Students will be able to describe the patterns associated with the number 1089 and informally describe the algebraic thinking around the pattern.

Connecting algebra to challenging number patterns.

15 minutes

I have this written on the board:

“Gather round for the magic show.”

I start this lesson immediately. Students don’t need to drop off their stuff or even take our their supplies (yet). I arrange a portion of the room for a brief interactive number magic show. Students gather around and we begin. I have several students model the trick for the class. Everyone is a part of the trick (since they are all observing) but the four volunteers will actually try the calculations.

Modeling this trick is very fun. I give the four volunteers four different color markers. The sit around the table and each observer watches the volunteers closely. I ask them to each pick one number from the magic number bin (a hat or box or something). As the magician, I walk them through the steps of the trick.

“Write your number down.”

“Reverse the order of the digits and write this number down. For example, 123 would now be 321.”

“Subtract the smaller number *from* the larger. For example, I would solve 321 – 123 and get 198.”

“Now reverse this number and write it down. For example, I would reverse 298 to get 891.”

“Finish by adding this two numbers. Write your result down. I would add 298 and 892.”

As we are working through the trick, the audience is both quiet and curious. I always talk about the importance of silence during the trick. We want each student to have a moment to think about the problem and wonder what is happening. I don’t say it explicitly, but I want them to each think, “does that always work?” Once the trick is completed, I ask the four volunteers to read out their answer. Of course they all read out 1089 (I look at their work as they go and make small tweaks if I see an issue). Then I follow with a question for the group: “What do you notice?”

We share some observations and I encourage students to give more than just, “they all got 1089.” I ask them to look closer at each step. Students notice that none of the numbers had the same unit and hundreds digits. Others notice that more than one student had a multiple of 99 pop up before they summed 1089. Its fun to share observations, because much like the patterns we observed the day before, there are many ways to think about the trick that is unfolding before them.

Its important to blur the line of what is true and what isn’t true. “It seems that we always get 1089. Does this always happen? For every three digit number? For every number ever? When will it work and when won’t it work? How do you know?” This type of questioning will create uncertainty and doubt, which is an important step to motivate the investigation ahead. This is much better than just saying “oh yeah, we get 1089 and must get it every time.” I like to push a little further and show them the importance of proof. “You noticed that many students got 189 or 99 in there work. Will this always happen too? There are a lot of things happening here and your job is to try and figure out what seems true and what *is* true. What are some ways we can prove if something will *always* be true?”

As a group students will bring up the idea of algebra if you give analogy and example. You could ask a variety of questions, but I like to keep it simple. “If I wanted you to give me numbers that add to 5, what numbers could you use? What if I wanted you to show me *every pair* of numbers that add to 5? How would you do this?” Asking questions like this will help the class realize that algebra is a great tool for generalizing a finite amount of cases to an infinite amount of cases.

30 minutes

I give the students 30 minutes to play certain properties of the number 1089. The standard I ask them to emphasize is about conjecture. I remind them that one important quality of a mathematician is having the ability to conjecture. Invariably they ask me what “conjecture” means. We talk about it and then I tell them that I am giving them 30 minutes work tinker with the number and form some reasonable conjectures around the worksheet I give out. I circulate and collect the conjectures they form around the number and use those quotes as a basis for conversation and eventually for the summary at the end.

15 minutes

The summary is essentially a moment to gather the conjectures from the class. I try to only share properties of the number that they were able to tap into during their investigation. I remind them of the purpose of this lesson, which is to get them thinking, “do I want to research this number and others like it for my Math Exploratorium?”

A common conjecture: “I think that this trick will work for all three digit numbers if the units and hundreds digits are different.” I respond by asking the class a question, “why can’t they be the same? What would happen?” And if they are unable to answer, “can you give a specific example?” We demonstrate and show that if the digits were the same you would get zero every time.

Another important conjecture, “when we subtract and get a number like 99, we need to treat this number like 099.” Again I ask them to show a specific example and/or explain why they think they are right. We might also ask what types of numbers result in 99 and what types of number don’t. We could sort examples into a table and look for examples.

Other interesting conjectures try and informally describe the relationship between all numbers that appear before 1089: “I think you will always get 99, 198 or some multiple of 99.” I follow up on this one lightly and just ask others to confirm that this did or did not happen.

The most complex conjecture we reach around this number is to touch upon the algebra. If the class reaches some type of conjecture, I follow up a bit on it, but I don’t push to hard here as this lesson is just meant to introduce the beginning ideas around the algebra that defines why we get 1089 for every 3-digit number that has different unit and hundred place digits.

It usually makes sense to only discuss the notion of place value in algebra and help them understand that the number

453 = 100(4) + 10(5) + (1)3 and then directly in line with it to show the pattern,

xyz = 100x + 10y + 1z

And then show the example of

453 -354 and mention what you will encounter in the algebraic representation, where you will probably need to represent the process of borrowing in algebra. I try an avoid getting too far into this discussion. It is really tough to talk through this without rushing. I want my students to have a chance to process this problem and decided if they are ready to attempt the algebra. Its too easy to talk about this, rush it and then ruin the teaching moment. I finish class by reminding them that this lesson was simply an introduction and that they can pursue this investigation for their math Exploratorium.