make a list of the first ten multiples the 3. I beg your pardon of the numbers in your list are multiples the 6? What pattern perform you check out in wherein the multiples the 6 appear in the list? Which number in the list room multiples the 7? deserve to you predict as soon as multiples that 7 will show up in the list of multiples of 3? define your reasoning.

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## IM Commentary

This task investigates divisibility properties for the numbers 3, 6, and 7. Students an initial make a list of multiples the 3 and then inspection this perform further, searching for multiples the 6 and also 7. In enhancement to noticing the every other multiple of 3 is a multiple of 6, student will watch that all multiples that 6 are likewise multiples the 3 due to the fact that 3 is a aspect of 6. Since the list of multiples the 3 is only long enough to present one many of 7, students will need to either continue the perform or generalize based top top their observations from part (b). Unequal 6, over there is no variable of 3 in 7 and so not every many of 7 has a variable of 3: in bespeak to be a many of both 3 and also 7, a number need to be a many of 21.

One important difference in the multiples the 6 and 7 that show up in the perform of multiples the 3 is the every multiple of 6 is also a lot of of 3. Therefore 6, 12, 18, \$ldots\$ all show up in the list of multiples the 3. Because 3 is no a variable of 7, no every multiple of 7 wake up in the perform of multiples the 3. The teacher might wish to direct or asking the students around this an essential difference in the multiples the 6 and also 7 i beg your pardon are additionally multiples of 3. The an initial solution additionally refers come the fact that an odd number times an weird number is odd and also the teacher may wish to go into this in higher depth as it is another great example the a sample exemplifying 4.OA.5.

The standards for mathematics Practice focus on the nature of the learning experiences through attending to the reasoning processes and habits that mind that students require to construct in order to obtain a deep and flexible expertise of mathematics. Specific betterworld2016.org loan themselves come the show of certain practices by students. The methods that are observable during exploration the a task depend on how instruction unfolds in the classroom. While the is feasible that jobs may be connected to numerous practices, only one practice connection will be questioned in depth. Possible secondary practice connections may be discussed yet not in the same degree of detail.

This particular task helps illustrate Mathematical exercise Standard 8, look at for and express regularity in repetitive reasoning. 4th graders do their list of multiples of 3. Climate they look for patterns and also connections come the multiples that 6 and also 7 as declared in the commentary. ÂThey deliberately look for patterns/similarities, do conjectures about these patterns/similarities, take into consideration generalities and limitations, and make connections about their principles (MP.8). ÂStudents an alert the repeat of patterns to much more deeply understand relationships in between multiples the 3 and also multiples that 6.Â They can then to compare this relationship to the relationship between multiples that 3 and also multiples that 7 and also look in ~ the differences between the two sets the multiples.Â By analyzing the repetitive multiples students can make conjectures and start to kind generalizations. ÂAs they start to explain their procedures to one another, they construct, critique, and compare debates (MP.3). Students would benefit from having access to \$frac14\$-inch graph document and colored pencils because that this task. The first solution mirrors some photos that students could easily generate with those tools.

## Solutions

Solution:1 Pictures

The first ten multiples of 3 are listed below:\$\$3, 6, 9, 12, 15, 18, 21, 24, 27, 30.\$\$

The multiples of 6 in the list room highlighted in larger, bolder face:\$\$3, flarge 6, 9, flarge 12, 15, flarge 18, 21, flarge 24, 27, flarge 30.\$\$It appears as if every various other number in the succession is a many of 6. In bespeak to check out why, below is a photo showing 10 \$ imes\$ 3:

Notice that 2 teams of 3 make 1 group of six. This deserve to be checked out in the photo as 1 team of 3 purple squares and 1 group of three white squares.

So through an even variety of threes, we can team them in bag to do sixes. As soon as there is one odd number of threes, there space some groups of 6 with a leftover group of three: in the picture, an odd number of threes leaves a purple team which go not enhance up with a white team (or evil versa).

The only number in the list that is a lot of of 7 is 21 which is \$7 imes 3\$. If we write the perform of multiples the 7:\$\$eginalign7, 14, flarge 21,& \28, 35, flarge 42,& \49, 56, flarge 63,& \70, 77, flarge 84& \endalign\$\$and then prolong the perform of multiples that 3:\$\$eginalign3, 6, 9, 12, 15, 18, flarge 21, & \24, 27, 30, 33, 36, 39, flarge 42, & \45, 48, 51, 54, 57, 60, flarge 63, & \66, 69, 72, 75, 78, 81, flarge 84endalign\$\$we can see the the an initial four multiples that 7 that show up in the list of multiples that 3 space 21, 42, 63, and 84.

21 is \$3 imes7\$.

We got 42 as a lot of of 7 because \$42=6 imes 7\$. We have the right to rewrite it as follows:\$\$6 imes7 = (2 imes3) imes7 = 2 imes(3 imes7) = 2 imes 21\$\$This is the same as 2 groups of 21. The following one they have actually in common is 63, which came from \$9 imes7\$. Together before, we can see the this is a many of 21:\$\$9 imes7 = (3 imes3) imes7 = 3 imes(3 imes7) = 3 imes 21\$\$In general, the multiples that 7 that show up in the perform of multiples of 3 are also multiples the 21, and these occur each 7th multiple of 3 since each seven teams of 3 make a multiple of 7.

Solution:2 arithmetic

The an initial ten multiples the 3 are detailed below: \$\$ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. \$\$

The multiples that 6 in the list are highlighted in larger, interlocutor face: \$\$ 3, flarge 6, 9, flarge 12, 15, flarge 18, 21, flarge 24, 27, flarge 30. \$\$ It shows up as if every various other number in the sequence is a lot of of 6. In order to view if this will certainly continue, keep in mind that the multiples the 3 could additionally be composed as \$\$ 1 imes 3, 2 imes 3, 3 imes 3, 4 imes 3, 5 imes 3, 6 imes 3, 7 imes 3, 8 imes 3, 9 imes 3, 10 imes 3. \$\$ The even numbers, 2, 4, 6, \$ldots\$ all have actually a aspect of 2 and when this is multiplied by 3 the product has actually a element of 6. This explains why the also numbered facets in the sequence are multiples the 6.

Alternatively, making use of 10 \$ imes\$ 3 as an example, we can write

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eginalign 10 imes 3 &= (5 imes 2) imes 3 \ &= 5 imes (2 imes 3) \ &= 5 imes 6 endalign

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so the 10 \$ imes\$ 3 is composed as a lot of of 6. The second equation supplies the associative residential property of multiplication. This argument works for any even number in ar of 8 because each even number has actually a variable of 2.

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On the various other hand, one odd number times an weird number is odd for this reason the 1st, 3rd, 5th, \$ldots\$ facets of this sequence space odd: due to the fact that 6 is a many of 2, any type of multiple of 6 is also a lot of of 2 and so must be even. This describes why the weird numbered elements of the sequence space not multiples of 6.

The just number in the list of multiples of 3 which is likewise a multiple of 7 is 21 = 3 \$ imes\$ 7. This is the saturday number in the sequence. We might guess that just as every second number in the succession is a multiple of 2 for this reason every seventh number in the succession is a multiple of 7. We can check that this is so by writing equations prefer in part (b). We usage 28 = 4 \$ imes\$ 7 as an example

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eginalign 28 imes 3 &= (4 imes 7) imes 3 \ &= 4 imes (7 imes 3) \ &= 4 imes 21 endalign