ALEX Classroom Resource

  

What are Perfect Squares?

  Classroom Resource Information  

Title:

 What are Perfect Squares?

URL:

 https://aptv.pbslearningmedia.org/resource/im20-math-ep19-711/what-are-perfect-squares/

Content Source:

 PBS
Type: Audio/Video

Overview:

This video 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 x2 + 2nx + n2.

The second aim is to help students see that perfect squares can be handy for solving equations because we can find their square roots. Recognizing the structure of a perfect square equips students to look for features that are necessary to complete a square (MP7), which they will do in a future video lesson.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 8
Accelerated
5. Use the structure of an expression to identify ways to rewrite it. [Algebra I with Probability, 5]

Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Rewrite algebraic expressions by combining like terms or factoring to reveal equivalent forms of the same expression.
Teacher Vocabulary:
  • like terms
  • Expression
  • Factor
  • properties of operations (Appendix D, Table 1)
  • Difference of squares
Knowledge:
Students know:
  • Properties of operations (including those in Appendix D, Table 1),
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression.
Skills:
Students are able to:
  • -Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
    For example, 3(x-5) = 3x-15 and 2a+12 = 2(a+6) or 3a-a+10+2and x2-2x-15 = (x-5) (x+3).
Understanding:
Students understand that:
  • Generating simpler, but equivalent, algebraic expressions facilitates the investigation of more complex algebraic expressions.
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 8
Accelerated
11. Select an appropriate method to solve a quadratic equation in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real. [Algebra I with Probability, 9]
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Select an appropriate method (taking square roots, factoring, completing the square, or quadratic formula) for solving a quadratic equation in one variable based on its original form.
  • Use completing the square to transform any quadratic equation into the form (x-p)2=q.
  • Derive the quadratic formula from (x-p)2=q.
  • Recognize that some solutions may not be real or namely that they may be imaginary or complex numbers.
  • Provide reasonable approximations when appropriate in a graph or table.
Teacher Vocabulary:
  • quadratic equation
  • Square root
  • Factoring
  • Completing the square
  • quadratic formula
  • Derive
  • Real numbers
  • Imaginary numbers
  • Complex numbers
Knowledge:
Students know:
  • Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
  • The method for completing the square.
  • A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when b2-4ac is less than zero.
Skills:
Students are able to:
  • Take the square root of both sides of an equation.
  • Factor quadratic expressions in the form x2+bx+c where the leading coefficient is one.
  • Use the factored form to find zeros of the function.
  • Complete the square.
  • Use the quadratic formula to find solutions to quadratic equations.
  • Manipulate equations to rewrite them into other forms.
Understanding:
Students understand that:
  • Solutions to a quadratic equation must make the original equation true and this should be verified.
  • When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
  • Different procedures for solving quadratic equations are necessary under different conditions.
  • If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
  • Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax2+bx+c=0).
Diverse Learning Needs:
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
5. Use the structure of an expression to identify ways to rewrite it.

Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful and more efficient ways.
Teacher Vocabulary:
  • Terms
  • Linear expressions
  • Equivalent expressions
  • Difference of two squares
  • Factor
  • Difference of squares
Knowledge:
Students know:
  • Algebraic properties.
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression.
Skills:
Students are able to:
  • Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
Understanding:
Students understand that:
  • Generating equivalent algebraic expressions facilitates the investigation of more complex algebraic expressions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.5.1: Define equivalent expressions.
ALGI.5.2: Rewrite an exponential expression in an alternative way.
ALGI.5.3: Rewrite a quadratic expression in an alternative way.
ALGI.5.4: Rewrite a linear expression in an alternative form.
ALGI.5.5: Understand that rewriting an expression in different forms in a problem context can shed light on the problem.
ALGI.5.6: Recall properties of exponents.

Prior Knowledge Skills:
    li>Give examples of the properties of operations including distributive, commutative, and associative.
  • Recall how to find the greatest common factor.
  • Combine like terms of a given expression.
  • Recognize the property demonstrated in a given expression.
  • Simplify expressions with parentheses (Ex. 5(4 + x) = 20 + 5x).
  • Simplify an expression by dividing by the greatest common factor (Ex. 18x + 6y= 6(3x + y).
  • Define linear expression, rational, coefficient, and rational coefficient.
    Example: See x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2
  • y2)(x2 + y2).
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.5 Solve simple algebraic equations using real-world scenarios with one variable using multiplication or division.

Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
9. Select an appropriate method to solve a quadratic equation in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Solve quadratic equations where both sides of the equation have evident square roots by inspection.
  • Transform quadratic equations to a form where the square root of each side of the equation may be taken, including completing the square.
  • Use the method of completing the square on the equation in standard form ax2+bx+c=0 to derive the quadratic formula.
  • Identify quadratic equations which may be solved efficiently by factoring, and then use factoring to solve the equation.
  • Use the quadratic formula to solve quadratic equations.
  • Explain when the roots are real or complex for a given quadratic equation, and when complex write them as a ± bi.
  • Demonstrate that a proposed solution to a quadratic equation is truly a solution by making the original true.
Teacher Vocabulary:
  • Completing the square
  • Quadratic equations
  • Quadratic formula
  • Inspection
  • Imaginary numbers
  • Binomials
  • Trinomials
Knowledge:
Students know:
  • Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
  • The method for completing the square.
  • Notational methods for expressing complex numbers.
  • A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when b2-4ac is less than zero.
Skills:
Students are able to:
  • Accurately use properties of equality and other algebraic manipulations including taking square roots of both sides of an equation.
  • Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations.
  • Factor quadratic polynomials as a strategy for finding solutions to quadratic equations.
  • Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions.
  • Make strategic choices about which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation.
Understanding:
Students understand that:
  • Solutions to a quadratic equation must make the original equation true and this should be verified.
  • When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
  • Different procedures for solving quadratic equations are necessary under different conditions.
  • If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
  • Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax2+bx+c=0).
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.9.1: Define quadratic equation and zero product property.
ALGI.9.2: Solve one-step equations using addition and subtraction that are set equal to zero.
ALGI.9.3: Solve two-step equations using addition and subtraction that are set equal to zero.

a.
ALGI.9.4: Define completing the square.
ALGI.9.5: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2= q that has the same solutions.
ALGI.9.6: Derive the quadratic formula from the form (x - p)= q.

b.
ALGI.9.7: Define quadratic formula, factoring, square root, complex number, and real number.
ALGI.9.8: Solve quadratic equations by completing the square.
ALGI.9.9: Solve quadratic equations by the quadratic formula.
ALGI.9.10: Solve quadratic equations by factoring.
ALGI.9.11: Solve quadratic equations by taking square roots.
ALGI.9.12: Recognize when the quadratic formula gives complex solutions.
ALGI.9.13: Write complex solutions as a ±bi for real numbers a and b.

Prior Knowledge Skills:
  • Identify perfect squares and square roots.
  • Define square root, expressions, and approximations.
  • Explain the distributive property.
  • Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
  • Recalving one-step equations.
  • List given information from the problem.
  • Identify the unknown, in a given situation, as the variable.
  • Test the found number for accuracy by substitution.
    Example: Is 5 an accurate solution of 2(x + 5)=12?
  • Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations.
  • Define equation and variable.
  • Set up an equation to represent the given situation, using correct mathematical operations and variables.
  • Recognize the correct order to solve expressions with more than one operation.
  • Calculate a numerical expression (Ex. V=4x4x4).
  • Choose the correct value to replace each variable in the algebraic expression (Substitution).
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.9 Identify equivalent expressions given a linear expression using arithmetic operations.
Tags: factoring, perfect squares, quadratic, solving equations, squared
License Type: Public Domain
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AccessibilityVideo resources: includes closed captioning or subtitles
Comments

There are student task statements and practice problems worksheets that accompany this resource.

  This resource provided by:  
Author: Kristy Lacks
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