#WithMathICan Explore Quadratic Models
Tell students they are to explore how to find maximum area. Have the following supplies available and out for students to use: scissors, string, markers, rulers, graphing calculators or access to Desmos. Put students into groups of three. Put the problems up on the SMART Board. Ask students to collaborate to determine a strategy to solve the problem in their groups for five minutes. Then hand out the problem set and ask students to solve the problems with justification.
As students are working, assist students by asking clarifying questions if they are stuck.
If they don't know how to start: Ask "How can you model the sides?"
If they are struggling finding area: Ask "What shape do you have? How do we find the area of that shape?"
If they are struggling determining the maximum area: Ask "Can you make a table with various values? Did you try some values in between what you have?"
If they have completed a model and a table: Ask "Can you write an equation to model the situation?"
If they get an equation: Ask "How can you find the maximum point?"
In a group, discuss and determine a solution method and justify using multiple methods. You may use the tools provided to assist you in your exploration.
2 Direct Instruction
Using the problems from the Exploration Activity, have students present multiple solution methods to finding the maximum area in each problem. Ask for a numerical solution, a graphical solution (highlight maximum on Desmos), and an algebraic solution. Allow students to share their thought process including misconceptions and how they changed their thinking. As needed, help students make connections between representations. Make sure students have access to numerical, graphical and algebraic solutions. Encourage students to ask questions on any points of confusion.
Students will present their solution methods and incorporate other solution methods into their notes.
3 Guided Practice
Students will complete a modeling problem as a member of a group. There are five different modeling problems so groups will compare and contrast their work at the conclusion of the problem solving.
Supplies needed: Markers, graph paper, posters or large sheets of paper.
Put students into small groups (3-4) and distribute the Quadratic Modeling Problems, one per group. Allow students to work together. Remind students to use the simpler problems from the exploration to help them.
As students are working, monitor groups and ask students clarifying questions. Sample questions:
If students are having trouble writing an algebraic equation, ask "How did you calculate the area numerically? or How is this problem similar to the exploration problems?"
If students are having trouble identifying an appropriate domain, ask "For what values of x can you find the area?, For what values of x can you not find the area?, "How could a different representation help you find the domain?"
If students are having trouble graphing, ask "What type of parent function do you have?, What are some important point to have for this particular type of function?, What tool do you have to help you graph?"
If students are having trouble finding other ways to calculate the maximum, ask "What does the maximum point represent on a quadratic function?, Do you know a way to calculate the maximum/minimum point?, Can you use another representation to help you solve this problem?"
In a group, discuss and determine a solution method and justify using multiple methods. You may use the tools provided to assist you in your problem.
4 Gallery Walk/Class Discussion
Have students post their posters. Give each student 4-8 sticky notes. Ask each student to write a question or an "ah-ha" moment they have for something on each group's poster as they do the gallery walk (they do not have to put their name on their question). After students have had an opportunity to post their questions and "ah-ha" moments, use the post-its to guide the class discussion specifically highlighting points of confusion that have been clarified, connections between multiple "ah-ha" moments.
Students will do a gallery walk comparing and contrasting the modeling problems from each group. Students will respond to the other group's problems by writing questions and "ah-ha" moments on sticky notes on at least one per poster. Participate in the class discussion.
5 Independent Practice/Extension
Ask students to complete independent practice items. For the extension problems, students will need access to Desmos or other graphing utility.
Complete independent practice items. For the extension problems, students will need access to Desmos or other graphing utility.
Key Standards Supported
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
Reasoning With Equations And Inequalities
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Solve quadratic equations in one variable.
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
(+) Represent a system of linear equations as a single matrix equation in a vector variable.
(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Graph the solutions to a linear inequality in two variables as a half- plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
The Real Number System
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.