Make a perform of the very first ten multiples of 3. Which of the number in her list room multiples that 6? What pattern perform you see in where the multiples that 6 show up in the list? Which number in the list space multiples that 7? deserve to you predict once multiples the 7 will show up in the perform of multiples the 3? describe your reasoning.

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## IM Commentary

This job investigates divisibility properties for the number 3, 6, and also 7. Students first make a list of multiples of 3 and then inspection this list further, in search of multiples of 6 and also 7. In enhancement to noticing that every various other multiple that 3 is a many of 6, college student will watch that every multiples the 6 are additionally multiples that 3 because 3 is a variable of 6. Because the perform of multiples that 3 is just long enough to display one lot of of 7, students will have to either proceed the list or generalize based upon their observations from component (b). Unlike 6, there is no factor of 3 in 7 and also so not every many of 7 has actually a element of 3: in bespeak to be a many of both 3 and 7, a number should be a many of 21.

One important distinction in the multiples that 6 and 7 that appear in the perform of multiples of 3 is that every multiple of 6 is likewise a many of 3. So 6, 12, 18, $\ldots$ all show up in the perform of multiples of 3. Due to the fact that 3 is not a element of 7, no every multiple of 7 wake up in the list of multiples that 3. The teacher might wish to direct or questioning the students around this vital difference in the multiples of 6 and also 7 which are likewise multiples the 3. The an initial solution also refers come the reality that one odd number time an odd number is odd and also the teacher might wish to get in this in greater depth as it is another an excellent example that a sample exemplifying 4.OA.5.

The criter for mathematics Practice emphasis on the nature the the finding out experiences by attending to the reasoning processes and habits that mind the students require to develop in order to attain a deep and also flexible understanding of mathematics. Specific benidormclubdeportivo.org lend themselves to the demonstrate of certain practices by students. The techniques that room observable during exploration that a task count on just how instruction unfolds in the classroom. While the is possible that benidormclubdeportivo.org may be linked to several practices, only one practice link will be debated in depth. Possible secondary practice connections may be discussed yet not in the same degree of detail.

This specific task helps illustrate Mathematical exercise Standard 8, look for and express regularity in repetitive reasoning. 4th graders do their list of multiples the 3. Then they look for patterns and connections to the multiples the 6 and also 7 as proclaimed in the commentary. ÂThey intentionally look because that patterns/similarities, do conjectures about these patterns/similarities, consider generalities and limitations, and make connections around their ideas (MP.8). ÂStudents notification the repeat of patterns to much more deeply recognize relationships in between multiples of 3 and also multiples of 6.Â They can then compare this relationship to the relationship between multiples of 3 and multiples that 7 and look at the differences between the two sets of multiples.Â By analyzing the repetitive multiples students have the right to make conjectures and also start to kind generalizations. ÂAs they begin to describe their procedures to one another, lock construct, critique, and also compare disagreements (MP.3). Students would advantage from having access to $\frac14$-inch graph document and colored pencils for this task. The first solution shows some photos that students can easily generate v those tools.

## Solutions

Solution:1 Pictures

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

The multiples that 6 in the list are highlighted in larger, bold face:$$3, \bf\large 6, 9, \bf\large 12, 15, \bf\large 18, 21, \bf\large 24, 27, \bf\large 30.$$It shows up as if every other number in the sequence is a lot of of 6. In bespeak to check out why, below is a picture showing 10 $\times$ 3:

Notice the 2 teams of 3 make 1 group of six. This can be seen in the snapshot as 1 team of 3 purple squares and 1 team of 3 white squares.

So v an even variety of threes, we can group them in pairs to make sixes. Once there is one odd variety of threes, there space some groups of 6 with a leftover team of three: in the picture, an odd variety of threes leaves a purple team which go not complement up with a white group (or evil versa).

The just number in the list that is a multiple of 7 is 21 i beg your pardon is $7 \times 3$. If we write the list of multiples that 7:\beginalign7, 14, \bf\large 21,& \\28, 35, \bf\large 42,& \\49, 56, \bf\large 63,& \\70, 77, \bf\large 84& \\\endalignand then expand the perform of multiples the 3:\beginalign3, 6, 9, 12, 15, 18, \bf\large 21, & \\24, 27, 30, 33, 36, 39, \bf\large 42, & \\45, 48, 51, 54, 57, 60, \bf\large 63, & \\66, 69, 72, 75, 78, 81, \bf\large 84\endalignwe have the right to see that the an initial four multiples of 7 that show up in the list of multiples the 3 room 21, 42, 63, and 84.

21 is $3\times7$.

We acquired 42 together a multiple of 7 since $42=6\times 7$. We deserve to rewrite it together follows:$$6\times7 = (2\times3)\times7 = 2\times(3\times7) = 2\times 21$$This is the exact same as 2 groups of 21. The next one they have in typical is 63, which come from $9\times7$. As before, we have the right to see that this is a many of 21:$$9\times7 = (3\times3)\times7 = 3\times(3\times7) = 3\times 21$$In general, the multiples that 7 that appear in the list of multiples that 3 are additionally multiples the 21, and also these take place each 7th lot of of 3 due to the fact that each seven teams of 3 make a multiple of 7.

Solution:2 arithmetic

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

The multiples that 6 in the list room highlighted in larger, bold face: $$3, \bf\large 6, 9, \bf\large 12, 15, \bf\large 18, 21, \bf\large 24, 27, \bf\large 30.$$ It appears as if every other number in the succession is a lot of of 6. In stimulate to view if this will certainly continue, keep in mind that the multiples that 3 could likewise be composed as $$1 \times 3, 2 \times 3, 3 \times 3, 4 \times 3, 5 \times 3, 6 \times 3, 7 \times 3, 8 \times 3, 9 \times 3, 10 \times 3.$$ The also numbers, 2, 4, 6, $\ldots$ all have a factor of 2 and when this is multiplied by 3 the product has a variable of 6. This explains why the even numbered facets in the sequence are multiples that 6.

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

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\beginalign 10 \times 3 &= (5 \times 2) \times 3 \\ &= 5 \times (2 \times 3) \\ &= 5 \times 6 \endalign

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so that 10 $\times$ 3 is written as a many of 6. The second equation offers the associative building of multiplication. This debate works for any even number in location of 8 because each even number has actually a factor of 2.

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On the other hand, an odd number time an weird number is odd so the 1st, 3rd, 5th, $\ldots$ aspects of this sequence room odd: because 6 is a multiple of 2, any kind of multiple that 6 is likewise a multiple of 2 and also so should be even. This describes why the strange numbered facets of the sequence room not multiples of 6.

The just number in the perform of multiples the 3 i beg your pardon is additionally a multiple of 7 is 21 = 3 $\times$ 7. This is the seventh number in the sequence. We could guess that just as every second number in the sequence is a multiple of 2 for this reason every saturday number in the sequence is a multiple of 7. We can examine that this is for this reason by writing equations prefer in part (b). We use 28 = 4 $\times$ 7 as an example

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\beginalign 28 \times 3 &= (4 \times 7) \times 3 \\ &= 4 \times (7 \times 3) \\ &= 4 \times 21 \endalign