Open Number Line - Decomposed Factors

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Objective

SWBAT model multiplication problems with decomposed factors on an open number line.

Big Idea

Open number lines are a model for mathematical thinking that can be used in multiplication and division, as well as addition and subtraction.

Opener

3 minutes

Students’ pathways to fluency include doubling (once is 2 facts, twice is a 4 facts, three times is 8 facts), using ten facts to figure out nine facts (10 groups less one group), and decomposing numbers into known facts (3.OA.7).

Today’s lesson focuses on using what you know, to figure out what you don’t. This strategy not only develops students’ flexibility in mathematical reasoning, it also supports an attitude of success. 

Direct Instruction

15 minutes

My ultimate goal is that students flexibly manipulate numbers, applying what they already know to solve what they do not (yet) know.

A tool that helps students to use what they know to model their thinking is an open number line. Today I will show my students how to use the open number line to decompose one of the factors to create multiplication problems they do know, rather than stopping when confronted with a multiplication problem with larger numbers. When students decompose numbers to manipulate them, they are reasoning abstractly and quantitatively (MP4).

Did you see the Math Cat? (Show Me Video)

There are different ways for students to use this strategy, depending on their mathematical skills.

It's important that this is taught very clearly and deliberately.  It is a useful strategy for many students, but it needs to be explained well so that it doesn't look intimidating or confusing.

After showing the student video, model 2 more examples on the whiteboard:

4 x 6 decomposed to (4 x 5) + (4 x 1) =

3 x 9 decomposed to (3 x 5) + (3 x 2) + (3 x 2) =

I use 2 or 3 different markers, in order to distinctly represent each parenthetical equation.

At the end of each example, I ask students to tell a partner:

  • Something you understood that was new to you
  • Something you almost understand, what else do you need to know to understand the idea?
  • Something you find confusing.  Why?

I listen to students' mini-conversations and then share out relevant/helpful/common questions or comments with the class.  In the interest of keeping the lesson moving, it works better for me to paraphrase and insert appropriate math vocabulary when needed than to have individual students share out to the class as a whole.

Interactive Modeling

30 minutes

I project problems onto the board and have most students copy it on their whiteboards.  Again, I prepare several paper copies printed for students whose learning styles are best addressed by using a more durable medium.

As I go through each model, I note (when I could) on a copy of the study page if there are students who seem particularly confused.  I write their initials next to the problem on which I note them struggling.  That way I can make sure to check on those students first, when we get to the independent/peer practice.

The page is designed as gradual release, with the decomposition fully written out at the beginning, partially written out in the middle, and at the end I ask students to give me all the information.  I wish I could print these in colors as it really seems to help some students keep track of the different steps of the process. 

Some students decomposed facts into other facts that they don't know - for example, 9 broken down into 6  +  3, but they didn't know their 6 facts.  Yes, I remind the student, these two numbers do add up to 9, but our goal is to break less familiar facts into 1, 2 and 5 because those are multiplication facts they all know by skip counting.

Independent Practice

15 minutes

The independent practice is scaffolded in the same way as the guided practice was.  Monitor the students to note if they've hit their learning threshold at a certain level. That way, instead of allowing it to go on to the frustration point, you can give them two more examples of the type of problem that is within their learning zone. 

The objective is for students to practice, and to be successful at their work. 

Wrap-up

3 minutes

Ask students about patterns they observed while using decomposed factors and the open number line to solve for the product of a multiplication equation.  Guide them towards making observations about 5s (5 or 0 in the ones place - or, if your students flipped things around, two different digits repeat in AB pattern in the ones place), 2s (always an even number in ones place).  If they make extensions to other patterns, support them!  Some will be starting to connect patterns with 2s to patterns with 4s and 8s, for example.