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 ÃÂ· 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: