b. Represent unknowns using symbols, such as □ and Δ.

c. Write and evaluate mathematical expressions using symbols and different

values.

M5A1. Students will represent and interpret the relationships between quantities algebraically.

a. Use variables, such as n or x, for unknown quantities in algebraic

expressions.

In the last entry, I discussed mathematical expressions as the language of mathematics and how important it is for students to learn to write and read mathematical expressions, starting when students start studying addition formally in Grade 1. I also discussed it may be possible for students to represent situations involving missing addend situations using mathematical expressions using a box, like 5+[ ]=8 (or 5+?=8, 5+__=8, etc.). In grades 1-3, those symbols are used as place holders for particular values. However, in Grade 4, students begin the next phase of using symbols to represent numbers and quantities, that is, the concept of variables.

As we consider teaching of this complex idea (variables), it may be worth noting a progression across grades:

Grade 4: writing math sentences with symbols like □ and Δ.

Grade 5: writing math sentences using letters as symbols

Now, there are a couple of implications from the previous paragraph. First, it is important that symbols like □ and Δ are familiar to the 4th grade students - that means they should be introduced to the use of those symbols in math sentences before Grade 4. The other implication is that the primary focus on Grade 4 is, then, on developing the concept of variables, not necessarily about using the symbols like □ and Δ. In fact, to develop the concept of variables, in some cases, you may not want to use symbols like □ and Δ. Instead, you may want to write mathematical expressions using words. For example, in the third grade, children learn about calculating the area of rectangles and squares. The GPS (M3M4) isn't quite clear whether or not the formulas should be developed in Grade 3. However, it may not be a bad idea to develop the formulas in the context of studying the concept of variables. In Grade 3, students learned that the area of rectangles and squares can be calculated by multiplying the lengths and the widths. Thus, we can express the relationship with a mathematical expression using words like this: Area = Length x Width. [Moreover, it is important to note that we cannot write the formula as A = lw yet since the use of letters is a fifth grade standards!] You can probably think of many other situations that will be appropriate for Grade 4 students, for example, Change = Amount Paid - Price, Number of Children = Boys + Girls, etc.. In fact, as students explore different patterns and rules to describe relationships (M4A1a), they can use mathematical expressions with appropriate words to represent the patterns and rules.

When students are comfortable with mathematical expressions with words, you may want to suggest using symbols like □ and Δ in some cases. In fact, having some experiences with mathematical expressions with words, may help students' transition to the use of letters as variables in Grade 5. In those situations, instead of using letters like x, y, a, b, etc., you may want to start with the initial of the words used in the expressions (thus A = lw).

Evaluating mathematical expressions (perhaps derived by students) by substituting different values (M4A1c) is also an important activity to help students understand the concept of variables. Again, it is important that we keep in mind that the main focus here is the concept of variables. By substituting different values, students are learning that the variables (words, symbols, or letters) stand for quantities that can vary.

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