Mondrian Factoring Models

Learning Resource Type

Learning Activity

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

Piet Mondrian is an artist famous for creating his masterpieces out of line art that utilized clean lines through rectangles. This activity will help us to create our own “Mondrian” by using our knowledge of factoring Quadratic trinomials through the use of Algebra tiles and area models.

This activity was created as a result of the ALEX Resource Development Summit.

Phase

During/Explore/Explain
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.4

Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.

UP:MA19.A1.4

Vocabulary

  • Linear expression
  • Quadratic expression
  • Exponential expression
  • Equivalent expressions

Knowledge

Students know:
  • How to recognize the parts of linear, quadratic and exponential expressions and what each part represents.
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.
  • That one or more parts of an expression can be viewed as a single entity.

Skills

Students are able to:
  • Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
  • Interpret expressions in terms of a context.
  • View one or more parts of an expression as a single entity and determine the impact parts of the expression have in terms of the context.

Understanding

Students understand that:
  • Making connections among the parts of an expression reveals the roles of important mathematical features of a problem.
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.6

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

UP:MA19.A1.6

Vocabulary

  • Quadratic expression
  • Zeros
  • Complete the square
  • Roots
  • Zeros
  • Solutions
  • x-intercepts
  • Maximum value
  • Minimum value
  • Factor
  • Roots
  • Exponents
  • Equivalent form
  • Vertex form of a quadratic expression

Knowledge

Students know:
  • Techniques for generating equivalent forms of an algebraic expression, including factoring and completing the square for quadratic expressions and using properties of exponents.
  • When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem.

Skills

Students are able to:
  • Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures.
  • Factor quadratic expressions.
  • Complete the square in quadratic expressions.
  • Use the vertex form of a quadratic expression to identify the maximum or minimum and the axis of symmetry.

Understanding

Students understand that:
  • Making connections among equivalent expressions reveals the roles of important mathematical features of a problem.
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.9

Select an appropriate method to solve a quadratic equation in one variable.

UP:MA19.A1.9

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).

Learning Objectives

I can factor a quadratic trinomial by using Algebra Tiles and area models. 

I can interpret the different parts of an expression such as factors, coefficients, and constants.

Activity Details

The instructions for students are included as part of the Google Doc. The teacher will need to provide a copy of the document to each student using Google Classroom, or by providing a hard copy. The teacher will provide Algebra Tiles. Other materials may be provided by the teacher or the students based on the teacher's discretion. 

Students will be able to factor quadratic trinomials using Algebra Tiles and will draw a representation of such models in order to create a Mondrian inspired piece of art. This activity is a great tool for having students independently practice factoring Quadratic trinomials through the use of Algebra tiles, therefore producing an equivalent form of the Quadratic expression. Students will factor given trinomials and draw the corresponding area models on graph paper to demonstrate an understanding of factoring. Students will be able to determine if factoring is correct by inspecting the model. Because students will be creating area models, they will be able to interpret the various parts of the expression easily, such as the coefficients, factors, and constants. The end product will be a creative representation of the factoring, which will also incorporate the artwork of Piet Mondrian.  Students will work independently.

Assessment Strategies

The final product can be used by the teacher to determine whether or not student mastered being able to factor quadratic equations by modeling.

Variation Tips

Could be adapted for lower grades by using area models rather than models using Algebra Tiles.

Quadratic trinomials with negative terms could be incorporated by changing the equations and adding red to represent the negative factors.

Background / Preparation

As an introduction, you may want to quickly show a few of Mondrian's works and relate those to models using the Algebra Tiles. 

You will need to provide a copy of the activity in Google Classroom or Docs. If students do not have access to technology, you can project problems on the projector, or give each student an individual copy.

Each student will need a piece of graph paper or a printed grid in order to draw models.

The teacher will need to have Algebra Tiles accessible.

The teacher may provide colored pencils or markers or have students bring their own coloring supplies.

https://docs.google.com/document/d/1JfU85PgPkqT_UWF58HXIINz3aGzE_SSGD8PsBm9XGfY…

Approved Date

Owner (Author)

auburntailgater
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