According to the CCSS, third-grade students “solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies.”

In the fourth grade, students “represent these problems using equations with a letter standing for the unknown quantity” (NGACBP & CCSSO, 2010). As they move along the learning progression, students learn to “write, read, and evaluate expressions in which letters stand for numbers” in the fifth grade and “write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “subtract y from 5” as “5 - y” in the sixth CCSSM. In the seventh grade, students “solve real-life and mathematical problems using numerical and algebraic expressions and equations.” And in the eighth grade, students “analyze and solve pairs of simultaneous linear equations.”

By having the students see both algebraic equations and inequalities as a kind of mathematical statement that represents the conditions of an unknown number or the variable x, students can see how algebraic equations and inequalities as related. In this way, the students’ understanding of algebraic equations and inequalities are taken to an even deeper level. The solution to an inequality usually has countless solutions, except in the case when a variable has special limits. The solution therefore can be expressed as a set of numbers in a fixed range, and this is a point where an inequality differs from a simple linear algebraic equation.

Through the collaborative planning and consideration of the multigrade learning goals, teachers were better able to discuss the meaningful distinctions, definitions, and multiple models, while laying out the pattern in the development of mathematical ideas as a concept becomes more complex allowing teachers to bridge the standards. While anticipating student-generated representations, they were able to engage in rich discussion of common misconceptions. Teachers with mathematical knowledge for teaching must have an extensive and complex set of knowledge and skills that facilitates student learning across the learning progressions so that they can respond to students’ responses and move the mathematical agenda forward.

As the students evolve through the learning progression in understanding algebraic equations or inequalities in one variable, it is essential to help them to develop the essential knowledge to model situations that can lead to equations in two or more variables to represent relationship between quantities and solve them in a variety of approaches. While it is important for teachers to expose students to solving such systems of equations at appropriate grade levels using formal approaches, it is very helpful for them to provide opportunities for students in elementary grades to try informal ways of solving such problems. Consider the following problem:

Text Box 6.2 A Math Happening 6b: Bikes and Trikes

John’s bike shop sells bicycles and tricycles. One day, they had a display outside. There were 12 total vehicles and 29 total wheels. How many bicycles and tricycles are on the display?

One way to model this could be to start solving this problem by splitting the number 12 into 6 bicycles and 6 tricycles as it is often a quick approach. But this would give a total of 30 wheels, which is not the same as that mentioned in the question. This may then prompt students to manipulate the numbers as they go through this “trial and error” approach. For example, they may consider 7 tricycles and 5 bicycles and quickly realize that this would lead to 31 wheels, which is more than the 29 wheels in the problem.

However, this gives them a clue to guess 5 tricycles and 7 bicycles and realize that the number of wheels match as given in the problem. While the approach is an informal trial and error approach, it does give an opportunity for students to reason up and down to solve the problem without having to formulate or solve system of equations formally. Another powerful approach in this problem is to employ the strategy of assuming all 12 vehicles were bicycles, then there would be 24 wheels that leave 5 wheels short. But these wheels can become the third wheel in the list of 12 bicycles which immediately yields 5 tricycles and 7 bicycles as the solution.

Teaching such powerful proportional reasoning strategies not only helps them to get ready for what is to come when they are exposed to system of equations but also helps them to become efficient problem solvers as they evolve through their learning progression. A video demonstration of this can be found at: http://www.pbslearning- media.org/resource/mgbh.math.ee.bikes/mike-and-padhu-bikes-and-trikes/.

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