# Stage 4 - algebra - number patterns

Continue a number pattern to match a table of values; determine the rule which involves more than one operation to match a number pattern

## Strategy

Students can:

- continue a number pattern to match a table of values
- determine the rule which involves more than one operation to match a number pattern

## Activities to support the strategy

### Activity 1

Provide students with a collection of matchsticks. Students build a sequence of squares with different side lengths using the matchsticks.

Here is a sequence of squares with sides measuring 1 matchstick, 2 matchsticks, 3 matchsticks, etc.

Now try this with your pattern! Make the next two squares of the pattern. The perimeter of a square is the distance all the way around. Complete the table to show the perimeter of each of the squares.

Length of one side of square (in matchsticks) | Perimeter of square (in matchsticks) |
---|---|

1 | 4 |

2 | ^{enter answer} |

3 | ^{enter answer} |

4 | ^{enter answer} |

5 | ^{enter answer} |

Predict the perimeter of a square with sides of six matchsticks.

#### Discuss

- If the square has sides 8 matchsticks long, what is the perimeter? 10 matchsticks long? Etc.
- Students determine the rule to describe their matchstick pattern.

2. Use coloured counters on an overhead projector to show other patterns for students to identify and describe, e.g.

#### Ask students to

- look at the pattern of counters and draw what they think the fifth and sixth shapes in the pattern would look like
- describe the patterns they can see
- develop an expression to show the number of counters needed for the n
^{th}shape

### Activity 2

A variable is a symbol or letter which represents a number in an expression or equation. For example, ** "b"** is a variable in the expression 3

*+ 5. This means,*

**b****can be equal to any number in this expression.**

*"b"*In an equation, variables can be independent or dependent. For example, in the equation * c* = 3

**+ 5, b is the independent variable (can be equal to any number) and c**

*b***is the dependent variable (the value of**

*is determined once we know the value of*

**c***).*

**b**For example, anthropologists have developed a formula to determine the height from femur length. In cm, a man's height is given as

- height = 2.59 x femur length + 66.4

Using pronumerals, we can use* f* to stand for femur length and

**to stand for the man's height. The formula may then be written as**

*h*- h = 2.59f + 66.4

The man's height depends on the length of the femur, so we say that **f** is the independent variable and ** h** is the dependent variable. The formula is written with the dependent variable

*as the subject.*

**(h)**1. Ask students to work through the following questions:

No | Equation | Independent variable | Dependent variable |
---|---|---|---|

1 | y = 3x |
^{enter answer} |
^{enter answer} |

2 | h = 3k + 2 |
^{enter answer} |
^{enter answer} |

3 | v = u + 5t |
^{enter answer} |
^{enter answer} |

4 | p = 3q - 2 |
^{enter answer} |
^{enter answer} |

5 | t = 3x2 + 2x - 5 |
^{enter answer} |
^{enter answer} |

6 | x + 3y = 4 |
^{enter answer} |
^{enter answer} |

7 | 3x + 2y = 9 |
^{enter answer} |
^{enter answer} |

8 | 3x + 2y - 9 = 0 |
^{enter answer} |
^{enter answer} |

9 | x - 3y - 5 = 0 |
^{enter answer} |
^{enter answer} |

10 | 12m = 4n + 6 |
^{enter answer} |
^{enter answer} |

Discuss the answers as a class, with particular emphasis on questions 6 -10.

2. As a class, determine the rule giving the relationship between x and y in each of the following and write the corresponding equation.

x |
-1 | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|---|

y |
-2 | 0 | 2 | 4 | 6 | 8 |

The rule is y is always double x.

The equation is y = 2x

a |
-1 | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|---|

b |
-4 | 0 | 4 | 8 | 12 | 16 |

The rule is __________________________

The equation is ______________________

3. Working in pairs, students determine the rule and write the equation for each of the following relationships.

x | -1 | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|---|

y | 3 | 5 | 7 | 9 | 11 | 13 |

- When
= 0,**x**= 5 = 0 + 5 = 2 x 0 + 5**y** - When
= 1,**x**= 7 = 2 + 5 = 2 x 1 + 5**y** - When
= 2,**x**= 9 = 4 + 5 = 2 x 2 + 5**y** - So the equation is
= 2**y**+ 5**x**

a |
-1 | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|---|

b |
-1 | 3 | 7 | 11 | 15 | 19 |

- When a = ______ b = ______________
- When a = ______ b = ______________
- When a = ______ b = ______________
- So the equation is _________________

## References

### Australian curriculum

ACMNA175: Introduce the concept of variables as a way of representing numbers using letters.

### NSW syllabus

MA4-10NA: Uses algebraic techniques to solve simple linear and quadratic equations.

## Teacher resources

### Lesson plans and activities

#### Explores triangular numbers

#### Exploring number patterns

- teachingideas.co.uk/maths/contents05nopatterns
- shodor.org/interactivate/activities/FunctionMachine
- nrich.maths.org

## Student resources

#### Number patterns which use a combination of operations

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