Starting through 9, list the first 10 multiples of 9. In the list in part (a) what trends do you check out through the digits in the 10"s place? What trends perform you check out via the digits in the 1"s place? Using pictures, words, or equations, explain the patterns you observed in part (b).

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## IM Commentary

This task examines the interesting fact that as soon as a number is divisible by 9 then the sum of its digits is likewise divisible by 9. Only one and also two digit numbers are examined below. A organic extension of this task would be to see if this pattern stays 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 could be encouraged to pursue even more this exciting fact about the amount of the digits in numbers divisible by 9. Understanding this rule gives great exercise in place worth and also properties of arithmetic.

Three services are offered for part (c), explaining the pattern. One offers a verbal description, the second supplies arithmetic manipulations, while the third uses a photo. Students functioning on this task will look for and make use of framework (MP7): in parts (b) and (c), students are looking both to determine a pattern and describe why the pattern holds. Students will additionally look for and also expush regularity in recurring reasoning (MP8), specifically in component (c); trends in basic call for identifying a process which is repeated multiple times.

## Solutions

Solution:1 Arithmetic Properties

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

For the tens locations, note that the number 9 have the right to be believed of as 0 10s and also 9 ones so the 10s place has a 0. The tens area of 18 is 1. Going via the list of multiples of 9, the tens digits are\$\$0, 1, 2, 3, 4, 5, 6, 7, 8, 9.\$\$For the ones area, we view that 9 has actually 9 ones, 18 has 8 ones, and going through the list we acquire ones worths of\$\$9, 8, 7, 6, 5, 4, 3, 2, 1, 0.\$\$So the 10s area starts at 0 and go up by 1 while the ones place starts at 9 and also goes dvery own by 1.

Another essential pattern that relates the ones and also tens locations is that if we include up the digit in the 10s location and also the digit in the ones location, we obtain 9 for each of these numbers.

We will certainly describe why the digit in the 10s location goes up by 1 and the digit in the ones area goes down by 1.

To go to the next multiple of 9, we add 9 each time. We understand that 9 = 10−1. So including 9 is like adding 10 and also then subtracting 1. Adding 10 rises the the digit in the 10s area by 1 (as lengthy as it is not 9). Subtracting 1 decreases the the digit in the ones place 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 describe why the 10s value increases by 1 and also the ones value decreases by 1 once adding 9. We examine the equation 18+9=27 as an example. We have the right to rewrite 18=1×10+8 and 9 = 10 - 1 to 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 created out every action to check out which rules of arithmetic are being provided. The initially line is arithmetic, the second line supplies the associative home of enhancement, the 3rd line offers the commutative residential or commercial property of addition, the fourth line offers the associative residential property of addition, and the fifth line supplies the distributive residential property of multiplication over addition. Notice that the very last line shows why the tens place has actually enhanced by one while the ones locations has decreased by one after adding 9.

The arithmetic measures above can be visualized in the table below: Each multiple of 9 is highlighted in green. To move from one multiple of 9 to the next, in the table, we move dvery own one (including 10) and to the left one (subtracting 1). The amount of the digits is still the same because the digit in the tens location has actually increased by 1 while the digit in the ones area has actually decreased by 1.

Solution:3 Picture for (c)

Here we give a visual depiction for the pattern in the multiples of 9. We first look at what happens with 9+9: To make a ten we have actually relocated one green block from the initially 9 to the second 9: this offers us a ten and also eight ones. So the 10s location has boosted by one and also the ones area has decreased by one. We deserve to watch the very same pattern going from 18 to 27 by including 9: Here as well we take away one from the 8 in 18 and also include it to the 9 to make a 10. The last picture looks a little out of order with the 7 ones between the 2 tens but we have the right to shift this approximately to put the two tens together. If we look at the set of equations in the second solution, the fact that the strips are switched roughly in order to have actually the 10s together reflects up as soon as the commutative home of addition is provided.

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This pattern will remain till we reach 90. When we include 9 to 90 tright here are no ones to take from in order to include to the 9 and make a 10.