Finding subtraction

Sometimes I have to go to work on days when I don’t have coverage for L – so she gets to come to meetings with me! What a lucky little duck huh?

Last week, I was going to an open writing support time for my college. I suspected that perhaps I would be the only one in the room at the time, but I wanted to be sure that if that wasn’t the case, I could have some things prepared for L that she could complete independently. I brought along some play-doh, some books she hasn’t seen in awhile, coloring, and I printed off a few Halloween-based pages for her. One of them was a math pyramid, which led to my realization that her math knowledge is really growing quickly!

A few months ago, we used a free trial of the Dreambox math app. While we loved the interface, she was frustrated by the concept she was working on, which was to build numbers through subtraction. In other words, to build 57, the app wanted her to move all 100 beads over, then subtract 4 tens, then subtract 3 ones. She wasn’t quite ready to think of numbers this way and we couldn’t get around that topic, so we didn’t extend our free trial.

Since that time, we’ve worked with single-digit subtraction, but have not talked about number building through subtraction. We have done lots of number building through the hundreds with number cards, base ten blocks, and base ten stamps.

In any case, we were using the fourth page of this pdf. I thought she could independently complete the first few rows. I packed up four tens of snap cubes and brought them along.

As I expected, she completed the first few rows independently. She’s got a good memory for addition facts and many of these were within her memory. She added 8 + 9 and 9 + 7 through counting on and correctly figured 17 and 16 as the answers. When it came to adding 17 and 16, though, she paused.

I had pulled out the snap cubes when she began her work and she eyed them at this point. She said, “I think I need to use my cubes.” I said ok, handed them to her, and sat back.

As I mentioned, the cubes were in lines of 10. She took two tens and put them in front of her. She stared at them for a minute and then said, “Three!”. She snapped three off of one of the tens and then placed the ten and the seven next to the number 17 on her paper. She then repeated the process with the other two tens, saying, “Four!” and making 16. Wow! I didn’t expect at all that she would work backwards from the number 20!!

She then took the ten from 17 and the ten from 16 and put them together. She held the 7 and the 6 and looked at them. “This makes more than a ten.” As I watched, she snapped three off of the 6 and added them onto the 7. She then moved the new ten over to the first two tens. I then heard, “Ten, twenty, thirty, thirty-one, thirty-two, thirty-three! It is 33!”

Ta da!

Ta da!

Yes, baby girl, it sure is.


Multiple representations of equivalence

A couple of math minutes today – one that laid out beautifully as I’d envisioned it, and one that gave me another reminder of how much I’m still chasing L’s abilities. Let’s start with the “fail,” shall we?

I’ve been keeping a clipboard in the car lately. Sometimes it has coloring pages (primarily from Dinosaur Train, but also color by number or other items I happen upon). Other times, it contains tasks that I estimate she can complete independently and quickly. This first math minute was one of those activities. It is a simple page I found online that is missing a sign (either subtraction or addition) in a series of number sentences. I passed the clipboard to L after she was buckled up in the Whole Foods parking lot. I pulled out of the parking lot, turned right, turned left, and entered the freeway. When I entered the freeway, I realized I hadn’t heard anything from L.

I asked, “So, what does that paper ask you to do?”

L replied, “I’m done.”

She passed me the clipboard. She was done.



Ok. Unexpected! I asked her how she knew what to do. She explained to me (as if speaking to a simpleton, by the way), that one could simply look at the answer and the first number. If the answer was larger, it would be an addition problem; a smaller answer would indicate a subtraction problem.

Fair enough.

Luckily, later this afternoon, I rebounded! I’ve been wanting her to think beyond simple mathematical equations (even multi-digit ones) because she’s pretty comfortable with the processes of addition and subtraction. I’d like her to begin thinking logically, rather than simply procedurally. I found this “balance the scale” page and decided to use it as the basis for this thinking. The goal of this paper would be to have kids identify which two addition sentences have the same answer and place them on either side of the balance.

I decided that I wanted her to explore this concept visually and kinesthetically, as well as have multiple ways to conceptualize the solutions. I pulled out three of our sets of math manipulatives:

All three models gave her the ability to combine terms as well as visually represent equivalence. She never saw the worksheet. I simply asked her to get the 2, 3, 4, and 5 inchimals. I prepared sets of 2, 3, 4, and 5 snap cubes. She got the 2, 3, 4, and 5 bananas. She then worked with her materials to determine that you could make two number sentences which equaled 7 (2+5; 3+4). Once she’d determined that, I asked her to show me on all three sets of materials so that she could physically see the balance points of the monkey and bucket balances, as well as see the same heights of the two stacks of inchimals.

Scene re-creation

Scene re-creation

We proceeded similarly for the next set. By the last two sets, I asked her to explain to me why the solutions made sense. She was able to verbalize that both sides equaled the same total (and give the accurate total). Job well done!

So, one very successful learning opportunity today, as well as another opportunity for mommy to remember that the tasks I anticipate as tricky are often not…

Bears in the cave

One of our favorite math games for the past few months has been the “bears in the cave” game. It’s a great, 5-10 minute game that helps kids think through how numbers are formed.

The idea of the game is simple – you have a total number of “bears” (in our house, we use unifix cubes but you could use any manipulative – this could easily be played at a restaurant with sugar packets or creamers when you’re waiting for food). The kiddo covers his/her eyes while you put some bears “to sleep in the cave” (tuck a few of the cubes out of sight), leaving some remainder of bears “playing in the field”. The kiddo’s job is then to figure out, based on how many bears are playing, how many bears must be sleeping. Here are the three iterations the game has taken in our house so far:

Version 1: Working to five

We introduced this game with five bears and played a few rounds. Pretty quickly it became obvious to me that L was adept at the number pairs 0,5; 1,4; and 2,3 as making five. I gave her control of the bears for a few rounds so I had to do the guessing. Then we upped the ante…

Version 2: Working to ten

Same as above, but with ten bears. This provided much more of a challenge. We played this game in this form on and off for 2-3 months. Each session was about 10 minutes in duration. We spent some time at the beginning talking through ways to solve the problem, including using your fingers, counting on from the bears you could see, and using a chart we’d made of number pairs to ten (* see bottom of this post).

How many bears are in the cave?

How many bears are in the cave?

We took turns being in charge of the bears. I had planned on continuing this game indefinitely because it’s hands-on and concrete and she wasn’t automatic with all of the number pairs yet. But then, a few weeks ago, she asked to play again and added, “Can we add more bears?” Ok, that brings us to…

Version 3: Working to twenty

We added a second set of 10 unifix cubes, but these of a different color. We played the game the same way, except the questions consisted of an extra level.

Now how many bears are in the cave?

Now how many bears are in the cave?

Whereas before, she simply had to count on to 10, now she had to count on to 20. In doing so, she has to consider the unifix cubes as the same units, regardless of color (so, there are 11 bears showing here, meaning that 9 bears are asleep in the cave). But then I followed up by asking how many green bears were sleeping and how many orange bears are sleeping. Here she had to then subdivide the bears by color to answer (so here, 6 green bears and 3 orange bears are sleeping). I think this version because she has to move flexibly among the thinking.

I’m not sure what’s next, but I do like that she asks for this game and even asks for new challenges with it. To me, that seems like it’s building a solid mathematical foundation while also being fun!


* Our number chart of how to make tens:

We began with 10 unifix cubes each – she had 10 purple and I had 10 red. We then colored in how many purple cubes she had to make ten (so the first line is 0 reds and 10 purples). We then traded – she got one of my reds and I got one of her purples. We colored in the next line, showing 1 red and 9 purples. We continued through until she’d traded all of her cubes for all of mine, making the final row all reds. We ended by counting the numbers of reds and writing those numbers down the left-hand side of the chart and doing the same on the right-hand side for the purple cubes.

In the early iterations of the bears in the cave game version 2, I would show her how to count how many bears she saw, locate that number on the left-hand side, then trace across to find the corresponding right-hand side number. Over time, the pairs have become much more automatic.

How to get to 10

How to get to 10