3.2: Hot Air Balloon
This activity is intended to be used with Algebra I, Chapter 2, Lesson 3.
ID: 8613
Time required: 30 minutes
Activity Overview
In this activity, students use a dynamic, electronic manipulative to perform integer addition and subtraction. The goals of the activity are to (1) provide students with a visual for adding and subtracting integers and (2) help students understand that subtraction can be thought of as “adding the opposite” or “adding the inverse.”
The model given in this activity shows a hot air balloon that begins at ground level with a certain number of helium bags (providing lift) and the same number of sand bags (providing weight). The vertical position of the hot air balloon is determined by adding or removing a number of helium bags (representing positive integers) and sand bags (representing negative integers).
Topic: From Arithmetic to Algebra
- Use technology to verify that adding the number \begin{align*}-x\end{align*} is equivalent to subtracting \begin{align*}x\end{align*}
- Solve one-step linear equations of the form \begin{align*}x + a = b\end{align*} where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are real numbers
Teacher Preparation and Notes
- This activity may be used to introduce or review integer addition and subtraction. You may choose to use the activity in its entirety or break it up into separate activities by sets.
- It is very important that you thoroughly describe the model to students prior to them exploring the model on their own. Use a projector or a real balloon to demonstrate the model in a whole class, teacher-led setting. Some students will catch on very quickly and wean themselves from using the model. Others will prefer and/or need to stay a longer time with the model.
- To download the calculator file, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=8613 and select "HOTAIR.8xp" to download.
Associated Materials
- Student Worksheet: Hot Air Balloon, http://www.ck12.org/flexr/chapter/9612, scroll down to the second activity.
- HOTAIR.8xp (program)
In this activity, students will explore:
- adding and subtracting integers using a model of a hot air balloon
- the relationship between addition and subtraction
Explain how the model of the hot air balloon works. Students will need to download the HOTAIR program to their handheld and work with a partner to complete the activity.
Things to Remember…
This model is similar to a hot air balloon and provides a way to visualize adding and subtracting integers.
- Positive integers are represented by helium bags. They raise the balloon.
- Negative integers are represented by sand bags. They lower the balloon.
- Addition is the operation of putting bags on the balloon.
- Subtraction is the operation of taking bags off of the balloon.
- Always reset the balloon’s vertical position to \begin{align*}0\end{align*} before beginning another calculation.
To begin, press PRGM and find the HOTAIR program in the list. Press ENTER to load the program.
Problem 1 – Integer addition
Students will now use the model to find the sum \begin{align*}2 + (-6)\end{align*}. Press ENTER to choose option 1, Integer Addition.
The program displays instructions. The right arrow adds a helium bag to the balloon, and the left arrow adds a sand bag.
Students can see the balloon at the bottom of the screen. The horizontal line represents ground level. The balloon’s vertical position is displayed at the bottom of the screen. Right now, the position is \begin{align*}-12\end{align*}.
To view this addition \begin{align*}a + b\end{align*} using the model, students are to first reset the balloon to ground level by pressing the right arrow.
To find the sum \begin{align*}2 + (-6)\end{align*}, students need to understand what the sum represents in terms of the balloon. The \begin{align*}2\end{align*} signifies adding two helium bags to the balloon, which raises it \begin{align*}2 \ units\end{align*}. They should press the right arrow twice to add \begin{align*}2\end{align*} helium balloons.
Adding \begin{align*}-6\end{align*} means you then add six sand bags, which lowers the balloon \begin{align*}6 \ units\end{align*}. Press the left arrow \begin{align*}6 \ times\end{align*} to add \begin{align*}6\end{align*} sand bags. The balloon’s resulting position is the sum.
Together with a partner, students need to translate the following expressions into “balloon language” and use the model to find the sum. Remind them to reset the balloon to ground level each time. When they are finished, they can press CLEAR to return to the menu.
- \begin{align*}-4 + 7 = \underline{3}\end{align*}
- \begin{align*}7 + 3 = \underline{10}\end{align*}
- \begin{align*}5 + (-7) = \underline{-2}\end{align*}
- \begin{align*}-5 + (-3) = \underline{-8}\end{align*}
Problem 2 – Missing addend
In this problem, students 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.
Students are to arrow down to option 2, Missing Addend, and press ENTER.
Suppose you want to find the value of \begin{align*}b\end{align*} such that \begin{align*}-4 + b = 3\end{align*}. Ask students to first think about what this means in terms of the balloon. They know that \begin{align*}4\end{align*} sand bags are added, and that the balloon ends up at \begin{align*}3\end{align*}. Ask students: How many and which type of bag do you need to add to have a resulting position of \begin{align*}3\end{align*}?
They are to input the value for \begin{align*}A\end{align*} and press ENTER. Then input the sum and press ENTER.
The program displays instructions. Remind students that the right arrow adds a helium bag to the balloon, and the left arrow adds a sand bag, as with Integer Addition.
They can see the balloon at the bottom of the screen. Now there is also a target balloon to the right. The target balloon is positioned at the value of the sum. The task for students is to 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.
To find \begin{align*}b\end{align*} using the model, students should first use the right arrow to reset the balloon’s vertical position to \begin{align*}0\end{align*}.
Add \begin{align*}4\end{align*} sand bags by pressing the left arrow \begin{align*}4 \ times\end{align*}.
The target balloon is above the balloon’s position, so students need to add helium bags. They should press the right arrow to add helium bags until the balloons line up. Instruct them to count as they press to find how many helium bags they added. For this example, \begin{align*}b = 7\end{align*}.
Together with a partner, students are to translate the following equations into “balloon language” and use the model to find the missing addend. When they are finished, they can press ENTER to return to the menu.
- \begin{align*}2 + b & = -3\\ b & = \underline{\;-5\;}\end{align*}
- \begin{align*}-6 + b & = -1\\ b & = \underline{\;5\;}\end{align*}
- \begin{align*}5 + b & = 1\\ b & = \underline{\;-4\;}\end{align*}
- \begin{align*}-2 + b & = 4\\ b & = \underline{\;6\;}\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.”
Students are to now arrow down to option 3, Integer Subtraction, and press ENTER.
Use the model in the same manner as in Problem 1, with one difference: In this model, the right arrow removes a helium bag, and the left arrow removes a sand bag.
The model is the same as before. The balloon is at the top of the screen.
Students need to reset the balloon at ground level by pressing the right arrow.
Now they can move the balloon to its starting point, \begin{align*}-2\end{align*}.
To remove \begin{align*}5\end{align*} helium bags, students should press the right arrow \begin{align*}5 \ times\end{align*}. Explain to students that 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*}.
Together with a partner, students are to translate the following expressions into “balloon language” and use the model to find the difference. When you are finished, press \begin{align*}\subseteq\end{align*} to return to the menu.
- \begin{align*}2 - 7 = \underline{-5}\end{align*}
- \begin{align*}-3 - 1 = \underline{-4}\end{align*}
- \begin{align*}5 - (-2) = \underline{7}\end{align*}
- \begin{align*}-4 - (-7) = \underline{3}\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.
Students can now 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*}. Explain to students that 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. They need to find how many and which type of bag they must remove to have a resulting position of \begin{align*}8\end{align*}.
Students need to input the value for \begin{align*}A\end{align*} and press ENTER. Then input the difference and press ENTER.
Use the model in the same manner as in Problem 2, with one difference: In this model, the right arrow removes a helium bag, and the left arrow removes a sand bag.
The balloon is at the top of the screen, and the target balloon to the right represents the difference.
Students need to first reset the balloon at ground level by pressing the right arrow.
Then they can move the balloon to its starting position, \begin{align*}-3\end{align*}.
The target balloon is above their balloon’s position, so they need to remove sand bags. Press the left arrow to remove sand bags until the balloons line up. Remind students to count as they press to find how many sand bags they removed. For this example, students should find that \begin{align*}b = -11\end{align*}.
Together with a partner, students are to translate the following equations into “balloon language” and use the model to find the missing subtrahend. When they are finished, they can press ENTER to return to the menu.
- \begin{align*}6 - b & = 9\\ b & = \underline{\;-3\;}\end{align*}
- \begin{align*}5 - b & = -3\\ b &= \underline{\;8\;}\end{align*}
- \begin{align*}-4 - b & = -1\\ b &= \underline{\;-3\;}\end{align*}
- \begin{align*}-2 - b & = 6\\ b &= \underline{\;-8\;}\end{align*}
Problem 5 (Extension) – Addition and subtraction exploration
Students are to arrow down to option 5, Addition and Subtraction, and press ENTER.
The balloon on the left is for addition and the one on the right is for subtraction. Students can use the up and down arrows to move between the two models.
For each of the following expressions, students are to use what they’ve learned from Problems 1 and 3 to translate into “balloon language” and then find each sum or difference.
1. \begin{align*}-2 - 4 = \underline{\;\;\;\;\;\;} \end{align*}
2. \begin{align*}-2 + (-4) = \underline{\;\;\;\;\;\;} \end{align*}
3. \begin{align*}5 - (-6) = \underline{\;\;\;\;\;\;} \end{align*}
4. \begin{align*}5 + 6 = \underline{\;\;\;\;\;\;} \end{align*}
For each of the following equations, students use what they’ve learned from Problems 2 and 4 to translate into “balloon language” and then find each missing addend or subtrahend.
5. \begin{align*}-1 - b &= 5\\ b &= \underline{\;\;\;\;\;\;}\end{align*}
6. \begin{align*}-1 + b &= 5\\ b &= \underline{\;\;\;\;\;\;} \end{align*}
7. \begin{align*}3 - b &= -4\\ & b = \underline{\;\;\;\;\;\;} \end{align*}
8. \begin{align*}3 + b &= -4\\ b &= \underline{\;\;\;\;\;\;} \end{align*}
Now students are to complete the following statements.
9. Taking off \begin{align*}8\end{align*} sand bags is the same as putting on \begin{align*}8\end{align*} ___________ bags.
10. Taking off \begin{align*}5\end{align*} helium bags is the same as putting on \begin{align*}5\end{align*} ___________ bags.
11. If \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are any two integers, then \begin{align*}a - b = a + \underline{\;\;\;\;\;\;} \end{align*}. That is, subtracting a number is equivalent to adding its ___________.
To exit the program, students can arrow down to option \begin{align*}6\end{align*} and press ENTER.
Solutions
- \begin{align*}-6\end{align*}
- \begin{align*}-6\end{align*}
- \begin{align*}11\end{align*}
- \begin{align*}11\end{align*}
- \begin{align*}-6\end{align*}
- \begin{align*}6\end{align*}
- \begin{align*}7\end{align*}
- \begin{align*}-7\end{align*}
- helium
- sand
- \begin{align*}-b\end{align*}; opposite (or additive inverse)
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