Oh, just flip it upside down and then multiply (“invert and multiply”).
But they are not sure which one to flip!!
Students learn this in late elementary, but when asked “Why?”,
they will say “Oh, it’s the rule”.
An example such as “How much is 12 divided by ½?” can illustrate the concept and learners can follow this example rather than a rule!!
How many half dollars are in $12?
How many half inches in 1 foot?
12 x 2/1 = 24!! ;)
Showing posts with label fractions. Show all posts
Showing posts with label fractions. Show all posts
Sunday, November 30, 2008
Wednesday, April 04, 2007
Sixth grade homework can sometimes look like this: 5/12 divided by 5/9. I wish fractions problems would use easier numbers so that the student can feel if their answer makes sense. I like to use 12 divided by 1/2 or perhaps the more difficult problem used by Liping Ma and/or Deborah Ball : 1 3/4 divided by 1/2.
Students can build intuition by asking themselves if the answer would be more than 1 or less than 1 by using 3/12 or 12/3 and seeing which of these examples their problem resembles.
These fraction problems are usually also related to attention to detail and reading. One of the other problems was you have 12 yards of fabric and will use 2/3 of a yard for each item -- how many items can you make? The student will often perform the multiplication of12 x 2/3 without asking themselves the question: Can I make more than 12 items or less? Once they have answered that questions, then they have a more clear sense of what to do. 1/2 vs. 2 can be used here.
1/2 x 1/2 is a great example as students often add instead of multiply and do not feel that multiplication should make their answer smaller!!
For division of fractions, why do we invert and multiply? If the student just memorizes the procedure, they will not know which procedure to use. It might (definite maybe) be ok if students were able to memorize these concepts and procedures but they get all jumbled!! Students will better comprehend the material and have the ability to more accurately retrieve it if they rely upon examples rather than memorizing procedures.
Students can build intuition by asking themselves if the answer would be more than 1 or less than 1 by using 3/12 or 12/3 and seeing which of these examples their problem resembles.
These fraction problems are usually also related to attention to detail and reading. One of the other problems was you have 12 yards of fabric and will use 2/3 of a yard for each item -- how many items can you make? The student will often perform the multiplication of12 x 2/3 without asking themselves the question: Can I make more than 12 items or less? Once they have answered that questions, then they have a more clear sense of what to do. 1/2 vs. 2 can be used here.
1/2 x 1/2 is a great example as students often add instead of multiply and do not feel that multiplication should make their answer smaller!!
For division of fractions, why do we invert and multiply? If the student just memorizes the procedure, they will not know which procedure to use. It might (definite maybe) be ok if students were able to memorize these concepts and procedures but they get all jumbled!! Students will better comprehend the material and have the ability to more accurately retrieve it if they rely upon examples rather than memorizing procedures.
Wednesday, March 28, 2007
Middle School Math is the Glue That Holds Everything Together
The middle school merry-go-round: Fractions, Decimlas, Percents
Which is bigger? .02 OR .059
As soon as the third decimal place is added, the difficulty goes up by a lot! When students compare and contrast these numbers (as opposed to learning them in isolation), they can find order and beauty in decimals.
The middle school merry-go-round: Fractions, Decimlas, Percents
Which is bigger? .02 OR .059
As soon as the third decimal place is added, the difficulty goes up by a lot! When students compare and contrast these numbers (as opposed to learning them in isolation), they can find order and beauty in decimals.
Friday, March 09, 2007
How Do You Say "1/4"?
1/4 to me is "one-quarter" and to most students, it seems to be "one-fourth".
A fifth-grader that I work with made me realize that I say "one-quarter" -- maybe from working on Wall Street. Parents and teachers should keep this in mind ;)
Borrowing with Fractions = Rewriting
Fractions are often seen as having their own rules -- which is true, however, subtracting one mixed number from another involves the concept of 'borrowing'. Because borrowing seems so mechanical, students may not analyze the process. If students see borrowing as 'rewriting', then the mixed fraction rewriting will make more sense. For example, 8 - 3 1/2 -- student will often make a mistake and get an answer of 5 1/2.
If they rewrite the 8 as
7+1
and then again as 7 + 2/2, then can then perform this operation with much greater ease and confidence.
1/4 to me is "one-quarter" and to most students, it seems to be "one-fourth".
A fifth-grader that I work with made me realize that I say "one-quarter" -- maybe from working on Wall Street. Parents and teachers should keep this in mind ;)
Borrowing with Fractions = Rewriting
Fractions are often seen as having their own rules -- which is true, however, subtracting one mixed number from another involves the concept of 'borrowing'. Because borrowing seems so mechanical, students may not analyze the process. If students see borrowing as 'rewriting', then the mixed fraction rewriting will make more sense. For example, 8 - 3 1/2 -- student will often make a mistake and get an answer of 5 1/2.
If they rewrite the 8 as
7+1
and then again as 7 + 2/2, then can then perform this operation with much greater ease and confidence.
Saturday, March 03, 2007
Non-Intuitive Answers
Multiplication of Fractions: Fractions get smaller and students are used to multiplication answers (products) being larger than whatever they started with. Mathematics Teaching in the Middle School has an excellent article update ("The Future of Fractions" was first published in 1978) by Zalman Usiskin from the University of Chicago -- the driving force behind the K-6 Everyday Math curriculum.
One of the features in the article was a NAEP (National Assessment of Educational Progress) items from 1978 for thirteen-year-olds.
Estimate the answer to: 12/13 + 7/8
You will not have time to solve the problem using paper and pencil (let alone a calculator!!)
The choices given -- and percents responding were
1 (7%)
2 (24%)
19 (28%)
21 (27%)
"I don't know" (14%)
Fractions have always been a sore spot for many and it is that foundational upper elementary/lower middle school Math that enables students to succeed in high school and beyond. It also contributes greatly to Math confidence or lack thereof.
Multiplication of Fractions: Fractions get smaller and students are used to multiplication answers (products) being larger than whatever they started with. Mathematics Teaching in the Middle School has an excellent article update ("The Future of Fractions" was first published in 1978) by Zalman Usiskin from the University of Chicago -- the driving force behind the K-6 Everyday Math curriculum.
One of the features in the article was a NAEP (National Assessment of Educational Progress) items from 1978 for thirteen-year-olds.
Estimate the answer to: 12/13 + 7/8
You will not have time to solve the problem using paper and pencil (let alone a calculator!!)
The choices given -- and percents responding were
1 (7%)
2 (24%)
19 (28%)
21 (27%)
"I don't know" (14%)
Fractions have always been a sore spot for many and it is that foundational upper elementary/lower middle school Math that enables students to succeed in high school and beyond. It also contributes greatly to Math confidence or lack thereof.
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