* lots of activities for beginning multiplication

> I am going to introduce multiplication; any advice on how --
> a good, easy way?
> kali, 11/08/00

Use oodles of manipulatives. Use them over and over and over again. It is impossible to do too much work with math manipulatives when you are introducing multiplication! Here are some inexpensive ideas (as well as ideas for manipulatives that cost \$\$). All of them reinforce the idea that multiplication is repeated addition.

1) Egg carton math. Have each child bring in an egg carton and a plastic container with some type of little objects. These could be pennies, beads, buttons, paper clips, raw macaroni, mini-pompoms... whatever. When you say and write a problem, such as 3 x 4, the children need to display this problem using different sections of the egg carton to hold each group.

By the way, I always told my students to think of the X in a multiplication problem as meaning "groups of." So 3 x 4 is "3 groups of 4."

Using the egg carton, then, they would only use 3 compartments, and they would put 4 items in each of those 3 compartments, counting as they go. And also after the problem is set up, they can count by those 4's: 4, 8, 12.

Then you could say, "4 x 3." Now they need 4 groups of 3, so they'll use 4 compartments and put 3 items in each, but they will still have 12. Count by those 3's: 3, 6, 9, 12.

2) Turn multiplication problems into rectangles. The cheapest way is just to use graph paper and color in squares. So again, with 3 x 4 (3 groups of 4), to show 3 groups of 4, students would color 3 rows with 4 squares in each row (4 + 4 + 4). To show 4 x 3 (4 groups of 3), students would color 4 rows with 3 squares in each row. They can then compare these two rectangles and see that in both cases, 12 squares are colored, but one looks like the other, turned 90 degrees.

The more expensive but more effective way to make rectangles is with Cuisenaire rods. This way the colors also help to reinforce the problems. To show 3 x 4, you'd need 3 purple (4) rods. To show 4 x 3, you'd need 4 light green (3) rods.

I wanted so badly for my students to each have a set of "Cuisenaire rods" to use at home that we made our own 2D version. I just made photocopies of centimeter grid paper onto stiff card stock. Then the students colored portions of the grid paper with crayons, using the 10 colors needed to match the rods and then cut them out with scissors. So, for instance, red rods are the length of two units (the white rods). So if they needed 10 red rods, they would need to color 20 squares red, then cut them out in clumps of two. They would need 30 light-green squares to make 10 light- green "three" rods. And so on. If you don't actually have the rods, the colors are:

 1: white 6: dark green 2: red 7: black 3: light green 8: brown 4: purple 9: blue 5: yellow 10: orange

Anyhow, the students colored and cut up all these strips and put them in an envelope with a clasp (actually, a ziploc bag would be better) for taking back and forth between home and school.

The next thing I did was to make a multiplication table for each child, also using centimeter graph paper. It had the numbers 1 - 10 running down the left side and running across the top, with a multiplication sign in the top left corner. Then there was a thicker line to separate out the answers. (Or you could write those numbers 1 - 10 in a different color from the answers. But it still helps to have a thicker line.) Either I or the students would fill in the answers, but it is essential that they are accurate. You can do this in stages. So while you are just working with the twos times table, just give them a grid where those facts are filled in. Gradually fill in more of the grid.

Now, let's pretend they are working on their 3's so we can use the same example as above. The grid is filled in with all of the 1's, 2's, and 3's running across as well as down: 3, 6, 9, 12, 15, 18...

If you ask them to show 3 x 4, they will think: Okay, 3 groups of 4, so I need 3 of my purples. They will lay these down on the multiplication grid, starting at the top left corner of the answer area. Then they will peek under the bottom right corner of their rectangle. They should find a 12 there. Now if you ask them, 4 x 3, they will think, Okay, 4 groups of 3, so I'll need 4 light-greens. They will lay these down in the same way and peek under the bottom right corner. Again they should find a 12.

I'm going to try to insert a few graphics here: first the grid (this one shows all the answers to 10x10):

And now the two problems showing 3 x 4 and 4 x 3 (with the grid numbers covered by the rods shown):

 3 x 4 = 12 (3 groups of 4 = 12) 4 x 3 = 12 (4 groups of 3 = 12)
Sorry this is getting way too long. Let me semi-briefly mention other ways to build the concept:

3) Lay Cuisenaire rods along a centimeter ruler, again to show multiplication as repeated addition. For 3 x 4, you would lay 3 purple rods end to end and see that they have reached 12 on the ruler.

4) Once the students become very good with the rods and multiplication grid, start turning the grid into a missing- number problem sheet. But scramble things up. Again use a partial grid while they are learning. But instead of writing the numbers in order across the top, write, for instance, 2, 5, 3, 1... and down the left side, 3, 1, 5, 2... and ask the students to fill in the answers. When they master that, sprinkle some answers around but take out some of the numbers from the outer edge, and the students need to figure out what number fits there. If they see a 15 on their grid, they will realize that the missing number at the top must be a 3 or a 5. If another number in that column is 6, then they know the number at the top must be a 3.

As I continued to create these missing-number grids for my students, they got better and better at their logical reasoning as well as their facts, and I began to remove more and more numbers. I got it down so that eventually I could put 10 well-placed numbers in the answer part of the grid, and they could complete the rest of the puzzle... 110 missing answers (90 more from the answers and the 20 missing numbers from the top and left edges)! It was a real challenge for them, logically, while reinforcing the facts. In fact, you could ask your students to create some of these puzzle sheets. Simply creating them is another exercise in logic.

Enough!! I just want to say that we used manipulatives throughout the year for multiplying. Drown them in manipulatives!! I also want to say that I have turned some of these activities into online activities at Math Cats (for instance, a multiplication grid activity... where each student originally had a magnetized set). I will try to post the Cuisenaire rods activity in the next day or so. (It automatically creates "Cuisenaire rod" rectangles to illustrate each problem you enter.) [It is now online at Math Cats in the MicroWorlds section:
multiplication rods (www.mathcats.com/microworlds/multiplication_rods.html). It requires the free MicroWorlds plug-in. A screenshot can be seen at www.mathcats.com/microworlds/multiplicationrods_overview.html]

Wendy P of Math Cats, 11/09/00 on teachers.net math board
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* How do you teach multiplication?

> Do you teach as a whole class (group) or teach individual
> (kids go at own pace)?

I like to start with the circles and stars approach found in Math by All Means: Multiplication, Grade 3 [at amazon.com] by Marilyn Burns. The hands on ideas/lessons in this book are great.

Colleen:)/k-3, 11/02/00 on teachers.net primary elementary board

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Using a Number Line

I used to post a huge number line across the top of the blackboard, with every 10 highlighted in yellow, and I'd use a yardstick to point to the numbers while we'd skip- count by a particular number. I wanted them to develop a mental number line for multiplication, to visualize the facts linearly. (It also helped them with their mental addition.)

Using Multiplication Grids

And then in terms of a 10 x 10 grid, we used to use magnetic number squares on metal boards, and the students would race to see how fast they could place all 100 products on the 10 x 10 board in the right locations. Inevitably some numbers got lost or mixed with another student's set, so eventually I developed the same game for computer. It is now online at Math Cats, specifically:

http://www.mathcats.com/microworlds/multiplication_grid.html

You need to download a MicroWorlds plug-in to use the project, but you can preview it (screenshot plus project description) at

http://www.mathcats.com/microworlds/multiplicationgrid_overview.html

Wendy P of Math Cats, 10/15/00 on teachers.net math board

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Sixes, Sevens and Eights

> I was wondering if someone could give me a better way of
> explaining the multiplication facts for six, seven, and eight.
> Jen, 11/08/00

We talk about sets, arrays, repeated addition, skip counting, and the family of facts. We use counters, spinners, graph paper, songs, and games. Relate the sixes to the threes, and the eights to the fours.

Pat, 11/08/00 on the teachers.net primary board

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On your site you mention that you reinforce the idea that multiplication is repeated addition. Unfortunately, it isn't.

There are occasions where it seems like that, but it will give the students the wrong impression when they later meet 5 * 0.316, for example. Adding 5 to itself 0.316 times?

Please see www.maa.org/devlin/devlin_06_08.html for further details.

Regards,
Paul Horth, Oracle Database Engineer, 11/5/08
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* Multiplication Songs

> I am starting to teach my children the times tables. I
> wanted to know if anyone knows any songs to teach them?
> jenn

People still teach the times tables? Be still my beating heart, and I wish I could get your students. My eighth graders have no idea how to multiply even simple numbers. But, I digress. May I suggest one of my favorites from my childhood that is now out on video. School House Rock ... the multiplication rock video. Sure, they're corny, but they might work. And, it's wonderful for me to watch (if we get the chance that is) and reminisce.

You can purchase the [DVD] through amazon.com:
Schoolhouse Rock Special 30th Anniversary Edition

Schoolhouse Rock - Multiplication Classroom Edition [Interactive DVD]

John, 5/24/00 on teachers.net math board
[links provided by Math Cats, updated 2010]

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I came to your site looking for suggestions. I home school my son and flash cards were just SO boring. I felt stuck. As I read through some great suggestions, I checked out the School House Rock. I got the Anniversary edition with every song ever made. My son LOVES it, I can do leave him in front of the tv and he is learning his tables, grammar, science and America facts!!!

He got out of bed last night complaining he couldn't turn the math off in his head! I can't say enough how great this is!

Schools should use this and forget the state testing garbage. Teach the essentials! Of my three children, THIS one is going to know his multiplication tables WELL, he will know how to spell and he will have a good vocabulary--unlike my older children who are going through public school. I missed my opportunity with them, I regret this immensely.

Anyway, THANK YOU and to anyone else looking, this DVD set at \$14.99 through Amazon is PRICELESS!!!!!!!!!!!

Diane Hall, 12/13/07, message to Math Cats

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Check out Hap Palmer's Multiplication Mountain CD at happalmer.com, featuring upbeat, original songs and a colorful activity guide designed to be used in the classroom or at home.

Hap is a pioneer in the area of movement and music in education. His inspiration for this new CD came from his own daughter's experience in learning multiplication in third grade. The "mountain" approach seems to be very effective.

Beth B., 2/18/09, message to Math Cats

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I found the book at school that has the multiplication songs in it (and many other great ideas to teach multiplication facts). The book is: Making Multiplication Easy: Strategies for Mastering the Tables Through 10/ Grades 2 - 4. It's a Scholastic book. The author is Meish Goldish.

LK, 11/30/00 on the teachers.net main board
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* Multiplication Rhymes, Chants, and Stories

> I am looking for ideas to teach multiplication facts to 4th
> can use, but have not come across any. Thanks!
> lateacher, 10/06/00

Multiplication Rhymes and Stories

We make up silly sayings to remember the hard ones. The kids really get in to making up ones for their hard ones. Ex:

"Had two 8's, dropped them on the floor, picked them up, had 64."

My third graders love to tell a new one they have thought up!

Jam, 10/06/00 on teachers.net math board

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I am teaching an undergraduate course, and several of the participants want those rhymes to help their students remember those troublesome facts. I have the following, but would be interested in hearing others, etc.

Multiplication Rhymes

4 x 4
A 4 by 4 is a mean machine
I'm gonna get one, when I'm 16.
4 x 4 = 16

6 x 6
Six cold 6 packs Thirsty-chicks will have for snacks. 6 x 6 = 36

7 x 7
Bend 'em up and down to make 49.
7 x 7 = 49

8 x 8
I 'ate' (8 times) 'ate (8) and
fell on the floor.
Couldn't get up 'till I was 64.
8 x 8 = 64

4 x 6
4 x 6 I'm hatching baby chicks
Ooooops!
I dropped them on the floor.
Would have been "eggs-actly" 24!
4 x 6 = 24

7 x 4
The animals are coming
7 x 4
Better open the gate
There's 28!
7 x 4 = 28

4 x 8
4 x 8
I ate the 'goo'
Now I'm thirsty-too! (32)
4 x 8 = 32

6 x 7
Happy Birthday to Kevin
He's 6 x 7
He blew and blew
'Cause he's 42.
6 x 7 = 42

7 x 8
7 packs of gum
Each with 8 sticks.
Open up, Big Mouth,
Can you chew all 56?
7 x 8 = 56

6 x 8
Flight 6 x 8! Don't be late!
Leaving at gate 48.
6 x 8 = 48

KathyB/1st/IA, 9/08/00 on the teachers.net math board

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I have seen where students write their own rhymes to the military march tune. Ex:

7 times 7 is 49, don't bend down you'll hurt your spine.

Think of the rhythm when soldiers are marching and have kids work in groups to come up with them, then class votes on best and can make a book of class best multiplication marches.

KAT, 9/08/00 on teachers.net upper elementary board

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Marching
I heard a long time ago that physical movement helps students learn math facts, including multiplication. We used to go out on the playground and march in a circle, skip-counting by a given number: (6) "6! 12! 18! 24! 30!" and so on. When they got good, we'd march and skip-count backwards. When they can do any table backwards and forwards, they won't have any trouble figuring out a specific fact. And they really did get so they could all skip-count by any number (10 or under).

Wendy P of Math Cats, 9/08/00 on teachers.net upper elementary board
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* multiplication Web sites

> Do you have any great lesson plans or sites for teaching multiplication?

Michele's Math Multiplication Tips
This site includes practical strategies for relating known multiplication facts to new facts.
http://www.redshift.com/~bonajo/mmathmult.htm

Multiplication: An Adventure in Number Sense
(step-by-step lessons by Natural Math and an online interactive multiplication grid)
http://www.naturalmath.com/mult/index.html

http://www.multiplication.com
[Note: This site offers some drill activities, online multiplication practice games, and links to multiplication resources, but Math Cats does NOT endorse the gimmicky methods for learning multiplication which are promoted at the multiplication.com site.]

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Check out Sum Fun at www.oswego.org/ocsd-web/games/SumSense/summulti.html. You need to arrange the number cards to make a true multiplication number sentence. Set the clock for how many problems you think you can complete in 1 or 2 or ... minutes.

Wendy P. of Math Cats

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Here is a listing of the other multiplication web sites and pages mentioned elsewhere on this page:

SchoolHouse Rock
Schoolhouse Rock Special 30th Anniversary Edition

Schoolhouse Rock lyrics and video links

another site for Schoolhouse Rock lyrics and video links

Multiplication Mountain at Hap Palmer's music site - - - - -

Multiplication activities here at Math Cats:

Interactive Multiplication Table
www.mathcats.com/explore/multiplicationtable.html
Visualize the table! Click on two numbers to be multiplied, then click the = sign, and you'll see the problem displayed as a rectangle (with row counts and column counts along the edges).

Multiplication Rods
(Enter two numbers to multiply and click to see a rectangle illustrating the problem.)
www.mathcats.com/microworlds/multiplicationrods_overview.html

Multiplication Grid
(Move 100 online number tiles onto a multiplication grid; a frog hops when you're right and keeps track of your score!) http://www.mathcats.com/microworlds/multiplicationgrid_overview.html
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* When do you teach multiplication?

> I am a first year third grade teacher who has to teach
> math. I am unsure of when to start multiplication. A few
> people said to wait until after X-mas and another told me I
> should have started it a month ago! HELP!!!
> Kim, 11/28/00

When isn't as important as how!

There's no rush. But do make sure you teach understanding not just the rote skill.

Before you even introduce times tables, teach and re-teach the adding of groups. Then work your way up to the idea that multiplication is the short cut to adding groups (e.g. two groups of two is the same as 2 X 2. Teach students to read the "X" sign as "groups of").

You can even make a class book of different groups which can be added together (3 + 3 + 3 = 9) and also shown as multiplication (3 x 3 = 9). Students can illustrate the groups (3 groups of 3 cats) and then make their individual pages into a class book which then can be borrowed by a different student each night. Get parents to write comments in the back. You could even do facts to 12 and have students work to make books for each group, i.e. adding groups of 1, groups of 2, etc.

Michael, 11/28/00 on the teachers.net math board

p.s. If you teach multiplication this way, you'll find they understand division better when you teach it as dividing into groups.

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In our curriculum, multiplication comes after graphing, building large numbers, and computation including addition and subtraction with lots of trading.

I do measurement and then time and then multiplication comes during January.

If you can find the series of books by Marilyn Burns you will get lots of hands on activites for multiplication and division. And I take all the time until March Break with multiplication.

Suzanne, 11/28/00 on the teachers.net math board

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I think that there are different reasons for teaching multiplication at different times. None are wrong. At my school, we start multiplication the first week of school (I teach 3rd also). The kids have had exposure to the basic concepts in 2nd grade and are excited about it. This also gives the kids who struggle with understanding how it works and memorizing the facts (do not forget to make sure that they understand) more time to learn the facts before the end of the year (and the high stakes testing). At the beginning of the year, we stick with 1 digit x 1 digit. At the end of the year (around March/April) we go back to multiplication and do multi-digit multiplication. This way, those kids who struggled have had time to work on the basics before getting to the harder parts. We also integrate teaching multiplication and division. We do a lot with the fact families-- it really helps. Our math books have multiplication/division in the middle of the year (if we went in order, we would probably start around Feb.-- but since I am about 2 weeks away from starting Chapter 1, you can see that we don't follow the order of the book all that well!) All 5 third-grade teachers follow the same basic concept order in math (for convenience mainly).

I would say to just pick when you think it best fits into your plan. I would suggest not leaving it for last if you have testing since it takes many kids some time for the concept to sink in. We do single-digit multiplication/division, then fractions/decimals, then place value, then multi-digit addition/subtraction, then multi-digit multiplication. We fit in the statistics, geometry, algebra, mathematical reasoning, and measurement wherever and whenever we can throughout the year.

T, 11/28/00 on the teachers.net math board

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Yes, read books by Marilyn Burns

Read anything she has written. She offers some very good activities, and, even better, she explains what results you can expect. My textbook seems to introduce multiplication late in the year, also. You can start some of her activities much earlier, giving you a head start. For example, my students love the circle and stars game and its follow-up discussions. They also liked looking for various patterns in a hundreds chart. They should have been introduced to the idea of multiplication last year, so don't feel like you are starting too early. My friend and I will soon have them copy those dreaded lists of facts. They will know that the facts will eventually have to be memorized. Come spring, they will start having times quizzes, which they love, stopwatch and all. In theory, I love all the activities that help them understand the concept of multiplication, but every year it comes down to memorization.

susan, 11/28/00 on the teachers.net math board

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Kim, I also teach third grade. As I introduce multiplication, I say, " 2 groups of 3" as I write 2 x 3. I do this for several weeks. I then have my children draw 2 groups of 3. etc. This third six weeks, we have started with the multiplication facts on a much tougher level. The students understand the concept of multiplication and we are just working on learning the facts. We will really hit them after Christmas though.

Vonda, 12/03/00 on the teachers.net math board
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* reteaching and reviewing multiplication

> I'm a 6th grade Resource teacher. I teach one Math class
> which has 9 students (LDs, EMDs, and BED). These students
> do NOT know their mult. tables. Is drill and practice
still the best way to learn?
skye/NC, 8/22/00 on teachers.net math board

You might try this: develop some pages with 100 multiplication facts on each one in no particular order. Make several different pages with the same facts but mixed up, and identify each different page with a letter to keep track more easily. Don't forget to use the zero and one facts too. Then use it like a contest, only each student competes against himself. The goal is to complete the sheet in five minutes or less with NO mistakes. Give each student a copy of sheet A and set the class timer for 5 minutes. The student tries one sheet a day (this makes good bell work, too!) and each day tries to better his time and his accuracy. When he's mastered sheet A (goal met 3 days in a row), give him sheet B, then when he's mastered that give him sheet C. You can make several versions, with the A page using lots of zeroes and ones, and the last pages using more 10, 11, and 12 facts. By the time the student has mastered all of the versions (five or six should do it -- if not, make up page G, H, I, J, K, however many it takes), he's mastered multiplication facts.

Try not to let them make it a competition among themselves, because doing it better than somebody else is not the important thing. Promote the idea of "personal best." Try to get them to encourage each other to simply do their best. You could provide a special reward of some sort (if you're allowed to use edibles, one small hard candy or a tiny tootsie roll works wonders! Here we have to give them an "IOU" until after lunch) each time a student meets a daily goal and moves on to the next sheet. (Getting 3 rewards in a row establishes an expectation.) Then do the same thing with division facts, then move on to equivalent fractions [1/4 = blank/8, blank/16 = 1/2, etc], and then to conversion of fractions to decimals and decimals to fractions when you get to that (you won't do a hundred of these on one page! Maybe start with 40, then 50 and then 60 max)[ex: 1/2 =.5 = 50%; 1/4 = 25% = .25, 3/4 = 75% = .75, and so on with fifths, sixths, eighths, tenths, and sixteenths - the most common fractional parts used in daily living]

Sorry this is so long. Let me know how this goes! Good luck!

Linda, 8/29/00 on teachers.net math board

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> HELP! 6th graders can't mult and div need help:
> There are a handful who cannot mult double digit numbers
> and forget division!! They are not serviced through content
> mastery and because we use Saxon math there is no such thing
> as "slow down and teach mult.& div"

You can't teach them anything past 2 digit multiplication and division unless you go back and teach it. I teach 5th grade and I spend a LONG period of time teaching multiplication and division. They need a lot of practice with these concepts to learn them. If your math book doesn't teach it, you're going to have to use your own resources. You need to start small and build up the concept. Go back to 2 digit x 1 digit. You also will need to spend time on it if they're going to learn it. Your administration may not like it, but they'll not be able to learn anything more advanced without the basics. I don't really know of any fun games.

You can use place value blocks with place value boards to teach the concept with manipulatives, first, but other than that, I use pencil/paper.

Also, try to incorporate as many real-world problems that will also make sure they master the concept and not just the rote skill of multiplying/dividing.

jody, 9/09/00 on teachers.net upper elementary board

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You need to cipher daily

What is cipher, you may ask? Ask your grandmother. She did it, and she probably still know how to multiply and divide.

It is just a game. Choose up sides any way you like. (I do it randomly the first time, then put winners at the top next time. We start the game at the bottom.) Start at the bottom of the 2 lists. Students have a best of three contest, in which the loser gets to choose the type of problem, ie add, subtract, multiply, divide, fractions, etc. I make up the problems as we go along and work them at the same time as the students. I also require all students at their desks to work the problems.

I do this 15 minutes per day, and the game is ongoing. When the game is over, we start a new one. There are several different versions of this game, but I can tell you that by the time my students get out of my class in May, they are so much more advanced than any of their peers in the other classes, that it isn't even funny. BUT, it takes a commitment of 15 minutes per day.

Phil, 9/10/00 on teachers.net multiplication board
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* 2-digit multiplication

> I need to teach 2-digit multiplication to a third grade
> class. How should I begin? Any Ideas? Hands on materials?
> Also, How can I relate multiplication to division? Please
> someone help me! I am student teacher.
> Wendy, 11/14/00

Please don't rush to teach 2-digit multiplication

From everything I know about third grade math, it is not appropriate to be teaching two-digit multiplication yet. Usually that would be done in grade 4, or at the end of grade 3 if the students are extremely solid with one-digit multiplication. From my experience, it takes months and months of hands-on work with one-digit multiplication facts for students to be really solid with the concepts (multiplication as repeated addition, etc.) as well as being really solid with the facts. Rushing into two-digit multiplication can backfire on you. You can end up with students who learn the steps for doing things without really "getting it."

If you are a student teacher, I am assuming that you have been instructed by your master teacher to introduce this material, so you may not be able to control the timing... but you can try. Have the students done lots and lots of hands- on work with one-digit multiplication and do they know all of those facts cold? If not, what is the rush? If you have been working with hands-on materials, then you can extend them for two-digit work if you absolutely must proceed with this lesson.

I highly recommend Cuisenaire rods for introducing even two-digit multiplication work. You could also use Base Ten Blocks. To do 2-digit work with Cuisenaire rods: Say you are multiplying 23 x 4. Well, I always tell my students to think of the "x" as "groups of," so this would be 23 groups of 4. However, it is much easier to examine 4 groups of 23. (Do your students know, from lots of earlier work, that a x b = b x a ? I hope so!!! If not, you are wasting your time to be doing 2-digit work.) Anyhow, the orange rods = 10, so you could set out 4 groups, each of which has 2 orange rods and 1 light green (3) rod. Then you can see that you have 8 orange rods (80) and 4 light green rods (4 x 3) or another 12, and 80 + 12 = 92.

If you did the same problem with Base Ten Blocks, you'd still have 8 orange rods and you'd have 12 single white cubes, still illustrating that you've got 80 + 12.

Now if you need to move right away into the algorithm (again, I hope you don't have to rush it), you could first break the problem up into expanded notation: 20 x 4 + 3 x 4 = 80 + 12 = 92.

I did a lot of this, breaking the problem up into two smaller problems, with my third-graders at the end of the year. A lot of them still struggle with multiples of 10. (They know that 2 x 4 = 8 but sometimes get stumped on 20 x 4. That is why they need to SEE it in rods, and count it up (repeated addition.))

After you do lots and lots of that expanded notation work, finally you can show them that the standard algorithm multiplies the ones digits first (4 x 3). You could show them about writing the two below the line and carrying the one above the 2 (the tens place in 23), but many people prefer another stage first, where you write the whole 12 below the line, then multiply 4 x 2 , which is really 4 x 20, and write 80 below the 12, and add them to get 92.

But anyhow, you just asked about introducing 2-digit multiplication, so I certainly hope you are not going to get into the standard algorithm for many weeks.

You also asked about relating multiplication to division. Again, this should be done extensively with single-digit work before ever considering getting into the 2-digit relationships. When you do hands-on work, the relationship is obvious. In my earlier post, I talked, for instance, about placing little objects into the sections of egg cartons. If you show 3 x 4 by placing four items into each of 3 sections (3 groups of 4), you see that you have 12 in all, and you also see that 12 divided into 3 groups gives you 4 in each group. And so on. The relationship between multiplication and division can (and probably should) be illustrated with virtually every hands-on problem every day. Don't even THINK about doing two-digit division in November of third grade, OK???

Wendy P of Math Cats, 11/15/00 on teachers.net math board
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