Algebra in the Early Grades (2008) by J. J. Kaput, D. W. Carraher, & M. L. Blanton. Lawrence Erlbaum Associates; New York (pub).

Curriculum and Evaluation Standards for School Mathematics, Addenda Series Grades 5-8: Patterns and Functions (1991) by E. Phillips. National Council of Teachers of Mathematics; Reston, Virginia (pub).

Hands On Equations (1996) by H. Borenson.  Borenson and Associates; Allentown, PA (pub).

Mathematics for Elementary Teachers: A Contemporary Approach (2014) by G. L. Musser, W. F. Burger, & B. E. Peterson; John Wiley & Sons (pub), 10th ed.

Mathematics In Context: Comparing Quantities (2006) by T. A. Romberg.  Encyclopedia Britannica (pub).

Mathematics In Context: Building Formulas (2003) by T. A. Romberg.  Holt, Rinehart, and Winston (pub).

Minnesota K-12 Mathematics Framework (1998) by W. Linder-Scholer. SciMathMN (pub).  Number Sense Activities section.

Navigating through Algebra in Grades 3-5 (2005) by G. J. Cuevas & K. Yeatts. National Council of Teachers of Mathematics; Reston, Virginia (pub).

Navigating through Algebra in Grades 6-8 (2007) by S. Friel, S. Rachlin, & D. Doyle with C. Nygard, D. Pugalee, & M. Ellis. National Council of Teachers of Mathematics; Reston, Virginia (pub).

Patterns and Functions Activities for Teachers by T. Frauenholtz, C. Rypkema; Bemidji State University.
Principles and Standards for School Mathematics (2000). NCTM; Reston, VA.

Department of Mathematics and Computer Science

8710.3320 MIDDLE LEVEL ENDORSEMENT LICENSE FOR TEACHERS OF MATHEMATICS

In this syllabus you will find the word TEACH. This will mean to:

1. Launch:  This is where the teacher sets the context of the problem or activity being worked on this day.  This involves making sure the students clearly understand the mathematical context and the mathematical challenge of the day’s activities.
2. Explore:  This is the time where the students get to work in pairs, individually, or as a class to solve problems presented by the lesson.
3. Share: This occurs when most of the students have made sufficient progress toward solving the problem presented with today’s lesson.  It is during this phase that the students learn how others approached the problem and possible solution routes.  Helps students deepen their understanding of the mathematical ideas presented in the day’s lesson.
4. Summarize:  During this phase the teacher concludes the lesson by clearly stating what the main idea was in the lesson, being sure to clear up any confusion that may arise during the “share” segment.  Helps students focus their understanding of the mathematical ideas presented in the lesson.

Standard

K/A

Activity

Assessment

C.A teacher with a middle level endorsement for teaching mathematics in grades 5 through 8 must demonstrate knowledge of fundamental concepts of mathematics and the connections among them. The teacher must know and apply:

(1)  concepts of patterns, relations, and functions:

(a) recognize, describe, and generalize patterns and build mathematical models to describe situations, solve problems, and make predictions;

K A

TEACH:
Problem solving
Understand the problem
Devise a plan
Carry out the plan
Revisit the problem
Examine different contextual problem situations, look for patterns, build models (tables, graphs, equations) for the patterns, generalize the patterns, solve the problem for the solution or extend the pattern to predict the solution. Encourage multiple approaches or solution paths.

Assignment:
Week: 1, 2, 6
Paul’s points – from SciMathMN Frameworks Number Sense Activities

The School Store – from Math In Context Comparing Quantities.

Relatives – from Musser, Burger, & Peterson; Section 9.3, #’s: 2,3,6,7,8.

Assessment:

• During class students will give presentations of models/patterns they constructed and the class will be expected to generalize patterns and make predictions.
• On a test students will recognize, describe, and generalize patterns and build mathematical models to describe situations, solve problems, and make predictions using tables, iterative functions, and explicit functions.

(b) analyze the interaction within and among quantities and variables to model patterns of change and use appropriate representations, including tables, graphs, matrices, words, algebraic expressions, and equations;

K A

TEACH:
Problem solving
Understand the problem
Devise a plan
Carry out the plan
Revisit the problem
Examine different contextual problem situations, look for patterns, build models (tables, matrices, words, graphs, algebraic expressions, equations) for the patterns, generalize the patterns, solve the problem for the solution or extend the pattern to predict the solution. Encourage multiple approaches or solution paths.

Assignment:
Week: 1, 2, 3, 4, 5, 9, 10, 11, 12, 13
Candy Boxes problem – from Algebra in the Early Grades, pp. 238-242;

Building Houses problem – from Navigations through Algebra in Grades 3-5;

Growing letters – Patterns and Function Activities for Teachers

Exploring Houses; Bouncing Tennis Balls; & Tiling Tubs – from Navigations through Algebra in Grades 6-8.

Assessment:

• Students will analyze the interaction within and among quantities and variables to model patterns of change and use appropriate representations, including tables, graphs, matrices, words, algebraic expressions, and iterative and explicit equations as they explain their answers orally and on a written test.

(c)  represent and solve problem situations that involve variable quantities and be able to use appropriate technology;

K A

TEACH:
Problem solving
Understand the problem
Devise a plan
Carry out the plan
Revisit the problem
Examine different contextual problem situations that involve variable quantities, look for patterns, build models (tables, matrices, words, graphs, algebraic expressions, equations) for the patterns, generalize the patterns, solve the problem for the solution or extend the pattern to predict the solution. Encourage multiple approaches or solution paths. Utilize technology when appropriate.

Assignment:
Week: 1, 2, 6, 7, 10, 11
Beams – from Math In Context Building Formulas;

Hands On Equations (1996) by H. Borenson.  Borenson and Associates, Allentown, PA (pub).

Assessment:

• Students will represent and solve problem situations that involve variable quantities and be able to use appropriate technology, manipulatives, graphing calculators, or computers, as they make oral presentations explaining their approaches and solution strategies.

(d)  understand patterns present in number systems and apply these patterns to further investigations;

K A

TEACH:
Problem solving

• Understand the problem
• Devise a plan
• Carry out the plan
• Revisit the problem

Examine different contextual problem situations that contain patterns present in number systems, build models (tables, matrices, words, graphs, algebraic expressions, equations) for the patterns, generalize the patterns, solve the problem for the solution or extend the pattern to predict the solution. Encourage multiple approaches or solution paths.

Assignment:
Week: 1, 2, 5
n-gon numbers – from Addenda Series Grades 5-8: Patterns and Functions (pg. 53);

n-gon numbers – from Algebra in the Concrete (1973) by M. Laycock & R. A. Schadler; Activities Resource Company (pub).  Pp. 7.

Assessment:

• Students will demonstrate they understand patterns present in number systems and apply these patterns to further investigations as they make oral presentations explaining their approaches and solution strategies and on a written exam

Introductions, values, and 88 problem

Continue introductions and wrap-up 88 problem. Examine Juicy Juice problem and student work.

Look at Juicy Juice solutions and view IMAP video #3

Week 2

Wrap-up solutions to problem solving -- look at Cryptarithms / Ninebl.sol

The Border Problem (& video)

LYNNE + LOOKS = SLEEPY ; USA + USSR = PEACE ; percentages

Introduce variables (candy boxes) and patterns

Week 3

Growing letters – construct and share

Growing letters – presentations

Growing letters – presentations

Week 4

Growing letters – presentations

Growing letters – analysis

Growing letters – analysis

Week 5

More patterns -- number of arms in classroom, growing letters, ... What is algebra? (NCTM: concrete / pictoral representation, graph, formula, table, and words)

Multiple representations on single page (from NW Service coop website)

Rectangles with a perimeter of 20

Examine recursive (Next = Now...) and explicit (y=ax+...) formulae
n-gons – sides, vertices, diagonals from a vertex, diagonals in n-gon, central angle, interior angle, exterior angle, triangular decomposition, sum of angles – n  = 3 .. 12 ,  get formulae for N from number patterns and geometric reasoning

Week 6

Paul’s Points - School store -- pencils 15 cents and erasers 25 cents; relatives, Skeeters (having four babies, poison, and simultaneous)

Using manipulatives - a balance scale approach. Solve: x+2=6, x-2=7, x+3=-8, x-4=-9, 2x+4=x+5, 3x+2x=x+8, 3x+-2x=-x+8, 2x+6=-x, 2x+3=2x-5, ...

Balance scales continued

Week 7

Midterm

Exam

Virtual balance scales

Virtual balance scales

Issues with balance scales: can you move from one side to the other?

Issues with balance scales: negatives?

Week 8

Alge-blocks

Algeblocks - multiplying terms

Algeblocks - factoring

Week 9

Algeblocks - what is a cubic

Algeblocks - maybe a surprise quiz?

Three island problem Building Houses and I Spy Patterns (from NCTM Navigations through Algebra in grades 3-5)

Week 10

Three island problem Building With Toothpicks and Exploring Houses (from NCTM Navigations through Algebra in grades 6-8)

Beams

Beams & Frogs (https://nrich.maths.org/1246)

Week 11

Review algeblocks, islands, balance scales, patterns, ...

Squares Cubed (from NCTM Navigations through Algebra in grades 3-5)

Bouncing Tennis Balls and Triangle Rule Machine (from NCTM Navigations through Algebra in grades 3-5 & 6-8)

Week 12

Review Houses on Islands problems

Relating intuition and algebra

Wrap-up Triangle rule machine

Week 13

Wrap-up bouncing tennis balls -- regression on calculators

Line of best fit -- slope and intercept

Wrap-up slope and intercept

Final Review

Final Exam – 2 Hours Comprehensive on Thursday, December 12th 10:30 am - 12:30