Last year, I developed a new strategy for teaching rounding that seemed to help more students remember the steps. Here is an example:
Round 32,759 to the hundreds' place:
Underline the digit to be rounded: 32,759
7's boss, 5, is looking over his shoulder to see if he is working. Since 5 is a big boss (5-9 are big bosses), he has the authority to give raises. He gives 7 a raise, so he is now worth 8. All numbers in front (32) get copied, and all numbers behind (59), become zeros.
copy +1 turn to zeros
Little bosses cannot give raises, so if 5 had been a 4, 7 would have no raise. 32,759
copy +0 turn to zeros
When subtracting with a series of zeros, I have the students ask themselves:
1) Can I take 7 from zero?
2) Can I borrow from a neighbor with nothing?
3) Can I borrow from two neighbors with nothing?
4) Can I borrow from 3 neighbors with 200?
Then I have them mark through the entire 200, borrow one from it leaving 199, then make my 0 in the one's place a ten. Subtract:
1 9 9 10
This is so much less complicated than borrow from the thousand's place, make it 1, now the hundred's place becomes ten, borrow from it, making it nine and making the ten's place......
When I published my new teacher web-page. I had the option of adding a blog. Blogging is something I hadn't done before, but I thought it might be useful as a tool to help parents help their children learn.
Today's blog is about tricks to help your child master multiplication facts. Below are the tricks I teach. Some may seem obvious to us (and even to most children), but
each year, I encounter children who are surprised to learn some of
2's If you know your addition doubles facts, you can quickly do 2's.
(Ex: 2x7= 7+7 (or 2 sevens added together) =14)
3's The distributive property teaches us that if we know 2x7 and 1x7, then we know 3x7. (2x7 =14
so use the doubles fact and add 7 more.
4's Again, the distributive property says we can double (x2) and double again to learn our 4's.
(EX: 7x4= 7x2=(14) x2 again =28)
5's While most kids think that counting by 5's on their fingers is the quickest way to do 5's, thinking of the numbers on the clock is the fastest way to learn them. Most kids know that the 3 represents 15, 6 is 30, and 9 is 45 on the clock, and it doesn't take long to fill in the others with a minimum of practice.
Like doubling the 2's to learn 4's, kids can double 3's
to find 6's. (Ex: 6x7 is the same as 3x7 (21) doubled, which yields 42.)
Another trick I teach for 6's is to take the 5 fact and add one more set of the number. (Ex: 6x7 is 5x7 (35) +1 more set of 7 to make 6 groups of 7 or 42)
7's By the time kids get to 7's, they know all of the
7 facts except 7x7. 8x7, and 9x7. Since they willl
soon have a trick for both 8's and 9's, the only one they will struggle with is 7x7. At this point, I would work with my child to memorize all the mutliplicaiton doubles (called square
8's Like 4, 8 is a power of 2, so we take the 4 rule and go one more step; double, double agian, and double once more (or double the 4 fact if they have mastered it). Like 7, kids should already have all the 8's up to 8x8 and 9x9.
9's There are so many tricks with 9's! 3 of my favorites are the finger trick (hold up all 10 fingers. Lower the finger corresponding to the number you are multiplying by your 9. All the fingers to the left represent the ten's place, the ones
on the right represent the one's place (the digits in all multiples of 9 add up to 9 or a mutliple of nine).
The second works on the same premise as the first: subtract 1 from 9's partner to get the ten's place, the remainder from 9
is the one's digit (Ex: 7x9: take one from 7 to get 6 in the ten's place. You need three more to make 9, so 3 is your one's digit.)
The third is using the distributive property (Ten 7's are 70, so 9 sevens would be 70-7, or 63.)
Of course, these are only crutches to help your child until he/she masters the facts by rote. The homework I have assigned and the games/computer activities/videos we do in class are all designed to master facts through repetition of auditory and visual stimuli. Encourage your child to do MathMagic at least 5 minutes each night and you will see a tremendous difference in their multiplicaiton skills. Don't forget to celebrate with them by showing joy with a high-five or a pat on the back when they master a new set. I have yet to meet the 4th grader who isn't elated to pass each level as though they mastered a new level on Mario Brothers!
For converting in the U.S. Customary system, we make a table of the 2 units being converted: (How many ounces in 5 pounds?)
ounces | pounds
so my answer is 80 ounces in five pounds
When teaching students to convert units in the metric system, we need to ask four questions:
1) is my unit changing to a larger unit or smaller unit?
(for this, they need to know the relative size of the units measured)
2) is my number going to get larger or smaller?
(if my unit gets larger, my number gets smaller, and vice versa)
* I remind the students you can get more ants in a room than elephants.
3) should my decimal move right or left?
(upright - or if my number is increasing, move the decimal right, if
smaller, move the decimal left)
4) how many places will the decimal move?
(for most questions in 4th grade, then answer will be 3 places UNLESS, I
am changing centimeters to meters (2 places) or to millimeters (1 place)
* I have the students count the number of zeros in the number of one
unit in the other (EX: 100 cm = 1 meter, so 2 zeros mean to move the
decimal 2 places).
The greatest difficulties I see for students in understanding fractions is in dealing with equivalent fractions, especially putting them into lowest terms. To make a larger equivalent fraction, I simply encourage the students to make a table and choose any column from the table:
2 4 6 8 10
3 6 9 12 15
Often, putting fractions in lowest terms is a matter of recognizing that the fraction is already in lowest terms. Four characteristics of a fraction in lowest terms:
1) The numerator is 1
2) The numerator and denominator are consecutive numbers.
3) The denominator is prime.
4) The numerator is prime BUT will not go evenly into the denominator. (3/8 is in lowest terms, but 3/6 is not)
Characteristics of fractions that are NOT in lowest terms:
1) Both numerator and denominator are even.
2) The numerator is exactly half the denominator.
This week we will begin multiplication by 2-digit numbers. I use a mnemonic device (or memory trick) that I call Tic-Tac-Toe to remind the students of the steps involved. Many students come back to school after the first night's homework and say, "My mom and dad didn't know anything about tic-tac-toe, so they just showed me the way they learned it." Tic-tac-toe is not a strategy for multiplying, but just a way of remembering the steps involved. Basically, after multiplying by the one's place (as we have done for the past week), students will TIC-TAC (cross out the one's place in the bottom number so they know that they are finished multiplying by it), and then TOE (place a zero in the lower level of the addition stage of the equation as a place-holder to remember to start in the ten's place). Since the game of tic-tac-toe is made of X and O, I use it to help students remember. See below:
X 15 Tic-Tac is placing an X over 5 after you mutliply both 4 and 2 by it.
120 This focuses the child's attention on the 1 for the next step
+ 240 Toe is placing this zero in the one's place
When we were young, there were really only three ways to practice your facts, and none of them were very fun: verbal drills, flash cards, and writing the facts. Those verbal drills were effective, but stale, and we know now were more effective with auditory learners than with visual or kinesthetic learners. Flash cards, again effective, and more appealing to the visual learner, but still usually no fun unless parents had the time to be creative in using them in fun ways. Writing the facts could be effective, so long as the child did not write the same fact repeatedly, in which case he/she simply copied the first number over and over, then the X sign, then the next, the =, then the answer, but let's be honest most of us did, just to survive the monotony of that task.
These days, kids have so many enjoyable ways they can work on mastering their facts. Flash cards have been converted into computer games that can provide not only needed repetition of the same facts, but timers to encourage automaticity (a fancy new education word for answering by reflex, rather than thinking about it, or counting on our fingers). I have included several of my favorites on my Links page, but there are so many out there that can be used to teach or to test mastery of the facts. There are apps for your iPhone, games for the kids' DS or other hand-held devices, multiplication rap songs to appeal to auditory learners, computer software games, and countless on-line activities.
Still, all the facts cannot be mastered overnight. Most kids need guidance in pacing themselves in working on the facts. Most teachers recommend working on one facts set at a time, with reviews built in for previous fact sets. I have seen other approaches on educational instruction web-sites, but haven't tried their strategies. In class, I like to use a combination of both the old-school, and the new technological approaches, but I always try to add some twist of competition with others or with one's self as a motivator that appeals to most kids.
I have known few kids who can master their facts without time and consistent practice. One of my own children learned them easily without my help, and the other needed to work with me every night until they were mastered. As you probably know, this is one of the most important math skills your child will undertake, and one that will be used throughout life, so take the time to make sure your child gets them. Work on a particular fact set in the car each time you go somewhere. Play games with them using playing cards, the computer, or inexpensive flash cards that can be purchased at the dollar store. Consistency is more important than total time spent. Don't try to work with your child for three hours the night before the timed test, but work with them 5 or 10 minutes every day leading up to the test, and you will see better results.
I will do my part at school, but children whose parents get involved in the process tend to learn them more easily and maintain them over time better.