Thought for the day: Many students find it easier to notice than to wonder. If they are doing more of one thing than the other, encourage them to try to balance it out.

Concepts: multi-digit multiplication; place value; arrays; distributive property

Beginning

I notice segments and dots of different thicknesses.
I notice a dot at every place that the segments intersect.
I notice that the dots make arrays.
I wonder why the dots are different sizes.
I notice that the number of each group of segments (3, 2, and 4) matches a digit.
I notice that 32's digits match with vertical segments, and 4's digit matches with horizontal segments.
I notice that there are 12 large dots and 8 small dots.

Exploring

I notice that the product of 32 and 4 is 128.
I notice that the number of large dots (12) matches the number of tens in the answer, and the number of small dots (8) matches the number of ones in the answer (128).
I wonder why the number of dots shows something about the answer.
I wonder if this kind of picture works for simpler (1-digit by 1-digit) multiplication.
I wonder if this kind of picture can help me find the answer to every multiplication problem.
I wonder if a 2-digit by 2-digit problem would include thick horizontal segments.
I wonder how I could show hundreds with the segments.
I wonder if I could use pictures like this for decimal multiplication.

Creating

As they notice and wonder, students may create their own diagrams like the one in this prompt. Many of their ideas will flow from things that they have wondered about. For example, they may

Create a diagram for 4 x 23, and compare it to the one in the prompt.
Create diagrams for 1-digit by 1-digit multiplication problems.
Create diagrams for the 2-digit by 1-digit problems.
Create diagrams for problems that have numbers in the hundreds or thousands.
Create diagrams for problems that have decimal place values.

Whenever they create a new diagram, students should look for patterns and explain how the diagrams show the answers.

Reflecting and Extending

I notice that light segments represent ones, and thick segments represent tens.
I notice that thin segments meet with thin segments in a small dot.
I notice that thick segments meet with thin segments in a larger dot.
I wonder if thick segments would intersect with think segments in even larger dots.
I wonder if I could let the segments stand for any place values I choose.
I notice that when I draw a diagram for 2-digit by 2-digit multiplication, there are two arrays of large (tens) dots.
I wonder if these diagrams can help me think of (new) strategies for multiplying multi-digit numbers.

Notes

You can learn more about using this type of image with decimals in Exploration 6: Visualizing Decimal Multiplication from my book, Advanced Common Core Math Explorations: Numbers and Operations.

The image shows an example of "cross-hatch" multiplication. Thin segments represent ones and thicker segments stand for tens. The segments for one number are shown perpendicular to the segments for the other. The segments intersect at points to form arrays, which students may have seen while learning about the meaning of multiplication.  The size of the dots also represent place value:

ones x ones = ones                                       (small dots formed where thin segments cross thin segments)
ones x tens = tens     tens x ones = tens     (larger dots formed where thin segments cross thicker segments)
tens x tens = hundreds                                 (even larger dots formed where thicker segments cross thicker segments)
etc.

Cross-hatch images are fairly abstract. They probably work best for older or more advanced learners. They offer a means to visualize the distributive property and may help students make sense of rules for multiplying multi-digit numbers. Things to try: (1) ask students to create examples of their own, first for simpler calculations, then for more complex ones, (2) ask students to write and justify statements such as

4 tens x 7 tens = 28 hundreds

as they draw their pictures. (This type of statement would come up in cross-hatch drawing for a calculation such as 43 x 72.)