Unit 8, Lesson 19
Quadratic Steps
Extra Support Materials for Algebra 1
19.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSA-REI.B.4.b
Solve quadratic equations by inspection (e.g., for ), 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 for real numbers and .
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this Warm-up, students are asked to spot a common error that often takes the form . The format of the question here closely resembles the values that students will see when working with the quadratic formula in the associated Algebra 1 lesson. Analyzing another’s work to find the error gives students an opportunity to construct viable arguments and critique the reasoning of others (MP3).
Launch
NoneActivity
NoneHere is Han’s work to solve an equation. Determine the error he made and be prepared to explain the correct way to solve it.
Activity Synthesis
The purpose of the discussion is to remind students to be careful about common errors when working with square roots. Here are some questions for discussion:
- “How can you remember where square roots fit into the order of operations?” (I like to think of the expression under the square root as a separate question to evaluate before doing anything else.)
- “Two common errors occur when using squares and square roots. First, is not correct in most cases. Next, is also not true in most cases. Choose values for and to assure yourself that these are not true.” (For the first one, but For the second one, , which is about 3.6, but .)
19.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students examine values that can be multiplied by a number to create a perfect square. In the associated Algebra 1 lesson, students prove the quadratic formula. As part of the process of proving the quadratic formula, students multiply the equation by so that the coefficient of the quadratic term is a perfect square. The work of this activity supports students in understanding that step.
Students construct viable arguments and critique the reasoning of others (MP3) when they analyze others’ answers. By generalizing a rule for a value that will guarantee a perfect square coefficient, students look for and express regularity in repeated reasoning (MP8).
19.3
Activity
Instructional Routines
MLR1: Stronger and Clearer Each TimeMaterials
NoneActivity Narrative
In this activity, students go through each step of completing the square. In the associated Algebra 1 lesson, students prove the quadratic formula by completing the square with the general form of a quadratic equation in standard form. By closely examining the steps for an actual example, students should be supported to follow the steps for a more abstract example in the associated Algebra 1 lesson.
When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.