IM Alg2.3.10 Lesson: A New Kind of Number

[size=150]Jada was helping her cousin with his math homework. He was supposed to solve the equation [math]8+x=5[/math]. He said, “If I subtract 8 from both sides, I get [math]x=5-8[/math]. This doesn’t make sense. You can’t subtract a bigger number from a smaller number. If I have 5 grapes, I can’t eat 8 of them!”[/size][br][br]What do you think Jada could say to her cousin to help him understand why [math]5-8[/math] actually does make sense?
[size=150]Numbers on the number line are often called[b] real numbers[/b].[/size][br][br][img]data:image/png;base64,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[/img][br][br]The equation [math]x^2=9[/math] has 2 real solutions. How can you see this on the graph of [math]y=x^2[/math]? [br]
Draw points on this real number line to represent these 2 solutions.
How many real solutions does [math]x^2=0[/math] have? Explain how you can see this on the graph of [math]y=x^2[/math].
Draw the solution(s) on a real number line.
How many real solutions does [math]x^2=\text{-}1[/math] have? Explain how you can see this on the graph of [math]y=x^2[/math].[br]
Draw the solution(s) on a real number line.
On the real number line:
[list][*]Draw an arrow starting at 0 that represents 3.[/*][*]Draw an arrow starting at 0 that represents -5.[/*][/list]
[size=100][size=150]This diagram shows an arrow that represents [math]\sqrt{-1}[/math].[/size][/size]
Draw an arrow starting at 0 that represents [math]3\sqrt{\text{-}1}[/math].[br]Draw an arrow starting at 0 that represents [math]\text{-}\sqrt{1}[/math].[br]Draw an arrow starting at 0 that represents [math]\text{-}5\sqrt{1}.[/math][br]
The absolute value of a real number is the length of the arrow that represents it.
What is the relationship between the absolute value of a real number and the absolute value of the square of that number?[br]
If we want [math]\sqrt{\text{-}1}[/math] and its square to have this same relationship, then what should the absolute value of [math]\sqrt{\text{-}1}[/math] be?[br]
What should the absolute value of  [math]3\sqrt{\text{-}1}[/math] be?
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Information: IM Alg2.3.10 Lesson: A New Kind of Number