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
Examples of noticing and wondering
I notice lines [actually segments] and dots of different thicknesses.
I notice a dot at every place that the segments meet (intersect) and that the dots make arrays.
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 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 how I could show hundreds with the segments.
I wonder if I could use pictures like this for decimal multiplication.
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)
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.)