This is a problem in the MCAS Grade 8 Practice Book.
The current topic is geometric patterns and the question has 3 answers. I got the first 2 but the answer to the 3rd baffles me.
There are 3 pictures in the pattern (triangles made up of hexagons, some shaded, some white).
c. If the pattern continues, find a rule that will tell you how many white hexagons will be in any future drawings. Explain how you found the rule.
This is the table I made showing the pattern. The left numbers are the drawing numbers. The right numbers are the white hexagons in the drawing.
1 - 9
2 - 12
3 - 15
I can see that the number of hexagons is increasing by multiples of 3 in each drawing but I am drawing a blank as to how to make this into a rule.
The answer is:
The rule is 3n+6. Based on the table, you can see that the numbers of white hexagons are increasing multiples of 3. If you add 2 to the drawing number and multiply by 3, (n+2)x3, you will get the pattern. The expression (n+2)x3 simplifies to 3n+6.
I understand that n represents the drawing number and that the x3 represents the multiples of 3.
When I replace the n with the next drawing number (4), it comes out correct at 18 white hexagons.
The part I’m lost on is the 2. Where the hell did the 2 come from? Why am I adding 2 to the drawing number?
Can someone please explain this to me in idiot terms?
Without seeing the picture you’re describing, I don’t know if there’s anything more to it than simply “Because that’s what works.”
If you’re looking for a function or formula that takes the numbers 1, 2, 3 and gives you the numbers 9, 12, 15, once you notice that 9, 12, 15 are all successive multiples of 3, you know that the formula must look like 3*(n+?). (Each time n goes up by 1, n+? will go up by 1, and 3*(n+?) will go up by 3.) A little simple trial and error reveals that the ? must be 2.
Carmady: Thanks. I feel better about myself now. I made it through 19 sections in this damn book before being completely hung up on that stupid question. The annoying thing so far is that everything else in the book has been representative of either a previous section, or the actual instructions for the current section. The part about making a rule is completely alien to me. I have no clue how to make a rule. Rule is such a definite word. I’m not comfortable enough with math to be making rules. Thudlow Boink (hehehehehehe boink…I love that word…boink…
Here are the pictures. represents a white hexagon and :eek: represents a shaded hexagon.
:)
:):eek:
:):):)
:)
:):eek:
:):eek::eek:
:):):):)
:)
:):eek:
:):eek::eek:
:):eek::eek::eek:
:):):):):)
You are overthinking this. The “drawing numbers” have nothing to do with it. It is supposed to be a geometrical problem, not an arithmetical one. Clearly the rule is to extend the horizontal and vertical sides of the triangle by one hexagon each time, then count the hexagons along the edges. Thus, the next picture is
:)
:):eek:
:):eek::eek:
:):eek::eek::eek:
:):eek::eek::eek::eek:
:):):):):):)
So the answer is 18.
Of course, the arithmetical rule is “add 3 each time,” but the rationale is in the geometry.
I think the point was to make a rule so that I could find out immediately how many white hexagons would be in, say, drawing 3,416 - without having to sit there adding all the little hexagons. I’m assuming that’s why the answer was the equation, instead of just adding 3 for each drawing. I just didn’t understand how the people who made the book decided to use a 2. Bastards.
I find it amusing that section 20 in this book is variables. I have been using variables pretty much all through the other 19 sections. Why did they suddenly feel the need to teach how to find out the value of the variables? Obviously the student in question would know how to do that if they made it to section 20 of the book.
And now, just for fun, here’s how to get that rule into the equation you came up with. The smallest possible triangle in that configuration would have 6 :)s (n=0). It increases by 3 each time. That may be intuitively enough for you to get to 3n+6, but I’ll do it the formal way.
Since the pattern increases by the same amount each time, we know it’s a linear equation. The form y = mx + b will work fine. We surmised above that, when x=0, y=6. Putting that into the equation, we get y = mx + b
6 = m(0) + b
6 = 0 + b b = 6
Now we pick one of the other examples, say, x=1,y=9. plug that in and we get y = mx + b
9 = m(1) + 6
9 - 6 = m m = 3
So plugging in all the numbers, we get y = 3x + 6, which, if you plug in n for x, is the same thing as your 3n + 6
As far as I can tell, your +2 just came in because you intuitively factored a 3 out of the expression.
There was no intuition involved. The +2 wasn’t mine. It was in the answer key.
I have to say, I’m very jealous of you guys who can look at this stuff and actually make sense of it without getting a headache.
I was never a math person. I was too busy dissecting things.
1 gives you 33, 2 gives you 43, 3 gives you 5*3, and so on. Or, in general, x gives you (x+2)*3 (since 3 = 1+2, and 4 = 2+2, and 5 = 3+2, and so on). That’s where that +2 came from.
Suppose I’m in a tall building, and I climb two flights of stairs every minute. In four minutes, what floor will I be on? You can’t answer that, unless you know what floor I’m on to start with. If I’m on the 20th floor now, in four minutes I’ll be on the 28th. If I’m on the 34th floor now, in four minutes I’ll be on the 42nd. The formula would be:
future floor = (2 * minutes) + starting floor
Lots of math problems seem to turn out that way; an initial state, plus an incremental change.
So here’s where the 2 comes from; your initial state is 6 (if there was a drawing number 0, it would have 6 white hexagons) and you factored out 3 as a common term.
It comes in handy having a grandpa who was a math teacher. Too bad he may be passing soon.
I just want to say I’m constantly impressed by your ability to simplify things down to where they can be understood. I’ve noticed it a lot in astrophysical threads, but apparently you do math, too.
You all actually helped quite a bit. Quite a few of the things said in this thread have actually popped up as I’ve continued working on the book. Your mention of linear equations went right over my head. Just 20 minutes ago, I learned about linear equations and I immediately recognized the equation you used.
I’m really enjoying this book. It’s nice to see what I remember from school. It’s also reassuring to see what I can pick back up again.
Now I just need to retain this until I take my assessment test on June 4.
That’s the smallest triangle you can get by using your hexagons or our smileys. And this:
:)
:):) Is the second smallest triangle.
And this:
:)
:):eek:
:):):) Is third.
But they don’t want to call it drawing 3. They’re calling it drawing 1. So in your equation, you have to add 2, so that what they’ve labeled “drawing 1” ends up being “the third smallest triangle”.
In other words, n represents the drawing number, while (n+2) represents the ordinal (1st, 2nd, 3rd, 3702nd, whatever) that the drawing would be, compared to the smallest triangle.
Also, here’s why there are an extra three per drawing. Let’s say the bottom row is:
:):):)
In the next iteration, the row replacates itself and adds one to the end. But some of those :)s have to stop being counted, and turn into :eek:s:
:):eek::eek:
:):):):)
So in the end, each replicates itself but changes to an :eek: and stops being counted. So they cancel out the :)s directly beneath them…except three of them. Which ones?
:):eek::eek:
:):):):)
^ _____^^
Those. The end one is new, and the parent never changed to an :eek:. In reality, it doesn’t matter which smilies you claim are “cancelled out”. But the point is, there are three that can’t be canceled by a :)-turned-:eek:
Yeah. they “simplified” by taking out the common factor.
y=3x+6=3(x+2) both expressions are equivalent.
I suppose in their anal way they will mark you wrong for not simplifying by taking out the common factor, unless “simplify” is one of the main lessons in the chapter too - in which case you must.
True. A lot of math books say you must simplify everything. Most teachers, though, are more lenient. I think both forms are simple enough, and that 3x + 6 is actually better, as one can more clearly see the process used to get there. Plus, it’s easier to graph as a linear equation.
represents a white hexagon and :eek: represents a shaded hexagon.
