Algebra Made Easy! Making Connections with Number Patterns

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Algebra Made Easy! Making Connections with
Number Patterns

• Dr. Dawn Parker
• Texas AM University
• Mathematics Methods

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Activities for Exploring Number Patterns

• Paper Folding
• Fold a piece of paper in half, and then in half
  again, and again, until you make six folds. When
  you open it up, how many sections will there be?
  Make a chart. Continue to fold as you look for a
  pattern.
• Paper Tearing
• Tear a piece of paper in half and give half to
  someone else. Each person then tears the piece
  of paper in half and passes half on to another
  person. How many people will have a piece of
  paper after 10 rounds of tearing paper like this?
  Continue tearing paper and record on a chart.
  Look for patterns as you complete each round.

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Algebra is a mathematical system that

• has a rich history of development
• uses symbols to represent numbers and operations
• allows us to graphically represent and analyze
  mathematical concepts
• has a logical foundation
• gives us a powerful problem solving tool

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Principles and Standards for School
Mathematics(NCTM 2000)

• The Algebra Standard emphasizes relationships
  among quantities and the ways in which quantities
  change relative to one another.
• To think algebraically, a student must be able to
  understand patterns, relations, and functions
  represent and analyze mathematical situations and
  structures using algebraic symbols use
  mathematical models to represent and understand
  quantitative relationships and analyze change in
  various contexts.
• Each of these basic components evolves as
  students grow and mature.

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Students in Grades 4-8

• Learn to represent patterns numerically,
  graphically, symbolically, and verbally
• Look for relationships in numerical and geometric
  patterns and analyze how patterns grow or change
• Use tables, charts, physical objects, and symbols
  to make and explain generalizations about
  patterns
• Use relationships in patterns to make predictions

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Mathematics TEKS

• Patterns, Relationships and Algebraic Thinking
  Grade 4
• (6)  Patterns, relationships, and algebraic
  thinking. The student uses patterns in
  multiplication and division. The student is
  expected to
• (A)  use patterns to develop strategies to
  remember basic multiplication facts
• (B)  solve division problems related to
  multiplication facts (fact families) such as 9 x
  9 81 and 81/9 9 and
• (C)  use patterns to multiply by 10 and 100.
• (7)  Patterns, relationships, and algebraic
  thinking. The student uses organizational
  structures to analyze and describe patterns and
  relationships. The student is expected to
  describe the relationship between two sets of
  related data such as ordered pairs in a table.

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Mathematics TEKS

• Patterns, Relationships and Algebraic Thinking
  Grade 5
• (5)  Patterns, relationships, and algebraic
  thinking. The student makes generalizations based
  on observed patterns and relationships. The
  student is expected to
• (A)  use concrete objects or pictures to make
  generalizations about determining all possible
  combinations
• (B)  use lists, tables, charts, and diagrams to
  find patterns and make generalizations such as a
  procedure for determining equivalent fractions
  and
• (C)  identify prime and composite numbers using
  concrete models and patterns in factor pairs.
• (6)  Patterns, relationships, and algebraic
  thinking. The student describes relationships
  mathematically. The student is expected to select
  from and use diagrams and number sentences to
  represent real-life situations.

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Mathematics TEKS

• Patterns, Relationships and Algebraic Thinking
  Grade 6
• (6.3)The student solves problems involving direct
  proportional relationships. The student is
  expected to (A)  use ratios to
  describe proportional situations
• (B)  represent ratios and percents with concrete
  models, fractions, and decimals and
• (C)  use ratios to make predictions in
  proportional situations.
• (6.4)   The student uses letters as variables in
  mathematical expressions to describe how one
  quantity changes when a related quantity changes.
  The student is expected to
• (A)  use tables and symbols to represent and
  describe proportional and other relationships
  such as those involving conversions, arithmetic
  sequences (with a constant rate of change),
  perimeter and area and
• (B)  use tables of data to generate formulas
  representing relationships involving perimeter,
  area, volume of a rectangular prism, etc.
• (6.5) The student uses letters to represent an
  unknown in an equation. The student is expected
  to formulate equations from problem situations
  described by linear relationships.

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Mathematics TEKS

• Patterns, Relationships and Algebraic Thinking
  Grade 7
• (7.3)  The student solves problems involving
  direct proportional relationships. The student
  is expected to
• (A)  estimate and find solutions to application
  problems involving percent and
• (B)  estimate and find solutions to application
  problems involving proportional relationships
  such as similarity, scaling, unit costs, and
  related measurement units.
• (7.4)  The student represents a relationship in
  numerical, geometric, verbal, and symbolic form.
  The student is expected to
• (A)  generate formulas involving unit
  conversions, perimeter, area, circumference,
  volume, and scaling
• (B)  graph data to demonstrate relationships in
  familiar concepts such as conversions, perimeter,
  area, circumference, volume, and scaling and
• (C)  use words and symbols to describe the
  relationship between the terms in an arithmetic
  sequence (with a constant rate of change) and
  their positions in the sequence.
• (7.5) The student uses equations to solve
  problems. The student is expected to
• (A)  use concrete and pictorial models to solve
  equations and use symbols to record the actions
• (B)  formulate problem situations when given a
  simple equation and formulate an equation when
  given a problem situation.

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Mathematics TEKS

• Patterns, Relationships and Algebraic Thinking
  Grade 8
• (8.3)  The student identifies proportional or
  non-proportional linear relationships in problem
  situations and solves problems. The student is
  expected to
• (A)  compare and contrast proportional and
  non-proportional linear relationships and
• B)  estimate and find solutions to application
  problems involving percents and other
  proportional relationships such as similarity and
  rates.
• (8.4)  The student makes connections among
  various representations of a numerical
  relationship. The student is expected to
  generate a different representation of data given
  another representation of data (such as a table,
  graph, equation, or verbal description).
• (8.5)  The student uses graphs, tables, and
  algebraic representations to make predictions and
  solve problems. The student is expected to
• (A)  predict, find, and justify solutions to
  application problems using appropriate tables,
  graphs, and algebraic equations and
• (B)  find and evaluate an algebraic expression
  to determine any term in an arithmetic sequence
  (with a constant rate of change).

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A Row of Triangles, Squares, or Pentagons

• A Row of Triangles
• If you line up 100 equilateral triangles (like
  the green ones in Pattern Blocks) in a row, what
  will the perimeter measure? If you think of this
  as a long banquet table, how many people can be
  seated? Create a chart to record the data and
  look for patterns as you add triangles.
• A Row of Squares
• What will the perimeter measure if you line up a
  row of 100 squares? If this is a long table, how
  many people can be seated? Create a chart for
  your data like the one in the Row of Triangles
  problem.
• A Row of Pentagons
• What will the perimeter measure if you line up a
  row of 100 pentagons? If this is a long table,
  how many people can be seated? Create a chart
  for your data like the one in the Row of
  Triangles and Row of Squares problem.

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Handshakes

• Suppose everyone in the room shakes hands with
  every other person in the room. How many
  handshakes will that be? (With one person, there
  will be no handshake. With two people, there
  will be one handshake. How many handshakes will
  there be with three people? Four? Continue by
  creating a chart of the data and look for
  patterns.

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Hundred Board Wonders

• Select a rule from the list below to explore
  number patterns using a hundred board
• Numbers with a two in them
• Numbers whose digits have a difference of 1 (Be
  sure the students always select numbers whose
  tens-place digit is 1 greater than the ones-place
  digit.)
• Numbers with a 4 in them
• Numbers that are multiples of 3
• Numbers with a 7 in them
• Numbers that are multiples of 5
• Numbers with a 0 in them
• Numbers that are divisible by 6
• Numbers with a 5 in the tens place
• Numbers that are multiples of 4
• Numbers having both digits the same
• Numbers that are both multiples of 2 and 3
• Numbers that are divisible by 8
• Numbers whose digits add to 9 (example 63)

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Tiling a Patio

• You are designing square patios. Each patio has
  a square garden area in the center. You use
  brown tiles to represent the soil of the garden.
  Around each garden, you design a border of white
  tiles. Build the three smallest square patios
  you can design with brown tiles and white tiles
  for the border. Record the number of each
  color tile needed for the patios in a table.
  Continue filling in the table as you design the
  next two patios.

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Whats the Best Deal?

• Your boss at the video game company where you
  work has given you two choices of salary schemes.
  You have to decide which of the choices will
  permit you to reach your goal of 1000 the
  fastest. You will need to support your choice by
  showing the data your collected and by describing
  the graphs for each situation.
• Choice 1 Your salary will be doubled each
  day. You will earn 1 the first day, 2 the
  second day, 4 the third day, 8 the fourth day,
  and soon.
• Choice 2 Your salary will increase 3 each
  day. You will make 3 the first day, 6 the
  second day, 9 the third day, 12 the fourth day,
  and so on.
• Which of these two ways to earn your salary will
  get you to 1000 the fastest?
• Complete a table for each plan. On graph paper,
  draw a graph for the total earnings for each
  salary plan.

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Amazing Calendars

• Calendars provide examples of number patterns to
  explore.
• Consider the following
• How does the calendar change as you look across a
  week?
• How does it change as you go up? Down? Across?
  Diagonal?
• What is the sum of all the numbers in a square?
  Find other sums? Do you notice any patterns?
• What other patterns do you notice?

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Triangle Patterns

• Use the triangle provided to explore number
  patterns.
• Consider the following.
• Do you see the perfect squares?
• Where are the odd numbers?
• Find the sum of the numbers in each row. See if
  you can find a shortcut.
• Take any hexagon like the one outlined. What is
  its sum? Can you find a shortcut?
• Take any two adjacent numbers in a triangle
  diagonal (for example, 4 and 7). Find their
  product. Where is it?

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Lets Go Fishing!

• Ten people are fishing in a boat that has 11
  seats. Five people are on one side and five are
  on the other side with an empty seat between
  them. What is the minimum number of moves it
  would take for the five people in the front of
  the boat to exchange places with the five people
  in the back of the boat? (You may only jump over
  one person at a time and one person can only be
  in a seat at a time.)