I want to bake blackberry cobbler. The recipe calls for a 9″ pie pan. All I have are rectangular ones.
How many of us remember the double-digit interest rates of the early 1980s? What do you notice in the story below? What are you wondering about? Leave a comment to tell us your thoughts!
One year, on December 31, Curtis, who doesn’t trust banks, put $1000 in a can and buried it in his back yard. He plans to continue adding $1000 to the can on the last day of each year until he’s ready to retire.
On the same day, Bill invested $1000 in a bank account that will pay 10% interest annually on the last day of the year. Unlike Curtis, he does not plan to continue investing more money each year.
The end of the summer is a great time for a ride around the neighborhood. What do you notice in the story below? What are you wondering about? Leave a comment to tell us your thoughts!
One fourth of the vehicles at Danielle’s Cycle Shop are tricycles. The rest are bicycles. Danielle counted a total of 45 wheels in her shop.
Raquel and Esperanza were asked to count Dr. Dolittle’s ostrichs and pushimi-pullyus. Raquel counted 67 heads, while Esperana counted 134 legs.
You don’t have to speak French in order to Notice and Wonder about this bit of architectural design! Leave a comment to tell us your thoughts (French is optional)!
A voussoir is a trapezoidal piece of stone often used to build arches.
We talk a lot about the problem-solving process here at the Math Forum and try to develop resources that will help teachers help their students get better at problem solving. We discuss how to encourage students to share their thinking (such as through Noticing and Wondering) and how to cultivate classrooms that value those thoughts and ideas as much as answers. But if we take a look at our own “problem solving” product, the Problems of the Week, we have to acknowledge that there isn’t so much support for process, starting with the “Compose Answer” button that appears at the bottom of each problem. Oops!
We have considered a number of possibilities, including an option (chosen by the teacher) to show just the scenario for a problem and then have fields in which students can submit their Noticings and Wonderings. That sort of thing would require some significant programming time, so while we are working on putting it in place (I’ll blog about it more before we get too far), we are first going to support the PoW process through some wording changes in the submission process. We’ve come up with some possibilities and wonder if anyone has alternative ideas.
On a problem page, it says, “Compose Answer”, which of course implies you have “an answer”. We’re thinking of changing that to “Submit Ideas”, which seems a bit more welcoming to submissions that might not actually contain an answer yet (or ever).
Once you get to the “submission” page, there are four spots we’re suggesting alternative wording:
What do you think? Would these sorts of changes convey “process” to your students? Do you have any other suggestions?
During the February 26th MoMath Masters Tournament, @MoMath1 tweeted, “No googling – how many sides on an enneagon?” We thought, “Hey! We know enneagons!” If you don’t, maybe this scenario from a problem we first used in 1998 will give you some hints (as well as some ideas for something you could do with one).
Extend the sides AB and ED of the regular enneagon ABCDEFGHI until they intersect.
In the Math Fundamentals problem Frog Farming, the goal is to make at least four different rectangular pens, each of which uses 36 meters of fence. Many students thought of this the same way I did, which was to consider half the necessary perimeter as the sum of two adjacent sides.
Rachel B, Seven Bridges Middle School
I know that the problem was finding perimeter. I know first you divide 36 by 2 and get 18. Then you find addends of 18 and they are the length and width. I added 12 plus 6 which equals 18 and 12 times 2 plus 6 times 2 equals 36.
Sarah G, Laurel School
First I decided come up with a length and width for a rectangle that would equal 18 because 18 is half of 36 and you have to multiply that number by two to get the perimeter. I decided on 2 and 16. I checked it by doing 16+16+2+2= 36. One could by length=16, width=2.
Rachel and Sarah and I were thinking about perimeter, in the context of this problem, like this:
Another way I thought of this was as 2(L + W). Hmm…..
Then I was mentoring a few students in this problem and noticed that they were thinking about the problem differently.
Ethan Z, Lorne Park Public School
I thought of a rectange which has 4 sides and 2 sides are equal and the other 2 sides are equal because Farmer Mead wants a rectangular pen that uses 36m of fencing. First I got the answer by thinking of 2 equal numbers that add up to less than 36. Then, the last 2 equal numbers are the difference of 36 to the first 2 equal numbers. That’s how I got all the numbers of the first question.
Emily G, Laurel School
I used 2 numbers, and doubled 1 number by two (ex. 6×2=12). 36-12 is 24. 24 is an even number that can be split into 2, which is 12 (ex. 24÷2=12.) 24+12 is 36!
Maybe because I had “seen” the problem differently, it took me a few minutes to figure out what these other kids were doing. Then I realized they were “seeing” the problem like this:
This seems more like 2L + 2W! These students tended to use more of a Guess and Check strategy to find solutions, whereas kids who used the first method were a little more systematic from the start. But it was fun to me to see these two different methods to what is a pretty simple idea. I like when simple things are done different ways.
I wonder how you “saw” the problem when I first described it. And how did your kids tend to see it?
Some Frog Farming links in case you are interested: