Keywords in Math

AHA! MOMENT

One year, about a month before the state test was to be administered, I gave my students one sheet of word problems from the released test items so that they could practice. One student, let’s call him Isaac, raised his hand. When I approached him he said, “Mrs. Hassan, there’s a mistake on this problem.” I quickly scanned the problem about currency conversions and asked him to explain the error he found. His response, “There are too many numbers. When I set up my proportion there is no blank so I don’t know how to solve this.” I probed further, “What do you mean there are too many numbers?” He showed me what he had done on the worksheet I had copied for all of my students. He circled all of the numbers that were in the problem, and continued his explanation, “See Mrs. Hassan? I know I need to set up a proportion. I looked at all of the numbers in the problem and there are four. That’s too many. There should only be three.” What I realized was that Isaac was using a keyword strategy that he had been taught at some point in his K-6 math education. He had no idea that circling the year that was given regarding the value of a dollar at a certain point in the past was not going to be helpful to solve the problem. This was the problem Isaac was working on:

http://www.cde.ca.gov/ta/tg/sr/documents/cstrtqmath7.pdf (#76 )

He circled 2002, 6, 1, and 300, all of the numbers given in the word problem. There was no blank in his proportion for which to solve. His favorite teacher from elementary school, the teacher he loved and trusted, told him that this strategy always works. “Look for the numbers, and the keywords that tell you what to do.” How could I possibly undo the damage of the keyword strategy? At this point, it became my mission to remove the ‘keyword’ strategy from the pedagogy of all teachers I encountered.

Consider the following problem:

Maria and Louisa were comparing scores from a recent math test. Louisa realized that her score was less than Maria’s. Instead of Maria telling Louisa how many points she earned she decided to tell her a math riddle, “Your score was 2 more than half of my score. “ Louisa earned 13 points on the test. How many points did Maria earn?

Now consider how a student who has been taught the keyword strategy may approach the problem. They see “less than” in the second sentence, the number 2 and “more than” in the third sentence and the number 13 in the fourth sentence. What does that mean to a keyword kid? I need to subtract (less than), then add (more than) 2 to 13? Or maybe I need to use inequality symbols? You think a keyword kid would even recognize the spelled out half as a number? Not likely.

If we don’t use the keyword strategy, what do we do? Teach a strategy that will help students tackle word problems, not ignore the words or context. A tool that I have found useful has been a tape diagram. Thinking aloud with students and modeling the use of the tape diagram has helped students like Isaac. Let’s see how the Maria and Louisa problem might play out in the classroom using a tape diagram.

“Louisa realized that her score was less than Maria’s.”

T: Who scored more points, Louisa or Maria? 

S: Maria. 

T: Can we draw something to show that Maria got more points than Louisa? 

S: Yes!

(Note that this first diagram may not be to scale, but it gets students engaged in the problem in a very simplistic way.)

“Your score was 2 more than half of my score.”

T: What does this say about Louisa’s score compared to Maria’s? 

S: Louisa got 2 more points. 

T: Look back at our tape diagram, who got more points? 

S: Maria. 

T: Let’s read that part of the problem again. It doesn’t make sense that Louisa just got 2 more points because her score was less than Maria. What else does the sentence say about Maria’s score? 

S: She got 2 more than half Maria’s score 

T: How can we show that Louisa had half of Maria’s score? <>S: We can divide Maria’s tape into two equal parts, then make Louisa’s tape the length of one part.

T: Now our diagram looks like Louisa got half the points of Maria. How can we represent the “2 more”? 

S: Add 2 to Louisa’s tape.

T: What else do we know about their scores? 

S: We know that Louisa earned 13 points.

T: What is the value of the unknown part of Louisa’s tape? 

S: 13–2=11. 

T: Let’s label it! 

T: What does Louisa’s tape tell us about the value of each part of Maria’s tape? 

S: That each part is 11. 

T: What was Maria’s score on the test? 

S: She earned 22 points.

The work with tape diagrams can be extended to algebraic equations. With a firm grasp of the situation, a teacher may continue the conversation:

T: If we let Maria’s score be represented by x, can we write an expression to represent Louisa’s score S: Yes. The part of the tape showing the value of 11 is 1/2 x therefore the total length of Louisa’s tape is 1/2 x + 2. 

T: What number were we given that represents the total length of Louisa’s tape? 

S: 13 

T: Since 13 and = 1/2 x + 2 are expressions representing the same length, we can write the expressions equal to one another. Write the equation 

S: 13 = 1/2 x +

Whether you choose to use tape diagrams to help your students understand word problems or not, think twice about using keywords. There is an Isaac in your classroom that needs something more.

This blog post was written by Stefanie Hassan, a writer for Eureka Math, A Story of Ratios (6–8) & A Story of Functions (9–12)