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Patterns: Applications of solving problems using patterns

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Intros
Lessons
  1. Introduction to Applications of Solving Problems Using Patterns:
    How to solve world problems for number patterns, function tables, and function formulas.
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Examples
Lessons
  1. Patterns - Word Problem 1
    Siobhan picks 3 flowers every year on her birthday to add to her book of dried flowers. Her collection has 12 flowers this year.
    1. Use the table to show many flowers she will have by the 5th year.
      Representing Numbers: Tally Marks
    2. Write the rule (formula) for the function table.
    3. How many flowers will she have in the 62nd year?
  2. Patterns - Word Problem 2
    Regular polygons follow a pattern for the variables: number of sides (s) and sum of total internal angles (A).

    Representing Numbers: Tally Marks
    1. Use the table to write a rule (formula) for the pattern. Create a new column for number of sides for each shape.
    2. What is the sum of angles for a regular octagon (8 sides)?
    3. What is the sum of angles for a regular dodecagon (12 sides)?
  3. Patterns - Word Problem 3
    Kelsey found a leak in a pipe in her basement and put a measuring jar under it to catch the water. She noticed that every hour, the water level increased by 2mL in the jar.
    1. There is 34mL of water in the jar after the 1st hour. Fill out the function table and write the rule (formula).

      Representing Numbers: Tally Marks
    2. How much water will there be in the jar after 1 week?
    3. Kelsey wants to know when she will have to dump out the jar (before it overfills). The measuring jar has a maximum capacity of 500mL. When will she have to dump out the water?
  4. Patterns - Word Problem 4
    At an airport, the pay parking costs a flat rate plus an additional cost for every hour you are parked there.

    Representing Numbers: Tally Marks
    1. The table shows how much people paid for the amount of time they were parked. What is the rule (formula) for the pattern?
    2. What was the flat rate? And what was the cost per hour?
    3. If someone left their car at the airport for a week, how much would they have to pay for parking?
  5. Patterns - Word Problem 5
    Nora's laptop is broken. She wants to compare the rates that are charged by two different computer repair stores. Store A charges a $50 service fee up front plus $30 per hour. Store B charges a $100 service fee up front plus $15 per hour.
    1. Write the rule and fill out a function table for both stores.
      Representing Numbers: Tally Marks
    2. Explain when Store A would be the better deal (lower cost). Then, explain when Store B would be the better deal.
    3. If it took 24 hours to fix the laptop, how much would it cost at each store? And what is the difference between the costs?
Topic Notes
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In this lesson, we will learn:

  • How to solve word problems for number patterns, function tables, and function formulas.

Notes:

  • Being able to identify and use patterns allows for better problem solving
    • You can use patterns as a shortcut to find the answers for other questions where the same pattern exists; the same method can help you find solutions for multiple problems
    • Using patterns can help you save time
      • Ex. If there are 2 red marbles for every 3 green marbles, how many green marbles would there be if there were 264 red marbles?
      • It would take a long time to draw all the marbles
      • By using a pattern (using a rule) you can find the number of green marbles in just one step
      • The rule is y=32xy = \frac{3}{2}x, so plugging in the number of red marbles y=32 y =\frac{3}{2} (264) gives yy =396. There are 396 green marbles when there are 264 red marbles.

  • When dealing with pattern word problems, rename the input and output as relevant variables (i.e. choose your variable as the first letter of the variable type)
    • ex. years (yy), hours (hh), water (WW), cost (CC)

  • Look for these common words in the pattern word problems
    • “every” means to multiply
    • Time units (such as hours, minutes, years) are usually inputs
    • “there is __ this time” surrounds the first output (ex. $12 after the first hour)

  • Recall that the formula for number patterns is given as y=mx+by = mx + b
    • Or, it can be thought of as:
      (outputvariable)=(multiplier/divisor)x(inputvariable)±(addend/subtrahend)(output variable) = (multiplier/divisor) x (input variable) \pm (addend/subtrahend)

    • Ex. “There are 3 frogs for every turtle at the pet store”
      • The input is “turtles” (tt) and the output is “frogs” (ff)
      • “every” means multiply with the multiplier “3” to the input (tt)
      • The formula is given as: f=3tf = 3t

    • Ex. “It costs $0.20 for every piece of gum”
      • The input is “pieces of gum” (gg) and the output is “cost” (CC)
      • “every” means multiply with the multiplier “0.20” to the input (gg)
      • The formula is given as CC = 0.20g