This expansion applies if [alpha] and [beta] are not zero or

negative integers and [gamma] is not an integer.

Since there are prime directed graphs of arbitrarily large size, Corollary 2.27 yields examples of directed graphs of polynomial growth of degree k for every non

negative integer k.

The teacher now works with the students to obtain well-definitions of [10.sup.n] where n is 0 or a

negative integer. The first definition is suggested by the statement 1 = [10.sup.n]/[10.sup.n] = [10.sup.n-n] = [10.sup.0].

Nevertheless, because the students were accustomed to thinking in terms of repeated addition (e.g., as shown in Figure 3) we maintained this model for multiplication involving positive and

negative integers. Thus, a positive number multiplied by a positive number was modelled as repeated steps to the right thus yielding a positive result; a positive number multiplied by a negative number as repeated steps to the left thus yielding a negative result; and a negative number multiplied by a negative number as repeated backward steps facing left resulting in a distance to the right of zero and hence a positive result.

So the adjustment from subtracting k cuts the last

negative integer out of the equation.

Negative integers aren't entirely out of the question.

Let x be the maximum absolute value of

negative integers. We can add x to the actual values v and handle natural numbers for negative values.

25), the number line provides a "straightforward image for us to associate with positive and

negative integers" (Haylock & Manning, 2014, p.

Mary, similarly, may report on her ability to solve 20 arithmetic problems that involve both positive and

negative integers: 9 + (-1) =_--, and -6 + --_= 11.

Her 10-year-old sister was quicker at answering Level 1 problems but noted, "I had to think!" Levels 2 (medium) and 3 (hard) present increasingly challenging problems using multiplication, division, factors, sequencing decimals, working with

negative integers and other tasks.