Equations in Early Grades
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The concept of an equation can be introduced early as follows: One equation consists of two configurations of counters on two boards. Some counter are variable and their values are unknown. We can ask the question, "For what values of variable counters do these two boards have the same value?" In math jargon, "to solve an equation" means to find such values of the variable counters. The more general question deals with cases when we have two or more pairs of boards and we look for the values of variable counters that solve all of these equations. In early grades, solving equations should be limited to problems that involve only small integers and rational numbers with small denominators. All such problems can be solved by the "guess and check" method that requires very little knowledge. But it may require a lot of inventiveness to make "smart" guesses that provide instant solutions to apparently difficult problems. One of the most common problems that leads to solving two equations that contain two variables is the "Heads and Legs" problem which has an unlimited number of simple instances. Heads and Legs In a park, some children were playing with some dogs. The total number of heads was 7, and the total number of legs was 20. What was the number of tails in the park?  Children work in groups of four. They use two variable counters:
The condition, the number of heads was 7, is shown on one pair of boards:  
The condition, the number of legs was 20, is on the second pair of boards:  
The Solution The boards 1 and 2 are twice subtracted from board 3 and 4 giving:  
Regrouping is performed to give:  
The values on boards 3 and 4 are halved, giving the answer: t = 3.  
Here is what Leslie Jackson, a kindergarten teacher, wrote in her journal after trying this problem in her class:
"The children used 7 circles of out construction paper for heads, 20 brown rectangles
for legs. A large green construction paper was used as a background to glue on circles
and rectangles so the children would not lose parts.
How it might be improved: There were 7 heads and 20 legs in the park. We told the children they went for a walk in the park. There were some children playing with dogs. If we counted all the heads in the park, we would count 7. So the children glued the seven heads onto the grass (green construction paper). The children then were told there were 20 legs. I asked the children how many legs the people had. The answer was 2. How many legs do dogs have? The answer was 4. I told the children they must use all of the legs. They were allowed to work with each other. Kinders like to have their own work, so they each had a paper, but did a lot of talking. Many of the kinders were able to complete it, others were distracted. Some of the children took the initiative to count and write totals of people and dogs on their paper."  Kindergardeners arranged them to make four children and three dogs.
"The children enjoyed figuring out this activity. Once they did they were really proud
of themselves. I would like to continue problem-solving projects like this with the
kinders. There is a lot of math involved in this activity for kinders."
My reactions to it:
"This project worked well. I did not use pennies and peanuts [one way that we had
solved the problem in the university class] because the kinders would become too
distracted with eating the peanuts and counting the pennies. It worked very well
with the shapes."
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