram and this process for talking through the problem on less complicated, result unknown problems extensively before moving on to unknown parts.

Another note—be careful of calling this a subtraction problem. Yes, most students will use subtraction to solve the problem, but they could also use a counting up strategy (18 and 2 more is 20. It’s 10 more to 30, and another 3 to 33. So her grandmother gave her $15). Allow for flexible strategies in solving ALL problems!

I hope that gives you some fresh ideas for tackling this tricky concept!