Unit 5, Lesson 11
What Are Perfect Squares?
Integrated Math 2
11.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSA-REI.B.4.a
Use the method of completing the square to transform any quadratic equation in into an equation of the form that has the same solutions. Derive the quadratic formula from this form.
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this Warm-up, students reason about quadratic expressions that are perfect squares. They look for and use structure to rewrite expressions (MP7). Though each expression appears to be more complicated or to have more pieces than the preceding one, the underlying structure of all the expressions is unchanged. The last expression is written in factored form, but because the factors are identical, students can see that it can also be written as . Viewing a complicated expression as a perfect square prepares students to consider completing the square later.
Launch
Give students a moment to observe the list of equations, and ask them what they notice and wonder about the equations. Solicit a few observations and questions. Tell students that their job is to think about what should be so that each equation is always true, regardless of what is.
Activity
NoneEach of these expressions is a perfect square, which means that each can be written as something multiplied by itself.
Rewrite each expression as something multiplied by itself in the form . For example, can be rewritten as .
- 100
Activity Synthesis
Invite students to share their responses and how they viewed the expressions. If not mentioned by students, point out that the question can be thought of as “When something is multiplied by itself, it is equal to something squared,” so the inside of the parentheses must be equal to that something.
Explain to students that perfect squares are related to what students learned in earlier grades about the area of a square. Multiplying the side length of a square by itself gives the area of the square, so we can say, for instance, that 25 is a perfect square because it is the area (in square units) of a square whose side length is 5 units. can be thought of as the area of a square with side length .
11.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
The goal of this activity is to illustrate that a perfect-square expression can take different forms, some of which may not look like or . The work here prompts students to recognize structure in perfect-square expressions, particularly when written in standard form, preparing them to complete the square in an upcoming lesson.
Students are given a series of expressions that are clearly perfect squares, for example or , and asked to rewrite them in standard form. As they repeatedly apply the distributive property to expand these expressions into standard form, students begin to recognize a pattern in how the two forms of expressions are related (MP8). They see that squaring an expression such as produces an expression in standard form in which the constant term is and the linear term is . Then, they use that insight to articulate why certain expressions in standard form (such as ) can be described as perfect squares.
11.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
Earlier in the unit, students solved equations with a squared variable on one side and a square number on the other, for example, or . This activity allows students to see that more complicated quadratic equations can also be solved relatively easily when both sides of the equal sign are perfect squares. The work here motivates upcoming lessons on completing the square.
Lesson Synthesis
Reinforce the key points of this lesson by displaying some quadratic expressions such as:
Discuss questions such as:
- “Which of these quadratic expressions are perfect squares? Which are not?” (The second and fourth expressions are perfect squares. The other two are not.)
- “How can you tell?” (When in standard form, an expression with perfect squares has a constant term that is a number squared and a linear term that is twice that number. In , we can see 144 is 12 squared, and 24 is twice 12. In , the term 400 is -20 squared and -40 is twice -20.)
- “Why isn’t the third equation a perfect square? Isn’t 16 a perfect square?” (16 is 4 squared or -4 squared, but for the expression to be a perfect square, the coefficient of the linear term should be 8 (twice 4) or -8 (twice -4).)
- “Suppose we have two equations: and . Which is easier to solve? Why?” (The second one, because the equation has a perfect square on each side. It can be rewritten as . There are two numbers that can be squared to make 9, which are 3 and -3, so or .)
- “Can’t we simply rewrite the left side of in factored form and solve?” (Rewriting the left side gives , but then we are stuck. We can’t use the zero product property because the expression does not equal 0.)
Student Lesson Summary
These are some examples of perfect squares:
- 49, because 49 is or .
- , because it is equivalent to or .
- , because it is equivalent to .
- , because it is equivalent to or .
A perfect square is an expression that is something times itself. Usually we are interested in situations in which the something is a rational number or an expression with rational coefficients.
When expressions that are perfect squares are written in factored form and standard form, there is a predictable pattern.
- is equivalent to .
- is equivalent to .
- is equivalent to .
In general, is equivalent to .
Quadratic equations that are in the form can be solved in a straightforward manner. Here is an example:
The equation now expresses that squaring gives 25 as a result. This means must be 5 or -5.