Digit Discoveries and Algebra for Middle School - An Investigation

The results below are well-known but, as usual, I am offering an investigation for the classroom that has many objectives:

(1) Digit properties of multiples of 9 (and their 'proofs')
(2) Review place-value and algebraic representation
(3) Investigate patterns based on data collection
(4) Develop inference and conjecture
(5) Introduce students to algebraic proof
(6) And, of course, practice for those open-ended questions we've all come to know and love...

Children are often fascinated by the discoveries they can make regarding 2- and 3-digit numbers. At some point in middle school all students should either discover on their own or be introduced to the remarkable properties of the number 9 in our base 10 number system. The investigation below will explore some of this.

Students are also fascinated by the results of taking a 2- or 3-digit number and reversing its digits. With or without calculators, students like to see how these numbers are related, particularly when they are added or subtracted. In this activity, they will have the opportunity to discover some of these properties and use basic algebra to explain why they work. Perhaps, this will also lead to questions about palindromes, but that's for another day...

The questions below are designed for middle schoolers through Algebra 1. The proofs require some basic algebra, so you can make those parts optional for the prealgebra group. For this group, having them state their conjectures and suggesting possible explanations are more than enough.

STUDENT/READER ACTIVITY/INVESTIGATION

(1) List all of the 2-digit multiples of 9. What do you notice about the sum of their digits?
(2) Using the fact that any 2-digit number can be represented algebraically as 10a+b, show/justify/explain/demonstrate/prove the following:

If a 2 -digit number is a multiple of 9, so is the sum of its digits AND
if the sum of the digits of a 2-digit number is divisible by 9, then the number is a multiple of 9.

(3) If you made sense of (2), why stop with 2-digit numbers! State and prove a similar result for 3- and 4-digit numbers!

Now for reversals:

(4) To be a mathematical researcher, one needs to do what the scientific researcher does. Collect lots of data first, then make conjectures and PROVE them! Choose at least 5 different 2-digit numbers, in addition to the examples below, and complete the table.

Number..........Reversal............Sum..........Difference (Larger-Smaller)
41.....................14.......................55...............27
33....................33......................66................0
72....................27......................99................45
Your turn - do this FIVE more times.

(5) Make conjectures about the how the sum and difference are related to the digits of the original number. Using the algebraic representation 10a+b for any 2-digit number, PROVE your conjectures (or disprove them!).

(6) 72 and 27 are not only reversals. They are are also both multiples of 9. Does this have to be true for any 2-digit multiple of 9? Explain! Further, is there a special property for the sum of the number and its reversal in this case. Make sure you verify conjectures for several cases before attempting to prove it.

(7) Make a similar table for 3-digit numbers. Is there an obvious relationship for the sum of the number and its reversal this time? The difference? Make conjectures and PROVE them!

If you feel this activity is useful, please comment, share it and rate it below. Enjoy!