starting with 9, perform the an initial 10 multiples the 9. In the list in part (a) what patterns execute you see with the number in the 10"s place? What patterns do you see with the digits in the 1"s place? utilizing pictures, words, or equations, define the patterns you it was observed in part (b).

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IM Commentary

This task examines the amazing fact that when a number is divisible by 9 climate the sum of its digits is likewise divisible by 9. Just one and two number numbers space examined here. A natural extension of this task would be to check out if this pattern continues to be true for three digit or bigger numbers. In ~ this age, students room multiplying huge numbers through a single digit number (4.NBT.5) and also so they could be motivated to pursue additional this amazing fact around the amount of the number in number divisible by 9. Knowledge this preeminence provides terrific practice in ar value and also properties of arithmetic.

Three services are detailed for component (c), explaining the pattern. One offers a verbal description, the second uses arithmetic manipulations, while the third uses a picture. Students functioning on this job will watch for and also make usage of framework (MP7): in components (b) and (c), students space looking both to recognize a pattern and explain why the sample holds. Student will likewise look for and also express regularity in repeated reasoning (MP8), especially in component (c); patterns in basic require identifying a procedure which is repeated multiple times.


Solutions

Solution:1 Arithmetic Properties

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

For the 10s places, note that the number 9 deserve to be believed of as 0 tens and 9 ones so the tens place has actually a 0. The tens place of 18 is 1. Going with the list of multiples the 9, the 10s digits are$$0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$$For the people place, we watch that 9 has 9 ones, 18 has 8 ones, and also going v the perform we gain ones worths of$$9, 8, 7, 6, 5, 4, 3, 2, 1, 0.$$So the tens location starts at 0 and go increase by 1 if the ones location starts at 9 and goes down by 1.

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

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

To walk to the following multiple of 9, we add 9 every time. We recognize that 9 = 10−1. So including 9 is like including 10 and also then individually 1. Adding 10 rises the the number in the tens place by 1 (as long as that is not 9). Individually 1 to reduce the the number in the ones place by 1 (as long as the is not 0).

Solution:2 Equations and snapshot for part (c)

Here we usage the nature of arithmetic to help explain why the 10s value boosts by 1 and also the ones worth decreases through 1 when adding 9. We examine the equation 18+9=27 as an example. We can rewrite 18=1×10+8 and also 9 = 10 - 1 come get


eginalign18+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 action to view which rules of arithmetic room being used. The very first line is arithmetic, the second line offers the associative building of addition, the third line offers the commutative residential property of addition, the fourth line uses the associative building of addition, and also the fifth line provides the distributive building of multiplication over addition. An alert that the really last line shows why the 10s place has actually increased by one while the ones locations has diminished by one after including 9.

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

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

Solution:3 picture for (c)

Here we give a visual depiction for the sample in the multiples the 9. We very first look in ~ what happens through 9+9:

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To make a ten we have actually moved one eco-friendly block from the an initial 9 to the second 9: this offers us a ten and also eight ones. For this reason the 10s place has actually increased through one and also the ones location has lessened by one. We can see the same pattern going from 18 to 27 by adding 9:

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Here as well we take away one native the 8 in 18 and include it come the 9 to do a 10. The final snapshot looks a little out the order with the 7 ones in between the two tens yet we can transition this approximately to put the two 10s together. If we look in ~ the collection of equations in the 2nd solution, the fact that the strips space switched around in bespeak to have the 10s together shows up as soon as the commutative residential or commercial property of enhancement is used.

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This sample will remain until us reach 90. As soon as we add 9 to 90 there are no persons to take it from in bespeak to add to the 9 and make a 10.