# Algebra Patterns and Functions

## Knowledge base & tools to develop mathematical literacy

### Knowledge base

Mathematical knowledge base with concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come. Particularly important for how learners develop their understanding for algebra, patterns, and functions are:

*Concepts related to:*

- Algebra & Patterns
- Ratio & proportion
- Concepts and Outcomes from Science for All Americans (SFAA) & National Assessment for Educational Progress (NAEP)

### Information to facilitate literacy in algebra, patterns, & functions & tools to achieve it

- Assessment
- Functions & patterns - Work sheet of a sample of Authentic Assessment for a Bridge pattern
- Functions & patterns - Work sheet of a sample of a Directed Assessment for a Bridge pattern
- Assessment - process proof and conjecture - content zero and properties of zero - addition and subtraction - with teacher reflection
*Scoring guides*- Equality & equal symbol - scoring guide or rubric across levels

- Classification or Sorting - Activities to assess by 1, 2, & 3 properties; use of Venn diagrams to represent intersecting & disjoint groups.
- Conjectures - Articulating, editing, & refining conjectures in a second or third grade class
- Equality and equal sign in equations - Discusses equality, equals symbol, its big ideas, supporting concepts, development, scoring guide, & ideas to facilitate understanding. Includes the use of an equal arm balance to discover equality. With photos and different models made from K'Nex.
- Graphing - Instructional Plan for teaching graphing & solving for two unknowns four ways. (8-12 Grade)
*Justification & proof*with 3 examples: concrete, pictoral, & symbolic levels for odd plus odd is even

- Multiples, common multiples, & FTA - Discussion script includes: least common multiple, the next common multiple, fundamental theory of arithmetic (FTA) see FTA chart in activities.
- Primes and composites - Diagram to use to explore the primes of whole numbers 1 - 40 & invent the Fundamental Theory of Arithmetic (FTA)
- Algebra, patterns, & function vocabulary - lists of vocabulary for primary, middle, & upper levels.
- Statistics timeline and history
- Reflections from ESU #1 workshop

*Activities* to facilitate algebra, patterns, & functions literacy

- Adding consecutive numbers - Work sheet challenge to find a sshort cut to add 4 consecutive numbers, hint, and verification proof.
- Arithmetic Properties - Chart with properties, conjectures, examples, & non examples of numbers and operations: addition, subtraction, multiplication, division, identity, inverse, zero, commutative, & distributive.
- Calendar math problems - List of challenges & hints to identify dates which are: factors, primes, relatively prime, pi, triangle numbers, square numbers, palindromes, perfect number, reciprocals, reciprocal numbers.
- Equality, relationships, & equal arm balances
- Expressions & formulas - Bus, train boarding and deboarding problems for problem solving, expressions & recursive formulas, problem solving strategies of working backwards& acting out.
- Factors - Planning notes, 4 Work sheets: table to list factors 1-50, table to sort factors by least to most for numbers 1-50, table to list factors 1-100, table to sort factors by least to most for numbers 1-100,
- Fibonacci Sequence explanation Slide show that explains the Fibonacci sequence and how to generate it. for generating the sequence & Fibonacci Numbers in Nature
- Function exploration - includes definitions with examples, diagrams, & representations including a function machine. Also activity with work sheet and charts to explore a function with a variable factor (3) & constant factor (27) & discover
*casting out nines* - Fundamental Theory of Arithmetic (FTA) - Chart with primes & prime factorization to 30 and blank to 40 to represent the FTA.
- Graph, the math representations behind the modern graph - Story of Nicole Oresme, Father of the modern graph, or as he would say:
*the Latitude of Forms* - Graphing Notes - Data sheet for coordinate graph, manipulated, responding, independent, dependent, variables, relationships, extrapolation, interpolation, continuous & categorical data
- The Königsberg bridges problem
- Linear equations - fact sheet for y=mx+b, slope, & y intercept
- Graphing Pictures plots - Work sheet to plot a picture and create your own points to plot. Source Johnnie Ostermeyer
- Growth - plants and the mathematics of growth
*Patterns*:- Even, odd, and super even patterns exploration and investigation activities
- Categorical sorting - Activities descriptions, Venn diagrams, sort by one, two, three properties, disjoint groups and intersecting groups
- Pattern activities - Investigatons to develop patterns & formulas. Starting with number strips, dot patterns & other growing patterns: v, w, pyramid, prism, tower, triangle, rectangle, log stack, can stack, hand shake, dancers, step pyramid, & disc
- Stairways and stacking blocks problems - Work sheet, also known as odd squares & tower problem
- Sixteen Colored Squares in an Eight by Eight square problem how many?
- Growth - plants and the mathematics of growth patterns
- Petals around the Rose - is a simulation to find a strategy to determine the number of petals around the rose as simulated by dice. Direcctions & hints.

*Probability*activities for see data analysis & probability- Magic phone number - Work sheet Magic explained with math, Inverse properties of numbers & operations.
- M&M's what colors in a package? - Work sheet Count, chart, determine percentage, mean , median, mode of M&M's in a package by color.
- Multiplication of whole numbers, multiplication of binomials (x+1)(x+1), & area patterns
- Necklace perfectly beaded - Work sheet to identify a perfect beaded necklace repeating pattern & create one
- Primes and composites. A study of whole numbers - Chart with prime factors for 2 - 40 to invent the
*Fundamental Theory of Arithmetic* - Pythagorean Theorem proof
- Toothpick challenges
- Growing squares challenge - Growing squares challenge slides for the challenge above. Slides step through adding toothpicks to make growing squares, record data in a table, and ind the a pattern of increasing squares and increasing number of toothpicks. v
- Growing squares worksheet - Use toothpicks to find a pattern & formulas to determine the number of toothpicks added to make larger squares. Toothpicks in the perimeter, toothpicks to make the squares in the squares, and the number of squares in the growing squares.
- Toothpicks in a rectangular pattern - Worksheet with hints to write equations for the sums of toothpick patterns in a rectangle and discussion ideas for what is an equation, function, and can they be both.

*Relationships*- Pools and sidewalks Worksheet growing squares and perimeter patterns, relationships, & functions .- Simultaneous Equations proofs
- Stem and leaf plot - Work sheet with problems and Java plotter available on the internet
- Variables. One variable activities - ping-pong ball race experiment, collect data, analyze (ping-pong ball race, how many stars in a minute).

#### Ratio & proportion

- Jo Tall and Jo Short - Work sheet ratio & proportion activity, historical research problem used to investigate the develop of reasoning as suggessted by Piaget
- Measuring heights of trees & other objects - clinometer, shadows, scale (Biltmore) stick
- Candle problems - Work sheet with sample rate & proportion problem and solution with follow up problem and questions.
- Sack Lunches - Teacher randomly passes out lunches. What are the odds? - Work sheet with hints & discussion ideas.
- Wages for Pet Care
- Acrobats Logic Problem - Work sheet draw a diagram, proportion & problem solving diagrams

*Page Overview*:

*Other math knowledge base & activity pages*

*Process dimensions *

*Content dimensions*

One of the universal customs that man has successfully established on earth is the

*Hindu - Arabic numerals*

to record numbers.

*Algebra* was greatly advanced by al-Khwarizmi’s book:

*Hisab al-jabr w'al-muqabala* From which the word *algebra* came.

*Read *...

Cell phone mathematics. *The story of how Brad Parkinson* used creativity and mathematics to enable the use of cell phones.