Subject:Math
Grade Level: High School
Materials: Pictures of Australian landmarks, graphing calculators, LCD projector or Smartboard with Internet access, laptops, pencils, rulers
About: Students explore the properties of quadratic functions and parabolas by looking at examples from Australian architecture and natural landscapes.
The students will find the equation of a parabola that models one of the arches of the Sydney Opera House.
Students make connections between mathematics and real-life situation. By using Internet websites, they learn mathematical concepts and also learn about Australia. They search for information that describes the history of Australia and conjecture about why there are so many parabolic shapes there, and in particular in Sydney. Students use graphing calculators and apply them to real-world mathematics problems.
Quadratic functions are an important part of a high school mathematics curriculum. These lessons look at parabolas in a different way than a traditional textbook would. Students may also find Australia's history as a penal colony interesting.
| Students will discover real-life applications of mathematics. |
| Students will learn about quadratic functions and their properties. |
| Students will graph quadratic functions by hand and using a calculator. |
| Students will use a graphing calculator to learn about regressions and lines of best fit. |
| Students will make connections to other content areas. |
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| Students solve problems that arise in mathematics and in other contexts. |
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| Students use multiple representations to represent and explain problems. |
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| Students understand and make connections among multiple representations of the same mathematical idea. |
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| Students recognize and apply mathematics to situations in the outside world. |
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| Students use mathematics to show and understand physical phenomena. |
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| Students analyze and solve verbal problems that involve quadratic equations. |
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| Students investigate and generalize how changing the coefficients of a function affects its graph. |
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| Students find the roots of a parabolic function graphically. |
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| Students determine the vertex and axis of symmetry of a parabola, given its graph. |
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| Day 1: Exploring Quadratic Functions |
| Students will graph a parabola by hand using its equation. |
| Students will learn about the axis of symmetry and vertex of a parabola. |
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| Worksheet "Graphing Quadratic Functions.doc" |
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| Start off the lesson by having students use the general form of a quadratic equation to identify the a, b, and c of several quadratic functions. |
| Discuss the differences between linear and quadratic functions. Ask questions like "What does the graph of a linear function look like?" "What do we call the graph of a quadratic function?" |
| Briefly talk about symmetry and explain what the axis of symmetry of a parabola is. Use one of the equations from the beginning of class to teach students how to find the axis of symmetry. The equation to find the axis of symmetry is x = -b/(2a). |
| Ask students where they have heard the word "vertex." Explain what the vertex of a parabola is. Use the axis of symmetry to find the vertex of the parabola. |
| Have students practice finding the axis of symmetry and vertex of the rest of the functions from the beginning of class. |
| Hand out the worksheet titled "Graphing Quadratic Functions." Instruct students to first find the axis of symmetry and vertex of the function from problem 1. Then students can complete the chart and graph the function. Students should identify the axis of symmetry and vertex of the parabola. Have them compare the vertex and axis of symmetry they found algebraically with those they found graphically. |
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| Complete graph number 2 on the worksheet. |
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| Give two quadratic functions and ask students to find the axis of symmetry and vertex of each. |
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| Day 2: Graphing Quadratic Functions with Calculators |
| Students will use graphing calculators to view the graph of quadratic functions. |
| Students will find the maximum or minimum of a quadratic function. |
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| Write two quadratic functions on the board, one with a positive a and one with a negative a, and have students find the vertex of each. |
| Go over the "do now." Ask students if they think the parabolas will open up or down. Talk about how they can tell if a parabola opens up or down using the "a" value in the equation. Relate a positive parabola to a smile and a negative parabola to a frown. |
| Give students a few examples and have them decide if the parabolas open up or down. |
| Talk about the turning point (vertex) of a parabola. Show students how they can tell if a parabola has a maximum or minimum. If the "a" in the equation is positive, the parabola has a minimum. If the "a" in the equation is negative, the parabola has a maximum. |
| You can use a basketball analogy to help students remember maximums and minimums. When you shoot a ball into the basket, there is a high point, or maximum, before the ball falls and goes in the basket. When you bounce a ball once, it hits the ground, or minimum, before it comes back up to your hands. |
| Have students tell if the parabolas from earlier will have a maximum or minimum. Hand out the graphing calculators while the students are working. |
| Show students how to enter the equations into the graphing calculator. Remember to show them how to enter the window values. Ask if they correctly identified which parabolas opened up or opened down. |
| Show students how to find maximums or minimums using the calculator. Have them verify if their answers for the parabolas were correct. |
| Give students several more functions to view. Have them find the maximum or minimum. |
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| Give students a few functions and have them decide if the parabola opens up or down and has a maximum or minimum. |
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| Have the students write down the vertex of each of the functions they graphed on the calculator. |
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| Day 3: Parabolas in Australia |
| Students will view pictures of parabolas in the real world. |
| Students will use the Internet to find information about Australia. |
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| LCD projector |
| Laptops with Internet access |
| Pictures of parabolas: London Bridge Great Ocean.jpg, store in Queen Victoria Building.jpg, and Opera House and Harbour Bridge.jpg |
| Websites listed above |
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| Write two equations on the board. Have students find the vertex and axis of symmetry of the function and tell if it opens up or down and has a maximum or a minimum. |
| Show picture of the London Bridge on the Great Ocean Road. Ask a student to point out where the parabola is. Talk about why the rock is shaped like that. Have students use a search engine to figure out how the water shaped the rocks along the Great Ocean Road. |
| Show a picture of the store in the Queen Victoria Building. Use link 4 to have the students search for other examples of parabolas in the Queen Victoria Building (i.e. the dome on the outside of the building). |
| Show a picture of the Sydney Opera House and the Harbour Bridge. Use links 2,3,5 and 6 to find out information about Sydney. Students can research the history of Australia as a penal colony and look at Sydney today. |
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| Use the Internet, newspapers, magazines, or books to find a picture of a parabola from the real world. Bring in the picture. |
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| Have students write down examples of where they saw parabolas in Australia and why they are used in architecture. |
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| Day 4: Finding the Equation of the Roof of the Sydney Opera House Part 1 |
| Students will use graphing calculators to perform linear and quadratic regressions. |
| Students will decide which line best fits data points. |
| Students will find the slope of a segment connecting two points. |
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| Graphing calculators |
| LCD projector |
| Worksheet: "Sydney Opera House Parabola.doc" |
| Colored pencils |
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| Hand out worksheet. Have students answer the questions in the "do now." |
| Go over the "do now" and have students start the class work section. Use coordinates (-2.5, 3) for point R, (0,2) for point S and (2,0) for point T. Students can use colored pencils to label their points. Hand out graphing calculators. |
| Talk about the answer to question number 4. Answers may vary, but lead students to figure out that the shape of the roof is not linear. |
| Have students answer questions 1 through 7 on the worksheet. Circulate and help students perform the regressions. |
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| Answer question number 8 on the worksheet. |
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| Have students write the equation of the parabola on the line of page 1 (picture of the opera house). Collect this page. |
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| Day 5: Finding the Equation of the Roof of the Sydney Opera House Part 2 |
| Students will use a picture of the Sydney Opera House to draw a parabola that models its roof. |
| Students will use attributes of quadratic functions to discover information about the roof of the Sydney Opera House. |
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| Graphing calculators |
| LCD projector |
| Worksheet: "Sydney Opera House Parabola.doc" |
| Colored pencils |
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| Hand out the students' graphs from the day before. Have them identify the a, b, and c of the function and the x and y intercepts. |
| Discuss what the relationship between the c and the y intercept of a function is. |
| Hand out graphing calculators and show students how to find the x intercepts of a graph. Ask them why they are called "zeros." |
| Show students how to use the calculate menu to find y values for specific x values. Have them write down the y values for the x values of -10, -6, -3, 0, 5, and 7. |
| Show students how to use the table to find x and y values. Have them confirm their coordinates from the calculate menu. |
| Have students plot their points. |
| Have the students find the axis of symmetry and vertex graphically. |
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| Have students algebraically confirm the axis of symmetry and vertex of the function modeling the roof. |
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| Have students use colored pencils to complete the drawing of the parabola. |
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Marisa Laks
mlaks@schools.nyc.gov
Louis D. Brandeis High School
145 W 84th St.
New York, NY 10024
Marisa is a third-year math teacher who is interested in connecting mathematics to other content areas. She will complete her master's degree in Mathematics Education at CUNY City College in May. In addition to receiving a Teacher's Network Gotham Gazette Grant, she has been awarded an NCTM Mathematics Education Trust Grant. Marisa is also a TNLI MetLife Fellow.
Important documents for this lesson plan.
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