Starting with 9, list the first 10 multiples that 9. In the perform in component (a) what patterns carry out you see through the number in the 10"s place? What patterns perform you see with the digits in the 1"s place? making use of pictures, words, or equations, explain the patterns you it was observed in component (b).

You are watching: What numbers are divisible by 9

## IM Commentary

This job examines the interesting fact that as soon as a number is divisible through 9 then the amount of its number is additionally divisible through 9. Only one and two digit numbers space examined here. A natural extension of this activity would be to view 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 so they could be motivated to pursue more this amazing fact about the sum of the digits in numbers divisible by 9. Expertise this rule provides fantastic practice in location value and properties that arithmetic.

Three options are provided for component (c), explaining the pattern. One gives a verbal description, the second uses arithmetic manipulations, while the third uses a picture. Students functioning on this task will look for and make use of structure (MP7): in parts (b) and (c), students are looking both to recognize a pattern and also explain why the sample holds. Student will also look for and express regularity in repeated reasoning (MP8), particularly in part (c); fads in general require identifying a process which is recurring multiple times.

## Solutions

Solution:1 Arithmetic Properties

The an initial ten multiples of 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 believed of as 0 tens and also 9 ones so the tens place has actually a 0. The tens ar of 18 is 1. Going with the list of multiples that 9, the 10s 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 also going through the perform we get ones worths of\$\$9, 8, 7, 6, 5, 4, 3, 2, 1, 0.\$\$So the tens place starts in ~ 0 and also go up by 1 while the ones place starts in ~ 9 and goes under by 1.

Another crucial pattern that relates the ones and tens places is the if we add up the number in the 10s place and also the digit in the ones place, we get 9 for each of this numbers.

We will define why the digit in the tens place goes up by 1 and the digit in the ones location goes under by 1.

To walk to the next multiple that 9, we add 9 each time. We understand that 9 = 10−1. So adding 9 is like adding 10 and also then individually 1. Adding 10 boosts the the number in the tens place by 1 (as lengthy as that is no 9). Individually 1 decreases the the digit in the ones place by 1 (as lengthy as the is not 0).

Solution:2 Equations and picture for component (c)

Here we usage the properties of arithmetic to aid explain why the tens value boosts by 1 and also the ones worth decreases by 1 when adding 9. We examine the equation 18+9=27 together an example. We have the right to rewrite 18=1×10+8 and also 9 = 10 - 1 come get

\beginalign18+9 &=(1×10+8)+(1×10−1) \\&=1×10+(8+1×10)−1\\&=1×10+(1×10+8)−1\\&=(1×10+1×10)+(8−1)\\&=(1+1)×10+(8−1).\endalign

We have actually written the end every step to watch which rules of arithmetic room being used. The an initial line is arithmetic, the second line offers the associative residential or commercial property of addition, the third line offers the commutative residential or commercial property of addition, the fourth line offers the associative residential property of addition, and also the 5th line provides the distributive residential property of multiplication over addition. An alert that the really last line shows why the 10s place has actually increased through one when the ones areas has diminished by one after adding 9.

The arithmetic steps above can it is in visualized in the table below:

Each many of 9 is emphasize in green. To move from one multiple of 9 come the next, in the table, we relocate down one (adding 10) and also to the left one (subtracting 1). The amount of the number is tho the same since the number in the tens place has actually increased through 1 while the number in the ones ar has decreased by 1.

Solution:3 picture for (c)

Here we offer a visual depiction for the sample in the multiples of 9. We an initial look at what happens through 9+9:

To do a ten we have moved one green block from the first 9 to the 2nd 9: this provides us a ten and also eight ones. So the tens place has actually increased through one and the ones location has decreased by one. We deserve to see the same pattern going from 18 come 27 by including 9:

Here also we take away one from the 8 in 18 and add it to the 9 to make a 10. The final picture looks a tiny out that order v the 7 ones between the two tens but we can shift this approximately to placed the two 10s together. If we look at the set of equations in the second solution, the reality that the strips room switched roughly in order to have actually the 10s together mirrors up as soon as the commutative residential or commercial property of addition is used.

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This pattern will continue to be until we reach 90. Once we add 9 to 90 there are no ones to take it from in order to add to the 9 and make a 10.