Problems In Our "Times"

On a recent test, students were asked to find an expression the would be equivalent to 25 X 41 .

I'd be willing to bet that you read that as 25 "times" 41.

The students were given four choices as a correct answer:

A) (20 x 40) + (5 X 1)
B) (20 X 40) + (40 X 1)
C) (25 X 40) + ( 25 X 1)
D) ( 25 X 40) + (26 X 1)

Now, before you put on your number crunching hat, consider the rereading the "times" sign in the original problem. If you read the X as "groups of", you become aware that we are looking for some combination that is equivalent to 25 groups of 41 or 41 groups of 25.

Do you see a choice that makes 41 groups of 25? I do! Decompose the 41 into 40 and 1. Does that help? The answer is C, and I'll C U L8TR.

T-Cubed

May seem so simple, but......WOW!

Sometimes the simple things in life are better! This student used simple multiplication problems to help solve his/her division situations. I am so pleased to see students working "smarter not harder"! Obviously, he/she really understands that division depends on a real understanding of the relationship between multiplication and division. After all, in division you start with a known product and a known factor, and you simply try to figure out logically what the unknown factor is. 745 divided by 16 is the same as 16 X n = 745 .

So, why not start out with 16 X 1 = 16 ? It's easy and it leads directly to figuring out 16 X 10 = 160 , and 16 X 2 = 32 , and 16 X 20 = 340 , and 16 X 40 = 680... (being able to double and halve is sooooo important!). By the way, I'd probably throw in a 16 X 5 = 80 , as it is just half of 16 X 10!

At any rate, this work really does show great "smart" thinking! Even with the error in problem number two, this really is great mathematical thinking. Can you find and fix the error in problem number two? I bet the student that did this will see it quickly and be able to revise this problem easily!

T-Cubed

Multiplication and Division RELATED? :-}

The two scans above, starting with the top scan (uh, not the pug), represent some of the most efficient strategies around when it comes to multiplication and division. Perhaps, the best thing about these strategies is that most of the calculations can be carried out mentally and do not rely on the rote memorization of many math "facts". Truly, this is how most mathematicians think, smarter not harder!

Parents, students, fellow countrymen, and all other living beings, please challenge yourself to see if you can find the beauty in this kind of thinking. You may just find your inner math geek after all!

As a footnote, I wish that I could say that I "hired" a student to write this for me, but that would not be honest. I wrote this myself! Bad hand writers of the world unite! We shall rule.

PLEASE COMMENT!

T-Cubed

Revsions: Not easy to do, but worth it!

Remember, you can click on the images to show them at full size.

These revisions are very well done. They really show an understanding of what was done incorrectly, and the new work shows that the concepts in question are, indeed, now understood.

I especially liked the explanation in the "money" problem that sated that the student in question "Didn't add back the 3 (cents) that I rounded to." This shows a pretty complete understanding of the error that was made. I can also accept the "blonde moment" explanation for multiplying by 7 instead of 6, as that is just a lapse in concentration, and it's funny!

I hope this shows that really revising problems is a bunch of work, and that it is not something to rely on as a way to overcome a lack of participation in the class or at home. Rather, revisions are a way to refocus on a few concepts or errors that were not quite clear at test time.

What do you think?

Click on "comments" below. Type your comments. Type in the funky letters in the funky letter box. Click on anonymous. Click publish. wait for me to approve and post your comment :-}

Homework That ROCKS!

The homework shown above (you can click on each image and it will enlarge) really shows what we are looking for in terms of effort in and out of the classroom!

The first thing that I notice is that ALL of the work is completed on separate paper. Bigger space means more ability to think outside of the tiny "box" provided on the page itself, and this is not optional. All students should be completing work on separate paper.

The second thing that I look for is work that was revised, enhanced, or merely recorded during our class review sessions. These sessions take about 20 minutes, and they are only valuable if the strategies shown are recorded. Again, this is not an option. Even if you had the same answer, you can record alternative strategies.

The last thing that I noticed was the level of detail in the review work. This student was so focused on our discussion that she recorded her answer in the same "surfer voice" that I was using during our review ("The length of Mrs. Phillip's porch is totally like 16 feet!") .

This work is totally rad and awesome! WORD!

Mr. R.

AKA T-Cubed

I should have mentioned that leaving a reply is pretty easy. All you have to do is click on the "comments" tab, type your reply (please don't use your name if you are a student), and then click on "anonymous" before you hit "publish". The post will come to me, and if it is appropriate, I will post it to the page.

Beijing Olympic Subtraction Example: Cool!

Take a look at this work from our Beijing Olympic Subtraction problem, and see if you are on the right track.

The work shown is very well constructed and efficient.

Please let me know what you think:-}

Splitting Up Work Makes Life Easy!

I wish that I could make this uploaded work a bit more clear, but it still is great to emphasise my point.

This work comes from our last Mini-Quiz, and it is a great example of how organizing your work in a very simple way can lead to a great outcome. K did an amazingly simple but powerful thing when she drew lines to separate her problems. She made it easier for her brain to focus on the tasks individually. Separate spaces for separate thoughts. Simple but very important.

I also noticed something else really cool. K did not use ultra-small writing. While I like saving trees as much as the next teacher, I know that simply writing bigger and using more space often creates better results.

I also love love love love love love the expanded notation in the addition problem, the open number line in the subtraction problem, the generic rectangle in the multiplication problem, and the misconception corrected in the division problem. This work ROCKS!!!!!!