Using Number Paths As A Linear Model

In this series of Math Is Visual Prompts, we slowly introduce the idea of a linear model (i.e.: number line) through the use of a number path. This is a much less abstract way to introduce the idea of linear measurement through counting to make it accessible by all.

In this series of Math Is Visual Prompts, students are given the opportunity to begin moving from counting sets (i.e.: individual objects like candies, toy cars, bugs, etc.) to the very early stages of linear measurement through the use of “number paths” on their developmental trajectory towards the eventual use of the much more abstract number line.

As always, the visuals provided as a consolidation are useful AFTER students are given the opportunity to work with these ideas concretely and after consolidating in the classroom through student work.

Let’s get started…

Introducing The Number Path

Note that a number path looks an awful lot like a number line, however you’ll notice that each “square” is labelled explicitly instead of at the “end” of each unit (on each ‘tick’). This is to promote students explicitly counting each square to make the thinking much more accessible for early counters. 

Number Path Linear Model - Cardinality and Composing Decomposing.024 Number Path Introduction

While this is slightly more abstract than counting say actual dinosaurs or donuts, it is much less abstract than looking at a line with ticks on it.

Show Me…

We then ask students to “show 4 on the number path”. What we are looking for is students showing that 4 includes 4 squares and not just the last square. This builds on the idea of cardinality.

Number Path Linear Model - Cardinality and Composing Decomposing.029 Show 4 Solution

We then ask to show 7 with the number path:

Number Path Linear Model - Cardinality and Composing Decomposing.035 Show 7 solution

Composing Numbers Through Early Addition

Then, we start asking students to show a number and then add some.

Number Path Linear Model - Cardinality and Composing Decomposing.041 Show 3 plus 2 more

After adding two more, we get these two groups of 3 and 2 on the number path:

Number Path Linear Model - Cardinality and Composing Decomposing.045 3 plus 2 more is equal to what

Finally, even a student who is still counting all can come up with the newly composed quantity of 5:

Number Path Linear Model - Cardinality and Composing Decomposing.050 3 plus 2 more is equal to 5

Something to note here is also the fact that there is this idea of hierarchical inclusion lingering plus the idea that when you put together a quantity of 3 and another quantity of 2, you get 5. You can also highlight that the quantity of 5 can be decomposed into a group of 3 and 2.

We then do this again for 2 plus 5 more:

Number Path Linear Model - Cardinality and Composing Decomposing.059 5 plus 2 more

Decomposing Numbers Through Early Subtraction

We then do something similar for early subtraction and decomposing numbers such as 8 remove 5:

Number Path Linear Model - Cardinality and Composing Decomposing.071 8 remove 5 is equal to 3

as well as 9 remove 3:

Number Path Linear Model - Cardinality and Composing Decomposing.078 9 remove 3 is equal to 6

So there you have it. We are well on our way to start thinking mathematically with linear models through the use of a very concrete and countable approach we call the number path. 

How’d It Go?

Thanks for watching and reading!

Did you use this in your classroom or at home? How’d it go? Post in the comments!

Math IS Visual. Let’s teach it that way.

kylepearce3

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