Rounding is just non-intuitive.
Not only do you have to wrap your head around the fact that you're going to pretend that a number is a different number, but you have to remember that even though it's called rounding DOWN, you don't round down by subtracting 1 the same way that you might think that you round up by adding 1 (that's not *really* how you round up, but if you're nine, that might be how you've explained it to yourself).
Oh, and rounding up if the digit is 5? That doesn't actually make sense; it's just a convention to memorize.
I wanted Syd to understand the concept of rounding, not just memorize a rule that doesn't make sense to her, so using this post on hands-on rounding as a spine, we went over it again from the beginning.
The post doesn't mention using a hundred board, but we've relied on our hundred board a lot throughout Syd's entire math education, so once again we began a lesson by having Syd build it:
I've got three sets of these 1-100 tiles, two DIY sets and one for the light table/overhead projector, so I could leave the 100 grid intact and give Syd a second set of tens to use in making a number line between two consecutive tens:
This part was all review, as she had no trouble with any of it.
Syd built a number line between two consecutive tens a few times, each time with me handing her the numbers in between at random. We then reviewed rounding to the nearest ten by noting on each number line which consecutive ten a number fell between, and which consecutive ten it was closer to.
Numbers ending in 5, of course, round up merely through convention, as we also discussed.
To end the activity on this day, Syd and I played a game with two 10-sided dice. One of us was "Rounded Up" and the other was "Rounded Down;" we each took turns rolling the two 10-sided dice, making a number from it, then deciding if that number rounded up or down and giving a point to the correct category.
Although the points go to the correct category regardless of whose turn it is, when it's your turn to roll, you can do your best to build a number that gives you the point. It's tricky, though, because you can't always build a number that goes your way!
I let this lesson settle for a few days while Syd worked on different concepts in her Math Mammoth, then we had another lesson on rounding to the nearest hundred and thousand.
Syd didn't seem confident in building a number line between two consecutive thousands, so I had her do so using the hundred flats from our Base Ten blocks as physical markers, with each hundred written on a torn piece of paper--use what you've got! I then used a domino to mark a place on the line--9,600, say--and gave Syd another domino to use as a game piece to hop to each thousand:
The thousand with the fewest number of hops to get to it, obviously, is the thousand that the number is closer to.
After this, we turned to our light table, and I asked Syd to build another number line between two consecutive thousands:
I used the dry-erase marker to modify the 0-100 tiles to make them into numbers in the thousands (30, for instance, became 5,300), then asked Syd to place them on the number line. She found this a LOT harder than working with a number line between two consecutive tens. Her number line got so wonky, in fact, that I abandoned the tiles work, gave her a different number line--9,000 to 10,000--and asked her to simply draw in the hundreds in between.
As she did so, I noticed that she did not have 9,500 in the middle--actually, it was very close to 9,000---so I asked her to check her spacing. She was VERY not happy with this, because she knew then that she'd made an error, and she HATES making errors.
Aside: That's an attitude that we work on a lot, by the way. I see it as a very destructive type of thinking that will keep her from establishing a strong and persistent work ethic, so I speak constantly of the fact that knowledge and skill do not enter one's brain by magic, but instead require a lot of training to achieve. When one has the correct answers immediately, one is not learning, but reviewing. Learning is the space in which we try to spend much of our school days, I tell her, and we know that we are in that space if we are struggling and making mistakes.
Hopefully one day that will sink in!
Because Syd was tipped off that she'd made an error, all of her will became focused not on finding and correcting her error, but instead on defending it to the death. She insisted her spacing was perfect. I noted that there was only a finger's width of space between each integer up to 9,500, but that the others on the way to 10,000 were really far apart. She decided to deliberately misunderstand me, and began screaming that I wanted her to write everything with one finger in between, and that was impossible. I excused myself to my room, and told her that she should choose another activity, and we'd finish the lesson later. I then read quietly for an hour while Syd screamed the house down in the next room.
Of course, eventually one must finish one's schoolwork if one wants an hour of screentime, and one DOES want this hour, so much, much, much later, Syd went back to work with a good attitude renewed:
I was still pretty over it, so Matt finished the review with her, and then they played the same dice game, this time with four 10-sided dice, with the goal, again, to decide if the number rounded up or down to the nearest thousand:
Afterwards, I quizzed Syd by throwing out more numbers, and she was able to round them all correctly! She was also SUPER happy to do so, because she knew that she was getting them all correct.
Have I mentioned that this child exhausts me? Oh, my word, she exhausts me!
But, of course, so does the other one...