Completing the square

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From users who are members of Transition to university :

Christian Lawson-Perfect	said	Ready to use	7 years, 8 months ago
Hannah Aldous	said	Needs to be tested	7 years, 8 months ago
Chris Graham	said	Has some problems	7 years, 8 months ago
Lauren Richards	said	Has some problems	7 years, 8 months ago
Elliott Fletcher	said	Has some problems	7 years, 9 months ago
Vicky Hall	said	Has some problems	7 years, 9 months ago

History

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Elliott Fletcher 7 years, 8 months ago

Published this.

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Christian Lawson-Perfect 7 years, 8 months ago

Gave some feedback: Ready to use

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Christian Lawson-Perfect 7 years, 8 months ago

Saved a checkpoint:

I've split the last part, which asked you to find solutions after completing the square, into a separate question.

I moved the "Write the following expression in the form ..." text into the statement, because it's repeated for every part.

The advice now begins with a concise description of completing the square, which means the advice for the individual parts can just show how it's done.

The marking uses string restrictions, which just about does the job - you'd have to give a pretty artifical answer to get away with not completing the square - but it would be better if it did pattern-matching.

Restore this checkpoint

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Hannah Aldous 7 years, 8 months ago

Gave some feedback: Needs to be tested

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Chris Graham 7 years, 8 months ago

Gave some feedback: Has some problems

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Chris Graham commented 7 years, 8 months ago

You seem to have sorted the pattern matching and there is a nice gradient of difficulty now.

In the part prompts, replacing e.g. "the expression" with "the following expression" will be clearer to the student.

I would suggest that part (d) becomes its own question, and then you could split this into two parts: the first part completing the square and the second finding the solutions.

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Hannah Aldous 7 years, 8 months ago

Gave some feedback: Needs to be tested

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Lauren Richards commented 7 years, 8 months ago

Great question but some minor changes:

Remember to full stop the end of your maths in each section of the advice.

Your advice doesn't match the parts. The advice has writing for a)i), ii) and iii) but there are only questions i) and ii). There is also advice for b)i), ii) and iii) and yet only one question for b).
a)ii) shouldn't be in the advice - that does not correspond to any question. b)ii) and b)iii) shouldn't be there either.

In d), the answer for x1 and x2 were switched around in the advice. The advice is the incorrect version.

Check the working for c)i), I am not sure that it is correct but it does end up with the right answer. Where does the last term from the original expression go and where do the two numbers in the penultimate line appear from? Same for c)ii).

part d) - is the square of the integer in the bracket supposed to be in the form 2^2 rather than 4? Everywhere else in the question it has been in the form 4.

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Lauren Richards 7 years, 8 months ago

Gave some feedback: Has some problems

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Christian Lawson-Perfect 7 years, 8 months ago

Gave some feedback: Has some problems

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Christian Lawson-Perfect 7 years, 8 months ago

Saved a checkpoint:

Why are there three of each type of equation? Unless there's some kind of development, you should have one of each instance; the student can retry the question to get more examples.

The marking doesn't check that the student has put the expressions in the desired form. Ask Elliott about pattern-matching.

You could scale things so that only integers are used in the answer to part d. (I got $0.5 \pm \sqrt{50.25}$ )

Restore this checkpoint

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Hannah Aldous 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Elliott Fletcher 7 years, 9 months ago

Gave some feedback: Has some problems

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Elliott Fletcher commented 7 years, 9 months ago

Main Parts

a) good

b) i) and ii) good

b) iii) I think there is an error here, i got $\simplify{x^2+29x-7}$ , so i calculated the correct answer as (x+29/2)^2-869/4, however the displayed correct answer is (x+29/2)^2-71.

c) i) I think there is an error here, i got $\simplify{6x^2+36x+13}$ , so i got the answer as 6(x+3)^2-41. The advice section also shows this answer, but the displayed answer 6(x+3)^2-51.

c) ii) There is an error here too, i got $\simplify{3x^2+60x-10}$ , so i think the answer should be 3(x+10)^2-310, but the displayed answer is 3(x+10)^2-59.

d) good, but i would relabel the two possible answers here, as you have in the advice section as x1 and x2 (x1 being the smaller one).

Advice

I think it's fine if you don't show the working out for every question (because i know how long that would take), but i think you should say at the beginning of the advice for each part something like "We will just do one question from each part, as the other questions follow the same pattern" or "This can be similarly done for ii) and iii)".

d) I would write the numbers as fractions instead of decimals, so it's easier for the students to follow. Also, you say in the question to give the smallest answer first, so i think x1 should be the smaller number instead of x2.

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Hannah Aldous 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Vicky Hall 7 years, 9 months ago

Gave some feedback: Has some problems

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Vicky Hall commented 7 years, 9 months ago

In all parts of the question, put $\=$ after the expression and before the answer box.

In the advice, make sure a), b) etc are in heading 4 and that your solutions begin on the next line down. Anywhere that you reference an earlier part, write it as a) rather than a. You should write solutions for all subparts, although I agree that only subpart i) needs to go into detail.

In part d) of the advice, make sure that when you take the square root you find $\pm$ answer, which you can do by using \pm. The last lines of this section aren't rendering so have a look at that.

When I tried part c), the solution in the advice came up with a recurring decimal, although this only happens some of the time. I'd fix this so that Numbas produces a fraction instead, which should happen if you use the \simplify command.

Also ensure all of your equations in the advice display in the centre of the screen.

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Hannah Aldous 7 years, 9 months ago

Gave some feedback: Needs to be tested

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Hannah Aldous 7 years, 9 months ago

Created this.

Name	Status	Author	Last Modified
Completing the square	Ready to use	Hannah Aldous	20/11/2019 14:34
Complete the square and find solutions	Ready to use	Christian Lawson-Perfect	20/11/2019 14:34
Inbbavathie's copy of Complete the square and find solutions (SP1)	draft	Inbbavathie Ravi	15/07/2017 22:57
Inbbavathie's copy of Complete the square and find solutions	draft	Inbbavathie Ravi	24/07/2017 03:48
Katherine's copy of Complete the square and find solutions	draft	Katherine Tomlinson	02/08/2017 12:47
Complete the square and find solutions [L4 Randomised]	Needs to be tested	Matthew James Sykes	25/07/2018 10:17
Ann's copy of Complete the square and find solutions	draft	Ann Smith	09/10/2018 12:27
Completing the square	draft	Adrian Jannetta	16/08/2023 14:56
Complete the square and find solutions	draft	Johnny Yi	02/12/2020 01:07
Simon's copy of Completing the square	draft	Simon Thomas	07/03/2019 10:08
Simon's copy of Complete the square and find solutions	draft	Simon Thomas	07/03/2019 10:10
JD's copy of Completing the square	draft	JD Ichwan	21/05/2019 05:37
Completing the square and finding roots of a quadratic equation	draft	Maria Pickett	11/09/2019 13:51
Andrew's copy of Complete the square and find solutions	draft	Andrew McKinley	25/09/2019 13:38
AD3 Completing the square	Ready to use	Andrew Neate	09/04/2024 15:14

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Question statement

We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a square plus another term - this is called "completing the square".

This can be useful when it isn't obvious how to fully factorise a quadratic equation.

Rewrite the following expressions in the form

$(x+b)^2-c$ or

$a(x+b)^2-c$

Give any introductory information the student needs.

			Name	Type	Generated Value

evens1	integer	12
evens2	integer	20
evens3	integer	22
odds	integer	23
odds2	integer	13
odds3	integer	27

			Name	Type	Generated Value

all	integer	2
all2	integer	6
multiall	integer	20
big	integer	36
sml	integer	6
multiall2	integer	24

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(10..30 #2)

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: `integer`

→ Used by:

evens2
evens3

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Mathematical expression
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Part b) Gap-fill
1. Gap Mathematical expression
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Part c) Gap-fill
1. Gap Mathematical expression
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Part d) Gap-fill
1. Gap Mathematical expression
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Gap-fill

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

$x^2+ \var{evens1}x =$ Unnamed gap

Advice

Completing the square works by noticing that

$(x+a)^2 = x^2 + 2ax + a^2$

So when we see an expression of the form $x^2 + 2ax$ , we can rewrite it as $(x+a)^2-a^2$ .

a)

We have $x^2+ \var{evens1}x$ , so we can replace it with $(x+\var{evens1/2})^2-\var{evens1/2}^2 = (x+\var{evens1/2})^2 - \var{evens1^2/4}$ .

Check that this is equivalent to the original expression by expanding the brackets:

$\begin{align} (x+\var{evens1/2})^2 - \var{evens1^2/4} &= \simplify[basic]{ x^2 + 2*{evens1/2}*x + {evens1/2}^2 - {evens1^2/4} } \\ &= x^2 + \var{evens1}x \text{.} \end{align}$

b)

Replace $x^2 + \var{odds}x$ by $\simplify[basic]{(x+{odds}/2)^2-({odds}/2)^2}$ , to obtain

$\begin{align} x^2 + \var{odds}x &= \simplify[basic]{(x+{odds}/2)^2-({odds}/2)^2} \\[0.5em] &= \simplify[basic]{ (x+{odds}/2)^2 - {odds^2}/4} \text{.} \end{align}$

c)

Replace $x^2+\var{evens2}x$ with $(x+\var{evens2/2})^2 - \var{evens2/2}^2$ . Remember to keep the $\var{evens2-evens1}$ term on the end!

$\begin{align} \simplify[basic]{ x^2 + {evens2}x + {evens2-evens1}} &= \simplify[basic]{ (x+{evens2/2})^2 - {evens2/2}^2 + {evens2-evens1} } \\ &= \simplify[basic]{ (x+{evens2/2})^2 + {evens2-evens1 - evens2^2/4} } \end{align}$

d)

First, notice that $\simplify[basic]{ {all}x^2 + {multiall}x } = \simplify[basic]{ {all}*( x^2 + {multiall/all} x)}$ .

Then, we can replace $x^2 + \var{multiall/all}x$ with $(x+\var{multiall/all/2})^2 - \var{multiall/all/2}^2$ .

$\begin{align} \simplify[basic]{ {all}x^2 + {multiall}x + {odds3-evens2}} &= \simplify[basic]{ {all}*( x^2 + {multiall/all} x) + {odds3-evens2}} & \text{Extract the common factor of } \var{all} \\ &= \simplify[basic]{ {all}*( (x+{multiall/all/2})^2 - {multiall/all/2}^2) + {odds3-evens2} } & \text{Complete the square}\\ &= \simplify[basic]{ {all}*(x+{multiall/all/2})^2 - {all}*{(multiall/all/2)^2} + {odds3-evens2} } & \text{Expand the constant term}\\ &= \simplify[basic]{ {all}*(x+{multiall/all/2})^2 + {odds3-evens2 - (multiall/2)^2/all}} & \text{Collect constants} \end{align}$

Give a worked solution to the whole question.

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Checkpoint description

Elliott Fletcher 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Hannah Aldous 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Hannah Aldous 7 years, 8 months ago

Lauren Richards commented 7 years, 8 months ago

Lauren Richards 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Hannah Aldous 7 years, 9 months ago

Elliott Fletcher 7 years, 9 months ago

Elliott Fletcher commented 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Vicky Hall 7 years, 9 months ago

Vicky Hall commented 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Hannah Aldous 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

d)

Generated value: `integer`