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Bounds and Error Intervals
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Foundational skills
Understand bounds vocabulary
\[ \text{Which is larger: UB or LB?} \]
Understand what upper bound and lower bound mean.
Identify if a value is within bounds
\[ \text{Could actual be } 7.18\text{?} \]
Determine whether a given value could round to a stated rounded value.
Bounds from rounding to place value
Bounds from rounding to nearest integer
\[ 47 \text{ cm} \rightarrow \text{LB, UB?} \]
Find bounds when rounded to the nearest whole number.
Bounds from rounding to nearest 10
\[ 350 \rightarrow \text{LB, UB?} \]
Find bounds when rounded to the nearest 10.
Bounds from rounding to nearest 100
\[ 4700 \rightarrow \text{LB, UB?} \]
Find bounds when rounded to the nearest 100.
Bounds from rounding to nearest 1000
\[ 84000 \rightarrow \text{LB, UB?} \]
Find bounds when rounded to the nearest 1000.
Bounds from rounding to decimal places
Bounds from rounding to 1 d.p.
\[ 5.4 \rightarrow 5.35, 5.45 \]
Find bounds when rounded to 1 decimal place.
Bounds from rounding to 2 d.p.
[ 3.72 \rightarrow 3.715, 3.725 ]
Find bounds when rounded to 2 decimal places.
Bounds from rounding to 3 d.p.
[ 0.847 \rightarrow \text{LB, UB?} ]
Find bounds when rounded to 3 decimal places.
Bounds from rounding to significant figures
Bounds from 1 s.f. (large numbers)
\[ 400 \text{ (1 s.f.)} \rightarrow 350, 450 \]
Find bounds when a large number is rounded to 1 significant figure.
Bounds from 1 s.f. (small decimals)
\[ 0.007 \text{ (1 s.f.)} \rightarrow ? \]
Find bounds when a small decimal is rounded to 1 significant figure.
Bounds from 2 s.f.
\[ 73 \text{ (2 s.f.)} \rightarrow 72.5, 73.5 \]
Find bounds when rounded to 2 significant figures.
Error intervals
Write error interval using inequality notation
\[ 6.25 \leq m < 6.35 \]
Express bounds as an error interval using inequality notation.
Truncation
Truncate a number
\[ 7.869 \xrightarrow{\text{trunc 2 d.p.}} 7.86 \]
Remove digits after a given decimal place without rounding.
Bounds from truncation
\[ 14.3 \text{ (trunc)} \rightarrow 14.3, 14.4 \]
Find bounds when a value has been truncated.
Error interval for truncated value
\[ 8.2 \leq L < 8.3 \]
Write an error interval for a truncated value.
Bounds calculations: Addition
Maximum value of A + B
\[ \text{Max}(p + q) = \text{UB}(p) + \text{UB}(q) \]
Calculate the maximum possible value of a sum.
Minimum value of A + B
\[ \text{Min}(p + q) = \text{LB}(p) + \text{LB}(q) \]
Calculate the minimum possible value of a sum.
Bounds calculations: Subtraction
Maximum value of A − B
\[ \text{Max}(x – y) = \text{UB}(x) – \text{LB}(y) \]
Calculate the maximum possible value of a difference.
Minimum value of A − B
\[ \text{Min}(x – y) = \text{LB}(x) – \text{UB}(y) \]
Calculate the minimum possible value of a difference.
Bounds calculations: Multiplication
Maximum value of A × B
\[ \text{Max}(a \times b) = \text{UB}(a) \times \text{UB}(b) \]
Calculate the maximum possible value of a product.
Minimum value of A × B
\[ \text{Min}(a \times b) = \text{LB}(a) \times \text{LB}(b) \]
Calculate the minimum possible value of a product.
Bounds calculations: Division
Maximum value of A ÷ B
\[ \text{Max}(m \div n) = \text{UB}(m) \div \text{LB}(n) \]
Calculate the maximum possible value of a quotient.
Minimum value of A ÷ B
\[ \text{Min}(m \div n) = \text{LB}(m) \div \text{UB}(n) \]
Calculate the minimum possible value of a quotient.
Bounds calculations: Combined
Bounds with multiple operations
\[ \text{Max of } (a + b) \times c \]
Calculate a bound for an expression with multiple operations.
Bounds in context: speed
\[ v = d \div t \text{ — find max speed} \]
Use bounds to calculate maximum or minimum speed.
Bounds in context: density
\[ \rho = m \div V \text{ — find min density} \]
Use bounds to calculate maximum or minimum density.
Bounds in context: area
\[ A = l \times w \text{ — find max area} \]
Use bounds to calculate maximum or minimum area.
Special cases
Bounds where one value is exact
\[ C = \pi d \text{ (}\pi\text{ exact)} \]
Calculate bounds when one value has no error.
Same quantity appears twice
\[ A = s^2 \text{ (same } s \text{)} \]
Calculate bounds when the same variable appears multiple times.
Appropriate degree of accuracy
\[ \text{Round to suitable precision} \]
Calculate a bound and round to a sensible level of precision.
Timer (Optional)
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