Let's have some fun and unpack this problem. I will be focusing our discussion on 12/17 and 12/18 on this problem. Please participate and add your thoughts to this problem.
Let's modify the Christmas carol The Twelve Days of Christmas for an algebraic reasoning task befitting the season:
How many gifts did your true love receive each day? If the song was titled “The Twenty-Five Days of Christmas,” how many gifts would your true love receive on the twenty-fifth day? How many total gifts did she or he receive in the first two days? The first three days? The first four days? How many gifts did she or he receive on all twelve days?
Be sure to explain your reasoning. Post pictures, instructions, numbers, whatever makes sense to you. If you solve it one way, solve it another way. Can you come up with a formula for the X days of Christmas?
Hello and season's greetings, all! May I contribute to Brooke's challenge about algebraic reasoning? I've drawn and pondered, and will try to share what I have so far. I'm at a point where I have more questions for you math pros out there; I'm hoping to learn more about equations, graphs, and contexts.
I am learning about quadratics, and am not sure if I'm on track with this, so any feedback is welcome. I'm wondering how or when to make an equation. I'm in the process of looking at quadratic stuff, but am hoping to really understand how those fancy equations work their way into stories and graphs.
I know the graphs are kind of blurry, but I hope you can see the idea.
YIKES! I just took a good hard look at my "computations" and table, and notice that I can't be answering the question accurately. I'm like, so as of day 3, there's 10 creatures. But, according to my table of values, I wasn't compounding adequately. Help! : )
Edited: Ok, so I just reviewed the others' posts, and am not up to that level of algebraic manipulation, but, I did end up with the same number as Josh after 25 days, (that is, after I realized I'd made an addition error at days 19 to 20) so maybe I'm not so far afield after all.
Another edit: I've tried using Misschristine's lovely table to try and add up the gifts. It's hard to present because I had to write so small. I will try to share it, anyway. I am still perplexed about my approach. On the table that I handwrote the days and numbers of gifts, I still wonder about its accuracy. Right now I'm more concerned with the idea than with the exact gift count.
I just created the table in a word document and then copied and pasted it and all my work from word. You could copy and paste a graph from excel.
I do math all the time but I didn't think "quadratic" when I first saw the problem. I do lots of tutoring in our "Math Literacy" course which has frequent exercises figuring out formulas in Excel, so I went to
1 + 1 + 2 + 1 +2 + 3 ...etc. Just a ton of repeated addition that could be set up in Excel.
So ... I didn't go to the more advanced thinking even though I'm perfectly capable of it. Is it better to think harder just because I can? I could get it into Excel and get your answer lickety-split.
Then I read Joshua's explanation that's an awesome example of how we can see patterns and things that *have to be true* for all examples of a pattern... otherwise known as algebra. MissChristine's was even clearer in making the transition...
First, I love that you jumped in with both feet and tried this problem. It makes me smile from ear to ear. Second, I love that you were able to read what other people posted and your understanding increased. Third, you have made my week because of your participation and engagement. It helps me to want to post another fun problem to unpack. Are you willing to participate again?
I appreciate the positive feedback, for sure! Yes, I'd like to explore more interesting math questions with the group. I did read the others' posts; algebraic breakdowns like a couple of them, are not so relatable to me which is why I try drawing and area tiles first, but I hope to get the hang of more algebraic displays eventually. I have to admit, I'm still pondering on the 12 days. Still exploring and questioning. Thanks for the inspiration!
Thanks for the fun problem. How many gifts? My personal thinking is that the answer is too many gifts on every day, especially since so many of the gifts are live animals!
Here's my Math thinking. You're counting down from a number and adding on every number you count. Let's try an example. For day 6 you get 6 + 5 + 4 + 3 + 2 + 1. 6 different numbers is messy, what if we pair the numbers at the extremes to even them out? We can rearrange and create three pairs that each add to seven. (6 + 1) + (5 + 2) + (4 + 3) = 7 + 7 + 7 = 3 * 7 = 21. Trying to generalize, we need to find the number of pairs and what they each add up to, so let's get out the algebra.
If we have x days, we'll have half that many pairs, x / 2. How much will the pairs add up to? Well, here's the sequence: x, x - 1, x - 2, x - 3, ....3, 2, 1 So our first pair would be (x) and (1) for x + 1. Since they all turned out to be 7s in our example, we can guess that they'll all be x +1, but we'll check another one to be sure. The next pair would be (x - 1) and (2) for x - 1 + 2 = x + 1. Yep it seems to work out. So our formula or (number of pairs) * (sum of each pair) becomes (x / 2) * (x + 1) which simplifies to (x^2 + x) / 2. To check our answer, we can see if our example of 6 days gives 21 gifts with this formula: (6^2 +6) / 2 = (36 + 6) / 2 = 42 / 2 = 21 YES! One more thing to check is that since our method relies on pairs, it doesn't seem intuitively like it would work for odd number where we'd be left with an extra number in the middle at the end that we can't pair up. So let's try an example: 5 + 4 + 3 + 2 + 1 = 15 and our formula says (5^2 + 5) / 2 = 15. It looks like it works for odd numbers too! Why? Well, my formula says I'll get x / 2 pairs. If I have 5 days, I should get 5 / 2 pairs or 2 1/2 pairs, I can pair day 5 and day 1 and day 4 and day 2 but end up with day 3 left over. So I have 2 pairs that adds to 6 and a solitary 3 left over. How many pairs is that? Well, it's 2 whole pairs of 6 and a leftover that's half that. It's 2 1/2 pairs. Because we're pairing from the outsides, the leftover number will always be the middle number which is half of what the pairs add up to.
So, if you had 25 days of Christmas you'd get (25^2 + 25) / 2 = 325 gifts
For a whole year of Christmas you'd get (365^2 +365) / 2 = 133,590 gifts
I really like your thinking and analysis - and it coul be broken down a little more so that students could follow the algebraic abstractions.
Welcome to the Math and Numeracy group, I don't think I have had the pleasure to read/see any of your other postings in this group! So first, welcome! Your post had me smiling from the moment it started. I love it when people are open to thinking and have FUN! I needed to have fun this week because I just finished up my final week of classes! Thank you for your participation and for your detailed explanation. Is this a problem you would consider using? What could be done to make it more interesting? I wonder what it would look like if it fractions...hmmmm.
To figure out how many of each type of gift use the formula:
G = d(13 –d)
Where G represents the total number of a gift and d represents the day the gift is given.
One way to find the total number of gifts given is explained below:
(Use the formula above for each gift given)
(1 x 12) + (2 x 11) + (3 x 10) + (4 x 9) + (5 x 8) + (6 x 7) + (7 x 6) + (8 x 5) + (9 x 4) + (10 x 3) + (11 x 2) + (12 x 1)
Use the commutative property of addition to get:
((1 x 12) + (12 x 1)) + ((2 x 11) + (11 x 2)) + ((3 x 10) + (10 x 3)) + ((4 x 9) + (9 x 4)) + ((5 x 8) + (8 x 5)) + ((6 x 7) + (7 x 6))
By combining like terms we get:
((2 x 12) + (2 x (11 x 2)) + (2 x (3 x 10)) + (2 x (4 x 9)) + (2 x (5 x 8)) + (2 x (7 x 6))
Using the distributive property we can simplify this to:
2 x ((12) + (2 x 11) + (3 x 10) + (4 x 9) + (5 x 8) + (6 x 7))
Factor out a 2 from each set of numbers:
2 x ((2 x 6) + (2 x 11) + (3 x 5 x 2) + (2 x 2 x 9) + (5 x 4 x 2) + (3 x 2 x 7))
Using the distributive property we can simplify this to:
2 x (2 x ((6) + (11) + (3 x 5) + (2 x 9) + (5 x 4) + (3 x 7))
Using the associative property, the expression becomes:
(2 x 2) x ((6) + (11) + (3 x 5) + (2 x 9) + (5 x 4) + (3 x 7))
Simplify the expression by doing the multiplication in the parentheses:
4 x (6 + 11 + 15 + 18 + 20 + 21)
Adding the numbers in parenthesis we get:
4 x 91
And finally 4 x 91 = 364
I could imagine starting with a calendar to show why the first thing gets 12, etc and a tape diagram to show that "13-d" idea.
I hadn't considered including all the mathematical properties!!! What a great piece you added to this discussion. Do you teach these properties to your adult learners? Do you find that it is important?
My initial perception of the story in the song was that every verse after the first just recapped gifts that had already been given instead of giving them again so that by the end the receiver had 12 partridges. (In 12 separate pear trees? Or could they all be in the same pear tree? I feel like that wasn't made clear.) In spite of the unwieldiness of this menagerie, this interpretation does seem mathematically interesting. Because I love my square inch tiles but didn't want to spend all night playing with them, I decided to model this as if all the gifts were tiles and there were only 6 days of gifts. You can see my exploration here: https://youtu.be/YJldGXHYq2o.
What do you notice? What do you wonder?
I love your tiles display, Sarah, but also the idea that the way someone interprets the story in the song would affect the gift count. I choose to complicate things by imagining the gifting repeats and adds up "literally." I also wanted to do a tile experiment but with all 12 days. This picture shows only a build-up on the simplest sense, that is, a count that doesn't reflect the mega-trend. I stopped building the mega-trend, not because it's lengthy, but because it doesn't progress as neatly as a 1 day through 6 gifts does: as I get to, say, 4 days of 9 gifts, or 5 days of 8 gifts, the sequence doesn't build in a nice, neat stair-step as in 1 day through 6 gifts. So I'm going to show you a little Gaussian pattern I observed with my first tile experiment.
Dear heroes of adult numeracy,
Some day I will learn how to post images on LINCS and not get frustrated that I have to relearn how to do it each time, but that day is not today.
Here's a link to a Google Drawing of my work: https://docs.google.com/drawings/d/1d7MPmrqMegJU3ntie0e6ALxKZR_u0J414i0OAwQF_TI/edit?usp=sharing
I started with a chart that is somewhat similar to what Sarah and Lori did, but with much less color and beauty. Then I added down my columns (12 ones, 11 twos, 10 threes...) I got 364 total presents.
When I started calculating the daily totals (1, 3, 6, 10,...) I recognized the Triangular Numbers, which is a sequence I feel like contains the secrets of the universe because of how often it comes up. From experience, I know the function n(n+1)/2 can be used to figure out the nth number in the sequence, which is the same thing Joshua came up with in a different form. In this case n is the day and the output is the total number of gifts, on that day. I've been noodling a little to figure out a function that will sum all of the previous days.
I figured it out for 12 days: (12n^2 - 120n + 440)/2
But I haven't generalized it yet for any number of days.
yours in productive struggle,
Very cool display of the days and gifts.
Thank you for jumping in and sharing your construct with us. You and all the others have made this week great for me, not only did I finish my classes this week but I got to interact with some of the greatest mathematical minds I know!
I was trying to think of a way to modify this problem to include perhaps the rational number system.
I keep viewing your video over and over, again and again. I love it because it models your thinking and also because I would love to learn how to make videos like this that are time lapsed. Your visual really helped me "see" the pattern of the "X" days of Christmas!
Thank you for posting,
Mark noticed the triangular numbers in the sums for each day. They are the sums of the consecutive whole numbers from one to some number. You can see the consecutive numbers in the rows of Mark's first illustration. Taking Mark's illustration and pushing the rows over so that it's aligned to the right instead of the left, I see consecutive numbers in columns as well. We know that the triangular numbers can be modeled with a quadratic function as Joshua and Mark have pointed out. What do we have here, though? The accumulated gifts on a given day are that day's triangular number plus all the triangular numbers less than it.... like a triangle of triangles. What kind of a function might model that? If a triangular number can be represented with a configuration of dots in a triangle, maybe we are looking for pyramidal numbers which can be represented by a configuration of objects (like tiles) in a pyramid with a triangular base. (See Mark's configuration on the left and my variation on it on the right.)
By the way, Brooke, my phone's camera has a time-lapse function built in. If yours doesn't, you can probably get an app that will do it. I just set up my phone on top of a cardboard box and put something on top of it to hold it in place. This is my "studio." The phone is not in the picture because I used it to take the picture, but the blue box on top is what I used to weigh down the phone so it wouldn't tip over. At the end I left the video running for a few minutes so the final product would be visible for a few seconds.
Hi Sarah and All,
I'm still investigating the growth of patterns, especially this one.. I notice you mentioned some quadratic-ness going on. And something triangular; does that imply cubic function? I know I'm in over my head, but I'm a good swimmer! LOL