Sunday, May 04, 2008

Using Hands-On Equations to Solve Verbal Problems

Word problems, also known as verbal problems, provide the students with the opportunity to think through a situation in depth. Short cuts such as "clue words" will not serve the students well, as every so called "clue word" is often not a clue at all. For example, teaching that the word "is" means "equal," presents a difficulty with a problem such as, "Four times a number is increased by 2...." Certainly no equal sign is involved here.

Hence the use of clue words, although designed to be helpful to the students and to serve as a crutch, is actually a disservice to the student since it does not convey to the student that there are no short-cut to doing verbal problems, rather the student must think!

There are methods, however, that the student can use to help break down the problem and represent the various elements. The following example shows how this can be done with one such problem using Hands-On Equations. This problem is taken from the Hands-On Equations Verbal Problems Book. Using Hands-On Equations, this problem is accessible to students as early as the 4th grade.

Theresa could purchase four small gifts and a $3 doll for the same price as three of the same small gifts and one $5 doll. What was the price of each of the small gifts?


We let the blue pawn represent the price of each of the small gifts. The price of four of the small gifts would therefore be represented by 4 blue pawns. The $3 doll would be represented by a red 3 cube. And likewise for the other side.

The setup for the problem therefore looks as follows:

From here, we can use legal moves (remove three blue pawns from each side) to simplify the setup.

From this simplified setup we can see that the blue pawn is worth 2. Hence, The cost of each small gift is $2.

Check: $11=$11


If you would like to provide these types of problems to your upper elementary and middle school students, you may wish to obtain the HANDS-ON EQUATIONS VERBAL PROBLEMS BOOK, which has more than 250 number, coin, age and distance problems, as well as general story problems, for all three levels of Hands-On Equations, along with solutions!

Additionally, if you are a teacher in grades 3 to 8 you may wish to attend a Making Algebra Child's Play workshop this season, In this workshop, you will learn how to use the Hands-On Equations program to solve equations, and also how to apply the concepts to verbal problems.

If you have already attended a Making Algebra Child's Play workshop, or are already using Hands-On Equations in your classroom or in your math program, and you are teaching in grades 6 and up, we encourage you to consider attending the Day2 Hands-On Equations Verbal Problems Workshop. In this workshop, you will review the key ideas of Hands-On Equations and you will also see how to apply these ideas to solve a wide variety of consecutive integer, age, coin and distance problems, including rowing up and downstream! This workshop will also be of interested to teachers of the gifted grades 2 and up, and teachers of low-achieving high school students.


Kira Brennan, age 8 presented the solution below to the above problem:

Kira's solution in her words:
"I solved the problem by drawing a picture of four presents and a person with a doll that has a $3 tag on it in her hands, and put an equal sign next to, and then I drew three presents and a girl holding a $5 doll in her hands.

"When I saw the picture, I saw that each present could be a blue pawn, and the doll could be a block (cube). So I put four blue pawns and a red 3 cube on the left hand side, and three pawns and the 5 cube on the right side. I guessed then that each present costs $2, but I took three pawns off each side anyway, and I could see you have to add $2 to 3 to equal $5 on the other side. Also, I counted 2-4-6-8-11 on the left, and 2-4-6-11 on the right, 11 equals 11, so each present must cost $2. It's harder if you just set up the equation, I think, but it was easy after I drew the picture."

Kira Brennan,
Age 8
(Note from Kira's mom: Kira has not tried verbal problems yet, so it was her idea to draw the picture first. I scanned her first sketch she did quickly to solve the problem, but then she insisted on drawing another one with the entire solution)


This problem is appropriate for students in grades 5 and up who have had Level I of Hands-On Equations

Pedro’s dad is three times Pedro’s age. In 10 years, Pedro’s dad will be twice as old as Pedro will be then. How old is each now?

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