Counting Numbers Whole Numbers Integers Rational Numbers -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 We don’t get all the way to the irrational numbers. That’ll be a topic in a future ppt—or an addition to this one.

Counting Numbers 1 2 3 4 5 6 7 … Generally, we count things we see, in order, one-by-one. Zero isn’t here because it’s abstract—not some “thing” that we can touch or see or itemize and count in this way.

Whole Numbers … 1 2 3 4 5 6 7 Whole numbers include zero. It’s an idea, a rather recent achievement for humankind. Zero is the starting point before you take your first step or count your first item. It’s what you have when everything you had is gone. The distance between zero and one is the unit segment that defines your number line.

Integers … 7 -7 6 -6 5 -5 4 -4 3 -3 2 -2 1 -1 Here are the units on the other side of zero. Zero is like the fulcrum of this balance going off in both directions to positive infinity and negative infinity. On each side of zero the numbers are mirror opposites of each other. And, since zero is in the middle, any number and its opposite will add to zero.

Rational Numbers 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7 … -1 -2 -3 -4 -5 -6 -7 … Fractions make the rational numbers. This slide is an attempt to begin to appreciate the multiple names of points on the number line and the tremendous, infinite density of numbers lying between the units.

Here is a way to begin to look at the rational numbers within a unit segment. Notice the various names for each point—common names like 2/2, 3/3, 4/4… 10/10 and beyond. Notice especially the common names for zero—0/2, 0/3, 0/4… 0/10, etc. Many students don’t think of zero as a possible numerator for a fraction or a quantity that can be equally and fairly shared (divided) among however many.

This is quite similar to the previous slide: it just builds up rather than converging. Probably many teachers will not want to show each of the three slides in this group. Instead, they’ll chose the one(s) they think their students will most easily understand.

This one shows the labels of the points just above the segments.

Families

Families

Families

Counting Numbers Whole Numbers Integers Rational Numbers -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 Slideshows end, but numbers keep on going… Infinitely left and right and infinitely in between.