Sunday, January 5, 2014

Help A Rookie Elementary Math Teacher Teach Addition And Large Numbers

A teaching rule I have for myself is: Always try to follow the units of a solid curriculum, even when I'm completely ignoring their individual lessons.

It's a good rule. For one, it imposes a structure on the year, a net for me to play tennis with. It also means that I get lots of the benefits of a curriculum without losing any flexibility. By sticking to the units I make sure that, on a macro level, the material that we're learning is connected and that topics are reviewed and previewed in a sensible fashion. (An added bonus is that if I use the big-level sequencing of a curriculum, and I can use a lot of their homework materials, cutting down my workload.)

(Another teaching rule: Homework is sometimes necessary, but its impact on my life should be minimal.)

Anyway: I'm prepping for the next few weeks of 4th Grade right now, and my usual rules and tricks are faltering, and I could use some help.

The TERC Investigations curriculum has, as its next unit, a series of lessons about place value, addition and subtraction and numbers up to 10,000.

The problem is that I'm pretty sure that this stuff is all going to be too easy for my kids. So, just skip the entire unit. Well, for a couple of reasons I try to avoid that. First, maybe my kids are 85% of the way to where they need to be. That's close enough that this unit might be too easy, but not a waste of their times. Second, I'm sure that I have some students who would super-duper benefit from this unit.

So, just make the unit harder. How? By bumping up the numbers that we're dealing with by several orders of magnitude, I guess.

But the other problem is that I don't really have any mathematically juicy lessons, either for addition or large numbers. After a bit of thinking, I was able to come up with some preliminary ideas:
  • Make a book with 100 numbers on each page. What page would 45,321 be on? What about 12,000,345?
  • If we could deal with some juicy visual contexts that involve huge quantities, then we could end up with contexts for lots of big-number math.
  • Pick random big numbers; pick random kids; ask them to add random numbers in their brains, share strategies; rinse; repeat.
  • More? Help?

I'm also still not really sure where my kids struggle. Like, they definitely sometimes get freaked out by large numbers, but what sorts of mistakes or misconceptions are they going to have with this stuff? What is it going to look like for a kid to mess up with large numbers or big addition?

Any help that you guys could provide here would be helpful. I'll hang out in the comments.


  1. Would this go well with an estimation unit?

    1. Yeah, for sure. I'll spend some time on 101qs and see if I can find some enormous quantities there.

  2. I'm interested in seeing how the comments play out here. I have a 4th grade son of my own. One of the things I've noticed with him and larger numbers is that he sometimes loses place value - especially when there are zeros in the middle of the number. A number like 1964 is no problem but he might slip up on a number like 19604 or 19064 in terms of just how large it is and what words to use to name it.

    Rather than just skipping, would it make sense to pre-test the unit and target any small areas where there are multiple kids with misconceptions?

    1. Thanks for that mistake. If only there were a site that collected these things that I could just refer to in my planning...

  3. Mental math, counting circles (up or down by some big number), explanation of mental strategies.

    Word problems, especially "make your own" word problems ("Pick a calculation from homework page X and make up a problem for it." Then compile the class problems and solve in groups or as a class.)

    Place value card game: Write __ , __ __ __ on paper. Take turns drawing a number card, place digit in one of the spaces. Can't move digit once placed. Vary target, so highest number wins sometimes, and lowest number wins other times.

    Patterns that need big numbers? The first one that comes to mind is "abacaba...":
    Or fill in a very large Pascal's Triangle, as many rows as will fit on your classroom wall.

  4. When I was trying to help my kids understand place value a little better, we played around with a "binary adding machine" made out of duplo blocks. They both seemed to like it a lot. You'll get the sense of it here:

  5. As a sixth grade teacher these strategies will pay off for the future:
    1) Have the students design subtraction problems that have a specific number of instances where they have to borrow (You choose how many instances) without using a number with a digit of zero and they have to have at least one digit of a 1 in the bigger number.
    2)Most kids/adults use calculators for numbers with more than 3 digits so begin to have them look for the patterns in numbers when adding. Example if you take 5,000 and add 6,000, make them change both numbers so that they have the same sum. After finding a successful solution make them subtract their numbers and look for the patterns between multiple solutions. Patterns, patterns, patterns...
    3)Start transitioning them from finding numbers for a blank space/shape to finding numbers represented by a picture/symbol then replaced by a letter/variable.
    4)Number Sense strategy for subtraction: Make them find the answer through estimation. Make them round each number to the nearest ten thousands and subtract, make them round to the nearest thousand and subtract, round to the nearest hundered and subtract,.... so on and so on. The important part of that activity is the reflection and recognition of patterns.... and what happens to their answers as the digit they are rounding to gets smaller in place value.

    Having a visual representation of a number might be a good strategy too. Find pictures of different stadiums/arenas and have the students estimate how many sets there are. Working with large numbers is difficult because they can't generalize just how big they are, and wouldnt be able to guess if their answer is right. Providing those mental strategies, knowledge of patterns that should happen can help make up for the lack of life experience with big numbers.

    1. Your first few suggestions remind me that algebra isn't that far away for these kids. Maybe this is an opportunity to start pushing them toward patterns and generalizations in their addition and subtraction.

  6. On twitter @FedericoChialvo mentioned James Tanton's Exploding Dots. I highly recommend it. About half way through lesson 1.4 he discusses how to add from left to right using his exploding dots. He continues to concept in the next lesson on subtraction. This would really solidify place value.

    1. The issue is that my kids can, basically, add and carry and all that stuff without too much trouble. What I'm looking for is some juicy math that will require addition, but won't be boring for kids who are computationally comfortable.

  7. Do they have a sense of how big 10,000 is? Drawings, maps, or number lines involving large numbers of reasonable scale. For example, place 110 & 2500 on a number line with a tick mark at zero and nothing else.

    Adding or subtracting three (or more) terms with incentive to think about it: 20,000 + 300 +40,000. Or: 40,000 + 37 - 10,000. Reasoning out whether you have to go in order is more like algebraic thinking, I think, than the x + 800 = 1100 type of thing.

    1. Your number line idea great. And I love the push toward articulating rules about computation.

  8. I struggle with this same exact thing in 5th grade. Our third unit in Investigations is a number unit with addition/subtraction and I have the mix of students who would really benefit from it while others I feel have a great grasp of place value within computation. I, like you, like to "stick to the curriculum" and don't like the feel of just skipping it. I also don't like just making the numbers larger bc generally it is just more cumbersome than difficult.
    I decided to try it this year as more of a math workshop model during the entire unit. I started with a number talk each day to be sure that students were all having a chance to hear each other talk about strategies and then students worked in groups. The groups were heterogeneous, however I worked with a different group of students each day to be sure the ones who needed extra assistance got it and the ones who needed challenging got the questions they needed to push them.
    I think for 4th grade, the students who are comfortable, could move on and play Close to 7500. I find that game challenging for the kids and they especially like trying the scoring with + and - scores...adds a different dimension.
    In our district, we also have an addition 50 minute block 2x/week for RTI and I worked with those students in Marilyn Burn's Do The Math Add/Sub so the Investigations unit did seem to fly by.
    I find it was more about the discussions and about the game strategies that benefited the students the most.
    So, that probably was not much help at all, but I just wanted you to know that I have the same problem each year!
    -Kristin @Mathminds

    1. It's helpful to hear how you deal with it, so thanks!

      And, just because I'm writing here now, a quick thought about "sticking to the curriculum." I use curricula to ground my work. It's there as an idiot check to make sure that I don't mess over the kids. It's there to be more thoughtful than I can be, and it's there to teach me everything the authors already know about this content.

      The first thing that I'll probably do is find a lesson for tomorrow that would give me a chance to take a close hard look at what these kids know. Then, ideally, I'll spend about a week doing some really excellent math involving huge numbers, while slowly getting an even better sense of their addition skills. We'll do mental addition in the first few minutes of those lessons. And then we'll make some strategies explicit, and call it a unit.

      That is, unless I or someone else can show me some really great math problems that would give everyone a chance to practice addition.

    2. Could not agree more about the curriculum. I find the Teacher's Notes the most overlooked piece by many and that is where I do a lot of learning, reading, and re-reading. I will be following along w/this post to see others' suggestions bc I am always up for a change to better my instruction! Thanks!

  9. I would explore using the skill (multi-digit add/sub) in a problem-solving context. Our fourth graders started the year with a similar unit. We found a problem (sorry it's in school right now) but along the lines of: You start with a given amount of $ (it was like $25,000) and there was a list of very expensive "gifts". You needed to buy gifts for four people and spend as much of the money as you could. This was a good activity because the kids who still needed help with the basic computation and place value concepts (lining up digits in the correct place values, regrouping properly) could focus on that (and not worry so much as long as they didn't go over-budget) while the others were challenged to find the best combination (and there were lots of combinations so you needed to persevere). Lots of strategies emerged, such as finding a total cost then subtracting, adding the prices in chunks, and repeatedly subtracting. We had good arguments about whether or not two people could share one gift, or one person go gift-less...the kids liked changing the parameters of the problem to fit their answers.
    Kids were then challenged spend as close to half the money as possible, and I think there were a couple things we threw in that I can't think of right now. We got lots of mileage out of the project, and the kids were engaged at many different levels, and we were able to really work with kids who still had misunderstandings.
    As for place-value, I have found that kids who still struggle benefit from those place value flip books (we have ones that are double-sided and go to millions one one side and from thousands through thousandths on the other, I think ours are from Lakeshore or ETA). Many of the place value games mentioned above are also good.
    In general, I'm skeptical about ramping up the skill difficulty; I think it's better to find creative ways for the kids to use the skill, in whatever way will engage them.

    1. Interesting.

      Writing good problems (and making good projects) is something that I absolutely need to get better at.

  10. I wouldn't skip this unit. I am always amazing about what I think my students versus what they really know. While the can probably add large numbers, the subtraction piece can be very challenging especially if you throw a couple of zeros on the larger number. Also, my 4th graders do not have a solid foundation of number sense. They do not truly understand what a large number is made up of. We did a unit similar to this but we focused on CCSS of knowing place value, expanded form, and written form. We did A LOT of practice with the written form with oral practice. We had to undo a lot of bad habits like tossing in the word AND. Also, they really struggled with being able to correctly say the hundred thousand period. They did well with the hundreds period and the millions period, but continued to misunderstand the hundred thousands. So... you have a lot to work with with this unit.

    I think a creative way to use this skill would be to have them go "shopping" for expensive things like, cars, motorcycles, televisions. Maybe give them a budget of $50,000. Just a quick idea.

    Also, I would second all those who mentioned this is a great unit to teach rounding and mental math. These are critical skills for all areas of math.


  11. Play a game that the students like-dice rolling, high card draw, or something else they enjoy playing in teams (even a video game might be fun). Make a point system that involves powers of 10 (10000 pts for rolling doubles, 1000 pts for rolling a sum above 7, snake eyes and pts are halved, etc). Play game and keep score and every now and then pause to see who is in lead, and how many pts. ahead each team is of the next, have surprise merges so two teams points are added together, etc. Eventually you could pair students up and have them play one on one, giving them a chance to practice keeping score on their own-and keeping an eye to make sure their opponent is keeping correct score. Good luck!

  12. Jo Ann beat me to it. When mathematicians talk about the problems of kids learning math and our failures in teaching math, they aren't ever thinking about the struggling kids who can barely add.They're talking about the kids who basically get it, that they have no understanding of the conceptual underpinnings of math. And top of the heap on their gripe list is place value. So your dismissal of it as "oh, no big deal, they get it" is probably something to examine. Yes, they are solid algorithmically. I very much doubt they understand place value. And yes, you'll have to come up with something intellectually challenging. But that will be fun. Just don't overestimate their abilities.

    More than one mathematician bewailing kids these days have mentioned the Peano axioms--not so much as something that kids should be taught, but that teachers should be teaching with an awareness of, if you follow. I have no idea what that means, though.

    1. You guys have me convinced. No skipping!

      Plan: A few stand-alone lessons that will give me time to carefully figure out where these kids stand. Then, I'll plan something cool, perhaps inspired by so many of the interesting ideas shared here.

  13. I wonder if some Fermi-problem estimation might be a fun fit for now? I find that the main things that trip me up as a mathematically literate adult working with place value are really understanding what order of magnitude my numbers have and my answer should have. Questions like if I add up a long list of thousands and ten-thousands, how many would I need to add to expect to have an answer in the one-millions?

    The other piece, and maybe you recall this from Triangleman's Decimal Course, is putting 1, 1000, and 1000000 on the same number line -- that's not really adding but it could be: how many 1000's do you need to add to make a million?

    Here's a link to some Fermi problem stuff, including a contest that starts 1/28/14:

  14. I read this blog post right before yours, from Math with Bad Drawings:

    That might be an interesting discussion if your students have any concept about exponents. (I know it's probably not standard fourth grade stuff, but my second-grade nephew understands the idea behind squaring...)

  15. I may have one helpful word: TrianglemanDecimalInstitute

    ... Make that two, add: Abacus

    There's an NCTM journal article titled "The Chinese Abacus: A window into standards-based pedagogy" by Jerry A. Ameis that you might find helpful... I thought it had some good lesson ideas anyway.