# Lesson 11

What are Perfect Squares?

### Lesson Narrative

This lesson has two key aims. The first aim is to familiarize students with the structure of perfect-square expressions. Students analyze various examples of perfect squares. They apply the distributive property repeatedly to expand perfect-square expressions given in factored form (MP8). The repeated reasoning allows them to generalize expressions of the form $$(x+n)^2$$ as equivalent to $$x^2 + 2nx + n^2$$.

The second aim is to help students see that perfect squares can be handy for solving equations because we can find their square roots.

Achieving these aims prepares students to solve quadratic equations by completing the square in upcoming lessons. Knowing that quadratic equations can be much more easily solved when one side is a perfect square and the other side is a number motivates students to transform expressions into that form. Recognizing the structure of a perfect square equips students to look for features that are necessary to complete a square (MP7).

### Learning Goals

Teacher Facing

• Comprehend that equations containing a perfect-square expression on both sides of the equal sign can be solved by finding square roots.
• Comprehend that perfect squares of the form $(x+n)^2$ are equivalent to $x^2+2nx+n^2$.
• Use the structure of expressions to identify them as perfect squares.

### Student Facing

• Let’s see how perfect squares make some equations easier to solve.

### Student Facing

• I can recognize perfect-square expressions written in different forms.
• I can recognize quadratic equations that have a perfect-square expression and solve the equations.

Building Towards

### Glossary Entries

• perfect square

A perfect square is an expression that is something times itself. Usually we are interested in situations where the something is a rational number or an expression with rational coefficients.

• rational number

A rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point on the number line that you get by dividing the unit interval into $$b$$ equal parts and finding the point that is $$a$$ of them from 0. We can always write a fraction in the form $$\frac{a}{b}$$ where $$a$$ and $$b$$ are whole numbers, with $$b$$ not equal to 0, but there are other ways to write them. For example, 0.7 is a fraction because it is the point on the number line you get by dividing the unit interval into 10 equal parts and finding the point that is 7 of those parts away from 0. We can also write this number as $$\frac{7}{10}$$.

The numbers $$3$$, $$\text-\frac34$$, and $$6.7$$ are all rational numbers. The numbers $$\pi$$ and $$\text-\sqrt{2}$$ are not rational numbers, because they cannot be written as fractions or their opposites.