Because of all of the math talk my students do in class every day, they are very comfortable (and flexible) in sharing multiple strategies and solution paths. They can explain others’ strategies in their own words and agree and disagree with one another beautifully, however when asked to make connections between two representations (numerical or visual), I feel like I get very “surface” connections. I will read things in their journals like “They are the same because they have the same numbers” or “They are different because we double and halved the other numbers instead” or “We both used an area model” Something like this…

After going through their journals the other day with Faith, we were thinking and questioning one another about how we, as teachers, can have students dig deeper into the connections. We obviously would like to them to notice them but if not put in a position to make connections, will they on their own? Is there a way to frame a task or question that would push them to think a little deeper about how and why the two representations are alike/different yet still arrive at the same answer? How do we encourage students to make deeper, more meaningful connections when we know they can, but just may not be sure of how to get there?

Instead of a number talk they other day, I did a math routine I named “Where is _____ in _____?” I was hoping the prompt would have them go beyond just looking “at” the representations and look “into” the meaning of each representation. On the board, I posted some examples of their representations from the day before in which they had done a surface job of connecting. I had them work independently for a few minutes and then talk as a table before the group share. I was much happier with the conversation and felt like asking them to look “into” the problems really got them thinking about what the representation was showing.

This an example of two area models in which students the day before had simply said, “We both used the area model” without thinking about how they were related. I love the (.3 x .2) in each of the quadrants of the first grid.

This student had a different take on how the two area models were alike, which led to such an interesting discussion! She also did some lovely work with showing how the two distributive properties were within one another through factoring.

This student showed the distributive property and double/halving in a wonderful way…

This one was the only student who connected the area model of .6 x .4 to the strategy of .6 x .5 – .06. He showed where the .6 x .5 would be in the model and then scratched out where the extra .06 were coming off to arrive at the answer.

Having students make connections in math is so incredibly important and so difficult to do, especially with so many variations in strategies and representations. I would love to hear other ways to encourage these connections!

-Kristin

Maya QuinnRE: “I would love to hear other ways to encourage these connections!”

If you are working with area models and factoring, one approach is to consider the four-quadrant area model for multiplying a pair of two-digit numbers, where numbers are decomposed by place value (i.e., tens place and units place).

A common task is to present something like 12×44, and have the student draw the corresponding four-quadrant area model.

An alternative task is to

startwith the four-quadrant area model, and then see if the student can work backwards to find what the numbers being multiplied are!For example, the quadrants for 12 (left side) times 44 (above):

|400_|_40|

|_80_|_ 8_|

Presented with this diagram: Can the student figure out how to label the sides?

This gives good practice with factoring (e.g., what could go in the ones places to get an 8 in the bottom-right corner?) and thinking about a representation from an inverted perspective.

Incidentally, some four-quadrant area models correspond to multiple products.

In the above case, both 11×48 and 22×24 produces the same four-quadrant model as 12×44.

Students can try and explore which solutions exist, and (possibly with scaffolding) realize that the same representation for multiple products indicates that they are all equal.

Indeed: 11×48 = 22×24 = 12×44 = 528.

There are interesting follow-up questions for this task; here are three:

1. Can the student create other four-quadrant area models that can be produced by more than one pair of products? (How does one think these up?)

2. Given a four-quadrant area model, if you fill in any three of the quadrants, then it uniquely determines the fourth one. Why?

3. How can the above task be adapted for the multiplication of positive mixed numbers? (i.e., where the tens place and units place are replaced by the whole number part and proper fraction part, respectively).

MQ

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