1.05 Using technology for operations with whole numbers
Technology can make doing mathematical operations quicker and easier, but there are some common mistakes and pitfalls to keep in mind.
The most common type of technology used for performing operations is the hand-held calculator, so let's run through the basics.
The buttons we will use at the moment are
- The number buttons: $\editable{0}$0 - $\editable{9}$9
- The operations: $\editable{+}$+, $\editable{-}$−, $\editable{×}$×, and $\editable{\div}$÷
- The equals sign $\editable{=}$=
- Parentheses $\editable{(}$( and $\editable{)}$)
To perform an operation between two numbers with your calculator,
- Press the buttons corresponding to the digits of the first number going from left to right
- Press the corresponding operation button
- Enter the second number the same way as the first
- Press the equals sign
The answer should now appear on the calculator display.
Worked examples
Example 1
Evaluate the expression $23+42$23+42 with your calculator.
Think: There are three parts in the expression: two numbers and the operation. The order we enter the numbers is sometimes important, but not in the case of addition.
Do: The sequence of button presses would be:
$\editable{2}$2$\editable{3}$3$\editable{+}$+$\editable{4}$4$\editable{2}$2$\editable{=}$=,
giving the answer $65$65.
To evaluate an expression involving more than one operation we need to pay attention to the order of operations. We always evaluate inside parentheses first and, working from left to right, do any multiplication or division before addition or subtraction. If you enter the expression exactly as it appears in the question, your calculator will automatically do the correct order of operations.
Sometimes it may be quicker to break up the expression and do each operation separately. To do this, perform each operation then press $\editable{=}$=, respecting the order of operations. The answer each time you press equals becomes the 'first number' that you can then perform further operations on.
It's always useful to check the reasonableness of your answer. If the answer doesn't seem right, there's a good chance it isn't!
Example 2
Evaluate the expression $\left(53+64\right)\div35\times23$(53+64)÷35×23 with your calculator.
Think: Be careful when entering the expression, putting parentheses in the right places and keeping everything in the correct order. Pay attention to the order of operations and the same considerations as the single operation example. Here we have a division, so order is important.
Do: You can either use the sequence of button presses:
$\editable{(}$($\editable{5}$5$\editable{3}$3$\editable{+}$+$\editable{6}$6$\editable{4}$4$\editable{)}$)$\editable{\div}$÷$\editable{3}$3$\editable{5}$5$\editable{×}$×$\editable{2}$2$\editable{3}$3$\editable{=}$=,
or the sequence of button presses:
$\editable{5}$5$\editable{3}$3$\editable{+}$+$\editable{6}$6$\editable{4}$4$\editable{=}$=$\editable{\div}$÷$\editable{3}$3$\editable{5}$5$\editable{=}$=$\editable{×}$×$\editable{2}$2$\editable{3}$3$\editable{=}$=,
both resulting in the answer $\frac{2691}{35}$269135.
There are many mistakes that can occur when using your calculator, here are some common ones:
- Not using the correct order of operations
- Missing a digit from a number
- Swapping the digits in a number
- Swapping numbers in the expression
- Using the wrong operation
- Adding an extra digit
Calculators are useful as long as they are used correctly and mistakes are avoided. Even so, you should always keep in mind that it might just be quicker and more reliable to work it out in your head or on paper.
Practice question
Question 1
Xavier used his calculator to evaluate $16\times85-2$16×85−2 and got the answer $926$926.
Is Xavier's answer correct?
Yes
ANo
BWhat mistake might Xavier have made when he entered the expression into his calculator?
He swapped the digits in a number.
AHe added an extra zero.
BHe swapped some of the numbers.
CHe forgot a zero.
DHe didn't use the correct order of operations.
E
Technology can make doing mathematical operations quicker and easier, but there are some common mistakes and pitfalls to keep in mind.
The most common type of technology used for performing operations is the hand-held calculator, so let's run through the basics.
The buttons we will use at the moment are
- The number buttons: $\editable{0}$0 - $\editable{9}$9
- The operations: $\editable{+}$+, $\editable{-}$−, $\editable{×}$×, and $\editable{\div}$÷
- The equals sign $\editable{=}$=
- Parentheses $\editable{(}$( and $\editable{)}$)
To perform an operation between two numbers with your calculator,
- Press the buttons corresponding to the digits of the first number going from left to right
- Press the corresponding operation button
- Enter the second number the same way as the first
- Press the equals sign
The answer should now appear on the calculator display.
Worked examples
Example 3
Evaluate the expression $23+42$23+42 with your calculator.
Think: There are three parts in the expression: two numbers and the operation. The order we enter the numbers is sometimes important, but not in the case of addition.
Do: The sequence of button presses would be:
$\editable{2}$2$\editable{3}$3$\editable{+}$+$\editable{4}$4$\editable{2}$2$\editable{=}$=,
giving the answer $65$65.
To evaluate an expression involving more than one operation we need to pay attention to the order of operations. We always evaluate inside parentheses first and, working from left to right, do any multiplication or division before addition or subtraction. If you enter the expression exactly as it appears in the question, your calculator will automatically do the correct order of operations.
Sometimes it may be quicker to break up the expression and do each operation separately. To do this, perform each operation then press $\editable{=}$=, respecting the order of operations. The answer each time you press equals becomes the 'first number' that you can then perform further operations on.
It's always useful to check the reasonableness of your answer. If the answer doesn't seem right, there's a good chance it isn't!
Example 4
Evaluate the expression $\left(53+64\right)\div35\times23$(53+64)÷35×23 with your calculator.
Think: Be careful when entering the expression, putting parentheses in the right places and keeping everything in the correct order. Pay attention to the order of operations and the same considerations as the single operation example. Here we have a division, so order is important.
Do: You can either use the sequence of button presses:
$\editable{(}$($\editable{5}$5$\editable{3}$3$\editable{+}$+$\editable{6}$6$\editable{4}$4$\editable{)}$)$\editable{\div}$÷$\editable{3}$3$\editable{5}$5$\editable{×}$×$\editable{2}$2$\editable{3}$3$\editable{=}$=,
or the sequence of button presses:
$\editable{5}$5$\editable{3}$3$\editable{+}$+$\editable{6}$6$\editable{4}$4$\editable{=}$=$\editable{\div}$÷$\editable{3}$3$\editable{5}$5$\editable{=}$=$\editable{×}$×$\editable{2}$2$\editable{3}$3$\editable{=}$=,
both resulting in the answer $\frac{2691}{35}$269135.
There are many mistakes that can occur when using your calculator, here are some common ones:
- Not using the correct order of operations
- Missing a digit from a number
- Swapping the digits in a number
- Swapping numbers in the expression
- Using the wrong operation
- Adding an extra digit
Calculators are useful as long as they are used correctly and mistakes are avoided. Even so, you should always keep in mind that it might just be quicker and more reliable to work it out in your head or on paper.
Practice question
question 2
Xavier used his calculator to evaluate $16\times85-2$16×85−2 and got the answer $926$926.
Is Xavier's answer correct?
Yes
ANo
BWhat mistake might Xavier have made when he entered the expression into his calculator?
He swapped the digits in a number.
AHe added an extra zero.
BHe swapped some of the numbers.
CHe forgot a zero.
DHe didn't use the correct order of operations.
E