starting with 9, perform the first 10 multiples the 9. In the perform in component (a) what patterns carry out you see with the number in the 10"s place? What patterns perform you see v the digits in the 1"s place? using pictures, words, or equations, describe the patterns you it was observed in component (b).

You are watching: What is a multiple of 9

## IM Commentary

This task examines the amazing fact that as soon as a number is divisible by 9 climate the sum of its digits is additionally divisible by 9. Just one and two number numbers room examined here. A natural extension of this task would be to see if this pattern remains true for 3 digit or bigger numbers. At this age, students are multiplying big numbers by a single digit number (4.NBT.5) and also so they might be encouraged to pursue more this exciting fact around the sum of the number in number divisible by 9. Knowledge this dominion provides terrific practice in ar value and properties of arithmetic.

Three solutions are detailed for part (c), explaining the pattern. One provides a linguistic description, the second uses arithmetic manipulations, while the third uses a picture. Students working on this job will look at for and make use of structure (MP7): in components (b) and also (c), students space looking both to identify a pattern and explain why the sample holds. Students will likewise look for and also express regularity in repeated thinking (MP8), specifically in component (c); patterns in general require identifying a procedure which is recurring multiple times.

## Solutions

Solution:1 Arithmetic Properties

The an initial ten multiples the 9 are\$\$9, 18, 27, 36, 45, 54, 63, 72, 81, 90.\$\$

For the tens places, keep in mind that the number 9 have the right to be assumed of as 0 tens and 9 ones therefore the tens place has a 0. The tens ar of 18 is 1. Going v the perform of multiples that 9, the tens digits are\$\$0, 1, 2, 3, 4, 5, 6, 7, 8, 9.\$\$For the persons place, we watch that 9 has 9 ones, 18 has actually 8 ones, and going through the perform we obtain ones values of\$\$9, 8, 7, 6, 5, 4, 3, 2, 1, 0.\$\$So the tens location starts at 0 and also go up by 1 when the ones place starts at 9 and goes under by 1.

Another important pattern that relates the ones and also tens areas is that if we add up the digit in the tens place and the number in the ones place, we obtain 9 because that each of this numbers.

We will define why the number in the tens ar goes increase by 1 and also the number in the ones place goes under by 1.

To go to the next multiple of 9, we include 9 each time. We know that 9 = 10âˆ’1. So adding 9 is like including 10 and then subtracting 1. Including 10 increases the the digit in the tens place by 1 (as long as it is no 9). Individually 1 reduce the the number in the ones place by 1 (as long as the is no 0).

Solution:2 Equations and snapshot for component (c)

Here we usage the properties of arithmetic to assist explain why the 10s value increases by 1 and also the ones worth decreases by 1 when including 9. We examine the equation 18+9=27 together an example. We can rewrite 18=1Ã—10+8 and 9 = 10 - 1 to get

eginalign18+9 &=(1Ã—10+8)+(1Ã—10âˆ’1) \&=1Ã—10+(8+1Ã—10)âˆ’1\&=1Ã—10+(1Ã—10+8)âˆ’1\&=(1Ã—10+1Ã—10)+(8âˆ’1)\&=(1+1)Ã—10+(8âˆ’1).endalign

We have actually written out every step to see which rule of arithmetic are being used. The first line is arithmetic, the 2nd line supplies the associative building of addition, the 3rd line supplies the commutative residential or commercial property of addition, the fourth line provides the associative property of addition, and the 5th line provides the distributive building of multiplication over addition. Notice that the very last line shows why the 10s place has actually increased by one when the ones locations has lessened by one after including 9.

The arithmetic steps over can be visualized in the table below: Each lot of of 9 is emphasize in green. To move from one lot of of 9 to the next, in the table, we move down one (adding 10) and to the left one (subtracting 1). The sum of the digits is tho the same due to the fact that the digit in the tens place has increased by 1 while the number in the ones ar has reduced by 1.

Solution:3 snapshot for (c)

Here we give a visual depiction for the sample in the multiples of 9. We an initial look at what happens with 9+9: To do a ten we have moved one eco-friendly block native the very first 9 to the 2nd 9: this provides us a ten and also eight ones. For this reason the 10s place has increased through one and also the ones place has decreased by one. We deserve to see the exact same pattern going native 18 to 27 by adding 9: Here as well we take away one native the 8 in 18 and include it come the 9 to do a 10. The final photo looks a little out that order through the 7 ones in between the two tens but we can change this roughly to put the two tens together. If us look at the set of equations in the 2nd solution, the reality that the strips room switched about in bespeak to have actually the tens together reflects up once the commutative home of enhancement is used.

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This sample will remain until we reach 90. As soon as we include 9 come 90 there space no people to take from in bespeak to add to the 9 and make a 10.