How do we teach a “Unit ratio” concept to 3^{rd} graders?
I taught algebra to middle and high school students in a transitional school for many years. A transitional school exists for the sake of dealing with various cases of rebellious children with so called “devious” behavior. I saw how easy they would get bored with routine and redundant math work. Some were very smart and had strong computational skills so I challenged them with nonroutine problems involving ratios and even those smart students would not know where to start. Of course they knew the crossmultiplication method that teachers taught when they learned fractions but that was all. It was evident that those children memorized the routine without understanding a very complex ratio concept at an early age.
Recently the department of education issued a long awaited universal curriculum in math in which a great portion is designated to mastering the “Unit Ratio” component in layers beginning with primary grades. In this article I would like to give an example from the Challenging Word Problems 3^{rd} grade book, a Singapore published work that can be used as educational entertainment with a child on the way to school or during other car trip intellectual conversations with your child. As an example: Laval is 18 years older than Chris. How old will Chris be when Laval is three times as old as Chris?
How do we approach this type of problem with a child who is still too young for Algebra? There are 2 other methods that we teach children to use. The most popular is “Guess and test intelligently”. What we mean by that is to have children list their guesses in a table and check that the numbers that they choose would obey the given condition. First, I would like to mention the common mistakes that most children make when they begin with their guesses. They often begin with “6” simply assuming that 18 is 3 times larger than 6. This mistake is one children make after they either only read a problem once, or do not pay attention to all words in the sentence assuming that Laval’s age is 18. Therefore, it is crucial that at a very early age children learn to carefully interpret the information in the problem, and only then attempt to solve it.
Helping children understand that the number they pick for Chris’s age not only needs to be 3 times smaller than Laval’s age but also that the difference between the 2 ages must be 18. Any child, more talented or less talented, (Klimenko V.V.: “What is talent? Is it God given talent, or a good teacher's luck, or is it hard work?”) can create a list of numbers, let us say, beginning with Chris’s birth! Check the child's understanding that when Chris was born his age was measured in minutes and hours and not years and it would be impossible to compare 2 numbers in a ratio when they are not measured in the same units. Then carefully have a child guess larger than zero numbers ending with 9 years old, with only one correct solution that would suit the given information.
The second method is described below and it is offered by the Singapore system of math education. It teaches children to draw models and label those models correctly to arrive to the evident (visual) solution. For example:
Chris
Laval 18 years
This diagram demonstrates that Laval’s age consists of 3 parts and Chris’s age is one, “unit”, part. Moreover, the difference between their ages, 18, is made up of two equal parts, and therefore 18 must be divided by 2 in order to obtain the answer. According to this method this problem is solved in one step! I am an adult and an experienced teacher and it took me a while to draw this diagram, where as I had no trouble solving this problem using an algebraic equation:
In conclusion, I would like to stress one more time the importance of teaching children how to obtain solutions to this type of problem logically and not mechanically! Teach them to think about the question first and only then allow them to solve the problem by a careful trial and error method.
