beginning with 9, perform the an initial 10 multiples of 9. In the perform in part (a) what patterns perform you see with the digits in the 10"s place? What patterns perform you see v the number in the 1"s place? making use of pictures, words, or equations, define the trends you it was observed in part (b).

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## IM Commentary

This job examines the amazing fact that once a number is divisible by 9 then the amount of its number is additionally divisible through 9. Just one and also two number numbers room examined here. A natural extension of this task would it is in to watch if this pattern stays true for 3 digit or larger numbers. At this age, students room multiplying huge numbers by a single digit number (4.NBT.5) and also so they might be motivated to pursue more this interesting fact about the amount of the digits in numbers divisible through 9. Knowledge this dominance provides fantastic practice in location value and also properties of arithmetic.

Three remedies are detailed for component (c), explaining the pattern. One gives a verbal description, the 2nd uses arithmetic manipulations, when the third uses a picture. Students working on this task will watch for and make use of framework (MP7): in components (b) and (c), students space looking both to recognize a pattern and also explain why the sample holds. College student will also look for and express regularity in repeated reasoning (MP8), particularly in component (c); fads in general require identify a process which is recurring multiple times.

## Solutions

Solution:1 Arithmetic Properties

The very first 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 deserve to be thought of together 0 tens and also 9 ones for this reason the tens place has a 0. The tens place of 18 is 1. Going with the perform of multiples the 9, the tens digits are\$\$0, 1, 2, 3, 4, 5, 6, 7, 8, 9.\$\$For the ones place, we view that 9 has actually 9 ones, 18 has 8 ones, and going through the perform we gain ones worths of\$\$9, 8, 7, 6, 5, 4, 3, 2, 1, 0.\$\$So the tens place starts at 0 and go up by 1 when the ones place starts at 9 and also goes down by 1.

Another crucial pattern that relates the ones and tens areas is the if we add up the digit in the 10s place and the digit in the people place, we obtain 9 because that each of this numbers.

We will explain why the digit in the tens place goes up by 1 and the number in the ones place goes down by 1.

To go to the next multiple that 9, we add 9 each time. We understand that 9 = 10âˆ’1. So including 9 is like adding 10 and then individually 1. Including 10 increases the the digit in the tens location by 1 (as lengthy as it is no 9). Subtracting 1 to reduce the the digit in the ones location by 1 (as long as it is not 0).

Solution:2 Equations and picture for part (c)

Here we use the properties of arithmetic to help explain why the 10s value rises by 1 and also the ones worth decreases by 1 when adding 9. We study the equation 18+9=27 together an example. We have the right to rewrite 18=1Ã—10+8 and also 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 the end every step to view which rule of arithmetic are being used. The first line is arithmetic, the 2nd line uses the associative home of addition, the third line supplies the commutative residential or commercial property of addition, the fourth line supplies the associative home of addition, and also the 5th line uses the distributive residential property of multiplication over addition. Notice that the very last line shows why the tens place has actually increased by one when the ones areas has lessened 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 move down one (adding 10) and to the left one (subtracting 1). The sum of the digits is quiet the same because the number in the tens place has actually increased by 1 if the number in the ones place has decreased by 1.

Solution:3 photo for (c)

Here we provide a visual representation for the sample in the multiples of 9. We first look at what happens with 9+9:

To make a ten we have moved one eco-friendly block native the an initial 9 to the second 9: this offers us a ten and also eight ones. Therefore the 10s place has increased by one and the ones place has reduced by one. We have the right to see the exact same pattern going native 18 to 27 by adding 9:

Here also we take far one native the 8 in 18 and add it come the 9 to do a 10. The final photo looks a tiny out of order through the 7 ones between the 2 tens however we can shift this around to put the two 10s together. If us look at the collection of equations in the 2nd solution, the truth that the strips room switched around in bespeak to have actually the tens together reflects up when the commutative residential property of enhancement is used.

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This sample will continue to be until us reach 90. When we include 9 come 90 there space no ones to take from in bespeak to add to the 9 and make a 10.