Completing the Square

Transcription

1 Completing the Square Student Probe Write this expression in vertex form:. Answer: Lesson Description This lesson uses an area model of multiplication and algebra tiles to develop the concept of completing the square for quadratic expressions. The lesson progresses from concrete to graphical to abstract representations of the process. In order to reduce any potential confusion caused by fractions, only even linear terms are used in this lesson. Rationale Students are frequently required to complete the square with no conceptual understanding, but rather a series of steps which make very little sense to them. This lesson uses an approach first developed by Pythagoras and made known by Euclid about 300 BCE. While solving quadratic equations using this method can be cumbersome, the quadratic formula is derived by completing the square. Additionally, using completing the square to change the form of expressions or equations can be particularly helpful in determining the vertex of a parabola without graphing. Preparation Provide a set of algebra tiles and a mat for each student, or for each pair of students. At a Glance What: Complete the square to change the form of a quadratic expression Common Core State Standard: CC.9-12.A.SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.* Matched Arkansas Standard: AR.9-12.QEF.AII.3.3 (QEF.3.AII.3) Analyze and solve quadratic equations with and without appropriate technology by: - - factoring, - - graphing, - - extracting the square root, - - completing the square, - - using the quadratic formula Mathematical Practices: Look for and make use of structure. Who: Students who cannot change the form of a quadratic expression by completing the square Grade Level: Algebra 1 Prerequisite Vocabulary: square, constant term, vertex, vertex form Prerequisite Skills: factoring quadratic trinomials, opposite numbers Delivery Format: individual, small group, whole group Lesson Length: 30 minutes Materials, Resources, Technology: algebra tiles, graphing calculator Student Worksheets: none

2 Lesson The Expect students to say or do If students do not, then the 1. What are perfect square Numbers that can be Refer to Factor Pairs. numbers? Give me some examples. represented as a square of tiles. 1, 4, 9, 16, etc. 2. Quadratic trinomials can be perfect squares, too. Model for students. Let s build with algebra tiles. What do you notice? What are its dimensions? It is a square. or So is a perfect square and its factored form is. 3. Build with algebra tiles. Is this a perfect square? Let s see if we can rearrange things to make it a perfect square. 4. What do we know about a square? So to make this a square, we need to make all sides the same length. 5. Let s start by splitting the green pieces into 2 equal groups. How many will be in each group? How do you know? No All sides are congruent. There are 6, so there will be 3 in each group. Model for students. Prompt students. Model by move the 6 green pieces together, then splitting into 2 groups of 3.

3 The Expect students to say or do If students do not, then the 6. Model this with your tiles. Is this a perfect square? Why or why not? Model for students. 7. How many unit pieces do we need? 8. But we cannot just add 4 more pieces, because that would change our expression. The only thing we can add is zero. Since, let s get the 4 we need this way. (See Teacher Notes.) 9. Now let s move the units into position to make a square. No, we need to fill in some unit pieces. 4 Model for students. Monitor students. Model if necessary. What are the dimensions of the square? How many negative units are left? 10. Write the algebraic expression for our square. 11. Does How do you know?? 4 Yes. or Model for students, pointing out how the diagram provides the expression. Refer to Multiplying Polynomial Expressions.

4 The Expect students to say or do If students do not, then the 12. This process is called completing the square. Why do you think it is called that? Because we filled in a square. 13. Now let s graph The vertex is. Monitor students to make on the (Students may notice that the sure they are graphing graphing calculator. x- intercepts are - 5 and - 1. See appropriately. What do you notice about Teacher Notes.) Ask students to find the the graph? vertex. 14. This is called the vertex form of the expression. The factored form makes the x- intercepts easy to find, the standard form makes the y- intercept easy to find, and the vertex form makes the vertex easy to find. Each form has advantages and disadvantages. 15. Repeat Steps 3-14 with additional quadratic expressions, where the linear term, b, is even. 16. Now that you have completed the square using the algebra tiles, let s try one without using the tiles:. 17. First, we must evenly split the linear term. How does that split evenly? That gives us dimensions of and How can we make equivalent to? and By subtracting 1 (or adding - 1) Refer to Factoring Quadratic Trinomials. Refer to previous models. Refer to previous models.

5 The Expect students to say or do If students do not, then the Verify that 21. What is the vertex of the expression? 22. Repeat Steps16-21 with additional quadratic expressions.. Students may verify algebraically, or by graphing both expressions on their calculator. Check on your calculator. Teacher Notes: 1. Students should have some experience using algebra tiles prior to this lesson. If this is not the case, spend some time explaining their use. 2. Use even linear (b) terms in this lesson to eliminate any confusion the use of fractions might cause. 3. Adding zeros in the form is very important for students to understand. 4. Students should be able to factor quadratic expressions and locate x- and y- intercepts. This would be a good time to reinforce those skills. 5. Students may realize that the x- coordinate of the vertex is always the midpoint of the x- coordinates of the x- intercepts. 6. This lesson should be done prior to the introduction of the quadratic formula. 7. Always relate the manipulative to the algebraic representation. Transition students away from the manipulatives once they have gained an understanding of the concept. This may take some time. Variations Once students have a conceptual understanding of completing the square, the quadratic formula may be introduced as a shortcut for finding the x- intercepts of a quadratic function. Instead of completing the square each time, it is performed once for the general solution. Formative Assessment Write this expression in vertex form:. Answer:

6 References Eves, H. (1990). An Introduction to the History of Mathematics. Philadelphia: Saunders College Publishing. Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide - Response to Intervention in Mathematics. Retrieved 2 25, 2011, from rti4sucess: on.pdf