7th Grade Chapter 2
The Commutative and Associative Properties of Addition and Multiplication apply when an expression has a single type of operation, either addition or multiplication. Remind students that subtracting a negative number is the same as adding a positive number. Show them subtracting a positive number is the same as adding a negative number. A good understanding of these properties is very useful for developing students’ mental math skills. With mixed operations, use the order of operations.
2-2 Distributive Property
The Distributive Property allows you to evaluate an expression in two ways. With 6(2+3), you can either add first and then multiply by 6, or multiply the 6 by every item inside the parentheses before finding the sum.. the multiplier is 6 is distributed to both the 2 and the 3. For some students, a diagram with arrows reminds them to multiply each part.
Addition and subtraction signs separate an expression into terms. Thus 2 + x - 8 has three terms. In contrast, xyz is just one term. The properties of rational numbers can be used to combine like terms, terms with identical variables.
2-4 Variables and Equations
An equation with no variable is either true or false. An equation with one or more variable is an open sentence. Once the variable is assigned a value, the resulting equation can be evaluated as either true or false. A solution of an equation is a value for the variable that makes the equation true.
2-5 Solving Equations by Adding or Subtracting
To solve an equation having one variable, you undo operations in order to get the variable alone on one side of the equation.. You must always do the same thing to both sides of an equation to keep the two sides equal. Addition and subtraction are the inverse operations focused on here.
2-6 Solving Equations by Multiplying or Dividing
The Multiplication Property of Equality states that you can multiply both sides of an equality by the same number, and the equality will stay true.
The Division Property of Equality states that you can divide each side of an equation by the same nonzero number. Divisors are restricted to nonzero values because division by zero is undefined.
2-7 Reasoning Strategy: Try, Test, Revise
The Try, Test, Revise problem solving strategy is common in the world outside the classroom. Everyone uses this strategy - from an infant just discovering his or her world to a scientist making discoveries that change the world. In this strategy, try a reasonable trial solution, not merely a random or wild speculation.
2-8 Inequalities and Their Graphs
The equation x = 1 has exactly one solution: 1. However, the inequality x<1 has an infinite number of solutions. They are all the numbers less than 1. For example: 1/2, 0, -0.7, -9, etc. In general, it is impossible to list all the solutions for an inequality, so we use a graph to represent all the solutions.
This lesson will extend students’ practice with the addition and subtraction properties of equality. The same number can be added to or subtracted from each side of an inequality to get a new inequality with the same solutions as the original. Graphing is another way to represent the inequality.
2-10 Solving One-Step Inequalities by Multiplying or Dividing
When you multiply or divide each side of an inequality by a negative number, you must reverse the direction of the inequality symbol. If you don’t reverse the direction of the symbol, the resulting inequality does not have the same meaning as the original inequality.
As with equalities, a solution’s reasonableness can be checked by evaluating the original inequality for that solution.