Thought for the day: Be sure to give your students plenty of time to observe and question before inserting your own ideas. Don't worry if they take a long time to get to the ideas that you are aiming for.

Concepts: factors and multiples, properties of multiples, multiplication facts, logical reasoning, Venn diagrams

Beginning

I notice four overlapping circles in different colors.
I notice that each circle and the region outside the circles is labeled with "M" and a number.
I wonder what the Ms mean.
I notice that the circles remind me of Venn diagrams that we use in language classes.
I notice that each whole number 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written somewhere in the diagram.

Exploring

I notice that the M2 circle has the even numbers, 2, 4, 6, 8, and 10.
I notice that every number is inside the M1 rectangle.
I wonder if “M” stands for “multiple.”
I wonder where the next ten whole numbers should go.
I wonder why the M4 circle is completely inside the M2 circle.
I wonder if every whole number belongs somewhere in the diagram.
I wonder if I could make a diagram like this using other multiples and/or more circles.

Creating

After they have had experience filling in numbers in the diagram, looking for patterns, making observations, and asking new questions, students can begin to create their own Venn diagrams using other combinations of multiples. Part of the challenge is to draw a correct arrangement of circles to fit the properties of the multiples that they choose. For example, if they include multiples of 3 and 6, the circle for 6 will need to be completely inside the circle for 3, because every multiple of 6 is a multiple of 3.

Reflecting and Extending

I notice that every whole number belongs in a single region in the diagram.
I notice that regions where the circles overlap often show patterns (certain types of numbers).
I notice that multiples of 2 and 3 are always multiples of 6 and that multiples of 2 and 5 are always multiples of 10.
I wonder if multiples of a and b are always multiples of a • b.
I wonder
if multiples of a • b are always multiples of a and b.
I notice
that most, but not all, of the numbers outside the circle are prime numbers. (1 and 49 are not prime.)
I wonder if I could draw a diagram like this for other multiples (like 1, 2, 3, 5, and 6 or 1, 2, 3, 5, and 7, for example).

Notes

This prompt is a Venn diagram that illustrates relationships between multiples of different numbers, where M1 stands for multiples of 1 (which applies to all counting numbers), M2 stands for multiples of 2 (even numbers), M3 represents multiples of 3, etc. For example:

• M4 is completely inside the M2 circle, because every multiple of 4 is a multiple of 2.

• The intersection of M2 and M3 (which includes four smaller regions) contains the numbers, 6, 12, 18, 24, 30, 36, etc., because every number that is a multiple of both 2 and 3 is always a multiple of 6.

• Numbers that are outside all four circles cannot be multiples of any of 2, 3, 4 or 5.

• The small region in the center that is contained with all four circles contains the multiples of 60 (numbers that are multiples of 2, 3, 4, 5, and 6).

Students who are familiar with least common multiples may make connections to this prompt, and those who are not familiar with the concept may begin to think about it informally as try to determine the type of number that belongs in each region.

The image below shows the placement for all whole numbers 1 – 60. Of course, students may continue beyond 60 if they like.

Example: the number 50 appears in the purplish region along with 10, because it is a multiple of 2 and of 5 but not of 3 or 4. Thus, it is contained within the M2 and M5 circles, but is outside of the M3 and M4 circles.