Modeling Prompt 6
Designing a Fountain
Integrated Math 2
In-Class Launch
Use after Unit 4, Lesson 17
To introduce the task, make sure students can easily visualize what shape is made by water coming out of a jet or fountain. Ask students what shape water makes when it comes out of a drinking fountain, and either sketch the shape for all to see or invite a student to sketch it. Ask if they have seen fountains with jets that shoot water. Consider showing examples of a few, including videos if possible. Here is an example image:
An important point to highlight is that a quadratic function models the curved path of the water through the air.
Take students through a simplified version of the task, with one statue and one water jet. Here is an image to use if needed. The trapezoid represents the statue and the red dot is the location of the water jet.
Ask students what they would need to know in order to write an equation for the path of the water that would make it go over the statue (the distance from the jet to the statue, the height of the statue). Fill in reasonable heights and distances.
Then ask: “What strategy would you use to find an equation for the path of the water?” (Use the given distances to represent the statue and the location of the jet on a coordinate grid, and then write a vertex-form equation by choosing a reasonable vertex and adjusting parameters as needed.) After some quiet think time, ask students to share their ideas with a partner before inviting them to share with the class.
If no student suggests estimating a reasonable vertex for the water’s path and writing a vertex-form equation, ask students to discuss this idea. Vertex form is a convenient way to write the equation if we know the size and the distance of the statue so we can represent it on a coordinate grid.
Teacher Instructions
If students aren’t sure how to get started, suggest that they sketch the pool and the statues from the side, like the example in the Launch. Students can then sketch a coordinate grid on top of it, choose an origin, and fill in the values they know. Sketching a reasonable path for the water should help them estimate the vertex.
Make technology available. Having access to graphing technology streamlines the process of testing different equations. Consider demonstrating how to use sliders in Desmos to adjust the parameters of a quadratic equation in vertex form. For example, if \(y = \text-a(x+k)^2 + h\) is used to represent a generic quadratic that opens down, students easily can experiment with its shape by moving sliders on \(a\), \(k\), and \(h\).
A nearby park has a decorative pool. The board of commissioners of the park is considering installing water jets in the pool to make it into a fountain. They have asked you to design the water jets. Here is a diagram of the pool from above:
Statues A and B are 4 feet tall each.
Statues C and D are wider and 5.5 feet tall each.
Statue E is 8 feet tall, with a vase on top.
The diagram is drawn to scale, and the pool is 15 feet wide.
The jets are at water level. You need to figure out the number of jets, where they should go, and what path the water should make. Your design needs to meet these criteria:
- The jets are placed only around the edge of the pool.
- There are at least two jets.
- The water doesn't hit any of the statues.
- Some water goes into the vase.
- The water doesn't go higher than 10 feet.
When you have your final design, create a presentation to explain it to the commissioners. The presentation should show the paths of the water and include your mathematical descriptions of the paths.
Teacher Instructions
If students aren’t sure how to get started, suggest that they sketch a coordinate grid on top of their sketch(es) of the pool from the side. They can then choose an origin and fill in the values they know. Sketching a reasonable path for the water should help them estimate the vertex.
Make technology available. Having access to graphing technology streamlines the process of testing different equations. Consider demonstrating how to use sliders in Desmos to adjust the parameters of a quadratic equation in vertex form. For example, if \(y = \text-a(x+k)^2 + h\) is used to represent a generic quadratic that opens down, students easily can experiment with its shape by moving sliders on \(a\), \(k\), and \(h\).
A nearby park has a decorative pool. The board of commissioners of the park is considering installing water jets in the pool to make it into a fountain. They have asked you to design the water jets. Here is a diagram of the pool from above:
Statues A and B are 4 feet tall each.
Statues C and D are wider and 5.5 feet tall each.
Statue E is 8 feet tall, with a vase on top.
The diagram is drawn to scale, and the pool is 15 feet wide.
The dots along the edge show where to place the jets. You don’t have to put jets in all five places. The jets are at water level.
dome shape with 2 red dots on curve and 3 red dots along bottom side. large circle E in center. circle A above and to the left of E. circle B above and to the right of E. circle C below and to the left of E. Circle D below and to the right of E.
You need to figure out the number of jets, where they should go, and what path the water should make. Your design needs to meet these criteria:
- The jets are only in the places indicated by the dots.
- There are at least two jets.
- The water doesn’t hit any of the statues.
- Some water goes into the vase.
- The water doesn’t go higher than 10 feet.
When you have your final design, create a presentation to explain it to the commissioners. The presentation should show the paths of the water and include your mathematical descriptions of the paths.
Before you begin your design, sketch a side view of the pool, showing the heights of the statues and the vase. This will help you figure out the paths of the water.
Teacher Instructions
If students aren’t sure how to get started, suggest that they sketch a coordinate grid on top of each sketch of the pool from the side. Then they can choose an origin for each grid and fill in the values they know. Sketching in a reasonable path for the water should help them estimate the vertex.
Make technology available. Having access to graphing technology streamlines the process of testing different equations. Consider demonstrating how to use sliders in Desmos to adjust the parameters of a quadratic equation in vertex form. For example, if \(y = \text-a(x+k)^2 + h\) is used to represent a generic quadratic that opens down, students easily can experiment with its shape by moving sliders on \(a\), \(k\), and \(h\).
A nearby park has a decorative pool. The board of commissioners of the park is considering installing water jets in the pool to make it into a fountain. They have asked you to design the water jets. Here is a diagram of the pool from above:
Statues A and B are 4 feet tall each.
Statue C is wider and 5.5 feet tall.
Statue D is 8 feet tall, with a vase on top.
The diagram is drawn to scale, and the pool is 15 feet wide.
The dots along the edge show where to place the jets. You don’t have to put jets in all four places. The jets are at water level.
rectangle. red dots on perimeter at top, left, bottom, right on center of each side. large circle in center labeled D. circle A to the left of D, circle B to the right of B, circle C above D.
You will need to figure out the number of jets, where they should go, and what path the water should make. Your design needs to meet these criteria:
- The jets are only in the places indicated by the dots.
- There are at least two jets.
- The water doesn’t hit any of the statues.
- Some water goes into the vase.
- The water doesn’t go higher than 12 feet.
When you have your final design, create a presentation to explain it to the commissioners. The presentation should show the paths of the water and include your mathematical descriptions of the paths.
Before you begin your design, sketch what the pool would look like from each side. Make sure to show the heights of the statues and the vase. This will help you figure out the paths of the water.
To help you get started, here is part of one sketch, showing the widths of Statues A, B, and D. It shows what the pool would look like if you were standing near the bottom of the diagram that shows the pool from above. It also shows the jet locations next to Statues A and B. Figure out the heights of Statues A, B, and D, according to the scale of this sketch, and then finish the sketch by representing the heights. Make your own sketches for the other sides of the pool.