 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Solutions for Session 2, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8    Problem C1

Some of the distinctive properties of 0 are as follows:

 • Any number added to 0 returns the original number (7 + 0 = 0 + 7 = 7). Zero is the only number that behaves this way. • Any number multiplied by 0 results in 0 (7 • 0 = 0 • 7 = 0). Again, 0 is the only number that behaves this way. • No number can be divided by 0 to return a unique real answer. • If two different numbers multiplied together make 0, one or the other must also be 0.   Problem C2 Answers will vary. Some important attributes may be that 0 is a placeholder that enables us to distinguish between different numbers and their place values, even if there is no value at a particular place; for example, 1,002 and 102. Without those zeros, we wouldn't be able to tell which number is one hundred and two and which is one thousand and two. Zero also separates the positive from the negative numbers.   Problem C3

 a. Yes. This point must be on the graph, since it satisfies the equation x • y = 12 (x = 2 and y = 6). b. Yes. This point also satisfies x • y = 12. c. No, this point is not on the graph because the product, x • y, yields -12 rather than 12. Similarly, any point in the coordinate system whose product of the x- and y- coordinates does not yield 12 will not be on the graph. d. For each value on the x-axis that you pick, you would have a corresponding y value such that the product of the two always equals 12. As the x values on the x-axis get larger and larger, the y values on the y-axis get smaller and smaller. And vice versa, as the x values get smaller and smaller (i.e., approach 0), the y values get larger and larger. So if the x value is infinitely large, the corresponding y value will be infinitely small, and the product will still be 12. For this reason, the curve will never touch either of the axes. If it did touch one of the axes -- let's say the x-axis -- it would mean that the y value is 0. Consequently, the product would equal 0, and that does not satisfy this equation.   Problem C4 The unique value is y = 3. We are looking to solve the equation 4 • y = 12. Since 4 has the inverse 1/4, we can multiply both sides by 1/4 to produce y. The solution is then y = 1/4 • 12 = 12/4 = 3.   Problem C5

 a. This is the point at which x and y are equivalent while still solving x • y = 12. Since x and y are equivalent, we can change the equation to x • x = 12, or x2 = 12. b. According to the graph, x and y must be between 3 and 4, since (3,4) and (4,3) are both on the graph. An estimate of (3.5,3.5) would be a good guess. From the equation above (x2 = 12), we know that the x- and y-coordinates of this point are both . The actual coordinates are, to three decimal places, (3.464,3.464). A decimal for the exact coordinates can never be fully written, since is an irrational number.   Problem C6 No, it cannot touch either axis. If it did, then x or y would be 0. In this case, it would be impossible to have x • y = 12, since multiplying by 0 always produces 0.   Problem C7 Look back at the solution for Problem C4; this solution depended on our ability to find a multiplicative inverse for the number 4. Zero is the only real number without a multiplicative inverse, so it is the only number where we cannot "divide" to find the other coordinate.   Problem C8 Yes, it should, as there are now two intersection points for these graphs. The other is a point that still satisfies x2 = 12. Since a negative number multiplied by itself equals a positive number, this suggests that the other x-coordinate is the opposite of the coordinate. To three decimal places, the coordinates are (-3.464,-3.464).     