Multiples Of 3, 6, And 7 - Illustrative Math Tasks

IM Commentary

This task investigates divisibility properties for the numbers 3, 6, and 7. Students first make a list of multiples of 3 and then investigate this list further, looking for multiples of 6 and 7. In addition to noticing that every other multiple of 3 is a multiple of 6, students will see that all multiples of 6 are also multiples of 3 because 3 is a factor of 6. Because the list of multiples of 3 is only long enough to show one multiple of 7, students will have to either continue the list or generalize based upon their observations from part (b). Unlike 6, there is no factor of 3 in 7 and so not every multiple of 7 has a factor of 3: in order to be a multiple of both 3 and 7, a number must be a multiple of 21.

One important difference in the multiples of 6 and 7 that appear in the list of multiples of 3 is that every multiple of 6 is also a multiple of 3. So 6, 12, 18, $\ldots$ all appear in the list of multiples of 3. Since 3 is not a factor of 7, not every multiple of 7 occurs in the list of multiples of 3. The teacher may wish to direct or ask the students about this key difference in the multiples of 6 and 7 which are also multiples of 3. The first solution also refers to the fact that an odd number times an odd number is odd and the teacher may wish to go into this in greater depth as it is another good example of a pattern exemplifying 4.OA.5.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This particular task helps illustrate Mathematical Practice Standard 8, Look for and express regularity in repeated reasoning. Fourth graders make their list of multiples of 3. Then they look for patterns and connections to the multiples of 6 and 7 as stated in the commentary.  They purposely look for patterns/similarities, make conjectures about these patterns/similarities, consider generalities and limitations, and make connections about their ideas (MP.8).  Students notice the repetition of patterns to more deeply understand relationships between multiples of 3 and multiples of 6. They can then compare this relationship to the relationship between multiples of 3 and multiples of 7 and look at the differences between the two sets of multiples. By examining the repeated multiples students can make conjectures and start to form generalizations.  As they begin to explain their processes to one another, they construct, critique, and compare arguments (MP.3). Students would benefit from having access to $\frac14$-inch graph paper and colored pencils for this task. The first solution shows some pictures that students could easily generate with those tools.