7th Grade Chapter 1

The variable h occurs twice in the expression 3h + 5 + 2h.  If a variable occurs more than once in an expression, you regard it as having the same value each time it occurs.

Ask students what would happen if there were no rule about whether to drive on the right side or the left side of the road.  Explain to students that the side to drive on is not important as long as everybody follows the same rule.  Similarly, the rules for the order of operations are neither right nor wrong, but simply a way of doing things that everybody follows to avoid confusion.

You may wish to introduce term and coefficient at this time.  In the expression (or term) 4y, 4 is the coefficient of y.  In any term with one variable, the coefficient is the number that multiplies the variable.  When you do not see the number, such as in x, it is understood to be 1.

Integers are Whole Numbers and their opposites.  Zero separates positive and negative integers on a number line.  Zero is neither negative nor positive.

You can model the sum of two integers.  Use one color to represent positive integers and a second color to represent negative integers.  Model each integer, then combine the models.  Group the zero pairs, then remove them.. The remaining units represent the integer sum.

You can read the expression - (-3) as the opposite of (the opposite of three).  It seems reasonable that the opposite of the opposite gets you back to where you started.  Mathematically, this means that - (-3) is the same as 3.  So you can read the expression 8 - (-3) as “8 plus the opposite of the opposite of 3”, which you can write as 8 + 3

Inductive reasoning assumes that a pattern will continue.  Sometime this is true, but sometimes it’s not.  When you make a conjecture based on a pattern, you are using inductive reasoning.  Because the conjectures you make are not always true, you should check your conjectures whenever possible.  All you need is one counterexample to prove a conjecture is not true.

Looking for a pattern is one of many problem-solving strategies.  See page 42 in your Red Math Book for a list of these strategies.

The product of two negative integers is positive.  This rule often seems counterintuitive.  You can use a pattern to suggest that multiplying two negative integers results in a positive answer.

Pierre de Fermat and Rene’ Descartes developed the Cartesian coordinate system in the seventeenth century.  This system uses two perpendicular number lines to identify the position of any point in a plane.  An ordered pair of numbers names the position of a point.  The pair is called ordered because the order of the numbers is crucial.  The first number always gives the horizontal distance from  zero and the second number always ives the vertical distance from zero.