Upper and lower bounds in multiplicative calculations | Maths

Starter quiz

The numbers either end of an ______ interval are the lower and upper bounds.

'error' ✓
The length of a leaf is measured as 6.7 cm correct to 1 decimal place. What are the lower and upper bounds?

6.2 and 7.2

6.75 and 6.85

6.65 and 6.74999999999

6.65 and 6.75 ✓
The length of a leaf is measured as 6.7 cm correct to 1 decimal place. The lower and upper bounds are 6.65 and 6.75. What is the error interval?

$6.65\leq{x}>6.75$

$6.65<{x}\leq6.75$

$6.65\geq{x}<6.75$

$6.65\leq{x}<6.75$ ✓

$6.65\leq{x}\leq6.75$
The mass of a kitten is 1.3 kg correct to 2 significant figures. What is the error interval?

$1.25\leq{x}\leq1.35$

$1.25\leq{x}<1.349999999999$

$1.25<{x}<1.35$

$1.25\leq{x}<1.35$ ✓
Here are the error intervals for $x$ and $y$ : $12.5\leq{x}<13.5$ and $3.5\leq{y}<4.5$ . What are the lower and upper bounds of $x + y$ ?

16 and 17

16 and 18 ✓

17 and 17

17 and 18
Here are the error intervals for $x$ and $y$ : $12.5\leq{x}<13.5$ and $3.5\leq{y}<4.5$ . What are the lower and upper bounds of $x - y$ ?

8 and 9

9 and 9

8 and 10 ✓

9 and 10

Exit quiz

The lower bound for a rounded number is the ______ value the number could have taken prior to rounding.

'smallest' ✓
$x$ = 4.5 correct to 1 decimal place. What is the error interval?

$4.45\leq{x}\leq4.55$

$4.45<{x}\leq4.55$

$4.45\leq{x}>4.55$

$4.45\leq{x}<4.55$ ✓

$4.45\geq{x}<4.55$
$y$ = 26 correct to 2 significant figures. What is the error interval?

$25.5\leq{y}<26.5$ ✓

$25.5 \geq{y}<26.5$

$25.5<{y}\leq26.5$

$25.5\leq{y}\leq26.5$

$25.5\leq{y}>26.5$
The error interval of $x$ is $4.45\leq{x}<4.55$ . The error interval of $y$ is $25.5\leq{y}<26.5$ . What are the lower and upper bounds of $xy$ ? (Give you answers to 3 decimal places)

'113.475 and 120.575' ✓
The error interval of $x$ is $4.45\leq{x}<4.55$ . The error interval of $y$ is $25.5\leq{y}<26.5$ . What are the lower and upper bounds of $x\over{y}$ ? (Give you answers to 3 s.f.)

'0.168 and 0.178' ✓
Which calculation will give the lower bound of $x={{a+b}\over{c}}$ ?

$LB_x={{LB_a+LB_b}\over{UB_c}}$ ✓

$LB_x={{UB_a+LB_b}\over{LB_c}}$

$LB_x={{LB_a+UB_b}\over{LB_c}}$

$LB_x={{UB_a+UB_b}\over{LB_c}}$

$LB_x={{UB_a+UB_b}\over{UB_c}}$

Worksheet

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Lesson Details

Key learning points

The biggest possible product comes from multiplying the upper bounds together.
The smallest possible product comes from multiplying the lower bounds.
The biggest possible quotient comes from dividing an upper bound by a lower bound.
The smallest possible quotient comes from dividing a lower bound by an upper bound.

Common misconception

Assuming that when dividing the UB will be the result of the UB ÷ UB.

Get the pupils to investigate dividing numbers. Keep the divisor the same and change the dividend. Ask the question, "What happens as the dividend increases?".

Keywords

Upper bound - The upper bound for a rounded number is the smallest value that would round up to the next rounded value.
Lower bound - The lower bound for a rounded number is the smallest value that the number could have taken prior to being rounded.
Error interval - An error interval for a number x shows the range of possible values of x. It is written as an inequality a ≤ x < b