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Topic: Number Sense and Computational Fluency
Replies: 79   Last Post: Nov 6, 2010 3:38 PM

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Cynthia Garland-Dore

Posts: 68
Registered: 12/3/04
Re: cluster problems for multiplying bigger numbers
Posted: Jul 27, 2002 5:58 PM
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Peg:

I am not really sure what you're asking regarding the order of cluster
facts and cluster problems. Do you mean going from one group of math
facts (e.g., from the 5's to the 6's etc.)?

I ask students to look for what they know first to solve a problem. If
it's a multiplication problem like 8 x 7, they may not know the answer
instantaneously. However, they do often know math facts or have some
knowledge of landmarks around the numbers that are related. Some
students may have to start with a version of repeated addition.

Perhaps
8 x 5 = 40
8 x 2 = 16

Add 40 + 16 together to get 56.

_______

or I know 7 x 7 = 49 so I have to add another 7 to find the answer to
7 x 8 =
49 x 1 is 50 plus 6 more is 56.

________
or perhaps a child beginning to look at multiplication ideas would
solve it through repeated addition

8 + 8 + 8 + 8 + 8 + 8 + 8 =

It's difficult for me to type out examples of ways students might
string these numbers together.

Some just start with the first number and add each number until they
have added seven 8's.

Others string numbers together 8 + 8 = 16 (3 times), etc. and then
string the bigger numbers until they have at total.

Even though this is look long and cumbersome, students learn so much
about what 8 x 7 means. It's a starting place for many, not an end
point. The goal is for 8 x7 to be a piece of knowledge the learner has
internalized enough to use from memory with fluency and automaticity.

Eventually problems like this do become math facts they just know.
However the knowledge base behind the math fact is so much stronger.



______________________________________
So, to get back to your question,

>I'm wondering about the order of the cluster facts given to help
students break apart larger numbers for multiplying
and dividing.

> Is the order random, or is it supposed to flow from one fact to the
next in a way that helps them put it all together?
______________________________________

I don't believe the process of learning basic multiplication is either
random or linear from one math fact to another. Instead students
explore number relationships and develop big ideas around
multiplication and division. Some of the knowledge is eventually put
to memory by the learner. I expect that students will have basic
multiplication facts accessible from memory in elementary school. They
will use this knowledge to think about the inverse (division) I know 8
x 7 = 56 so 56 ÷ 7 has to be 8. How they develop the knowledge and how
they access the facts is different for everyone.

Some of the ideas that are useful when students think like this are:

- using knowledge of common facts (2's, 5's, 10's) to solve other
problems

- using knowledge of multiples of 10's or 100's to solve problem
- using the idea of either doubling and/or halving numbers in the
problem
- understanding multiplication and division place value ideas (does
the student know the value of the numbers in the problem?)

Students practice these ideas through a variety of experiences
- creating visual models (e.g., array cards)
provide lots of resources (gridded paper, dotted paper, tiles,
cubes, etc.)

- solving real life problems

- using cluster (or related) problems

Questions or statements teachers can ask students as they explore
these ideas are:

- What do you know that will help you solve this problem?

- Can you draw a picture or diagram that will help tell others how
you solved your problem?

- Is there a landmark problem you know that will help you solve this
problem?


After a student has solved it one way:

- Is there another way you can solve this problem?

- Can you use your strategy again and solve the problem in fewer
steps?

- Share your solution with another students. Compare your methods to
see what is similar and different.


If you want to practice new methods that have emerged from the
students:

- Let's use _______ strategy (a strategy that's emerged from the
students in the classroom) to solve this problem).

- Do you think ________'s strategy will work with all numbers? Try 3
new problems to check out the strategy.


Students start with what they know. At first they might need to split
numbers into smaller parts or use repeated addition. They learn from
one another as they discuss their methods. The teacher guides,
facilitates and challenges the students to develop a few solid
strategies. With practice, students do become more efficient at
solving multiplication and division problems.

_______________________________________
An article you might find helpful is:

Developing Computational Fluency with Whole Numbers in the Elementary
Grades
Susan Jo Russell
http://www.terc.edu/investigations/relevant/html/CompFluency.html


I hope this information helps in some way. Please continue to share
how things are going.


Cynthia Garland-Dore






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