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# Colin Foster's Mathematics Education Blog

## 28 April 2022

### Tangible contexts for mathematics

Do contexts help students to understand mathematics or do they just make it harder for them to untangle the mathematics from all the extraneous information? I think the answer is yes – both of these happen on different occasions. So, what is it that gives some contexts the potential to be powerfully illuminating?

I think the answer is not ‘relevance’ to a student’s personal interests. Relevance might be motivating, possibly, but it doesn’t necessarily make the context more illuminating of the mathematics. That way lies a ‘learning styles’ kind of fallacy, that every student needs a different context that is just right for them, and the magical right context will somehow make everything clear to them. I don’t think that’s right. And anyway, students often seem more switched on by contexts that take them out of their existing worlds (e.g., spaceships, dinosaurs, unicorns) than those which merely reference things they are already familiar/bored with. So I don't think matching personal interests is the most helpful approach. I think it's more likely that generally most students are helped by the same illuminating, well-chosen contexts, and not really so much by others.

#### Ratio and multiplicative/proportional reasoning

Let’s take ratio or proportional/multiplicative reasoning as an example. This is widely acknowledged to be a (or possibly ‘the’) central concept in lower secondary mathematics. And something that many students really have a weak grasp of. If you wanted a concrete context to help students make sense of this area, what would you pick? If you opened a textbook at the ‘ratio’ chapter, what contexts would you expect to find?

Of course, ratio can be applied to all sorts of contexts, and it is important to do this and let students see how ratio can be relevant and important in a wide range of areas. That is fine. But what I am thinking about here is contexts that are deliberately used to try to develop students’ understanding of what ratio is and how it operates.

The problem for me with, for example, money as context is that if the ratio of money spent by, say, Usha and Sam is 3:1, and the ratio of money spent by Dave and Priya is also 3:1, it is quite hard to capture in words (or in pretty much any other way) what specifically it is about Usha/Sam and Dave/Priya that is the same, given that these ratios are the same. The ‘same ratio’ is a highly abstract concept here. So, although I think that money might at some point be a worthwhile context for using ratio, I don’t think it’s helpful for understanding what ratio is. My test is that I need to be able to complete the sentence: “When the ratios are the same, the _____ is the same” with something highly tangible and familiar (not mathematical) going into the blank space. For this reason, I think that most discrete ratios (money, different coloured beads on a string, different kinds of animals on a farm, boys and girls in a class, etc.) are not so useful.

#### Tangible context: paint

Instead, I think the ratios of continuous quantities are much more useful to begin with, and, in particular, my go-tos are always drinks (Foster, 2007) and paint. The fact that most students probably never mix their own drinks, and even professional decorators rarely mix pots of paint together to make new colours (and when students mix their paint in art, this would be by eye) is irrelevant. The point of the context is not that it’s something students do every day, or even ever. The point is that it’s easy to imagine (what Realistic Mathematics Education calls 'realistic', and which means something closer to 'realisable').

The reason that I think these contexts are useful is that:

• “When the ratios of red paint to white paint, say, are the same, the paint is the same colour.” and
• “When the ratios of orange juice to lemonade, say, are the same, the drinks taste the same.”
And everyone knows what these things mean. This means that you can have a discussion about various hypothetical mixtures of red and white paint, or fizzy orange, and you can initially completely avoid the word ratio and any 'rules' about when 'ratios' are or aren’t equal. You can just ask: “Would they be the same colour?” or "Would they taste the same?", and everyone knows what you mean and can engage in the thinking that you want them to do.

With paint, I find that having the two colours as red and white is particularly useful, because you then have the word ‘pink’ available, in addition to talking about ‘redness’ and ‘darker/lighter’. This all helps the discussion to focus initially on the mathematical thinking, rather than terminology. Once students appreciate that 2:3 and 20:30 and 1:1.5 and 4:6 are all ‘the same colour’, then it is natural to try to capture this ‘sameness’, and we can use a word like ‘ratio’ to do so. But doing it the other way round, beginning by stating that 'We say that' 2:3 and 20:30 and 1:1.5 and 4:6 are all ‘the same ratio’ invites students to ask, “What do you mean?” And that puts the teacher in the position of having to do the justifying, whereas really you want the students to be doing this, based on something that they have already gained a sense of.

#### Tangible context: fizzy orange

For the same reason, making fizzy orange using orange juice and lemonade can be another really illuminating context (and you could possibly even do this one for real in the classroom, Foster, 2007). Lemonade is better than water, I think, not just because the mixture tastes better, but because then you can ask, “Which mixture will be fizzier?” as well as “Which mixture will be more orangey?” Really tangible contexts like these do a lot of the work for you. Every child knows that adding more orange juice won’t necessarily make the mixture taste more orangey, if you are also adding more lemonade.

I would often begin a discussion of this scenario by suggesting a few possible mixtures of orange juice and lemonade (as in the table below), and asking students which mixtures would taste the same, and which would taste different. For any ones that they think would taste different, I would ask them which would taste more orangey, and I find that that sometimes causes them to change their minds. You often get to a situation where they think one mixture would taste more organgey, but also more fizzy, and so that causes them to go back and think again.As the discussion progresses, further possible mixtures are usually suggested by the students, and I would add these to the list. The point is to avoid telling students whether they are right or wrong, but to draw on their common sense and life experience to let them figure it out. They know everything they need to know to do this. This then forms a really good basis for more formal teaching of ratio.

For example:

Teacher: Would any of these mixtures taste the same? Are there any you’re sure would taste different?
Student 1: D and E would taste different.
Teacher: Why do you say that?
Student 1: D would taste stronger than E because there’s less lemonade in it.
Teacher: But D and E have the same amount of orange, don’t they, so shouldn’t they be equally orangey?
Student 2: No, because the orange is spread out in more lemonade in E.
Teacher: Can someone else explain what S2 is saying?
...
Teacher: Would any of these mixtures taste the same as each other?
Student 3: A and B would taste the same.
Teacher: Why do you say that?
Student 3: Because they both have 1 more lemonade than orange.
Teacher: Are there any other mixtures with 1 more litre of lemonade than orange?
Student 4: Mixtures C and D.
Teacher: So, would mixtures A, B, C and D all taste the same?
Students: Yes.

It’s likely at this point that some student will raise some doubt, perhaps relating to C being ‘nearly fifty-fifty’. Multiplicative language or thinking tends to appear around this point, if it hasn't already, which can then develop into getting the students to order A, B, C, D and E by ‘orangeyness’.

If this doesn't happen, then the teacher can be more proactive:

Teacher: Suppose I took two containers of Mixture A. How many litres would there be in each?
Student 3: 5 litres.
Teacher: What would happen if I mixed them together?

Every student will appreciate that mixing identical mixtures will lead to twice as much mixture, but that it will taste exactly the same. So this gives us Mixture E. And students will have already agreed that Mixture E must be less orangey than Mixture D, so this provides the nudge for everyone to think more deeply. Mixtures A and D can't taste the same if mixtures A and E taste the same and mixtures D and E don't! The idea that mixing 'identically-tasting mixtures' (still avoiding the use of the word ‘ratio’) will lead to a new mixture with exactly the same taste is highly intuitive, and nobody will ever doubt this. And that kind of knowledge is all that is needed to develop all the necessary ideas of ratio through this kind of discussion.

#### Tangible context: chromatography

Finally, I think a really helpful science context is chromatography and retardation factor ($R_f$) values (Note 2). There could be potential for some cross-curricular practical work with chromatography paper and water-soluble marker pens. Different inks dotted along a pencil line at the bottom of a sheet of chromatography paper will move at different rates as the solvent soaks up the sheet (Figure 1). Each component will travel at a fixed fraction of the speed of the solvent, and the $R_f$ value of each is defined as

$$R_f = \frac {\text{distance travelled by the substance}} {\text{distance travelled by the solvent}}$$

 Figure 1. Calculating the $R_f$ for the red substance.

This seems to me like a perfect, dynamic scenario for understanding ratio, because molecules of a substance are highly obliging, and obey the rules perfectly (unlike, say, two runners in a race, running at different speeds, who need to negotiate bends and are likely to get tired at different rates). Here, when the ratio is the same, the height above the baseline on the chromatogram is the same (and the substance is likely to be the same). I would be keen to hear from anyone who has used this context as a way to explore ratio with students.

### Questions to reflect on

1. What examples of illuminating contexts do you use - for ratio, or for other topics? What is so good about them?

2. When do you feel that contexts do and do not work well? Why?

### Notes

1. For a free lesson plan based on the fizzy orange idea, see https://www.map.mathshell.org/lessons.php?collection=8&unit=6230

2. People's recent familiarity with Covid lateral flow tests may also make this easier to grasp.

### Reference

Foster, C. (2007, May 24). Make maths sparkle. SecEd, 12. https://doi.org/10.12968/sece.2007.5.902. Available at https://www.foster77.co.uk/Foster,%20SecEd,%20Make%20Maths%20Sparkle.pdf


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# Devil's Dozen!!!

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19 Mar 2012, 06:54
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The next set of tough and tricky DS questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers. Good luck!

1. Jules and Jim both invested certain amount of money in bond M for one year, which pays for 12% simple interest annually. If no other investment were made, then Jules initial investment in bond M was how many dollars more than Jim's investment in bond M.
(1) In one year Jules earned $24 more than Jim from bond M. (2) If the interest were 20% then in one year Jules would have earned$40 more than Jim from bond M.

Solution: devil-s-dozen-129312.html#p1063846

2. If n is a positive integer and p is a prime number, is p a factor of n!?
(1) p is a factor of (n+2)!-n!
(2) p is a factor of (n+2)!/n!

Solution: devil-s-dozen-129312.html#p1063847

3. If x and y are integers, is y an even integer?
(1) 4y^2+3x^2=x^4+y^4
(2) y=4-x^2

Solution: devil-s-dozen-129312.html#p1063848

4. Of the 58 patients of Vertigo Hospital, 45 have arachnophobia. How many of the patients have acrophobia?
(1) The number of patients of Vertigo Hospital who have both arachnophobia and acrophobia is the same as the number of patients who have neither arachnophobia nor acrophobia.
(2) 32 patients of Vertigo Hospital have arachnophobia but not acrophobia.

Solution: devil-s-dozen-129312.html#p1063863

5. If at least one astronaut do NOT listen to Bach at Solaris space station, then how many of 35 astronauts at Solaris space station listen to Bach?
(1) Of the astronauts who do NOT listen to Bach 56% are male.
(2) Of the astronauts who listen to Bach 70% are female.

Solution: devil-s-dozen-129312.html#p1063867

6. Is the perimeter of triangle with the sides a, b and c greater than 30?
(1) a-b=15.
(2) The area of the triangle is 50.

Solution: devil-s-dozen-129312.html#p1063871

7. Set A consists of k distinct numbers. If n numbers are selected from the set one-by-one, where n<=k, what is the probability that numbers will be selected in ascending order?
(1) Set A consists of 12 even consecutive integers.
(2) n=5.

Solution: devil-s-dozen-129312-20.html#p1063874

8. If p is a positive integer, what is the remainder when p^2 is divided by 12?
(1) p is greater than 3.
(2) p is a prime.

Solution: devil-s-dozen-129312-20.html#p1063884

9. The product of three distinct positive integers is equal to the square of the largest of the three numbers, what is the product of the two smaller numbers?
(1) The average (arithmetic mean) of the three numbers is 34/3.
(2) The largest number of the three distinct numbers is 24.

Solution: devil-s-dozen-129312-20.html#p1063886

10. There is at least one viper and at least one cobra in Pandora's box. How many cobras are there?
(1) There are total 99 snakes in Pandora's box.
(2) From any two snakes from Pandora's box at least one is a viper.

Solution: devil-s-dozen-129312-20.html#p1063888

11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is$45 enough to buy 27 muffins and 27 brownies?
(1) $15 is enough to buy 7 muffins and 11 brownies. (2)$15 is enough to buy 10 muffins and 8 brownies.

Solution: devil-s-dozen-129312-20.html#p1063892

12. If x>0 and xy=z, what is the value of yz?
(1) $$x^2*y=3$$.
(2) $$\sqrt{x*y^2}=3$$.

Solution: devil-s-dozen-129312-20.html#p1063894

13. Buster leaves the trailer at noon and walks towards the studio at a constant rate of B miles per hour. 20 minutes later, Charlie leaves the same studio and walks towards the same trailer at a constant rate of C miles per hour along the same route as Buster. Will Buster be closer to the trailer than to the studio when he passes Charlie?
(1) Charlie gets to the trailer in 55 minutes.
(2) Buster gets to the studio at the same time as Charlie gets to the trailer.

Solution: devil-s-dozen-129312-20.html#p1063897
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23 Mar 2012, 03:53
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10. There is at least one viper and at least one cobra in Pandora's box. How many cobras are there?

Quite tricky.

(1) There are total 99 snakes in Pandora's box. Clearly insufficient.

(2) From any two snakes from Pandora's box at least one is a viper. Since from ANY two snakes one is a viper then there can not be 2 (or more) cobras and since there is at least one cobra then there must be exactly one cobra in the box. Sufficient.

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23 Mar 2012, 03:55
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11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is$45 enough to buy 27 muffins and 27 brownies?

700+ question.

Given: $$11m+7b\leq{15}$$, where $$m$$ and $$b$$ are prices of one muffin and one brownie respectively.
Question: is $$27m+27b\leq{45}$$? --> $$9m+9b\leq{15}$$. Question basically asks whether we can substitute 2 muffins with 2 brownies.

Now if $$m>b$$ we can easily substitute 2 muffins with 2 brownies (since $$2m$$ will be more than $$2b$$). But if $$m<b$$ we won't know this for sure.

But consider the case when we are told that we can substitute 3 muffins with 3 brownies. In both cases ($$m>b$$ or $$m<b$$) it would mean that we can substitute 2 (so less than 3) muffins with 2 brownies, but again we won't be sure whether we can substitute 4 (so more than 3) muffins with 4 brownies.

(1) $15 is enough to buy 7 muffins and 11 brownies --> $$7m+11b\leq{15}$$: we can substitute 4 muffins with 4 brownies, so according to above we can surely substitute 2 muffins with 2 brownies. Sufficient. (1)$15 is enough to buy 10 muffins and 8 brownies --> $$10m+8b\leq{15}$$: we can substitute 1 muffin with 1 brownie, so according to above this is does not ensure that we can substitute 2 muffins with 2 brownies. Not sufficient.

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23 Mar 2012, 03:56
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12. If x>0 and xy=z, what is the value of yz?

(1) $$x^2*y=3$$. If $$x=1$$ then $$y=z=3$$ and $$yz=9$$ but if $$x=3$$ then $$y=\frac{1}{3}$$, $$z=1$$ and $$yz=\frac{1}{3}$$. Not sufficient.

(2) $$\sqrt{x*y^2}=3$$ --> $$x*y^2=9$$ --> $$(xy)*y=9$$ --> since $$xy=z$$ then: $$z*y=9$$. Sufficient.

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23 Mar 2012, 03:57
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13. Buster leaves the trailer at noon and walks towards the studio at a constant rate of B miles per hour. 20 minutes later, Charlie leaves the same studio and walks towards the same trailer at a constant rate of C miles per hour along the same route as Buster. Will Buster be closer to the trailer than to the studio when he passes Charlie?

(1) Charlie gets to the trailer in 55 minutes. No info about Buster. Not sufficient.

(2) Buster gets to the studio at the same time as Charlie gets to the trailer --> Charlie needed 20 minutes less than Buster to cover the same distance, which means that the rate of Charlie is higher than that of Buster. Since after they pass each other they need the same time to get to their respective destinations (they get at the same time to their respective destinations) then Buster had less distance to cover ahead (at lower rate) than he had already covered (which would be covered by Charlie at higher rate). Sufficient.

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23 Mar 2012, 03:58
Just posted the explanations. Kudos points given to everyone with correct solutions. Let me know if I missed someone.
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23 Mar 2012, 05:25
Bunuel wrote:
11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is$45 enough to buy 27 muffins and 27 brownies?

700+ question.

Given: $$11m+7b\leq{15}$$, where $$m$$ and $$b$$ are prices of one muffin and one brownie respectively.
Question: is $$27m+27b\leq{45}$$? --> $$9m+9b\leq{15}$$. Question basically asks whether we can substitute 2 muffins with 2 brownies.

Now if $$m>b$$ we can easily substitute 2 muffins with 2 brownies (since $$2m$$ will be more than $$2b$$). But if $$m<b$$ we won't know this for sure.

But consider the case when we are told that we can substitute 3 muffins with 3 brownies. In both cases ($$m>b$$ or $$m<b$$) it would mean that we can substitute 2 (so less than 3) muffins with 2 brownies, but again we won't be sure whether we can substitute 4 (so more than 3) muffins with 4 brownies.

(1) $15 is enough to buy 7 muffins and 11 brownies --> $$7m+11b\leq{15}$$: we can substitute 4 muffins with 4 brownies, so according to above we can surely substitute 2 muffins with 2 brownies. Sufficient. (1)$15 is enough to buy 10 muffins and 8 brownies --> $$10m+8b\leq{15}$$: we can substitute 1 muffin with 1 brownie, so according to above this is does not ensure that we can substitute 2 muffins with 2 brownies. Not sufficient.

Thanks for the solutions Bunuel. How to get this qn solved in this way. I mean, i tried solving this qn in the usual algebraic way, as the qn sounded more that way n i got D. i'm not getting why its wrong that way. can you please help in this. Thanks for explaining.
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23 Mar 2012, 05:28
priyavenugopal wrote:
Bunuel wrote:
11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is$45 enough to buy 27 muffins and 27 brownies?

700+ question.

Given: $$11m+7b\leq{15}$$, where $$m$$ and $$b$$ are prices of one muffin and one brownie respectively.
Question: is $$27m+27b\leq{45}$$? --> $$9m+9b\leq{15}$$. Question basically asks whether we can substitute 2 muffins with 2 brownies.

Now if $$m>b$$ we can easily substitute 2 muffins with 2 brownies (since $$2m$$ will be more than $$2b$$). But if $$m<b$$ we won't know this for sure.

But consider the case when we are told that we can substitute 3 muffins with 3 brownies. In both cases ($$m>b$$ or $$m<b$$) it would mean that we can substitute 2 (so less than 3) muffins with 2 brownies, but again we won't be sure whether we can substitute 4 (so more than 3) muffins with 4 brownies.

(1) $15 is enough to buy 7 muffins and 11 brownies --> $$7m+11b\leq{15}$$: we can substitute 4 muffins with 4 brownies, so according to above we can surely substitute 2 muffins with 2 brownies. Sufficient. (1)$15 is enough to buy 10 muffins and 8 brownies --> $$10m+8b\leq{15}$$: we can substitute 1 muffin with 1 brownie, so according to above this is does not ensure that we can substitute 2 muffins with 2 brownies. Not sufficient.

Thanks for the solutions Bunuel. How to get this qn solved in this way. I mean, i tried solving this qn in the usual algebraic way, as the qn sounded more that way n i got D. i'm not getting why its wrong that way. can you please help in this. Thanks for explaining.

Can you please show your algebraic solution to point out possible flaws in it?
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23 Mar 2012, 06:28
Bunuel wrote:
priyavenugopal wrote:
Bunuel wrote:
11. Alice has $15, which is enough to buy 11 muffins and 7 brownies, is$45 enough to buy 27 muffins and 27 brownies?

700+ question.

Given: $$11m+7b\leq{15}$$, where $$m$$ and $$b$$ are prices of one muffin and one brownie respectively.
Question: is $$27m+27b\leq{45}$$? --> $$9m+9b\leq{15}$$. Question basically asks whether we can substitute 2 muffins with 2 brownies.

Now if $$m>b$$ we can easily substitute 2 muffins with 2 brownies (since $$2m$$ will be more than $$2b$$). But if $$m<b$$ we won't know this for sure.

But consider the case when we are told that we can substitute 3 muffins with 3 brownies. In both cases ($$m>b$$ or $$m<b$$) it would mean that we can substitute 2 (so less than 3) muffins with 2 brownies, but again we won't be sure whether we can substitute 4 (so more than 3) muffins with 4 brownies.

(1) $15 is enough to buy 7 muffins and 11 brownies --> $$7m+11b\leq{15}$$: we can substitute 4 muffins with 4 brownies, so according to above we can surely substitute 2 muffins with 2 brownies. Sufficient. (1)$15 is enough to buy 10 muffins and 8 brownies --> $$10m+8b\leq{15}$$: we can substitute 1 muffin with 1 brownie, so according to above this is does not ensure that we can substitute 2 muffins with 2 brownies. Not sufficient.

Thanks for the solutions Bunuel. How to get this qn solved in this way. I mean, i tried solving this qn in the usual algebraic way, as the qn sounded more that way n i got D. i'm not getting why its wrong that way. can you please help in this. Thanks for explaining.

Can you please show your algebraic solution to point out possible flaws in it?

Sure. Thanks. I just tried it in the vry usual algebraic eqn solving way.

Given,
From qn stem - 11m + 7b = 15 (i took the 'enough' word as 'equal to'..as the 'enough' keyword can be rephrased as '<=', i thought that,considering the max limit '=' would do in such price lmt case..not sure, whthr thrs any trap thr..)
From (1) stmt - 7m + 11b = 15
From (2) stmt - 10m + 8b = 15
Solving qn stem n stmt 1 eqns, got m and b as 5/6 (didnt get an int val thou..not sure whether thrs a trap)
Solving qn stem n stmt 2 eqns, got m and b as 5/6

Having m and b values, we can find whthr 45$is enough to buy 27m + 27b or not. (i.e. applying m n b in eqn 27m + 27b = 45 => 2 * 27 * 5/6 = 45). So, arrived at D. Math Expert Joined: 02 Sep 2009 Posts: 38918 Followers: 7742 Kudos [?]: 106323 [0], given: 11621 Re: Devil's Dozen!!! [#permalink] ### Show Tags 23 Mar 2012, 06:36 priyavenugopal wrote: Sure. Thanks. I just tried it in the vry usual algebraic eqn solving way. Given, From qn stem - 11m + 7b = 15 (i took the 'enough' word as 'equal to'..as the 'enough' keyword can be rephrased as '<=', i thought that,considering the max limit '=' would do in such price lmt case..not sure, whthr thrs any trap thr..) From (1) stmt - 7m + 11b = 15 From (2) stmt - 10m + 8b = 15 Solving qn stem n stmt 1 eqns, got m and b as 5/6 (didnt get an int val thou..not sure whether thrs a trap) Solving qn stem n stmt 2 eqns, got m and b as 5/6 Having m and b values, we can find whthr 45$ is enough to buy 27m + 27b or not. (i.e. applying m n b in eqn 27m + 27b = 45 => 2 * 27 * 5/6 = 45). So, arrived at D.

Not even to analyze the rest of it, I must say that there is a huge difference between $$11m+7b\leq{15}$$ and $$11m+7={15}$$. You can not just write = sing instead of <= just because it's more convenient and "enough" should be translated as <= only.
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23 Mar 2012, 10:18
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Bunuel wrote:
2. If n is a positive integer and p is a prime number, is p a factor of n!?

(1) p is a factor of (n+2)!-n! --> if $$n=2$$ then $$(n+2)!-n!=22$$ and for $$p=2$$ then answer will be YES but for $$p=11$$ the answer will be NO. Not sufficient.

(2) p is a factor of (n+2)!/n! --> $$\frac{(n+2)!}{n!}=(n+1)(n+2)$$ --> if $$n=2$$ then $$(n+1)(n+2)=12$$ and for $$p=2$$ then answer will be YES but for $$p=3$$ the answer will be NO. Not sufficient.

(1)+(2) $$(n+2)!-n!=n!((n+1)(n+2)-1)$$. Now, $$(n+1)(n+2)-1$$ and $$(n+1)(n+2)$$ are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1. So, as from (2) $$p$$ is a factor of $$(n+1)(n+2)$$ then it can not be a factor of $$(n+1)(n+2)-1$$, thus in order $$p$$ to be a factor of $$n!*((n+1)(n+2)-1)$$, from (1), then it should be a factor of the first multiple of this expression: $$n!$$. Sufficient.

I didnot understand why 1 is not sufficient, can you elaborate on that ?

what i did is - I took n! common * coprime number

So if we have n! everytime the factors for n! and n!* coprime should always be same.. What am i missing?

Will Vote for A

Given
n = +ve Int
P = Prime

(A)

p is a factor of (n+2)!-n!

Let n = 4

= 6! - 4!
= 4![(6*5) - 1] = 4! (29)

therefore n! = 4! will have sufficient factors as 4! (29)

data sufficient

Let n = 17

= (19)! - 17!
= 17! [(19 * 18) - 1]

data sufficient

Let n = 11

= (13!) - 11!
= 11! [(13*12) - 1]

data sufficient

(B)

p is a factor of (n+2)!/n!

Let n = 2

= (4!) / 2! = 12 = 2*2*3 = data sufficient

Let n = 4

= (6!) / 4! = 6 * 5 = 2*3*5 = data not sufficient
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23 Mar 2012, 10:43
1.

Vote for D

find y-x = ?
(A)
Jules = y + 12%(y) = 1.12y
Jim = x + 12%(x) = 1.12x

1.12y = 1.12x + 24
1.12y - 1.12x = 24
1.12(y - x) = 24
(y-x) = 24/1.12

sufficient

(B)
Jules = y + 20%(y) = 1.20y
Jim = x + 20%(x) = 1.20x

1.20y = 1.20x + 40
1.20y - 1.20x = 40
1.20(y - x) = 40
sufficient

-------------------------------------------------

3.

will vote for A

Given

x & y = INT

(A)

4y^2 + 3x^2 = x^4 + y^4

y^2(4-y^2) = x^2(x^2-3)

y X y^2(4-y^2) x^2(x^2-3)
E e e(e-e) = e e(e-o) = e
E o e(e-e) = e o(o-o) = e
o o o(e-o) = o o(o-o) = e
O e o(e-o) = o e(e-o) = e

(B) y = 4 - x^2

y = (2-x)(2+x)

if x = odd then y = odd
y = (2-odd)(2+odd) = odd * odd = odd

if x = even then y = even
y = (2-even)(2+even) = even * even = even

not sufficient
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23 Mar 2012, 18:39
Bunuel wrote:
7. Set A consists of k distinct numbers. If n numbers are selected from the set one-by-one, where n<=k, what is the probability that numbers will be selected in ascending order?

(1) Set A consists of 12 even consecutive integers;
(2) n=5.

We should understand following two things:
1. The probability of selecting any n numbers from the set is the same. Why should any subset of n numbers have higher or lower probability of being selected than some other subset of n numbers? Probability doesn't favor any particular subset.

2. Now, consider that the subset selected is $$\{x_1, \ x_2, \ ..., \ x_n\}$$, where $$x_1<x_2<...<x_n$$. We can select this subset of numbers in $$n!$$ # of ways and out of these n! ways only one, namely $$\{x_1, \ x_2, \ ..., \ x_n\}$$ will be in ascending order. So 1 out of n!. $$P=\frac{1}{n!}$$.

Hence, according to the above the only thing we need to know to answer the question is the size of the subset (n) we are selecting from set A.

I am not sure if I understand correctly.

There are a total of k integers. So probability of selecting first integer 1/k, now probability of second integer out of k-1 integer is 1/(k-1) and so on... so selecting n integers would be 1/(k*(k-1)*(k-2)*...(k-(n-1)) so we need both k & n. Am I missing something here?
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23 Mar 2012, 18:44
Bunuel wrote:
10. There is at least one viper and at least one cobra in a Pandora's box. How many cobras are there?

Quite tricky.

(1) There are total 99 snakes in Pandora's box. Clearly insufficient.

(2) From any two snakes from the a Pandora's box at least one is a viper. Since from ANY two snakes one is a viper then there can not be 2 (or more) cobras and since there is at least one cobra then there must be exactly one cobra in the box. Sufficient.

How do we know that there are no other varieties of snakes other than cobra and viper. This is precisely why I chose E.
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24 Mar 2012, 03:16
kuttingchai wrote:
Bunuel wrote:
2. If n is a positive integer and p is a prime number, is p a factor of n!?

(1) p is a factor of (n+2)!-n! --> if $$n=2$$ then $$(n+2)!-n!=22$$ and for $$p=2$$ then answer will be YES but for $$p=11$$ the answer will be NO. Not sufficient.

(2) p is a factor of (n+2)!/n! --> $$\frac{(n+2)!}{n!}=(n+1)(n+2)$$ --> if $$n=2$$ then $$(n+1)(n+2)=12$$ and for $$p=2$$ then answer will be YES but for $$p=3$$ the answer will be NO. Not sufficient.

(1)+(2) $$(n+2)!-n!=n!((n+1)(n+2)-1)$$. Now, $$(n+1)(n+2)-1$$ and $$(n+1)(n+2)$$ are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1. So, as from (2) $$p$$ is a factor of $$(n+1)(n+2)$$ then it can not be a factor of $$(n+1)(n+2)-1$$, thus in order $$p$$ to be a factor of $$n!*((n+1)(n+2)-1)$$, from (1), then it should be a factor of the first multiple of this expression: $$n!$$. Sufficient.

I didnot understand why 1 is not sufficient, can you elaborate on that ?

There are two examples given which give two different answers to the question whether p is a factor of n!.

(1) p is a factor of (n+2)!-n!

If $$n=2$$ and $$p=2$$ (notice that in this case $$(n+2)!-n!=22$$ and $$p=2$$ is a factor of 22), then since $$p=2$$ is a factor of $$n!=2!$$ the answer to the question is YES;

If $$n=2$$ and $$p=11$$ (notice that in this case $$(n+2)!-n!=22$$ and $$p=11$$ is a factor of 22), then since $$p=11$$ is NOT a factor of $$n!=2!$$ the answer to the question is NO.

Two different answers, hence not sufficient.
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24 Mar 2012, 03:22
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mithun2vrs wrote:
Bunuel wrote:
7. Set A consists of k distinct numbers. If n numbers are selected from the set one-by-one, where n<=k, what is the probability that numbers will be selected in ascending order?

(1) Set A consists of 12 even consecutive integers;
(2) n=5.

We should understand following two things:
1. The probability of selecting any n numbers from the set is the same. Why should any subset of n numbers have higher or lower probability of being selected than some other subset of n numbers? Probability doesn't favor any particular subset.

2. Now, consider that the subset selected is $$\{x_1, \ x_2, \ ..., \ x_n\}$$, where $$x_1<x_2<...<x_n$$. We can select this subset of numbers in $$n!$$ # of ways and out of these n! ways only one, namely $$\{x_1, \ x_2, \ ..., \ x_n\}$$ will be in ascending order. So 1 out of n!. $$P=\frac{1}{n!}$$.

Hence, according to the above the only thing we need to know to answer the question is the size of the subset (n) we are selecting from set A.

I am not sure if I understand correctly.

There are a total of k integers. So probability of selecting first integer 1/k, now probability of second integer out of k-1 integer is 1/(k-1) and so on... so selecting n integers would be 1/(k*(k-1)*(k-2)*...(k-(n-1)) so we need both k & n. Am I missing something here?

We are not interested in the probability of selecting first, second, etc numbers from the set. We are interested in the probability that numbers selected will be in ascending order.

Now, any group of n numbers is equally likely to be selected and these n numbers can be selected in n! ways, out of which only 1 will be in ascending order so the probability is 1/n!.

Hope it's clear.
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24 Mar 2012, 03:35
1
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Expert's post
mithun2vrs wrote:
Bunuel wrote:
10. There is at least one viper and at least one cobra in a Pandora's box. How many cobras are there?

Quite tricky.

(1) There are total 99 snakes in Pandora's box. Clearly insufficient.

(2) From any two snakes from Pandora's box at least one is a viper. Since from ANY two snakes one is a viper then there can not be 2 (or more) cobras and since there is at least one cobra then there must be exactly one cobra in the box. Sufficient.

How do we know that there are no other varieties of snakes other than cobra and viper. This is precisely why I chose E.

Answer to the question is B, not E.

If there is some other snake, then group of {other, cobra} will be possible and the statement (2) will be violated (basically the possibility of other variety of snake is ruled out by the same logic as the possibility of second cobra).

Hope it's clear.
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24 Mar 2012, 06:43
1
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Bunuel wrote:
mithun2vrs wrote:
Bunuel wrote:
7. Set A consists of k distinct numbers. If n numbers are selected from the set one-by-one, where n<=k, what is the probability that numbers will be selected in ascending order?

(1) Set A consists of 12 even consecutive integers;
(2) n=5.

We should understand following two things:
1. The probability of selecting any n numbers from the set is the same. Why should any subset of n numbers have higher or lower probability of being selected than some other subset of n numbers? Probability doesn't favor any particular subset.

2. Now, consider that the subset selected is $$\{x_1, \ x_2, \ ..., \ x_n\}$$, where $$x_1<x_2<...<x_n$$. We can select this subset of numbers in $$n!$$ # of ways and out of these n! ways only one, namely $$\{x_1, \ x_2, \ ..., \ x_n\}$$ will be in ascending order. So 1 out of n!. $$P=\frac{1}{n!}$$.

Hence, according to the above the only thing we need to know to answer the question is the size of the subset (n) we are selecting from set A.

I am not sure if I understand correctly.

There are a total of k integers. So probability of selecting first integer 1/k, now probability of second integer out of k-1 integer is 1/(k-1) and so on... so selecting n integers would be 1/(k*(k-1)*(k-2)*...(k-(n-1)) so we need both k & n. Am I missing something here?

We are not interested in the probability of selecting first, second, etc numbers from the set. We are interested in the probability that numbers selected will be in ascending order.

Now, any group of n numbers is equally likely to be selected and these n numbers can be selected in n! ways, out of which only 1 will be in ascending order so the probability is 1/n!.

Hope it's clear.

I think I will still disagree, it is a classic case of choosing numbers after replacement or without replacement. You interpret as earlier option and I later. Your thots pls
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24 Mar 2012, 06:51
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mithun2vrs wrote:
I think I will still disagree, it is a classic case of choosing numbers after replacement or without replacement. You interpret as earlier option and I later. Your thots pls

I think you misinterpret the question. We are not talking about the replacement at all, we are talking about the probability of selecting some group in certain order. Just try to calculate the probability of choosing 5 numbers out of 12 distinct numbers in ascending order and compare it to the probability of choosing 5 numbers out of 24 distinct numbers in ascending order.
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24 Mar 2012, 08:41
Bunuel wrote:
mithun2vrs wrote:
I think I will still disagree, it is a classic case of choosing numbers after replacement or without replacement. You interpret as earlier option and I later. Your thots pls

I think you misinterpret the question. We are not talking about the replacement at all, we are talking about the probability of selecting some group in certain order. Just try to calculate the probability of choosing 5 numbers out of 12 distinct numbers in ascending order and compare it to the probability of choosing 5 numbers out of 24 distinct numbers in ascending order.

Sorry to keep on with the problem,

Probability is defined as No of fav outcomes / No of Total possible outcomes

in both cases No of favorable outcomes is 1 and in 1st case No of total possible outcomes is 12c5 and in the second case No of total possible outcomes is 24c5. Pls correct me where I am going wrong.
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24 Mar 2012, 14:52
1
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mithun2vrs wrote:
Bunuel wrote:
mithun2vrs wrote:
I think I will still disagree, it is a classic case of choosing numbers after replacement or without replacement. You interpret as earlier option and I later. Your thots pls

I think you misinterpret the question. We are not talking about the replacement at all, we are talking about the probability of selecting some group in certain order. Just try to calculate the probability of choosing 5 numbers out of 12 distinct numbers in ascending order and compare it to the probability of choosing 5 numbers out of 24 distinct numbers in ascending order.

Sorry to keep on with the problem,

Probability is defined as No of fav outcomes / No of Total possible outcomes

in both cases No of favorable outcomes is 1 and in 1st case No of total possible outcomes is 12c5 and in the second case No of total possible outcomes is 24c5. Pls correct me where I am going wrong.

Again: you count the probability of selecting 5 numbers out of 12 or 24, whereas the question asks about the probability of selecting some group of numbers in a certain ORDER. You just interpret the question in a wrong way.
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# Devil's Dozen!!!

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## Algebra 1: Common Core (15th Edition)

a) Find the slope of each line by writing their equation in slope-intercept form: $4x-3y=9$ $3y=4x-9$ $y=\frac{4}{3}x-3$ The slope of the graph of $y=\frac{4}{3}x-3$ is $\frac{4}{3}$ The slope of the graph of $y=\frac{3}{4}x+7$ is $\frac{3}{4}$ The slopes are not the same, so the lines cannot be parallel. Multiply the slopes: $\frac{4}{3}\times\frac{3}{4}=1$. The slopes are neither opposite reciprocals nor perpendicular. b) Find the slope of each line by writing their equation in slope-intercept form. $6y=-x+6$ $y=\frac{-1}{6}x+1$ The slope of the graph of $y=\frac{-1}{6}x+1$ is $\frac{-1}{6}$ The slope of the graph of $y=-\frac{1}{6}x+6$ is $\frac{-1}{6}$ The slopes are the same, so the lines will be parallel.


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# Virtual reality advertising with brand experiences: the effects of media devices, virtual representation of the self, and self-presence

Accepted 05 Oct 2020
Published online: 27 Oct 2020

Abstract

As virtual reality (VR) devices are becoming more advanced and affordable, VR content with 360-degree views is gaining more attention and popularity. Both YouTube and Facebook have been supporting the adoption of 360-degree videos since 2015, and more than one million 360-degree videos have been posted on Facebook (Ayrey and Wong 2017). In addition, New York Times adopted VR technology and started releasing VR stories as part of the newspaper’s daily news feeds, beginning with free distribution of one million Google Cardboards for VR viewing to its subscribers (Hanlon 2016). Moreover, the worldwide spending on VR content/apps and location-based VR was predicted to rise from $2.54 billion in 2019 to$3.77 billion in 2021 (Petrov 2019). These statistics show that using VR could be a real life experience for ordinary people in the near future, given that most people have their own smartphone through which they can easily download and use VR content and apps. Further, in the wake of COVID-19 pandemic, people may have to stay at home more, and therefore it is likely that VR content and advertisements are consumed in their lives (Taylor 2020).

## The effect of different media devices on VR ad experiences

A variety of media devices are available for users to have access to 360-degree VR content. For instance, VR content can be experienced via smartphones or computers with which users can view VR videos from various angles by moving around their smartphone or computer mouse. Recently, HMDs are considered a sophisticated VR device because they display a simulated world much like a real world by responding to the user’s physical head movements (Psotka 1995). They also enhance spatial knowledge thanks to kinesthetic and vestibular feedback (Ruddle and Péruch 2004).

Researchers have attempted to understand the effects of media devices, yet the findings are somewhat inconclusive. Some studies report that simple devices tend to negatively affect users’ immersive media experiences in the virtual environment compared to HMDs (MacQuarrie and Steed 2017; Yim et al. 2018). For example, Adachi, Song, and Cramer (2019) found that individuals who watched a 360-dgree tourism video with a HMD reported a more positive destination image including cognitive, affective, and overall evaluations of the promoted location, compared to those who watched a video with a computer. Individuals who used the HMD reported that they felt as if they became the traveler and were actually in the promoted location.

However, other studies reveal insignificant differences in users’ physiological responses (Wiederhold, Davis, and Wiederhold 1998) and task performance (Slater, Alberto, and Usoh 1995) across different media devices. In the literature, some evidence suggests that media devices may work differently on different aspects of VR experiences. For instance, Santos et al. (2009) compared different types of media devices including high-end VR systems (i.e. HMDs) and low-cost devices (i.e. traditional desktops) in terms of users’ virtual experiences and task performance levels. They uncovered that users favored overall VR experiences through HMDs, but their performance was better when using a desktop.

Similarly, several studies showed that 3D ads are not necessarily more effective than 2D ads. Yim et al. (2018) explored how stereoscopic 3D dimensionality affected the process by which viewers’ memory of brand names embedded in a soccer game was formed compared to the memory process in the traditional 2D display. They found that as viewers paid more attention to the sports game in stereoscopic 3D display, the viewers were less likely to remember the brands embedded in the stadium, while the opposite pattern was found in the 2D display. Further, Breves and Schramm (2019) analyzed the impact of stereoscopy on brand placement effectiveness. They revealed that the viewers of 2D movies were more likely to remember the brands placed in the movies and to critically reflect the persuasive intentions behind the placements, compared to those of 3D movies. In addition, Visinescu et al. (2015) compared 2D and 3D shopping websites in terms of cognitive absorption—a state of deep involvement with technology (Agarwal and Karahanna 2000)—, perceived ease of use, and intentions to buy online using the website. They found that shopping websites using 3D environments were associated with lower perceived ease of use and lower cognitive absorption. Overall, these findings indicate that 3D technologies or high levels of vividness require more cognitive resources.

In sum, the extant body of literature has explored the effectiveness of media devices on VR experiences. However, the findings are somewhat indecisive. Further, in the context of advertising, little is known about the effect of using different media devices to view VR ad. Thus, the current study sets forth the following set of research questions:

RQ1a-c: How does viewing VR ad through different devices (smartphone vs. computer vs. HMD) influence VR ad experiences including (a) liking the ad, (b) favorable attitude toward the advertised brand, and (c) intention to purchase the advertised product?

## Brand experiences through virtual representation of the self

The present study also aims to investigate how the concept of virtual representation of the self (VRS: Bailenson, Blascovich, and Guadagno 2008) could be applied in the context of VR ad. Bailenson, Blascovich, and Guadagno (2008) conducted an experimental study where participants interacted with virtual representations of themselves or of others in an immersive virtual environment. They found that participants maintained less distance between their physical selves and VRS than they did between their physical selves and virtual representations of others (VRO). This finding demonstrates that participants display more intimacy with VRS than with VRO. In a similar vein, Fox and Bailenson (2009) revealed that participants in the VRS-running condition showed significantly higher levels of exercise than those in the VRO-running condition and in no virtual representation condition. In an advertising context, Ahn and Bailenson (2011) uncovered that endorsing products using the customer’s own appearance is more effective for enhancing favorable brand attitude and stronger purchase intention than using VRO. These studies collectively imply that VRS has positive impacts on customers’ evaluation of and purchase intention toward products advertised via VR. The present study seeks to support these findings in the context of VR ad.

Virtual environments can be experienced as similar as real environments thanks to technologically synthesized sensory information (Blascovich et al. 2002). VR ad allows viewers to engage in content control and information adjustment based on their own preferences, by turning the device, dragging their finger, or moving the computer mouse to look around and to explore every angle within the content. The interactive process shifts the role of users from being passive to being active in a mediated environment as their decision and behavior in VR change the environment they view. Consequently, the level of control handed over to viewers offers content versatility, which enhances their feeling of connection and presence during VR experiences (Biocca 2006).

The ability to control the view in VR provides a strong connection with VRS. Unlike traditional advertising where a model is the center of the narrative, users are at the core of VR ad. In particular, structural features of VR support a strong feeling of identification with VRS, which can be similarly understood as a model in traditional advertising. That is, through VRS, VR ad provides customers with an opportunity to learn about the product and engage in experiential marketing.

Extant research drawn from social cognitive theory has documented the important relationship between customers and models. Social cognitive theory suggests that human beings have the ability to learn through observation (Bandura 1977, 2001), and behavioral modeling can be influenced by observers’ level of identification with models (Bandura 1977, 2001). Viewers should notice an adequate level of similarity with the models to believe that they can achieve the same anticipated results by engaging in the observed behavior. That is, to the extent customers perceive similar features (e.g. age, gender, race, ethnicity, etc.) to a model, they are more likely to be persuaded by the content and to show favorable responses to the ad (Andsager et al. 2006; Torres and Briggs 2007). For this reason, a significant number of studies have addressed the importance of tailoring ad content toward the target customers to enhance the feeling of identification with a model (e.g. Meyers-Levy 1988; Whittler 1991; Williams and Qualls 1989).

Persuasive effects of similarity have been well documented in the context of VR such that when VRS is perceived to be similar to users, the impact of persuasion is influential (Ahn and Bailenson 2011; Fox and Bailenson 2009; Fox, Bailenson, and Binney 2009). For example, individuals who viewed the VRS-featuring their own face in an exercise simulation reported more exercise than those who viewed the virtual self-featuring a stranger’s face (Fox and Bailenson 2009). In the same manner, the effectiveness of similarity has been reported in diverse contexts (e.g. Fox, Bailenson, and Binney 2009; Hershfield et al. 2011; Song et al. 2013). In all, empirical findings suggest that resemblance of VRS to a customer’s appearance would facilitate more favorable experiences in VR ad.

RQ2a-c: Between absence and presence of a non-personalized VRS, which will lead to more favorable VR advertising experiences, such as (a) liking the ad, (b) favorable attitude toward the advertised brand, and (c) intention to purchase the advertised product?

## The role of self-presence in the virtual context

VR ad naturally increases the feeling of being in the virtual environment, namely the feeling of presence, and consequently leads to positive psychological consequences. Tussyadiah et al. (2018) showed that, in a tourism context, the feeling of being in the virtual environment enhanced enjoyment of VR experiences. Further, the study revealed that the heightened feeling of being there resulted in stronger liking and preference in the destination. The study also found that positive attitude change led to a higher level of visiting intention. Also, Kerrebroeck, Brengman, and Willems (2017b) investigated the impact of VR in the context of transformational brand experience appeals. They found that VR ad generated higher perceptions of vividness and presence than a regular two-dimensional video, with vividness positively affecting attitude toward the ad, both directly and indirectly via presence. They also showed that vividness in turn elicited a positive effect on brand attitude which stimulated consumers’ purchase intention. Although these studies have offered some insights about the effects of VR ad, more studies are called for in order to pinpoint the impacts of VR ad on the feeling of presence and other psychological outcomes.

When experiencing VR from the first-person perspective, a user’s VRS may or may not be present in the virtual environment. Even when VRS is not present on screen, users may assume that their virtual self is present in the environment (Kilteni, Groten, and Slater 2012; Lee 2004) by observing and experiencing changes in the virtual environment corresponding to their own actions. For example, when users are walking around in the virtual environment, the view and environment change according to their actions. In this respect, self-presence is an important concept in understanding how one’s virtual self can be experienced in VR. Lee (2004) classified three types of presence based on the three domains of virtual experiences: physical, social, and self-presence. Although all three types of presence can be experienced in the context of VR ad, self-presence is of particular interst in the present study, given that VRS is closely related to the concept of self-presence.

Self-presence occurs when technology users do not notice the artificiality of the representation of the self in the virtual environment (Lee 2004). In other words, users may experience illusion that there is no divergence between their virtual self and their own physical self (Eastwick and Gardner 2009; Gee 2008; Lee 2004). Thus, self-presence is an inherent characteristic of a successful representation of the virtual self.

Two theoretical mechanisms may work to illuminate self-presence in virtual contexts (Behm-Morawitz 2013; Tamborini and Skalski 2006). The first mechanism refers to a connecting experience, which explains that users are mapping their actual self onto their virtual self to the extent that they feel present with the virtual self. The second mechanism concerns an evaluation process made by users in comparing and contrasting the virtual self with their physical self, especially when the virtual self adopts a dissimilar social role from that of the physical self. For example, in a virtual environment where a female user is assigned to a male virtual self, the first mechanism commences when the user starts to attribute the male virtual self to her physical body. In the second mechanism, the user may make an evaluation on the similarities and differences in terms of experiences and behaviors between her physical self (i.e. female) and virtual self (i.e. male) in her social environment. Users encounter a sense of self-presence at varying degrees while interacting with the virtual self (Lee 2004). Depending on the degree of self-presence experienced, the impact of virtual experiences may vary (Tamborini and Skalski 2006). When users put themselves in the role of a virtual self, they tend to let the mental presentation of their actual self be reshaped by the mental representation of their virtual self (Biocca 2006), which ultimately influences user experiences.

A significant body of research has suggested that self-presence contributes to favorable experiences in diverse virtual contexts (Jin and Park 2009; Song, Kim, and Lee 2014; Song, Peng, and Lee 2011). For example, Behm-Morawitz (2013) examined how self-presence experienced in a virtual environment affected users’ healthy behaviors in the physical world. The study found that users who experienced a stronger sense of self-presence tended to engage in more physical activities online and healthier eating offline than those who reported a lower level of self-presence.

In the understanding of the way technology use influences user experiences, self-presence, or presence from a broader perspective (see Lee 2004), has been identified as a significant mediator that facilitates effective virtual experiences. Supporting the argument, extant research has documented the mediating role of self-presence or other types of presence (e.g. physical and social presence) in diverse contexts (Jin 2011; Kim and Song 2016; Kim, Song, and Lee 2018; Kim and Timmerman 2018; Song, Kim, and Lee 2014; Song, Kim, and Park 2019). The present study examines the mediating role of self-presence between VRS and VR ad experiences.

It is assumed that seeing VRS, compared to not seeing, would lead users to feel stronger self-presence. However, it may not always be the case if a virtual self is not customized to reflect features of the user’s physical self and does not look similar to the user. As claimed earlier, observing a virtual self may not be effective when VRS is perceived as dissimilar to the user. In fact, some studies suggest that seeing VRS does not always lead to a strong feeling of self-presence (Hou et al. 2012; Jin 2010; Song, Peng, and Lee 2011). For example, Song, Peng, and Lee (2011) found that, when people see VRS during an exercise gameplay, those with high body image satisfaction experienced strong self-presence, but people with low body image satisfaction expressed weak self-presence. This finding implies that seeing VRS is not necessarily channeled into enhanced self-presence in some circumstances. In sum, the current study sets forth the following set of research questions.

RQ3a-c: How does self-presence mediate the association between VRS and VR ad experiences including (a) liking the ad, (b) favorable attitude toward the advertised brand, and (c) intention to purchase the advertised product?

## Method

### Participants

The experiment employed a 2 (VRS: presence vs. absence) × 3 (media device: smartphone vs. computer vs. HMD) using a between-subjects design. In total, 203 participants were recruited from a private university located in a metropolitan area of Seoul, South Korea. The sample included more females (n = 107; 52.7%) than males (n = 89; 43%). Seven individuals (3.4%) did not indicate their gender. Participants were randomly assigned to one of the six groups: VRS presence in smartphone (n = 36), VRS presence in computer (n = 35), VRS presence in HMD (n = 26), VRS absence in smartphone (n = 38), VRS absence in computer (n = 35), and VRS absence in HMD (n = 33). Some participants did not fully complete the survey questionnaire after the experiment, and those unfinished responses were not included in the final dataset. This resulted in unbalanced cell sizes.

A majority of participants (n = 183, 90.1%) reported that they had not seen the VR ad, which was utilized for this experiment, before study participation. Neither previous experience with the VR ad (yes/no), χ2(2) = .47, n.s., nor previous experience with the brand advertised in the VR ad (yes/no), χ2(2) = .49, n.s., was significantly different across the six conditions. Thus, group equivalence was ensured. Power analysis indicated that the power was .99 indicating that the hypothesis test is good at detecting a false null hypothesis.

### Materials

With respect to the two conditions of VRS, in the absence condition, the VR ad did not display any part of the VRS. The absence condition could be likened to experience a floating head. The scene was viewed from the first person perspective and the participant could control the view. However, the participant’s body was not seen in the ad. In the presence condition, the VR ad displayed part of the VRS on the screen. Viewers became part of the content such that they could see their virtual body from neck to toe within the VR ad. In addition, the part of the body in the ad was a pre-set feature (see Figure 1).

Figure 1. VR ad used in the VRS condition. The users could see their VRS’s hands and legs. Source: Dr. Hayeon Song.

### Apparatus

#### Smartphone

Samsung Galaxy S7 smartphones were used in the smartphone condition. Participants were instructed to watch the VR ad on YouTube in a full-screen mode.

#### Computer

A desktop computer installed with a VR media player named “GOM player” was used in the computer condition. Participants were instructed to watch the VR ad on this particular media player in a full-screen mode.

#### HMD

Commercial VR headsets produced by Samsung, called Samsung Gear VR, were used in the HMD condition. The VR ad was played on the Oculus mobile platform, which was made available when a smartphone and the VR headset were connected.

### Procedure

After approval from the Institutional Review Board (IRB), participants were invited to an experiment lab in the university. Upon arrival in the lab, participants were assigned to one of the six conditions, and research assistants provided a brief description of the study and procedure. Research assistants followed a script to ensure that each participant received the same instruction. Participants were instructed regarding how to use the assigned media device to watch the 360-degree view of the VR ad in their assigned condition. To help participants view the ad in various angles, they were instructed to move the smartphone itself (the smartphone condition), to drag with a mouse (the computer condition), or to turn around their head (the HMD condition). Participants were also instructed to stay seated while the ad was playing and asked to raise their hand if they had any questions or problems. After watching the assigned VR ad, they were asked to complete a questionnaire about their experiences.

## Results

### The effects of media devices and VRS

A series of ANCOVA tests were conducted to examine the effects of media devices (RQ1a-c) and the effects of presence/absence of VRS (RQ2a-c) on VR ad experiences. Because the virtual self in the VR ads was set as a male, participants’ gender was selected as a covariate to prevent any potential gender effects. The presence/absence of VRS was dummy coded (0 = presence of VRS, 1 = absence of VRS).

The first analysis was conducted on the outcome variable of liking the ad (RQ1a, RQ2a). The results showed no significant difference across the three media device conditions, F(2, 189) = .52, n.s., [smartphone (M = 5.28, SD = 1.55); computer (M = 4.92, SD = 1.50); HMD (M = 5.15, SD = 1.61)]. However, there was a significant main effect of VRS, F(1, 189) = 10.68, p < .001, ηp 2 = .053. Individuals in the VRS-absent condition (M = 5.41, SD = 1.44) reported stronger liking than those in the VRS-present condition (M = 4.81, SD = 1.60). No significant interaction effect was detected, F(2, 189) = .30, n.s. (see Figure 2).

Figure 2. The effect of media device and virtual self on liking the ad.

Another ANCOVA test was performed on attitude toward the advertised brand (RQ1b, RQ2b). Results demonstrated no significant difference across the media devices, F(2, 186) = .60, n.s., [smartphone (M = 5.15, SD = 1.18); computer (M = 4.92, SD = 1.14); HMD (M = 5.18, SD = 1.05)]. However, there was a significant main effect of VRS, F(1, 186) = 4.80, p < .05, ηp 2 = .025. Individuals in the VRS-absent condition (M = 5.23, SD = 1.08) reported a more positive attitude toward the brand than those in the VRS-present condition (M = 4.92, SD = 1.16). Additionally, no significant interaction effect was found, F(2, 186) = .94, n.s. (see Figure 3).

Figure 3. The effect of media device and virtual self on brand attitude.

The last ANCOVA test was conducted on intention to purchase the advertised product (RQ1c, RQ2c). No significant difference was found across media devices, F(2, 189) = 2.10, n.s., [smartphone (M = 4.37, SD = 1.74); computer (M = 3.93, SD = 1.71); HMD (M = 4.58, SD = 1.55)]. Also, there was no significant main effect of VRS, F(1, 189) = 1.16, n.s., between the VRS-absent condition (M = 4.39, SD = 1.71) and VRS-present condition (M = 4.16, SD = 1.67). There was no significant interaction effect, F(2, 189) = .13, n.s. (see Figure 4).

Figure 4. The effect of media device and virtual self on intention to purchase.

### The mediation effects of self-presence

RQ3a-c investigated how self-presence would mediate the association between VRS and VR ad experiences: liking the ad (RQ3a), attitude toward the advertised brand (RQ3b), and intention to purchase the advertised product (RQ3c). In order to answer RQ3a-c, a series of PROCESS (model #4) (Hayes 2013) were performed. This method uses a bootstrapping approach, and the procedure is based on 5,000 bootstrap sample. Results were interpreted based on the 95% confidence interval (CI). As mentioned earlier, participants’ gender was controlled for in the analyses.

As for liking the ad (RQ3a), results showed that self-presence mediated the association between absence of VRS and liking the ad (indirect effect = .36, Boot SE = .13; CI = [0.13, 0.63]). Absence of VRS, compared to presence of VRS, fostered stronger self-presence (a = .73), with self-presence being associated with more liking of the ad (b = .50).

With respect to attitude toward the advertised brand (RQ3b), results revealed that self-presence mediated the association between absence of VRS and attitude (indirect effect = .23, Boot SE = .09; CI = [0.08, 0.42]). Absence of VRS led to stronger self-presence (a = .76), with self-presence being associated with more favorable attitude (b = .30).

As for intention to purchase the advertised product (RQ3c), results demonstrated that self-presence mediated the association between absence of VRS and intention (indirect effect = .33, Boot SE = .12; CI = [0.12, 0.58]). Absence of VRS led to stronger self-presence (a = .73), with self-presence being associated with stronger intention (b = .45). In all, the findings indicate the mediating role of self-presence. Specifically, absence of VRS increased the feeling of self-presence, and it led to stronger levels of liking the ad, favorable attitude toward the advertised brand, and intention to purchase the advertised product.

## Discussion

The present study investigated the effects of media devices and VRS on VR ad experiences and the underlying mechanism in this association via a theoretical lens of self-presence. We discuss primary findings of the study, implications and contributions, and future research directions below.

### Primary findings

The study found that media devices for viewing VR ad did not matter in participants’ VR ad experiences. As the results for RQ1a-c indicate, viewing with a smartphone, computer, or HMD did not reveal significant differences on liking the ad, brand attitude, and purchase intention. Some may assume that VR headsets (i.e. HMD) compared to smartphones or computers would lead to more favorable experiences, as they are more advanced technology and often presumed to provide more immersive media experience. Yet, the current study’s findings demonstrate that such assumption is not true. In fact, although the outcome variables are different, this finding is in line with previous studies which showed that media devices did not create any significant differences on users’ physiological responses (Wiederhold, Davis, and Wiederhold 1998) and task performance (Slater, Alberto, and Usoh 1995). It is true that extant literature has been inconclusive about the effects of media devices as conflicting findings have been documented in diverse sets of studies (Fonseca and Kraus 2016; MacQuarrie and Steed 2017; Yim et al. 2018). In this respect, a contribution of the present study to the literature is that it offers more sophisticated evidence suggesting that viewing a 360-degree video itself may provide immersive experiences, yet the device to view the 360-degree video is not as important as it is commonly assumed.

It is notable that the effects of media devices were somewhat different across the outcome variables. Although the score differences between the devices were not significant as described above, it is possible to see a general pattern (see Figures 2–4). In all cases, the computer condition had the lowest score, but for the outcome variable of liking the ad, the score of the smartphone condition was higher than that of the HMD condition. In contrast, for the outcome variable of intention to purchase, the score of HMD condition was higher than that of the smartphone condition. In the case of the outcome variable of brand attitude, the magnitude of the two conditions was different depending on the existence of VRS. These findings show that there could be still some room for VR headsets to further increase intention to purchase along with the technology’s future development and utilization.

Another primary finding of this study is that the use of VRS may not be an effective strategy in some cases. As found in the results for RQ2a-b, absence of VRS induced more liking of the ad and more favorable attitude toward the advertised product compared to presence of VRS. Bandura (1977, 2001) claimed that observational learning could be influenced by the observer’s perceived level of identification with the model. Specifically, when people perceive the model to be similar to themselves, they tend to have effective learning experiences (Fox and Bailenson 2009). However, in the present study, VRS was not tailored for each individual, which might create psychological distance between the individual and VRS. In this scenario, the study’s finding is in line with previous studies which demonstrated that customers in a non-targeted group were difficult to persuade and they tended to show negative experiences such as distraction, irritation, or offense (Lipman 1991; Star 1989).

Most VR ads create engaging content and experiences, which help users feel integrated into the experience rather than passively watching the content or others’ experiences. In this sense, one may consider inviting users into VR ads by incorporating VRS in the ad content. However, the current study suggests that if VRS cannot be personalized for each user to look similar to the user, it would be better not to use it. Although technology allows to create personalized VRS, this task takes much time, effort, and cost. Therefore, using a standardized VR ad without VRS would be a cost-effective marketing strategy, especially when targeting mass audience. It should be noted, however, that the present study is not incongruent with previous studies’ finding that resemblance of VRS to a customer’s appearance would facilitate more favorable experiences in VR content (Ahn and Bailenson 2011; Fox, Bailenson, and Binney 2009; Hershfield et al. 2011; Song et al. 2013). Rather, the present study highlights that VRS could be effective only when VR ads are personalized to each customer.

Lastly, the present study found that self-presence had a significant mediating effect on VR ad experiences, suggesting that the reason for VRS’ effect on people’s VR ad experiences is related to the feeling of self-presence. In fact, the mediating role of self-presence has been documented in the extant research, which addresses the connection between VRS and virtual experiences (Song, Kim, and Lee 2014). What is unique in the present study’s findings is that absence of VRS fostered to stronger self-presence, which consequently enhanced positive VR ad experiences. Put another way, the finding indicates that presence of VRS interrupted the feeling of self-presence. The finding can be understood by the association between self-presence and embodiment. Self-presence is the feeling of one’s self being located in a virtual environment, and it does not require a body representation in the form of an avatar. Lee (2004) indicates that virtual self can be either physically manifested (whole or partial body) or psychologically assumed (without body). When the virtual self is psychologically assumed, “a virtual environment reacts to users as if they were in there (e.g. first-person perspective game, other people greeting you by name)” (Lee 2004, p. 40). In this respect, the present study’s finding provides an important implication that the association between the feeling of self-presence and having a physically manifested VRS may not be always positive, especially, if VRS was not tailored toward the user. It is in parallel with the findings of previous studies (Hou et al. 2012; Jin 2010; Song, Peng, and Lee 2011).

### Implications and contributions

The aforementioned findings have important implications. First, the finding that media devices do not matter in VR ad experiences suggests that ad companies may not need to invest in the advanced technology to promote their products in 360-degree VR ads. Although VR campaigns that provide customers with VR experiences with high quality HMDs may attract people’s attention, the present study’s findings show that VR experiences with a smartphone or computer would be as effective as VR experiences with a HMD, but at a lower cost.

The finding on VRS also implies that promoting in-store VR ad campaigns, as conducted by a shoes company, TOMS, could be a worthwhile strategy in order to maximize the positive effects of non-personalized VRS. That is, making an effort to be part of an in-store promotional event itself may create a stronger sense of involvement and engagement compared to watching an ad at home. Thus, although VRS might not be personalized to each individual, inviting people to a store for VR ad experiences would be an effective marketing strategy.

Additionally, a notable contribution of the present study is that it utilized various media devices enhancing ecological validity in conducting the experiment. There have been few studies thus far which employed available media devices to look into the impacts of VR ads. By providing ad practitioners with first-hand evidence regarding the effects of VR ads, the present study offers future directions of utilizing VR technology beyond untested assumptions in implementing VR ad campaigns.

### Limitations and directions for future research

There are a few limitations to be noted. First, the current study examined VR ad with a 360-degree feature that can be viewed with different devices including smartphone, computer, and HMD. However, it is unclear whether the study’s findings can be also applicable to VR ad with a highly advanced technology that allows interactive tailoring for each user. Another important point to note is that VRS tested in this experiment was an actual human model in the pre-recorded 360-degree video. It is close to passive watching as a user had no control over movements or behaviors of VRS. However, having an animated avatar as VRS may lead to different experiences than passively watching a pre-recorded human model, as the user can often control an animated avatar’s body movement. Thus, it should be noted that the findings of the present study might be limited to human VRS in a 360-degree video only, not to animated avatars in VR content created with computer graphic design.

Second, the study’s findings are limited to the first-person perspective only. Past studies indicate that experiences with the third-person perspective and the first-person perspective would provide different virtual experiences (Petkova, Khoshnevis, and Ehrsson 2011). Thus, VR ad using the third-person perspective may not show the same patterns of the findings identified in the present study.

Lastly, it is important to consider potential individual differences drawn from experiences with 360-degree videos. Although varying degrees of technology have been incorporated in people’s daily life, some may not be still familiar with 360-degree videos and they may not be exposed to VR ads quite often. Thus, in-store, in-person based marketing strategies can be a good way to provide customers with opportunities to view 360-degree videos and to develop familiarity with this new method of experiencing ads. Perhaps, in-store campaigns can further incorporate a smartphone option in addition to a HMD to allow customers to use their own smartphone to view VR ads. It implies that media competency should be considered when VR ads are used. Given that little is known in this area, the current study calls for more empirical studies.

## Conclusion

Despite these limitations, the present study contributes to the study of VR ads in relation to the effect of media devices and the theoretical concepts of VRS and self-presence. Unlike the common assumption that advanced technology, VR ad via HMD in the present study, would bring positive effects in promoting effectiveness of ads, the present study revealed that it might not be a universal proposition. More detailed customization, by and large, could play a more essential role in enhancing liking the ads, brand attitude, and intention to purchase than the advanced technology itself. These findings offer both theoretical and practical implications. In order to validate and elaborate the present study’s findings, more empirical studies are called for.

## Disclosure statement

No potential conflict of interest was reported by the authors.

### Notes on contributors

#### Hayeon Song

Hayeon Song (Ph.D., University of Southern California) is a professor in the Department of Interaction Science at Sungkyunkwan University in Seoul, South Korea. Her primary research interest is to investigate ways to use new media (e.g., artificial intelligence, robots, games, social media, mobile phones, etc.) as persuasive and educational vehicle for health promotion. Her broader research areas include media psychology, health communication, and human-computer interaction.

#### Jihyun Kim

Jihyun Kim (Ph.D., University of Wisconsin-Milwaukee) is an Associate Professor in the Nicholson School of Communication and Media at the University of Central Florida, USA. Her primary research is focused on the effects and implications of new media/communication technologies for meaningful outcomes (e.g., education, health). Her research also explores human-machine communication in diverse contexts.

#### Thao P.H. Nguyen

Thao P. H. Nguyen (M.S., Gachon University) is a Ph.D. student in the Human Resource Studies program at Cornell University. Her major interests focus on discovering the impact of technology in the workplace on HR practices and employee experiences. She is particularly interested in virtual teams and leadership, especially how geographical dispersion and reliance on technological communication shape team dynamics and performance, and what virtual leaders should do to lead effectively.

#### Kwan Min Lee

Kwan Min Lee (Ph.D., Stanford University) is the inaugural Korea Foundation Professor in Contemporary Korean Society and New Media, and the Director of UX (User Experience) Lab at the Wee Kim Wee School of Communication and Information at Nanyang Technological University (NTU). His research interests are UX (User Experience) innovation, social and psychological effects of ICT (Information and Communication Technologies), and human-machine interaction. He is an associate editor of Computers in Human Behaviors.

#### Namkee Park

Namkee Park (Ph.D., University of Southern California) is a professor in the Department of Communication at Yonsei University, South Korea. His research interests include social psychological implications of communication technologies including human-computer interaction (HCI) and computer-mediated communication (CMC).

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Appendix

VR ad with absence of VRS:

VR ad with presence of VRS:


text:
# What is the difference between continuity and differentiability of functions of 2 or more variables, intuitively?

I was reading my calculus book. For continuity, I was first told to imagine a surface to be a mountain; where there is a large rock, for ex, whose side rise vertically upward, or where there is a sudden change in altitude, there is dis-continuity. Another case of dis-continuity is where there is a hole, or chasm, like the one in the graph of $z=\frac{-1}{(x-y)^2}$. I was confused. I thought those places of sudden changes in altitude are places of non-differentiability (I was taught that for functions of 1 variable). The book then says in symbols, continuity is (say there is 2 points $P(x_0, y_0, z_0)$ and $Q(x, y, z)$), $lim_{x\rightarrow x_0, y\rightarrow y_0}f(x,y)=f(x_0,y_0)$. But I thought this limit expression doesn't say anything about the things like sudden changes in alitutide.

Then I got to part about differentiability. Here, the book explains that a surface may indeed be continuous but may contain crags, which have sharp points or edges. This is where I suddenly realize the striking, yet confusing resemblance between what continuity is and what differentiability is. Does sharp points just places of sudden change in altitude (z) or other variables?

If a function $f: A \subset \Bbb R^n \to \Bbb R$, where $A\neq \varnothing$ is an open set, is discontinuous at a point, it's automatically non-differentiable at the point. Continuity is a necessary condition for differentiability; one first restricts their attention to continuity, then worries about differentiability.
For $n=1$, such a function, intuitively, is continuous at a point if it gets arbitrarily close to its value at this point when $x$ gets near the point from either directions; in other words, it doesn't have a "jump" at the point neither from the right nor from the left. For $n=2$, the concept is nearly the same, however we are concerned here with infinitely many directions through which the point might be approached, so for example, a function might be continuous at $(0,0)$ if one restricts the attention to the paths $y = 0$ and $x=0$, however discontinuous otherwise. Consider, for instance, $f(x,y) = \frac{xy}{x^2 + y^2}$ if $(x,y) \neq (0,0)$ and $0$ otherwise. If one computes $\lim_{(x,y) \to (0,0)} f(x,y)$ over the paths $x=0$ and $y=0$, one gets $0$. However, over the path $y = x$, the limit is $1/2$. To sum up, for $n=2$, a function is continuous at a point if it gets arbitrarily close to its value at the point as $(x,y)$ approaches the point from any possible direction.
Roughly speaking, points of non-differentiability of a continuous function, for $n=1$, are points where the graph has a "corner", and a "corner" is a point where the function "suddenly" changes its "trajectory", but still doesn't "jump" (imagine the trajectory of a diagonally thrown ball and the point at which it hits the ground vs the trajectory of a skateboarder over a skate-park). For $n=2$, it is rather a set of points over which the surface doesn't "flow smoothly" or has an "edge" (imagine a "smooth flow" as that of the surface of a skate-park or a saddle). This is because, intuitively, the derivative of $f$ should give information about the "rate of change" of the function. So for instance, consider two planes which intersect diagonally at a line $\ell$. The upper part corresponds to a function $f$ which is continuous but non-differentiable over the set of points $(x,y)$ for which $f(x,y) \in \ell$ (a concrete example would be $f(x,y) = x$ for $x\ge 0$ and $-x$ for $x < 0$ - the function is continuous on $\Bbb R^2$ and non-differentiable over the $y$-axis).


text:
## Cryptology ePrint Archive: Report 2018/331

Estimate all the {LWE, NTRU} schemes!

Martin R. Albrecht and Benjamin R. Curtis and Amit Deo and Alex Davidson and Rachel Player and Eamonn W. Postlethwaite and Fernando Virdia and Thomas Wunderer

Abstract: We consider all LWE- and NTRU-based encryption, key encapsulation, and digital signature schemes proposed for standardisation as part of the Post-Quantum Cryptography process run by the US National Institute of Standards and Technology (NIST). In particular, we investigate the impact that different estimates for the asymptotic runtime of (block-wise) lattice reduction have on the predicted security of these schemes. Relying on the LWE estimator'' of Albrecht et al., we estimate the cost of running primal and dual lattice attacks against every LWE-based scheme, using every cost model proposed as part of a submission. Furthermore, we estimate the security of the proposed NTRU-based schemes against the primal attack under all cost models for lattice reduction.

Category / Keywords: post-quantum cryptography, public-key cryptography, cryptanalysis, learning with errors, NTRU, NIST

Original Publication (with minor differences): Conference on Security and Cryptography for Networks (SCN'18)

Date: received 9 Apr 2018, last revised 3 Dec 2018

Contact author: benjamin curtis 2015 at rhul ac uk

Available format(s): PDF | BibTeX Citation


text:
# pfgplots plot in degrees

Everywhere on the web, and in this forum, tells me pfgplot plots trig functions in degrees, which is what I'm after.

Here is the code for my graph:

\begin{figure}[H]
\centering
\begin{tikzpicture}
\begin{axis}[
domain=1:20000,
xmode=log,
xlabel=Frequency (Hz),
ylabel=Phase shift (\degree),
]
\end{axis}
\end{tikzpicture}
\caption{Transfer function with multiple values of $RC$}
\end{figure}

And the result, which as you can see, is in radians (otherwise it would stretch to minus 90).

How can I plot in degrees?

• Your arguments for atan make little since. – John Kormylo Jan 24 '15 at 16:42
• @JohnKormylo they make a lot of sense when you're analyzing the phase of a passive filter... – ACarter Jan 24 '15 at 16:54
• Right, the 2\pi is to convert frequency from Hz to redians. So the ratio of imaginary and real amplitudes is actually S for this case (pure inductor). – John Kormylo Jan 24 '15 at 21:09
• Nah 2\pi is angular frequency to Hz :P – ACarter Jan 25 '15 at 12:12

When you use \addplot [...] function {...} it uses gnuplot on the terminal, which uses radians for trigonometric functions, as noted in Section 4.3.5 of the pgfplots manual.

To use the pgf computation engine, remove the word function (or replace it with expression):

\documentclass{standalone}
\usepackage{pgfplots,siunitx}
\pgfplotsset{compat=1.11}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
domain=1:20000,
xmode=log,
xlabel=Frequency (\si{\hertz}),
ylabel=Phase shift (\si{\degree}),
]