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Decomposing numbers is the pathway to flexibility and fluency in all operations. Knowing how a number is made gives you the knowledge you need to add, subtract, multiply and divide with greater ease.

Giving students the opportunity to build numbers in multiplication teaches them how to count up to larger numbers, understand place value, use addition, and develop number sense.

Envision solving 328 x 3 using base ten blocks or actually try it out for yourself.
Take notice of all of the steps required to accomplish this task. Take notice of the many skills that can be developed by engaging in this task. Imagine how much your students’ mathematical skills can grow with this experience.

In a short amount of time, students can go from building with actual tools, to drawing a pictorial representation to writing out numerical equations.

Connect the work students do with multiplication to division. Ask students to multiply 328 x 3, get their total, then divide it by 3. The results may seem obvious to you, but the relationship between multiplication and division is not obvious to many children.

Give students any number, ask them to build it, then divide it by another number. What you will see happen is that they decompose larger numbers in far more efficient ways. Students shift from equally sharing ones and tens to equally sharing thousands, hundreds, tens, and ones. They also understand place value and the distributive property, and often use them both unknowingly and voluntarily.

Try dividing 428 by 4. Where do you start? You are probably going to start with the hundreds, dividing 400 by 4 and then 28 by 4, or something close to it. When dividing and decomposing numbers, students move from saying and thinking 4 goes into 4 one time to 400 divided by 4 is 100. This is a big and important idea, that often gets overlooked or mentioned briefly rather than deeply discussed and practiced.

When building and decomposing numbers, students begin to see that numbers are flexible, equivalent, and interchangeable.

You can get younger kids started in a couple of different ways.  One of the most under utilized problem types is the Part-Part-Whole, both parts unknown type. “There are 10 flowers in a vase. How many could be red? How many could be pink?”

This problem type allows for exploration, flexibility, and community building. If you were to pose this problem to students and allow them to experience building or drawing different combinations, you will find that in a short period of time, your students realize that numbers can be composed and decomposed in many different ways, 10 = __ + __.  Each individual student may not know all of the combinations, but as a community of learners, all students can contribute their part to the whole.
This work can be addressed or connected to the work students do with equal share problems, but is extremely useful when regrouping during subtraction. Consider the problem 72 – 8.  Students often decompose the 70 into 7 tens and subtract 8 from a 10, never touching the 2 in the ones place. This is just one type of thinking that results from a knowledge of and comfort with composing and decomposing. Sadly, most textbooks and curriculums do not develop this type of thinking or provide limited practice with it.

Though much work is required of teachers, you are really the only ones who are capable of designing and implementing the kind of work that develops the real mathematical talent in your students. No textbook could ever compete with you.