As you probably know if you have been following our discussions, many people, like me, advocate combining teaching math(problem solving, reasoning through rules and formulas, critical thinking, proof of solution, etc...) with literacy (writing problems, reasoning in writing, explaining solutions, defining terms, etc...).
To test the practice, solve the problems below and share your solutions with the rest of us. If you post the correct answers, I will send you the URL with many more similar problems with solutions included for your students as long as you give me your email with permission for me to contact you.
If you cannot solve the problems, email me and I'll send you the solution with ideas on how to implement matched reading and writing activities to expand the activity among students, as long as you give me your email with permission for me to contact you.
- Can you arrange 9 numerals - 1, 2, 3, 4, 5, 6, 7, 8 and 9 - (using each numeral just once) above and below a division line, to create a fraction equaling to 1/3 (one third)?
- What mathematical symbol can be placed between 5 and 9, to get a number greater than 5 and smaller than 9?
- How can you measure 6 liters of water using only 4 and 9-liter bowls?
4. Move one matchstick to get 4 identical triangles.
5. Move one matchstick to get the correct equation.
I would love to hear from you directly as well!
Thanks Leecy, for posting these fun problems! I want to encourage all my fellow "Mathletes" out there to try to work figure them out. Remember, to post your thoughts on how you solved them - because we can ALL learn from each other. This is a safe place for us (instructors, administrators, researchers, etc.) to post our thoughts even though we may be wrong. Thank you in advance for participating!
P.S. If you don't want to post your answers to the board directly - remember you can always just send them to me via my email. Let's talk math!
:) Absolutely, Brooke. Thanks for clarifying! Our CoPs are not only safe but also a lot of fun. Risk-taking is encouraged. I look forward to hearing different approaches from everyone on solving these. Feels like a party getting started! Leecy
By the way, the link to "comment on the discussion" at the bottom gets me to a page that doesn't have an option to do that. More "security," I guess.
And I can't see the actual problems any more in "add comment" zone (security?) but I do remember some of them :)
Looking at the "math symbol" to make something between five and nine by putting something between five and nine I say to myself "Puzzles are usually about hoping you assume that the word means what it means 99 44/100 percent of the time" and so "math symbol" could mean anything ever written in a math equation. So, a dot between 'em makes 5.9 . Fun language puzzle...
I've been mucking with Java programming so I started imagining a program to loop through all the possibilities for the first one ... and then applied some logic and trial and error ... and then went to Dr. Google to see if I'd missed anything; I hadn't gotten quite as far as this guybut here's how he limited the possibilities (from http://brainden.com/forum/index.php?/topic/113-fraction/&page=2 ) ... I'd realized they had to be 4 over 5 digits, and that the denominator had to be divisible by 3 so the digit sum had to be 3 or 9 (and the guy below talked about that as "modulo 0" so I think he had 'electronic assistance' since that's how you'd program to find divisibility). I hadn't done the rule out for the biggest digits, in each one, though.
Here's most of his post:
"If one tries to solve this problem without electronic assistance, one can reduce their trial and error attempts with the following information.
As the ratio of the numerator to the denominator is 1/3, our denominator needs to have the same number or one more digit than the denominator, and because there are nine digits this means the numerator will have four digits and denominator will have five. The max carry for 3n, where n is a digit 1 to 9, is 2, therefore the denominator must begin with a 1 or 2. And, by the same token, the smallest number the numerator can begin with is 4. (It can not begin with 3 as 3×3 equals 9, the largest decimal digit, and adding only 1 or 2 will bring the sum to only 10 or 11. There can neither be a 0 (zero) or two 1's in a zeroless pandigital common fraction of 9 digits, therefore the numerator must begin with a 4 or greater.)
5823/17469 and 5832/17496
Sue, I am not sure about your statement, "By the way, the link to "comment on the discussion" at the bottom gets me to a page that doesn't have an option to do that. More "security," I guess." Where do you see that broken link?
I love that you used wonderful language to define your reflection and solutions to problems. That is certainly the idea of having numeracy and language join hands!
I'll wait a little while to post the answers from the site. Send me your email, and I'll get it to you right away. Nice work!
When we get an email update, at the bottom of the email is a link with the words "comment on this discussion." It would be at the end of the email.
Actually, even if I"m all properly logged in, when I click that I go to the discussion but there is no place to comment there.
I figured out by doing the "just click everywhere" method of dealing with non-designed web pages that if I click on the title of an individual message, **then** I will have a "reply" option and luckily the first link I clicked that said "reply" actually meant it.
If a link just goes to a place to view a discussion, then frankly, that is what the link should say -- not that I should click to comment, and then let me try to figure out how to get to the page to *really* comment which *doesn't* have any indication that clicking on it would open a page where I could comment.
I'm sure that designing the page is part of somebody's job who actually has at least 1.67 other full-time jobs to do in a regular week, and yea, I probably post too often, So... I *really* don't expect things like having the site navigate me back to where I was, once I log in, like I see on other discussion sites. On the other hand, I don't think it's unreasonable (silly me, I know!) to expect that if a link says "click to comment" and I've gone through the onerous login maze, that ... Ill get to a page where I can comment.
Hi, Susan. I hear your justified frustration and will try to address it. Just to be sure I get it right, let me restate the issue.
1. You said, "When we get an email update, at the bottom of the email is a link with the words "comment on this discussion." It would be at the end of the email." So you are talking about the notification in your own email letting you know that there is something posted in one of your CoPs. Correct?
I'll ask other moderators what they suggest here and post the response.
2. Then you said "even if I"m all properly logged in, when I click that I go to the discussion but there is no place to comment there."
If you click on the Discussion Tab when you are logged into the CoP in which you want to participate, you will see a list of the discussions posted. Click on the discussion that you want to read. The initial post has a link that says, "Add a new comment." You can click that to respond to the initial prompt. If others have responded, you can click the "reply" link at the bottom of that person's comment, as I assumed you have done in posting in this discussion.
3. You said, " I figured out by doing the "just click everywhere" method of dealing with non-designed web pages that if I click on the title of an individual message, **then** I will have a "reply" option and luckily the first link I clicked that said "reply" actually meant it." I'm not sure about where you are at this point. I'm going to post your questions in our moderator group and see if we can get some help for you here.
Would you like to join me in a Web Conference, by phone (My Skype ID is leecyw), or through email? I'm glad to work this out with you. Thanks, Sue. Leecy
As I said, I have managed to figure out the exact path that I have to follow to get to where I can comment; clicking on the "discussions" tab also works.
I've "worked out" how to get here -- talking to me won't change the "click here to do this" links that take you on side trips.
It's sad that it's unreasonable to be expecting links to do what they say.
Susan, I don't think it is unreasonable at all to expect things to do what they say they are going to do!
I will assume that you contacted the LINCS support group through the Contact us" button at the top of the Community pages to receive technical support. If not, I encourage you to make use of that contact tool for two reasons: (1) to help solve issues and (2) to alert technicians to problems that need to be addressed.
In responding to the issue you reported, I received the following message:
"The login security features of the Community site require you to be logged in before participating in a discussion. If you are not logged in you can view a discussion but cannot comment. This is for the safety of the site and also other users of the site. The easiest way to participate in a discussion via an email notification message to go to the home page and log in, first. Once logged in, go to the notification email and select "Comment to discussion" or "Reply to comment" links within that email, you are then taken directly to the discussion or comment and can participate in the appropriate place."
Of course, it sounds like you resolved the issue and are able to manage things right now although they may not work ideally. So sorry you faced this frustration! I really appreciate that you are participating and helping to promote group interaction on important topics to all! Thanks. Leecy
Yes, whoever or whatever automated response can only say that I need to log in, and there's a way to do it.
That's not what frustrates me.
The links that say "click here to log in"
should, in my humble opinion, take me to a page to log in.
That's not security; that's web design to discourage participation. Should somebody have to send a tech email to ask why links aren't working to find out that well, they aren't supposed to work -- you are supposed to have to ask them the procedure to log in?
I realize that such a sophisticated, complicated task as removing links that don't work is probably 'way too much to expect -- heaven forfend the links be directed to the home page.
Hi, Susan. We have now gone beyond my perceived expertise, so I can only encourage you and others who are facing the frustration to those who can better handle the issue, if they can.
To One and All: For anyone facing technical issues within the LINCS site, I encourage you to connect to help through one of both links from the menu you find at the top of the page in your CoP: (1) Contact Us, which allows you to submit a ticket on any issue you are facing and (2) the Help link, which takes you to some pretty nifty videos on how to manage different tools in LINCS.
Susan, if you do find resolutions to your questions, please feel free to share them here, with our thanks. Leecy
The four and 9 liter thing doesn't say one of each so you can have a bunch of 'em and you can subtract four from nine or 8 from nine. . So if you fill the 9 liter bowl and then fill two four liter bowls from it, you'll have 1 liter left in it.
You could just pour that into a big enough jug and do that process six times.
Or you could do that twice and then add a four-liter jugful.
Or, you could Fill the nine liter jug an then pour out four, so you'd have five left ... and put that in another nine-liter jug and then do the process to get one liter and put those two together.
As I notice the variety of sentence structure you are using naturally to reflect on the problem, I'm delighted. Language is so intuitive. If you were to teach those sentence structures through grammar, students would be befuddled! Stay tuned in the next few days, and I'll post the site with its solutions. As with so much in life, there are different ways of approaching problems. Thanks, Sue. Leecy
If you move one of the matches from the 13 and place it over the equal sign, the equation becomes
XXII ≠ II VII
Yup, that's it, Miss Christine! I'll give others a little more time to jump in with descriptions of solutions to other items, and then I'll post the site. Thanks. Re the use of language, notice how easy it was to use a complex sentence naturally to explain a process. Complex sentences abound in math and science! :)
Another way I thought to do this was to push one of the I's in XXIII over to make it XXIV. Then you get XXIV divided by VII = II
This might be cheating, though, since the V would be a little lopsided...
Heh, heh. Now that's what I call creative, Rachel. We'll just call it a solution for a windy day? :) Leecy
Similarly, I only have a snarky solution to #4, which is to move the matchstick on the right of the first triangle so that it turns into a 4, and then you have 4 (triangle).
Based on the rest of these problems, that might be what we're going for, though.
Right on, Rachel! Maybe we should have taken a clue from the number of the problem: 4! The solution does push the envelope a bit, but it also pushes our creativity. Thanks! Leecy
That's an interesting idea, but note that the question requires one to "Move one matchstick to get the correct equation" (emphasis mine). What you produced was an inequality, not an equation per se.
That leads us back to literacy issues, reading comprehension, and being able to recognize semantic boundaries!
Hmmmmmm.... There you go, Robert. :)
Moderator, Reading and Writing, and
Diversity and Literacy Communities
The ones making the questions get to make the rules. Think about that...
I was not allowed to post images in replies to a discussion or comment, so I have posted the solutions to the problems and questions posed into a new forum. Please check there to find the link to the site, along with their proposed solutions.
Thanks to all of you who have contributed wonderful approaches to solving the questions raised. I also appreciate that you have shared language structures that are typical in explaining one's reasoning to solving problems. Bowing in your direction, Leecy
I would love to hear from you directly as well!