1. Whole Numbers

Middle Years

Lesson

Whenever you want to describe something, the hardest part can be finding the right words, or, in the case of mathematics, the right symbols. The language of mathematics is used all around the world. However, instead of writing sentences with words, we write mathematical sentences using numbers and symbols. In order to translate between our language and the language of mathematics, we will need to get familiar with some common expressions.

Let's start with the four basic operations. These symbols tell us what to do with our numbers and are usually referred to as: plus, minus, times and divide.

Word Description | Mathematical Operation | Mathematical Symbol |
---|---|---|

Sum of | Addition (plus) | $+$+ |

Difference between | Subtraction (minus) | $-$− |

Product of | Multiplication (times) | $×$× |

Quotient of | Division (divide) | $\div$÷ |

However, there are other ways we can refer to them.

Here are a few ways that we can refer to the same operation using different words:

- $4+9$4+9 can be described by: "the sum of $4$4 and $9$9" or "$4$4 plus $9$9" or "$9$9 more than $4$4".
- $8-3$8−3 can be described by: "the difference between $8$8 and $3$3" or "$8$8 minus $3$3" or "$3$3 less than $8$8".
- $5\times6$5×6 can be described by: "the product of $5$5 and $6$6" or "$5$5 times $6$6" or "$6$6 groups of $5$5".
- $9\div3$9÷3 can be described by: "the quotient of $9$9 and $3$3" or "$9$9 divided by $3$3" or "$9$9 shared between $3$3".

Careful!

Notice that when writing "$8-3$8−3" as "$3$3 less than $8$8" the order of the numbers switches around. This is important to remember because "$8-3$8−3" and "$3-8$3−8" have different solutions!

Understanding how to translate problems from words into mathematics can make them easier to solve.

A class of twenty-two students wants to buy their teacher a present with a card for the end of the year. If each student contributes $\$4$$4 and the present costs $\$75$$75, how much money do they have left over to buy the card?

**Think:** Let's start by figuring out what steps are needed to solve the problem, then translate that into the mathematical language so we can find a solution.

**Do: **We know from the question that there are $22$22 students who each contribute $\$4$$4. After finding how much money they have altogether, we then take away the money needed for the present to find out how much is left over for the card.

This means that we have $22$22 groups of $\$4$$4, from which we then take away $\$75$$75.

This is equivalent to: $22$22 times $4$4, minus $75$75.

We then translate this into the mathematics language to get: $22\times4-75$22×4−75.

Now, following the order of operations, we can solve to get:

$22\times4-75$22×4−75 | $=$= | $88-75$88−75 | (Evaluate the multiplication) |

$=$= | $13$13 | (Evaluate the addition) |

We now know that the students will have $\$13$$13 left over to buy a card for their teacher.

**Reflect:** We started by identifying the steps needed to solve the problem, then identifying which basic operations were happening at those steps. In this example we noticed that we could use multiplication to find the total money (since each student contributed the same amount) and then use subtraction to find the left over amount (since the cost of the present was subtracted from the total).

Did you know?

The earliest documented use of the plus sign "$+$+" was in 1360 by Nicole Oresme. He wrote it as an abbreviation of the word "*et*", which means "and" in Latin.

Which of the following is described by 'five groups of six'?

$6$6$-$−$5$5

A$6$6$\times$×$5$5

B$6$6$+$+$5$5

C$6$6$\div$÷$5$5

D

In addition to the four basic operations, we also have some symbols to describe the relationship between numbers which are called "inequality symbols".

These are:

Word Description | Symbol | Example |
---|---|---|

Greater than | $>$> | "$5$5 is greater than $2$2" can be written as "$5>2$5>2" |

Less than | $<$< | "$3$3 is less than $7$7" can be written as "$3<7$3<7" |

Greater than or equal to | $\ge$≥ | "$5$5 is greater than or equal to $4$4" can be written as "$5\ge4$5≥4" |

Less than or equal to | $\le$≤ | "$6$6 is less than or equal to $7$7" can be written as "$6\le7$6≤7" |

Equal to | $=$= | "$4$4 is equal to $4$4" can be written as "$4=4$4=4" |

Not equal to | $\ne$≠ | "$4$4 is equal to $5$5" can be written as "$4\ne5$4≠5"not |

Match the mathematical symbols to their word descriptions:

What is the mathematical symbol for "is greater than or equal to" ?

$=$=

A$>$>

B$\le$≤

C$<$<

D$\ne$≠

E$\ge$≥

FWhat is the mathematical symbol for "is not equal to" ?

$=$=

A$\ne$≠

B

Write the following word statement using mathematical symbols:

"nineteen is greater than eleven plus six".