IM 7.6.9 Lesson: Dealing with Negative Numbers

Which equation doesn’t belong? Explain your reasoning.
[table][tr][td][math]15 = \text- 5 \cdot \text- 3[/math][br][/td][td][math]4 - \text- 2 = 6[/math][/td][/tr][tr][td][math]2 + \text- 5 = \text- 3[/math][/td][td][math]\text- 3 \cdot \text- 4 = \text- 12[/math][/td][/tr][/table]
Solve each equation. Explain your reasoning.
[math]x+6=4[/math]
[math]x-\text{-}4=\text{-}6[/math]
[math]2(x-1)=\text{-}200[/math]
[math]2x+\text{-}3=\text{-}23[/math]
Here are some equations that all have the same solution.
[center][/center][left][/left][math]\begin {align} x &= \text-6\\ x - 3 &= \text-9\\ \text-9 &= x - 3\\ 900 &= \text-100(x - 3)\\ 900 &= (x-3) \cdot (\text-100)\\ 900 &= \text-100x + 300\\ \end {align}[/math][br][br]Explain how you know that each equation has the same solution as the previous equation. Pause for discussion before moving to the next question.
Keep your work secret from your partner. Start with the equation [math]\text{-}5=x[/math]. Do the same thing to each side at least three times to create an equation that has the same solution as the starting equation. Write the equation you ended up with on a slip of paper, and trade equations with your partner. Also, write the equation here.
See if you can figure out what steps they used to transform [math]\text{-}5=x[/math] into their equation. When you think you know, check with them to see if you are right. Briefly explain what your partner did to get their equation.

IM 7.6.9 Practice: Dealing with Negative Numbers

Solve each equation.
[math]4x=\text{-}28[/math]
[math]x-\text{-}6=\text{-}2[/math]
[math]\text{-}x+4=\text{-}9[/math]
[math]\text{-}3x+7=1[/math]
[math]25x+\text{-}11=\text{-}86[/math]
Here is an equation [math]2x+9=\text{-}15[/math]. Write three different equations that have the same solution as [math]2x+9=\text{-}15[/math]. Show or explain how you found them.
Select [b]all [/b]the equations that match the diagram.[br][img]data:image/png;base64,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[/img]
There are 88 seats in a theater. The seating in the theater is split into 4 identical sections. Each section has 14 red seats and some blue seats. Draw a tape diagram to represent the situation.
What unknown amounts can be found by using the diagram or reasoning about the situation?
Match each story to an equation.
A stack of nested paper cups is 8 inches tall. The first cup is 4 inches tall and each of the rest of the cups in the stack adds [math]\frac{1}{4}[/math] inch to the height of the stack.
A baker uses 4 cups of flour. She uses [math]\frac{1}{4}[/math] cups to flour the counters and the rest to make 8 identical muffins.
Elena has an 8-foot piece of ribbon. She cuts off a piece that is [math]\frac{1}{4}[/math] of a foot long and cuts the remainder into four pieces of equal length.

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