# Properties of Real Numbers 1Lesson Plan

 Developed by David McAdams Last update 29 Dec 2006 Grade Level 10 Subject Intermediate Algebra Online HTML http://www.lifeisastoryproblem.org/lesson/lppropertiesofnumbers1.html Copyright Unpublished copyright work © 2006, David McAdams, Orem Utah. This document may be duplicated for non-commercial educational use only. Contact Contact the author at DEMcAdams@usa.net.
1. Lesson Model

2. Goal(s)/Standards
• From National Council of Teachers of Mathematics
• Number and Operations Standard: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
• Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
• compare and contrast the properties of numbers and number systems, including the rational and real numbers, and understand complex numbers as solutions to quadratic equations that do not have real solutions;
• Compute fluently and make reasonable estimates.
• develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases.
• Algebra Standard: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
• Represent and analyze mathematical situations and structures using algebraic symbols.
• write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency = mentally or with paper and pencil in simple cases and using technology in all cases;
• use symbolic algebra to represent and explain mathematical relationships;
• judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
• Geometry Standard: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
• Use visualization, spatial reasoning, and geometric modeling to solve problems.
• use geometric models to gain insights into, and answer questions in, other areas of mathematics
• Problem Solving Standard: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
• apply and adapt a variety of appropriate strategies to solve problems;
• monitor and reflect on the process of mathematical problem solving.
• Reasoning and Proof Standard: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
• develop and evaluate mathematical arguments and proofs;
• select and use various types of reasoning and methods of proof.
• Communication Standard: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
• use the language of mathematics to express mathematical ideas precisely.
• Representation Standard: Instructional programs from pre-kindergarten through grade 12 should enable all students to:
• create and use representations to organize, record, and communicate mathematical ideas;
• select, apply, and translate among mathematical representations to solve problems.
• From Utah State Core
• Representation: Use a variety of visual representations including patty paper, graph paper, manipulatives and technology

3. Specific Objectives
At the end of the lesson the student will be able to:
• Identify equations representing and draw geometric representations of:
• the commutative property of addition: a + b = b + a;
• the associative property of addition: a + (b + c) = (a + b) + c;
• the commutative property of multiplication: a · b = b · a;
• the associative property of multiplication: a · (b · c) = (a · b) · c;
• the distributive property of multiplication over addition: a(b + c) = ab + ac.
• Demonstrate master of these properties by simplifying expressions and solving equations.

4. Materials and Preparation
• Each student will need a Properties of Numbers 1 Quick Review sheet.
• Each student will need a Properties of Numbers 1 Quiz for pretest.
• Each student will need a Properties of Numbers 1 Quiz for master assessment.
• This lesson requires the use of a straight edge and drawing surface for the teacher. This can be a white board, an overhead transparency, or other.
• Each student will need a straight edge and a drawing surface such as paper or personal whiteboard to draw figures.
• Each student will need a Properties of Numbers Worksheet 1.
• The teacher can use an overhead transparency of the Properties of Numbers Worksheet 1 while guiding students through the process.
• The teacher will need a large-size copy of the expression 2+3+5 for the association to present to the students.

5. Prerequisite Vocabulary
• associative property of multiplication
• commutative property of multiplication
• distributive property of multiplication of addition and subtraction
• geometric representation
• identity
• multiplicative identity
• visualization

6. Prerequisite Methodology
none

7. Instructional Procedure
1. Pre-Quiz
Have the students write in their journals how they might represent the equation 3 + 5 = 8 using a drawing. After a few minutes, have the students pair up and compare their representations. Have them discuss the relative merits of the various representations. Have each pair share with the class their representation and its merits.

2. Internet Activity (optional)
Explore representations using Introduction to Geometric Representation in Algebra. This activity can be used in conjunction with the worksheet.

3. Review - Measurement Activate prior knowledge of measurement of lengths by discussing various ways to measure the length of a line. Then ask the class to identify the area of a 3 x 5 rectangle. Ask if this is a reasonable representation of the algebraic fact 3 · 5 = 15.

4. State the problem and objective
1. State that geometric representations of algebraic principles can help us understand.
2. State and write on the board that the student will be able to:
• Represent addition and multiplication with arrays of dots and with lines.
• Identify strengths and weaknesses of geometric figures.
• Identify parts of a graph in Cartesian Coordinates.
• Show an emerging ability to represent three dimensional mathematics operation by building and explaining a three dimensional model of (a + b)3.

5. Guided Practice - Visualization Worksheet

6. Introduce the idea of three dimensional models of algebraic relationships.
1. Refer to the one dimensional model of a + b and the two dimensional model of (a + b)2 = a2 + 2ab + b2 in the worksheet.
2. Ask the students how they might represent (a + b)3 = a3 + 3a2b + 3ab2 + b3.
3. Pass out the template for (a + b)3. Have the students assemble the three dimensional figure, and then explain it to their partner. When they are ready, have them explain it to the teacher.

8. Differentiation for Diverse Student Needs
• Blind students will need accommodations to understand, interpret, and draw geometric figures.
• Students with cognitive difficulties will need additional assistance bridging from the concrete figure to the abstract algebraic meaning.
• Students with difficulty in fine motor skills may need assistance assembling their model of (a + b)3
• Some students may want to orally describe lines and points rather than in writing.

9. Assessments
• An informal assessment for learning will be performed by listening to students' contributions to discussion.
• An informal assessment for learning will be performed by reviewing the students' notes.
• A formal assessment for learning will be performed by grading the students' worksheets.
• A formal assessment of learning will be performed by listening to the students' explanations of the model representing (a + b)3.

• Other Resources
Representation of the Distributive Property

• Materials Masters
Download Template for (a + b)3 = a3 + 3a2b + 3ab2 + b3