3.2: Hot Air Balloon
This activity is intended to supplement Algebra I, Chapter 2, Lesson 3.
Problem 1 – Integer Addition
Press PRGM and find the HOTAIR program in the list.
Press ENTER to load the program.
Use the model to find the sum \begin{align*}2 + (-6)\end{align*}.
Press ENTER to choose option 1, Integer Addition.
Note: The right arrow adds a helium bag to the balloon, and the left arrow adds a sand bag.
Press ENTER to start and press \begin{align*}\rightarrow\end{align*} to move the balloon to ground level, the horizontal line.
The sum \begin{align*}2 + (-6)\end{align*}, in terms of the balloon:
- \begin{align*}2\end{align*} signifies adding two helium bags, which raises the balloon \begin{align*}2\ units\end{align*}. Press the right arrow twice.
- \begin{align*}-6\end{align*} signifies adding six sand bags, which lowers the balloon \begin{align*}6\ units\end{align*}. Press the left arrow six times.
The balloon’s resulting position is the sum.
With your partner, translate the following expressions into “balloon language” and use the model to find the sum. Remember to reset the balloon to ground level each time. When you are finished, press ENTER to return to the menu.
- \begin{align*}-4 + 7 = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}7 + 3 = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}5 + (-7) = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}-5 + (-3) = \underline{\;\;\;\;\;\;\;}\end{align*}
Problem 2 – Missing Addend
Find the value of \begin{align*}b\end{align*} such that \begin{align*}-4 + b = 3\end{align*}.
Arrow down to option \begin{align*}2\end{align*}, Missing Addend, and press ENTER.
You are given the value of \begin{align*}a\end{align*} and the value of the sum \begin{align*}a + b\end{align*}, but the value of \begin{align*}b\end{align*} is unknown.
Input the value for \begin{align*}A\end{align*} and press ENTER. Then input the sum and press ENTER.
Press ENTER to start and press \begin{align*}\rightarrow\end{align*} to move the balloon to ground level, the horizontal line.
In terms of the balloon, there are four sand bags. Press the left arrow four times.
You can see the balloon at the bottom of the screen. Now there is also a target balloon to the right. The task is now to find the value of \begin{align*}b\end{align*} that is needed to move the balloon on the right to the same position as the target balloon on the left.
The target balloon, on the right, is positioned at the value of the sum.
The target balloon is above your balloon’s position, so you need to add helium bags. Press the right arrow to add helium bags until the balloons line up. Count as you press to find how many helium bags you added. For this example, \begin{align*}b = 7\end{align*}.
With your partner, translate the following equations into “balloon language” and use the model to find the missing addend. When you are finished, press ENTER to return to the menu.
1. \begin{align*}2 + b & = -3 \\ b & = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}-6 + b & = -1 \\ b & =\underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}5 + b & = 1 \\ b & =\underline{\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}-2 + b & = 4 \\ b & =\underline{\;\;\;\;\;\;\;}\end{align*}
Problem 3 – Integer Subtraction
The model also provides a way of visualizing the subtraction of integers. As with addition, positive integers are represented by helium bags and negative integers by sand bags. However, subtraction is the operation of taking off a bag. For example, the expression \begin{align*}-2 - 5\end{align*} translates to “put on \begin{align*}2\end{align*} sand bags and then take off \begin{align*}5\end{align*} helium bags.”
Arrow down to option 3, Integer Subtraction, and press ENTER.
Note: The right arrow removes a helium bag, and the left arrow removes a sand bag.
Reset the balloon at ground level by pressing the right arrow.
Move the balloon to its starting point, \begin{align*}-2\end{align*}.
To remove \begin{align*}5\end{align*} helium bags, press the right arrow \begin{align*}5\ times\end{align*}. The balloon falls \begin{align*}5\ units\end{align*} to end at \begin{align*}-7\end{align*}, which is the difference \begin{align*}-2 - 5\end{align*}.
With your partner, translate the following expressions into “balloon language” and use the model to find the difference. When you are finished, press ENTER to return to the menu.
- \begin{align*}2 - 7 = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}-3 - 1 = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}5 - (-2) = \underline{\;\;\;\;\;\;\;}\end{align*}
- \begin{align*}-4 - (-7) = \underline{\;\;\;\;\;\;\;}\end{align*}
Problem 4 – Missing Subtrahend
This model shows two balloons side by side—like the model from Problem 2, except that it is used to find a missing subtrahend rather than a missing addend.
Arrow down to option 4, Missing Subtrahend, and press ENTER.
For example, find the value of \begin{align*}b\end{align*} such that \begin{align*}-3 - b = 8\end{align*}. In terms of the balloon, this means that the balloon ends up at \begin{align*}8\end{align*}, and you have used \begin{align*}3\end{align*} sand bags. You need to find how many and which type of bag you must remove to have a resulting position of \begin{align*}8\end{align*}.
Input the value for \begin{align*}A\end{align*} and press ENTER. Then input the difference and press ENTER.
Your balloon is at the top of the screen, and the target balloon to the right represents the difference.
Reset the balloon at ground level by pressing the right arrow.
Note: The right arrow removes a helium bag, and the left arrow removes a sand bag.
Move the balloon to its starting position, \begin{align*}-3\end{align*}.
The target balloon is above your balloon’s position, so you need to remove sand bags. Press the left arrow to remove sand bags until the balloons line up. Count as you press to find how many sand bags you removed. For this example, you should find that \begin{align*}b = -11\end{align*}.
With your partner, translate the following equations into “balloon language” and use the model to find the missing subtrahend. When you are finished, press ENTER to return to the menu.
1. \begin{align*}6 - b & = 9 \\ b & = \underline{\;\;\;\;\;\;\;}\end{align*}
2. \begin{align*}5 - b & = -3 \\ b & =\underline{\;\;\;\;\;\;\;}\end{align*}
3. \begin{align*}-4 - b & = -1 \\ b & =\underline{\;\;\;\;\;\;\;}\end{align*}
4. \begin{align*}-2 - b & = 6 \\ b & =\underline{\;\;\;\;\;\;\;}\end{align*}
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