I asked some students today to add the numbers from 1 through 100. After fielding some questions and making sure the students knew we were adding 1 + 2 + 3 + … + 99 + 100, they were off!

A few tried the obvious… 1 + 2 + 3..the long way. Luckily I had only given them about 1/8 of a page to work on!

The way I tried to show was using some skills we learned back in elementary school…looking for groups of numbers that add up nicely. For example, add the list: 2 + 7 + 4 + 3 + 6 + 8. Did you get 30? I did…but I did some rearranging (2 + 8) + (3 + 7) + (4 + 6) and said 10 + 10 + 10 is 30. I used that, along with an abbreviated list of numbers, to get the sum:

By adding groups of 100, I was able to get 4900! Then there’s the extra 100 sitting on top, which brings us to 5000. The missing 50 brings us right up to 5050.

A student, though, took a different approach. Same idea, different way of looking at it. I was impressed!

I presented a modified version…he wrote out a bunch more work. I presented only the pattern that he figured out from looking at it.

# 4. Lattice Multiplication

We’ve made it…we can multiply small numbers using area, we can multiply small numbers pretending to do area, and we can keep pretending as we use bigger and bigger numbers.

Now comes full-blown lattice multiplication. Here we go.

How it looks at the end:

What IS that, you may wonder. It’s a mix of area model, with some old-school place value thrown in. Let’s break it down, and start over.

That looks just like the area model…except no zeroes! Ok, we can deal with that. We’ll just use place value later on. Let’s multiply like area model (but we’ll do it a little off-center).

It looks more or less like area, only with the answers in corners, and no extra zeroes. That’s basically it. Now we just need to add place values just like we used to. Only instead of straight up and down, we need to go at an angle. So…let’s add imaginary lines in red to cut each box in half.

Now we just need to add inside each band…they all correspond to the same place value, and we can carry (in red) to the next line.

Finally, the answer is there and we can read it off: 33,813!

It’s not intuitive, but if you take a little bit from all the other methods, then you end up with lattice.

Good luck!

**note: if you multiply, and the answer to go in a box is only one digit, put a zero in the upper corner:

# 3. Expanded Area Model for Multiplication

We saw previously that drawing pictures is a good way to think about multiplication.

Multiplying is really just adding a whole bunch of things together, as long as they’re the same size. The little boxes showed that, but it got a little crazy once we used bigger numbers.

So, let’s keep on with the pattern of pretending, by trying out 14 x 7. Since I don’t actually know 14 x 7, I will break it down: 14 is really just 10 and 4. So I’ll split that side length in my picture. Then I have two little rectangles…10×7 and 4×7! I can do that!
(and if you want to get really fancy/technical, we did just demonstrate the distributive law!)

Wow, that was WAY easier than expected! Maybe we can apply that same logic to our original problem?

So…
867 = 800 + 60 + 7
and
39 = 30 + 9

BAM! Still annoying…but addition is better than multiplication.

Now…how does this all help with the lattice model? That ugly beast of multiplication…

# 2. Area Model of Multiplication

Another way to do multiplication is to think about it as finding the area of some rectangles.

Now, if we draw pictures of area for multiplication, we can start to visualize it! Let’s pretend, over the next few posts, that I can draw squares and rectangles. And that I can take good pictures.

For example, 3 x 2 = 6 in the area model is…

Not too bad. But when we get to bigger problems, we don’t really want to draw out every box. It gets tedious even for small numbers, and worse as they get bigger. Who’s going to count?!? Instead, we rely on our basic math facts, and pretend that there are little boxes!

It seems like we’re just reviewing our basic multiplication facts, right? Well, yes.

But the interesting part is in the next post, when we see how larger numbers (bigger than 10) mix with the area model to simplify things.

# 1. Lattice Multiplication: Introduction

Remember the good old days of long multiplication? You multiply, bring down, and carry. There are so many new methods, and lattice is one. It makes complete sense logically once you understand it, but it is one of the least intuitive structures. Check it out:

It’s crazy looking, and definitely not natural. Let’s get there one step at a time. Let’s solve a problem using our old-school math, and then figure it out using a few different methods. Eventually we’ll do it using lattice.

Problem:

867 x 39

Not too bad…so familiar and comfortable. Next we’ll look at basic multiplication using the area model.

# Arithmetic with Dots

Ben Orlin’s recent post on using arrays with multiplication made me reminisce on days past, in which I talked to other educators about foundations of mathematics. I’ve also started watching a great youtube lecture series on the history of math, which also made me rethink how I should be teaching math.

In tutoring, I’ve spent a lot of my time showing students how their current work is just their old work with a twist. Algebra is just arithmetic with a sprinkle of magic. (I’ve seen first graders do 1 + __ = 4). That’s a one-step equation in algebra, and students are forced to subtract one from both sides. WHY?!? Calculus is just algebra with very BIG and very small concepts added in. I spent a while showing a student that the equation of a tangent line to a curve is just point-slope form of a line. Oy. Then she got it.

So let’s look at how arrays can give us the basic operations, without needing any variables.

Addition: Hopefully obvious. 3 + 2 = 5. Count the dots.

Subtraction: A bit harder, because it involves “take away”. But otherwise obvious (once you know it, anyway).

That’s it for 1-dimensional. Now we move into 2D.

Multiplication can be defined as repeated addition (3 + 3 + 3 + 3 + 3 = 3×5) or as equal groupings (5 groups, each one has 3 in it). But instead of drawing numbers on a line, we go into arrays.

I’ll leave the details to Ben’s post.

Division is where I haven’t seen too much. We pester kids with the fact that it’s the inverse of multiplication using families (“If 2 x 5 = 10, then 10/5 = 2”). But where’s the JUSTIFICATION for it? Where can I SEE this happen?

The struggle later turns into remainders, and then remainders as decimals and fractions.
There are many great division strategies: standard long-division, partial quotients (one of my favorites), and many others.

But how do we show it? Well, just do multiplication backwards. Multiplication = repeated addition or equal groups. So division = repeated subtraction, or equal groupings.

Then let’s try 12 / 3.

So, we’re left with remainder 1. That seems reasonable, right? But what about when we want students to represent the remainder as a *gasp* fraction? We can tell them that since it’s in 3 equal groups, we should put 3 as the denominator and then the remainder 1 becomes the numerator. But…WHY?!?

Well, we’re counting rows. How many rows did we get? 4. How many more ROWS do we have? Well…1 piece left over. OK, so if we JUST LOOK AT THE ONE ROW left there, all of a sudden we see it: there is 1/3 of a row! So 13/3 = 4 1/3.

Cheers.
Jeff

# Motivation

The longer I work in education, the more I notice that what people are really discussing under different guises is the use of motivation. How do we motivate students? How do we motivate teachers and administrators?

Everyone responds differently to different things – I am motivated by academic success, making my close friends and family happy, and food.

What motivates you? Have these motivations played any part in your education?

[more detailed explanations and analyses of what I’m talking about to come later]

# Math Motivation

Lately, a lot of focus in what I’ve seen in math has been around motivation. It’s a key term in education, and more or less a standard to which people are help – who is supposed to motivate kids to learn, and who is held accountable for it? Some say parents, teachers, districts, kids themselves, etc. So how do we motivate them? I’ve seen a few ways:

1. Make the work more interesting/relevant/practical/etc.

This method relies on the kid engaging with the work because it just makes sense to engage with it. It’s interesting, fresh, and usually much more concrete than solving a number problem.

2. Make their lives dependent on it.

You know those posters…”Think You Won’t Need Math?” then it lists jobs that need math…right? I agree that all jobs, in fact all areas of life rely on math in some way. However, I think that motivation stops being relevant around middle school. I’m an avid math educator, and I think many of the skills you learn in math could help in life. I really do. But who needs trig unless they plan on going into math or science fields? Why does everyone need to know trig? Why not teach them how to work collaboratively, so that the kid who knows trig and hates writing can work with the writer who hates trig?

**Side note: I hate these posters.

3. Miscellaneous

Some teachers make the subject interesting by showing their own interest. Some just engage students personally (or in a similar way), and motivate them to get an education (and students accept that math is part of that).

Is just one of these ways enough? Do students need a blend, or can we assess the levels of motivation they have coming in and work off those? Some students will show up and learn no matter what you do…do they need to be “motivated” further?

# Math Example – The Chinese Room

Dan Meyer recently posted an article entitled The Chinese Room. The moral of the story is: if I give you something written in Chinese and a manual for writing down other Chinese characters, are you literate in Chinese? Now, you have to write the actual response (assuming the manual is correct), and to someone who is actually literate the answers must be correct. So…are you?

It’s an interesting debate, and one I’ve seen many time working with students. The idea is a bit abstract at first, but this post is dedicated to the same idea, outlined in a particular example: Linear Equations
Consider the two equations:

(1) y = -(3/4)x + 24

and

(2) 3x + 4y = 24

Mathematically, they’re equivalent. But do they mean the same thing? Same slope, same intercept…same interpretation?

When I’ve seen it taught, one of the fundamentals is changing between the 2 primary forms of equations (slope-intercept, standard form). But why?

Equation (1), how we often write them, to me signifies change over a given interval. You start with 24, and every 4 units it decreases by 3. Dollars per hour? Cookies per day? The graph ultimately shows how much you have (or owe, if you like negatives). I think of it as filling in the blank “after ___ days you will have ___ dollars” (change units as you please). There is an answer to be found, depending on what you’re looking for. Traditional algebra.

Equation (2), though, is something totally different. I see that, usually, as comparing costs. I sold x-ray goggles and yams, making \$24 total. The goggles cost \$3, and the yams \$4. So, how much did each cost? Here, the 2 variables each represent their own object. There is no change over time. So, what does the slope mean? For every goggle, the price of yams goes down? Not quite. It shows how much each could cost. Each ordered pair is a possibility. There is no one answer you’re seeking. Uncertainty, conditional information, no answer.

Reflecting on my own experience, I have basically taught students to turn all equations into form (1) and go from there (unless you’re graphing using intercepts – but that’s another show). So, I’ve been teaching them to read and write Chinese according to the post title. I haven’t promoted true understanding.

Have you done what I’ve done, or how have you been able to get students to the heart of it?