Introduction to solving patterns in T-tables and equations

Introduction to solving patterns in T-tables and equations

Lessons

In this lesson, we will learn:

  • How to determine and write one-step and two-step relationships in T-tables as equations with variables
  • How to solve equations: isolating variables in two-step relationships (opposite operation to both sides)

Notes:

  • Recall that T-tables keep track of patterns and allow calculation of subsequent terms
Introduction to Solving Patterns in T-tables and Equations
    • What is a fast way to find the 100th term? Adding by 2 each time is not efficient.
  • T-table columns have a relationship between them. By writing this relationship as an equation, you can put in any number and find information about that term quickly.
    • Ask: how do you go from the left column to the right column?
      • Find one-step relationships through trial and error (if numbers are getting bigger it is + or ×; if numbers are getting smaller it is - or ÷)
    • Assign variables to each column and use these variables in your equation
Introduction to Solving Patterns in T-tables and Equations

  • For more complicated patterns (two-step relationships) the equation will have two operations: the first step will be [× or ÷] and then the second step will be [+ or -]
    • Similar to one-step relationships, they can be found through trial and error

  • Isolating variables in two-step equations by “doing the opposite to both sides” in the backwards order of operations (solve by doing + or - first and then × or ÷ after)
Introduction to Solving Patterns in T-tables and Equations
  • Introduction
    Introduction to Solving Patterns in T-tables and Equations:
    a)
    Finding one-step relationships between columns in a T-table

    b)
    Writing equations for T-table relationships (one-step)

    c)
    Writing equations for T-table relationships (two-step)

    d)
    How to isolate variables in two-step equations (solving algebra equations)


  • 1.
    T-tables and one-step equations
    Write the equation for the relationship between the two variables. Follow the pattern to complete the T-table.
    a)
    variables aa and bb

    a

    b

    1

    14

    2

    15

    3

    16

    4


    20



    b)
    variables ff and gg

    f

    g

    50

    28

    49

    27

    48

    26

    47


    25



    c)
    variables qq and rr

    q

    r

    8

    32

    9

    36

    10

    40

    11


    1



    d)
    variables vv and ww

    v

    w

    27

    9

    24

    8

    21

    7

    18


    3




  • 2.
    T-tables and two-step equations
    Write the equation for the relationship between the two variables. Follow the pattern to complete the T-table.
    a)
    variables ii and jj

    i

    j

    1

    3

    2

    5

    3

    7

    4


    10



    b)
    variables dd and ee

    d

    e

    5

    13

    6

    16

    7

    19

    8


    1



    c)
    variables xx and yy

    x

    y

    20

    13

    18

    12

    16

    11

    14


    100



    d)
    variables xx and yy

    x

    y

    24

    4

    21

    3

    18

    2

    15


    300




  • 3.
    T-tables and equations: word problem - 1
    It costs $8 to rent a bicycle for the first hour. It costs $2 every hour after that.
    a)
    Fill out a T-table for the cost of the first five hours.

    hour

    cost












    b)
    Write an equation with variables for the relationship between hours (hh) and the cost (CC).

    c)
    How much would it cost to rent a bicycle for 24 hours?

    d)
    If someone paid a total of $26, how many hours did they rent a bicycle for?


  • 4.
    T-tables and equations: word problem - 2
    After the first month, seed A grew into a 2cm plant and continued to grow 5cm each month after that. Seed B grew to 6cm and continued to grow 3 cm each month after that.
    a)
    Fill out a T-table for the cost of the first three months for both seeds.

    month

    heightA







    month

    heightB








    b)
    Write an equation for each seed's growth, relating the number of months (mmA, mmB) and the height (HHA, HHB).

    c)
    After 1 year, what is the height difference between the two plants?

    d)
    When will each plant reach 42cm in height?