IM 7.6.18 Lesson: Subtraction in Equivalent Expressions

Find each sum or difference mentally.
[math]\text{-}30+\text{-}10[/math]
[math]\text{-}10+\text{-}30[/math]
[math]\text{-}30-10[/math]
[math]10-\text{-}30[/math]
Lin and Kiran are trying to calculate [math]7\frac{3}{4}+3\frac{5}{6}-1\frac{3}{4}[/math]. Here is their conversation:[br][list][*][b]Lin[/b]: “I plan to first add [math]7\frac{3}{4}[/math] and [math]3\frac{5}{6}[/math], so I will have to start by finding equivalent fractions with a common denominator.”[br][br][/*][*][b]Kiran[/b]: “It would be a lot easier if we could start by working with the [math]1\frac{3}{4}[/math] and [math]7\frac{3}{4}[/math]. Can we rewrite it like [math]7\frac{3}{4}+1\frac{3}{4}-3\frac{5}{6}[/math]?”[/*][/list][br][list][*][b]Lin[/b]: “You can’t switch the order of numbers in a subtraction problem like you can with addition; [math]2-3[/math] is not equal to [math]3-2[/math].”[br][br][/*][*][b]Kiran[/b]: “That’s true, but do you remember what we learned about rewriting subtraction expressions using addition? [math]2-3[/math] is equal to [math]2+\left(-3\right)[/math].”[br] [/*][/list][br]Write an expression that is equivalent to [math]7\frac{3}{4}+3\frac{5}{6}-1\frac{3}{4}[/math] that uses addition instead of subtraction.
If you wrote the [b]terms [/b]of your new expression in a different order, would it still be equivalent? [i]Explain [/i]your reasoning.
Write two expressions for the area of the big rectangle.[br][br][img]data:image/png;base64,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[/img][br]
Use the distributive property to write an expression that is equivalent to [math]\frac{1}{2}\left(8y+\left(-x\right)+\left(-12\right)\right)[/math]. The boxes can help you organize your work.[br][br][img]data:image/png;base64,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[/img][br]
Use the distributive property to write an expression that is equivalent to [math]\frac{1}{2}\left(8y-x-12\right)[/math].
Here is a calendar for April 2017. Let's choose a date: the 10th. Look at the numbers above, below, and to either side of the 10th: 3, 17, 9, 11.
Average these four numbers. What do you notice?
Choose a different date that is in a location where it has a date above, below, and to either side. Average these four numbers. What do you notice?
Explain why the same thing will happen for any date in a location where it has a date above, below, and to either side. 
Close

Information: IM 7.6.18 Lesson: Subtraction in Equivalent Expressions