# Numberless Word Problems

Students who read a word problem they don’t understand will often solve it using addition or any operation that looks easy to use based on the numbers in the question. Numberless word problems provide students with an opportunity to fully understand a word problem before attempting to solve it.

#### Exploring through an example:

*There are some mice in a field. Some more mice come.*

Questions to ponder:

- What do you notice?
- What do you know to be true?
- What do you think to be true?

What are the mathematical phrases here? What does *some* mean? Could be 5, 1000, 1 000 000. We know it is at least two. There are at least two mice in the field and at least two more mice come. There are at least four mice on the field now. You’ll also notice that a question is not included.

*There are 8 mice in a field. Some more mice come.*

Questions to ponder:

- What do you notice?
- What do you know to be true?
- What do you think to be true?

We now know that there are exactly eight mice in the field. Since at least two more mice come, there are at least ten mice on the field now.

*There are 8 mice in a field. 5 more mice come.*

We now there are eight mice in the field and 5 more mice come. There is a total of thirteen mice in the field now.

What do you think the question will be? *How many mice are in the field now?* Solve the question (if you haven’t already done so.)

#### Creating a numberless word problem

Start with a word problem you would normally give to students to do:

*Johnny’s garden has 4 rows of carrots. There are 7 carrots in each row. How many carrots are there altogether?*

First, remove the question. Students will naturally create the question on their own, when the rest makes sense to them.

*Johnny’s garden has 4 rows of carrots. There are 7 carrots in each row.*

Second, replace the last number with a “soft” word: some, a few, lots of, many, etc.

*Johnny’s garden has 4 rows of carrots. There are some carrots in each row.*

In this particular case, *there are some carrots in each row* will create some questions. Does each row have the same number of carrots or could each row have a different number? These are important questions that should arise, as we should never assume anything.

Third, replace the first number with a “soft” word. It can be the same or different from the other soft word.

*Johnny’s garden has a few rows of carrots. There are some carrots in each row. *

Students will bring different perspectives to the definitions of the soft words. It is important to talk about what they might mean and differentiate from “what they must mean”. A “few” doesn’t mean exactly three. A “few” isn’t necessarily more or less than “some”. The English language is not precise enough to assign a specific number to a soft word but allows for rich conversations.

Once you have removed the question and replaced numbers with soft words, you are ready to present your numberless word problem to students. Use the same questions and process as listed above to reveal each statement for discussion. By the time you reach the final statement *“Johnny’s garden has 4 rows of carrots. There are 7 carrots in each row. How many carrots are there altogether?”*, students have developed a deep understanding of what the word problem is talking about and how they will solve it.

#### Examples of Numberless Word Problems:

- Mice in a field: Single digit + single digit
- Numberless Graph: Reading a simple bar graph
*(Originally blogged about by Brian Bushart. Recreated with permission)* - Ice Cream – Created by Brian Bushar