Fun with Four: Decomposing to Known Factors of 1, 2, and 5

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Students will decompose equal group and array multiplication models into known facts (1, 2, and 5 as factors) using the associative property. Updated 7/24/15.

Big Idea

Students act as problem solvers to take an unknown fact and solve it by breaking it down into pieces that are known.


5 minutes

Students count by 1, 2, 5 and 10 chorally, in their head, or with a partner.  We all agree that these are numbers we can skip count confidently, which means we can use them for multiplication.  If students aren't accustomed to skip counting, model some skip counting strategies. We use skip counting throughout our school day, in a variety of ways, so students are used to doing it. 

After our skip counting experience, I explain that decomposing factors of 3, 4, 6, 7, 8 and 9 to known factors of 1, 2, and 5 is a helpful stepping stone en route to the end goal for 3rd grade -fluent knowledge of multiplication facts 0-10.   I also explain that decomposing an unknown factor to known pieces is a strategy that they will use in math beyond 3rd grade.

We discuss/define the word decompose, in a mathematical and general sense.  We discuss the word distribute and I usually distribute something to the class (markers, paper) as a way to help make the word stick. As it is easy to keep using it, I make sure to include it in my general classroom vocabulary use.

It is important to always review and connect the mathematical vocabulary you'll be using in the context of the lesson. For example, rather than merely saying, "Six times five", I will say "Six groups with five things in each of those groups --- or, six times five". I review the terminology used in today's lesson: factor, product, equal group, and array


Guided Practice

25 minutes

The primary goal for the guided practice is that SWBAT articulate their thinking about visual models of multiplication equations with 4 as a factor, and how what they think about how these equations are decomposed. The secondary goal is to increase students’ familiarity with multiplication equations in which 4 is a factor.  Both of these goals are directly linked to MP8:  Look for and express regularity in repeated reasoning.  Now that I have implemented the Common Core in my classroom, I allow students to discover that whatever numeral is decomposed in the equation is broken down into combinations with 1, 2, 5 or 10.  While it seems obvious to an adult, it’s something students need to experience on their own and then take ownership of when they comment upon it or explain it.  In the past, I would have told them this and felt pressured to move further, faster instead of the wiser, current approach of developing deeper understanding. 

I created a PowerPoint Fun Fours so that students can focus exclusively upon examining how the equations are decomposed.  This is a problem from the Fun with Fours activity:

Using this presentation frees me up.  The time I would have spent writing up questions can instead be given to strategically questioning the children so that they can have productive peer to peer mini-discussions.   

I have the children write out their responses on their whiteboards but this can be done one on paper as well.  I chose the whiteboards because, again, I want the students focused on thinking about the process of decomposing numerals.  I have found that students get very involved in making sure they copy everything correctly when they have a formal paper on which to write. This can sap their attention for actual critical thinking.



Independent Practice

25 minutes

Students can work on Fun with Fours with a partner or independently.


I monitor students and ask guiding or extension questions, based on students' individual needs. 

Example Guiding Question:  If  a child decomposed a factor of 7 into 4 + 3, I might then ask them how this made the problem easier for them to solve, to lead them to a recognition of the fact that while it a correct decomposition, it may not be one that lends itself to a solution that is any easier.  

I also do a lot of guiding and reteaching about the distributive property.   At the beginning of the learning curve, often students will distribute on of the decomposed factors but not the other. Watch for this!

Example Student Error w/Distributive Property:  7 x 6 =  (5 x 2) x 6 = (5 x 6) + 6

Instead of using a checklist, I sometimes find it helpful to take anecdotal notes on the Fun with Fours teachers page about student progress.

Taking notes using the format above makes it easier for me when I review the students' papers. I can more easily group them for reteaching, enrichment, or clarification.  I can also then transfer the relevant information to my teacher log more neatly than if I enter directly into a teacher record while I'm also monitoring the room.


5 minutes

Call the students together and ask them to (silently) think about one or more of the following questions, and then share their answers on paper/with a partner/with the class.

1. What is a reason for decomposing a factor in a multiplication problem? 

2. What are the easy factors into which you may decompose a larger number? (simple)

3. What is one equation you solved today by decomposing a factor into known factors?

4. Do you find it helpful to decompose a larger factor into known factors?  Why or why not?

5. How would you explain this strategy to someone at home?