Algebraic Thinking: Strategies

• Word problems often represent linear equations verbally, requiring translation into algebraic equations.

Steps to Translate Verbal Equations

• Identify unknowns and assign variables.
• Identify important values like rates and constants.
• Determine independent and dependent variables.
• Pinpoint words indicating operations, equality, and groupings.
• Formulate the equation.

Slope-Intercept Form

• In slope-intercept form, rates typically are the equation's slope, often using "per."
• Constants or starting values usually represent the y-intercept.

Example 1: Keisha's Savings

• Keisha has $45 in savings.
• She earns $13.50 per hour.
• Equation for total money (T) after working (h) hours.
• Unknowns: T = Keisha's total money, h = number of hours.
• Rate/slope: $13.50 per hour.
• Constant/y-intercept: $45 (savings).
• T is the dependent variable; h is the independent variable.
• "Total" indicates addition.
• The equation: T = 13.50h + 45.
• Units check: $ = ($/hr) * hr + $.

Standard Form

• In standard form (Ax + By = C), rates are variables (x, y) multiplied by constants (A, B).
• The constant (C) typically represents the total.

Example 2: Hernandez Family Fruit Purchase

• The Hernandez family spends $9.50.
• They buy six nectarines and five peaches.
• Equation in standard form.
• Unknowns: N = cost per nectarine, P = cost per peach.
• Constants: 6 nectarines, 5 peaches, $9.50 total (C).
• "And" suggests addition.
• Equation: 6N + 5P = $9.50.

Writing Linear Equations from Word Problems

• Identify unknowns and assign variables.
• Identify rates and constants.
• Determine dependent and independent variables.
• Identify operation-related words.
• Write the equation.

Looking for a Pattern

• Math is about pattern finding.
• Formulas can arise from pattern identification.

Example: Polygon Sides and Angle Sums

• Relate polygon sides to total angle measurements.
• Number of Sides: 3, 4, 5, 6, 7
• Sum of Angles: 180, 360, 540, 720, 900
• Angle sum increases by 180 for each added side.
• Angle sum of a polygon with n sides: 180(n – 2).

Encourage Multiple Approaches

• Expose students to problems with varied solutions.
• Provide materials: manipulatives, paper, and technology.
• Positively receive diverse, mathematically sound solutions.
• Multiple concept representations clarify for new learners.
• Students may reason differently, offering valid approaches.
• Upper elementary students should justify methods mathematically.

Modeling and Evaluating Real-World Situations

• Equations can model real-world situations.
• Related factors can form a model equation.

Modeling Linear Equations

• Consider the slope: Slope = rise/run = change in y / change in x.
• Use y = mx + b to find the y-intercept with a known data point.

Example 1: Melody's Pool Guests

• Temperature increases, and pool guests increase.
• Data: (70, 0), (75, 10), (80, 20), (85, 30), (90, 40).
• Slope: (10-0) / (75 - 70) = 2 (two new guests per degree).
• Y-intercept: 10 = 2(75) + b, b = -140.
• Equation: y = 2x - 140 (a general model).

Example 2: Student Absences and Temperature

• Temperature decreases, student absences increase.
• Equation: a = -2t + 100.
• Absences at 25 degrees: a = -2(25) + 100 = 50.
• Absences at 60 degrees: a = -2(60) + 100 = -20 (unrealistic).
• Validate models by testing low, mid, and high values.

Composition and Decomposition

• The base-ten system uses place values to show digit value by location.
• Expanded notation breaks numbers to show each digit's true value.
• Example: 8,436 → 8,000 + 400 + 30 + 6.
• Writing standard form from expanded form involves combining values.
• Example: 50,000 + 6,000 + 900 + 80 + 3 → 56,983.

Number Lines

• Number lines are straight lines with equally spaced numbers.
• Can start/end at any number and include negatives, fractions, or decimals.
• Numbers increase from left to right.
• Left is lesser, right is greater.
• Helpful for addition, subtraction, multiplication, and division.

Number Lines - Addition and Subtraction

• Addition: "Hops" from the starting number to the right.
• Subtraction: "Hops" from the starting number to the left.

Example 3 + (-6)

• Start at positive 3.
• Count six units left.
• The solution is -3.

Number Lines - Multiplication and Division

• Multiplication: Repetitive addition from zero.
• Division: Repetitive subtraction from the given number.

Example -10 ÷ 5

• Start at -10.
• Hop by 5s towards zero.
• Two hops, so -10 ÷ 5 = -2.

Divisibility Rules

• Simplify math with shortcuts for divisibility.
• Divisible by 2: even last digit.
• Divisible by 3: digit sum is a multiple of 3.
• Divisible by 4: last two digits a multiple of 4.
• Divisible by 5: last digit is 5 or 0.
• Divisible by 6: even and a multiple of 3.
• Divisible by 7: (other numbers - doubled last digit) = 0 or multiple of 7.
• Divisible by 8: last three digits divisible by 8.
• Divisible by 9: digit sum is a multiple of 9.
• Divisible by 10: ends in 0.

Algorithms

• An algorithm is a specific, step-by-step process for accurate results.

Addition Algorithms

• Partial Sums Addition: Add each place value separately, then total.
• Column Addition: Align by place value and add each column.
• Opposite Change Rule: Add to one addend, subtract from the other to reach zero.

Subtraction Algorithms

• Trade First: Ensure top numbers are greater/equal to bottom numbers.
• Counting Up: Count up from subtrahend to original amount.
• Left-to-Right Subtraction: Subtract each place value from left to right.
• Same Change Rule: Add or subtract the same amount to simplify.

Multiplication Algorithms

• Partial Products: Expand factors into place values and multiply each pair.
• Modified Repeated Addition: Use base-10 numbers for multiplication and addition.
• Lattice Multiplication: A grid breaks up multiplication by place value.

Division Algorithms

• Partial Quotients: Estimate with base-10 numbers until the remainder is less than the dividend.
• Column Division: Separate the problem into columns and work with each place value.

The Order of Operations

• Simplify expressions by following PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
• Multiplication and division precede addition and subtraction (left to right).

Example 1

Simplify 8 × (15 - 7) - 2

1. Parentheses: 8 x 8 - 2
2. Multiplication: 64 - 2
3. Subtraction: Answer = 62

Order of Operations - Complex Parentheses

• Resolve inner groupings first, working left to right.
• Number next to parentheses means multiplication.
• Simplify terms inside parentheses before multiplying.

Example 1 - Simplify [4(3+5)+2]{6[3+7(−8)]+182}

1. Inner Grouping 4(8) and 7(-8)
2. Next Grouping [32 + 2] = 34 and [3 - 56] = -53
3. Next Grouping {6[-53] + 182} = {-318 + 182} = {-136}
4. Divide: 34 / -136 = -1/4

Models for Multiplication and Division

• Visual models enhance understanding using strategies like rectangular arrays, equal groups, repeated addition/subtraction, number lines, and strip diagrams.

Rectangular Arrays

• Factors arranged in equal columns and rows.
• First factor: rows; second factor: columns.
• Multiplication: Total objects in the array is the product.
• Division: Divisor determines rows; columns are the quotient.

Equal Groups

• Visualize problems with pictures.
• First factor: number of groups; second factor: dots in each group.
• Multiplication: Total dots are the product.
• Division: Divisor is the number of equal groups; dividend is the dots distributed.

Repeated Addition and Subtraction

• Multiply: One factor is the addend; the other is the number of times to add.
• Divide: Subtract the divisor from the dividend until a remainder of 0.

Number Lines

• Show multiplication and division as repeated addition and subtraction.
• Multiplication: Hop forward the number of times.
• Division: Hop backward by the divisor until zero is reached.

Strip Diagrams

• Multiplication: One factor is the number of boxes; the other factor is in each box.
• Division: The larger number is in the top box; the divisor creates bottom boxes.

Area Models

• Visualize multiplication and division with whole numbers and decimals.

Example 12 x 25

• Expand both factors:
• 12 = 10 + 2
• 25 = 20 + 5
• Multiply the row and column header in each box
• [10 * 20] + [2 * 20] + [10 * 5] + [2 * 5] = 200 + 40 + 50 + 10 = 300

Box Method

• Area models can use base-10 numbers or values divisible by the divisor.

Example 825 ÷ 4

• Expand the dividend into multiples of the divisor plus remainder 825 → 400 + 400 + 20 + 4 + 1
• Divide 1st number
• multiply the result by 4 and carry the remainder to the next division
• Continue the process until the final divison with a remainder
• Add all quotients for the overall resultant answer with final remainder total

Patterns to Linear Equations

• The relationship is linear when the change in y / change in x is constant throughout.

Example: Find equation for the points (0, 4), (1, 2), (2, 0) and (3, -2)

• Rate of Change Formula: change in y / change in x = -2 / 1 = -2
• If the relationship between point is linear, the equation would be (y = mx + b)
• Slope (m) = -2 and given first point (0, 4) show the 'b' value is 4
• So, y = -2x + 4

Example: Find equation for the points (1, 5), (2, 8) and (3, 11)

• Rate of Change Formula: change in y / change in x = 3 / 1 = 3
• If the y intercept isn't included, working backwards to an x value of 0 is used
• Subtract 1 from x *Subratct 3 from y
• the result from (1.5) would be (0, 2)
• So the equation is (y = 3x + 2)
• Solving Algebraically
• y = mx + b where y=5, m=3 and x =1 solving for b
• 5 = 3 * 1 + b
• 5 = 3 + b
• b = 2

Patterns to Formulas

Explicit formula allows to find terms without knowing previous value.

Patterns to Formulas - Arithmetic

• The equation for nth term of arithmetic sequence is written as
• tₙ=a₁+d(n - 1),
• t ₙ means nth Therm, a₁ means first term, d means common difference, n means term number
• Example
• 3, 5, 7, 9
• So, a₁ would be 3, d would be 2 and the number of blocks shown in the (nᵗʰterm) + (n-1) group of 2 blocks )
• so tₙ=3+2(n - 1) and for the 100 term will become ₙ=3+2(100 - 1)= 201.

Patterns to Formulas - Geometric

• The equation format for the nth term of a geometric sequence is often written as: tₙ=a₁r⁽ⁿ⁻¹⁾
• Where tₙ means the nth term, a₁ means the first thermo, r means common ratio, and n means therm number
• Example
• Researcher records hourly bacteria seen in time periods 5,10,20,40,80. what will the bacteria consist of after 30
• The equation is for tₙ=a₁r(ⁿ⁻¹),so a₁ = 5, r=2 the therm equals:
• tₙ=5x2(ⁿ⁻¹), the therm equals : t= 5 x 2 (³⁰⁻¹)
• After 30 hours, there are 2,684,354,560 bacteria in the colony.

Equivalence of Expressions

• In the terms of fractions like appearing differently but having the same value
• Expressions appearing different but still having the same value.
• Manipulating the equation will yield simpler equations and results.
• 5x = 2x + 3x

Equivalence of Expressions- Combinig Lke Terms

• By adding or subtracting the coefficients of two or more like therms and keeping the variables and exponents the same
• A Label = The equation (3dogs + 2 dogs = 5 dogs will label the therm)
• In Algebra 3x + 2x simplifies to 5x
• Common Errors
• Adding wrong values
• Missing values

Equivalence of Expressions - Use Paremeters = Distribute

• Parameters are used to group part of an expression together
• Parameters are an indication that an operation should be applied to the entire expression
• Distribute the amount on the outside, to the inside the parentheses for each value
• Order matters to make things easier
• In an addition and subtraction sign is the same as -1, every therm inside the parameter must become negative
• when an addition to the parameter is the same as +1 every therm inside the parameter stays the same and its dropped for simpler coding

Symbolic to Verbal

• Operational symbols will need youto be able to identify a matching erbal description of the scenrio
• Additions, subtractions, multiplications and divisions all have there own keyworkds and values.

Symbolic to Verbal - Equation or Inequality?

• An equals sign or inequality symbol can be translated into certain keywords or phrases.
• Equality and inequlity symbols can be translated into key phrases and words.