Unit 3, Lesson 13
A Radical Identity
Integrated Math 3
13.1
Warm-up
10 mins
Quadratic Zeros
Standards Alignment
Building On
HSA-SSE.B.3.a
Factor a quadratic expression to reveal the zeros of the function it defines.
Addressing
HSA-SSE.A.2
Use the structure of an expression to identify ways to rewrite it. For example, see
Instructional Routines
Materials
None
Activity Narrative
In this Warm-up, students take the first steps to build a quadratic function in standard form from two known zeros. Students should recognize the structure (MP7) of a family of quadratic expressions that have the same two zeros. The family of expressions differ only by a constant multiple.
Launch
Arrange students in groups of 2. Give students 2–3 minutes to quietly read the task and make a plan, then ask students to work through the task with a partner, and finally finish the Warm-up with a whole-class discussion.
Activity
NoneStudent Task Statement
- Let
- For what values of
is the function equal to 0? Recall that these values are called zeros. - Rewrite the equation in standard form,
- For this function, how are
and related to the zeros? - Is this the only quadratic function with zeros at 3 and 4?
- For what values of
- Let
- What are the zeros of
? - Rewrite the equation in standard form.
- What are the zeros of
Activity Synthesis
The purpose of this discussion is for students to see the connection between the factored and standard forms and recognize that the family of functions that have the same two zeros have equations that differ only by a constant multiple.
Select groups to share their solutions. Display the equation
Highlight the solution to the last question and note that all quadratic functions with zeros at 3 and 4 are of the form
Display the functions
- “Do the two functions have the same zeros? Explain your reasoning.” (Yes. Each function is 0 when
- “What is another quadratic function that has zeros when
, such as - “Are the functions equivalent?” (No, because when
13.2
Activity
15 mins
An Identity from Zeros
Instructional Routines
Materials
None
Activity Narrative
In this activity, students reconstruct the standard form of a quadratic equation from the solutions guaranteed by the quadratic formula. The activity gives students an opportunity to practice rewriting expressions that include a square root and see how some complex-looking expressions can be rewritten into simpler forms.
Students must use the structure of the zeros to find ways to rewrite the expressions in simpler forms (MP7).
13.3
Activity
10 mins
Lesson Synthesis
The purpose of the discussion is to highlight the identities used in this lesson and show some uses of the identities. Ask students to describe the identities they used in this lesson. Display the identities for all to see.
-
-
Display the expression
Ask students how they can use these identities to rewrite the expression so that there are no square roots in the denominator. (Multiply by
Student Lesson Summary
Mathematical identities can be useful in a variety of situations.
In an earlier course, we learned how to derive the quadratic formula using the method of completing the square on a quadratic equation. We can check the formula by reconstructing the standard form of a quadratic equation using the zeros guaranteed by the quadratic formula. We can start with writing the factored form using the zeros as
After distributing the left side of the equation, it nicely becomes
Identities can also be used to rewrite expressions to be in a form that is clearer to understand. For example, the value of the fraction
In this form, we can estimate that
Have feedback on the curriculum?
Help us improve by sharing suggestions or reporting issues.