Rounding and estimating calculations 2

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Christian Lawson-Perfect	said	Should not be used	7 years, 8 months ago
Stanislav Duris	said	Needs to be tested	7 years, 9 months ago

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Christian Lawson-Perfect 7 years, 8 months ago

Saved a checkpoint:

I've split parts a and b into separate questions. I'm not sure about part c - is it obviousthat you'd always want to underestimate the distance? Maybe it needs a sentence like "you want to know if you'll meet your target in the next five minutes".

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Christian Lawson-Perfect 7 years, 8 months ago

Gave some feedback: Should not be used

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Stanislav Duris 7 years, 9 months ago

Published this.

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Stanislav Duris 7 years, 9 months ago

Saved a checkpoint:

Vicky suggested making this question and these parts because she thought the other question about estimation was not applicable to real life estimation enough. If this question still needs some stuff to do and someone else picks it up after I leave, you can maybe change some of the scenarios or adjust the question so there is no specific rounding students need to do and they can just choose one they want as Christian suggested. I sadly do not have time for this.

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Stanislav Duris 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Christian Lawson-Perfect 7 years, 9 months ago

Gave some feedback: Has some problems

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Christian Lawson-Perfect 7 years, 9 months ago

Saved a checkpoint:

There are too many items in part a - I can't add them all up in my head. Just two or three items, plus the ice cream, would be enough to get the idea across.

Both parts a and b have this problem - even after rounding, I can't do these calculations in my head, so I might as well use a calculator.

"The dimensions of the room are 6.49m×2.88m×3.39m where 3.39m is the height." is oddly worded. I'd say "The room measures 6.49m×2.88m, and 3.39m high."

In part b, I'd round to the nearest metre and do the calculation in my head. My maximum error when rounding to the nearest metre is $4 \times \left(\frac{1}{2}(\text{width}+\text{depth}+\text{height})+\frac{1}{4} \right)\text{m}^2$ , which will be around one bucket's worth.

By the way, the height I got, 3.39m, is completely unrealistic! In the UK the minimum ceiling height is 2.1m and the standard is around 2.4m.

The thesis of this question is that sometimes rounding to 1 sig fig is too blunt, but I don't think the given scenarios make a great case for it: they're all 'mental maths' scenarios. If I'm going to use a calculator, I might as well type in 2 decimal places rather than rounding to 1. Can you think of scenarios where you've got some very precise measurements, but only want a moderately precise answer?

You could use the separate minimum and maximum accepted value settings for number entry parts in a question like this - accept anything within a generous bound of the true answer, and let the student perform whatever estimation they like, maybe guiding them towards one.

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Stanislav Duris 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Stanislav Duris 7 years, 9 months ago

Created this.

Name	Status	Author	Last Modified
Rounding and estimating calculations 2	Should not be used	Stanislav Duris	20/11/2019 14:42
Rounding and estimating calculations - adding up prices in a shop	Ready to use	Christian Lawson-Perfect	20/11/2019 14:42
Rounding and estimating calculations - painting a room	Ready to use	Christian Lawson-Perfect	20/11/2019 14:42
Consumer Arithmetic - Rounding and estimating calculations - adding up prices in a shop	draft	Paul Hancock	24/07/2018 03:04
David's copy of Rounding and estimating calculations - painting a room	draft	David Rickard	07/11/2018 21:04

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Question statement

Sometimes, rounding to one significant figure is not enough as this estimate may often be very far from the actual value.

In this question, you will explore some real life scenarios where estimating a value can be helpful. In these kinds of situations, it is often better to decide whether we are going to always round up or always round down, to ensure we don't go below/above the true value. Decide whether it is better to round up or round down and estimate the calculation:

Give any introductory information the student needs.

			Name	Type	Generated Value

colour	string	blue
height	number	2.78
width	number	3.88
length	number	4.93
wall1	number	10.7864
wall2	number	13.7054
ceiling	number	19.1284
all	number	68.112
h	number	3
w	number	4
l	number	5
rwall2	number	12
rwall1	number	15
rceiling	number	20
rall	number	74
buckets	number	5
buckets1	number	4.9333333333

			Name	Type	Generated Value

p	list	[ 61/100, 99/50 ]
total_rounded_up	decimal	dec("7.6")
total_rounded_down	decimal	dec("6.9")
actual_total	number	7.48
ice_cream	number	4.89
cash	decimal	dec("6")
flavour	string	raspberry

			Name	Type	Generated Value

speed	number	2.32
rspeed	number	2
dist	number	600

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random("green", "red", "orange", "yellow", "blue", "purple", "pink")

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `string`

blue

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Choose one from a list
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. Choose one from a list
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Part b) Gap-fill
1. Gap Gap 0. Choose one from a list
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2. Gap Gap 1. Number entry
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Part c) Gap-fill
1. Gap Gap 0. Choose one from a list
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2. Gap Gap 1. Number entry
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1. Step Information only

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Imagine you are shopping at the supermarket. You only have £{cash} in cash. There are two items in your basket so far, costing £{p[0]} and £{p[1]}.

Just before checkout, you notice a tasty {flavour} ice cream on the shelf. It costs £{ice_cream}. Can you put this in your basket without going over your limit?

i) If we don't want to underestimate the total price of these 10 items, do we round the individual prices up, or down?

Gap 0.

ii) Estimate the total price if we buy the ice cream, rounding the price of each item to 1 decimal place.

£ Gap 1.

iii) Can we be sure that we have enough cash to pay for this?

Gap 2.

Advice

a)

We want to ensure we won't go over the limit, so it is better to overestimate. If we underestimated, we could potentially think we have enough money while we don't.

To overestimate our total, we round up.

ii)

We round up all our values to 1 decimal place:

Original prices	$£\var{p[0]}$	$£\var{p[1]}$	$£\var{ice_cream}$
Rounded up	$£\var{precround(p[0]+0.05,1)}$	$£\var{precround(p[1]+0.05,1)}$	$£\var{precround(ice_cream,1)}$

Now we calculate the total of these rounded numbers:

$£\var{precround(p[0]+0.05,1)} + £\var{precround(p[1]+0.05,1)} + £\var{precround(ice_cream,1)} = £\var{total_rounded_up}$

iii)

As the estimated total $£\var{total_rounded_up}$ is higher than $£\var{cash}$ , we may not have enough money to purchase the {flavour} ice cream.

b)

It is much better to have residual paint than to not have enough of it. That is why it is better to overestimate the area.

Therefore, we round up.

ii)

We round up all our values to the nearest integer:

Length: $\var{length}\,\mathrm{m} = \var{l}\,\mathrm{m}$ .

Width: $\var{width}\,\mathrm{m} = \var{w}\,\mathrm{m}$ .

Height: $\var{height}\,\mathrm{m} = \var{h}\,\mathrm{m}$ .

The total area consists of 3 areas:

Two walls of $\var{l}\,\mathrm{m} \times \var{h}\,\mathrm{m}$ (length by height), two walls of $\var{w}\,\mathrm{m} \times \var{h}\,\mathrm{m}$ (width by height) and a ceiling of $\var{l}\,\mathrm{m} \times \var{w}\,\mathrm{m}$ (length by width).

$\begin{align} \var{l}\,\mathrm{m} \times \var{h}\,\mathrm{m} &= \var{l*h}\,\mathrm{m}^2 \\ \var{w}\,\mathrm{m} \times \var{h}\,\mathrm{m} &= \var{w*h}\,\mathrm{m}^2 \\ \var{l}\,\mathrm{m} \times \var{w}\,\mathrm{m} &= \var{l*w}\,\mathrm{m}^2 \end{align}$

Therefore, the total area we need to paint is

$\begin{align} 2 \times \var{l}\,\mathrm{m} \times \var{h}\,\mathrm{m} + 2 \times \var{w}\,\mathrm{m} \times \var{h}\,\mathrm{m} + \var{l}\,\mathrm{m} \times \var{w}\,\mathrm{m} &= 2 \times \var{l*h}\,\mathrm{m}^2 + 2 \times \var{w*h}\,\mathrm{m}^2 + \var{l*w}\,\mathrm{m}^2 \\ &= \var{2*l*h}\,\mathrm{m}^2 + \var{2*w*h}\,\mathrm{m}^2 + \var{l*w}\,\mathrm{m}^2 \\ &= \var{rall}\,\mathrm{m}^2 \text{.} \end{align}$

iii)

$\var{rall}\,\mathrm{m}^2 \div 15\,\mathrm{m}^2 = \var{rall/15}$

We can only buy a whole number of buckets, so we need to decide between {buckets-1} and {buckets} paint buckets. As it is better to buy more paint than not buy enough, we should buy {buckets} buckets of {colour} paint.

c)

In this case, it is better to underestimate. It is less of an error to assume we ran less than the actual value. If we overestimated, we would assume we ran more than we actually did and would do less exercise than we initially wanted.

Therefore, we round down.

ii)

To calculate the distance we use the formula

$\mathrm{Distance} = \mathrm{Speed} \times \mathrm{Time}$

We round down the speed:

$\var{speed} \, \mathrm{m/s} = \var{rspeed} \, \mathrm{m/s} \text{.}$

We convert time into seconds:

$\begin{align} 5 \, \mathrm{min} &= 5 \times 60 \\&= 300 \, \mathrm{s} \text{.} \end{align}$

Estimate distance:

$\begin{align} \mathrm{Distance} &= \var{rspeed} \, \mathrm{m/s} \times 300 \, \mathrm{s} \\&= \var{dist} \, \mathrm{m} \text{.} \end{align}$

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This question is used in the following exam:

Transition Rounding by Chris Templet in Chris's workspace.

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Checkpoint description

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Error in variable testing condition

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Generated value: `string`

Parts

Gap-fill

a)

b)

c)

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Rounding and estimating calculations 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Stanislav Duris 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: string

Parts

Gap-fill

a)

b)

c)

Generated value: `string`