In a previous unit, students studied quadratic functions in some depth. They built quadratic expressions to represent situations and wrote equivalent expressions. They also graphed, analyzed, and evaluated quadratic functions to solve problems. In some cases, they investigated and interpreted the outputs of the functions. In others, they looked for the input values that produce certain outputs, and they found these values mainly by reasoning with graphs.
This unit picks up on where that unit left off. Sometimes we have a relationship that can be expressed with a quadratic function, and we want to know what input generates a particular output. How do we find out other than by graphing and estimating, or by guessing and checking?
In this lesson, students encounter a problem that cannot be easily solved by familiar strategies, which gives them a chance to persevere in problem solving (MP1). They write a quadratic equation and interpret what a solution means in the given situation. The work here motivates the need to solve quadratic equations. The formal definition of a quadratic equation will be introduced until the next lesson, after students have seen some variations of such equations and worked with them in context.
- Explain (orally and in writing) the meaning of the solution to a quadratic equation in terms of a situation.
- Write a quadratic equation that represents geometric constraints.
- Let’s find some new equations to solve.
Print copies of the blackline master, one copy for every student. Note that the images need to be printed on 8.5" by 11" paper for the indicated dimensions to be accurate. Printing on paper of another size (such as A4 paper) may distort the image. (The goals of the activity don't depend on the dimensions being exactly right, however.)
- I can explain the meaning of a solution to an equation in terms of a situation.
- I can write a quadratic equation that represents a situation.
A quadratic expression in \(x\) is one that is equivalent to an expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
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