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Year 10
Higher

Upper and lower bounds in multiplicative calculations

I can calculate upper and lower bounds for calculations involving rounded numbers.

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New
New
Year 10
Higher

Upper and lower bounds in multiplicative calculations

I can calculate upper and lower bounds for calculations involving rounded numbers.

Download all resources
Share activities with pupils

Lesson details

Key learning points

  1. The biggest possible product comes from multiplying the upper bounds together.
  2. The smallest possible product comes from multiplying the lower bounds.
  3. The biggest possible quotient comes from dividing an upper bound by a lower bound.
  4. 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

Pupils who struggle with the misconception can be reminded that they can try the different combinations and choose the appropriate one.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Video

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Starter quiz

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6 Questions

Q1.
The numbers either end of an interval are the lower and upper bounds.
Correct Answer: error
Q2.
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
Correct answer: 6.65 and 6.75
Q3.
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?
$$6.65\leq{x}>6.75$$
$$6.65<{x}\leq6.75$$
$$6.65\geq{x}<6.75$$
Correct answer: $$6.65\leq{x}<6.75$$
$$6.65\leq{x}\leq6.75$$
Q4.
The mass of a kitten is 1.3 kg correct to 2 significant figures. What is the error interval?
$$1.25\leq{x}\leq1.35$$
$$1.25\leq{x}<1.349999999999$$
$$1.25<{x}<1.35$$
Correct answer: $$1.25\leq{x}<1.35$$
Q5.
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
Correct answer: 16 and 18
17 and 17
17 and 18
Q6.
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
Correct answer: 8 and 10
9 and 10

Exit quiz

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6 Questions

Q1.
The lower bound for a rounded number is the value the number could have taken prior to rounding.
Correct Answer: smallest, lowest, small, low, least
Q2.
$$x$$ = 4.5 correct to 1 decimal place. What is the error interval?
$$4.45\leq{x}\leq4.55$$
$$4.45<{x}\leq4.55$$
$$4.45\leq{x}>4.55$$
Correct answer: $$4.45\leq{x}<4.55$$
$$4.45\geq{x}<4.55$$
Q3.
$$y$$ = 26 correct to 2 significant figures. What is the error interval?
Correct answer: $$25.5\leq{y}<26.5$$
$$25.5 \geq{y}<26.5$$
$$25.5<{y}\leq26.5$$
$$25.5\leq{y}\leq26.5$$
$$25.5\leq{y}>26.5$$
Q4.
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)
Correct Answer: 113.475 and 120.575, 113.475 & 120.575, 120.575 and 113.475, 120.575 & 113.475, 113.475 120.575
Q5.
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.)
Correct Answer: 0.168 and 0.178, 0.168 & 0.178, 0.178 and 0.168, 0.178 & 0.168, 0.178 0.168
Q6.
Which calculation will give the lower bound of $$x={{a+b}\over{c}}$$?
Correct answer: $$LB_x={{LB_a+LB_b}\over{UB_c}}$$
$$LB_x={{UB_a+LB_b}\over{LB_c}}$$
$$LB_x={{LB_a+UB_b}\over{LB_c}}$$
$$LB_x={{UB_a+UB_b}\over{LB_c}}$$
$$LB_x={{UB_a+UB_b}\over{UB_c}}$$