Seventh grade teachers do some heavy lifting. They begin to move students from arithmetic to algebra or, to put it another way, from learning about numbers to learning the mathematical reasoning that explains the relationships between numbers.

Students continue learning procedures, or how to solve problems, as well as the concepts behind those procedures, which is why they work.

Students need to understand the fundamentals of rational numbers. (Let’s set aside that the words middle schoolers and rational aren’t usually found in the same sentence.) Seventh graders need to know how to apply the four operations (+,-,x,÷) to all types of numbers within the rational number system, which includes any positive or negative whole number or decimal that can be expressed as a simple fraction. (Knowing what the rational number system includes helps you sound like an expert when badgering your children about their homework!)

Here’s where it gets a little complicated — no, make that exciting: a decimal that can be expressed as a fraction (including repeating decimals such as .333, which is ¹⁄₃) is rational, but a decimal that goes on forever (non-repeating) and can’t be expressed as a fraction, such as the square root of 2 or the mighty pi, Π, are known as irrational, but more on that in eighth grade. For now, focus on rational numbers.

In seventh grade, students continue to add, subtract, multiply, and divide, but now they are expected to be able to do this with all rational numbers — positive and negative numbers, fractions, and decimals. Students also have to be proficient at understanding and finding different representations of rational numbers, including converting decimals to fractions, fractions to decimals, and converting both fractions and decimals to percents. Students then need to show their understanding of rational numbers by solving word problems involving rational numbers.

Watch how seventh graders add and subtract with rational numbers.

In seventh grade, students learn new rules, procedures, and properties that they use to create and solve equations with negative numbers and variables. For example, adding a negative number to its positive (e.g., -4 + 4) always equals 0; the product of two negative numbers is always a positive number (-6 x -7 = 42); and the product of a negative number times a positive number is always a negative number (-8 x 6 = -48).

In sixth grade, students learned that a ratio is a way of comparing numbers, units, or quantities, such as running one mile in 6 minutes, 2 miles in 12 minutes, etc. These are known as proportional relationships. Seventh graders learn to use both fractions and decimals to display these relationships. They learn to test these relationships by modeling them on a chart, number line, table, or graph to show whether they really are proportional. If they are proportional, kids should be able to create an equation that represents the relationship.

Here’s an example of a proportional relationship turned into an equation. It’s based on running 1 mile in 6 minutes. If x equal miles and y equal minutes, this relationship would be written as 6x = y.

In addition to writing and solving equations with variables (ex: 2x + 4 = 12), seventh graders work with inequalities using greater than (>), greater than or equal to (≥), less than (<), and less than or equal to (≤). For example, say a salesperson is paid $50 per week plus $3 per sale, and this week the salesperson wants his pay to be at least $100. With x representing the number of sales, an inequality showing this might be $50 + $3x ≥ 100.