# Hidden in plain sight

I was doing a math moment with the kiddos at Ethel Streit this week. We have a couple who have had negative experiences with math, some who have only experienced math in traditional public school settings, and some who are quite gifted in mathematical thinking but have little facility with math facts or algorithms. To try to set the stage for the idea that we all explore big math ideas together, I made a “simple” drawing on the whiteboard:

I asked the kids how many squares there were.

The answers were immediate: 9. No, wait! 10! As the kids were thinking, I heard L say”9 times 5…?” I have to admit: I had no idea what she was talking about. She tried to explain to me that there were 9 on top and 9 on the right and 9 on the bottom and 9 on the left. She said that wasn’t quite it, and I chose to move on.

Eventually, the kids figured out there were 14 squares (for those of you following along at home, that’s 9 1×1 squares, 4 2×2 squares, and 1 3×3 square).

L piped up again with “times 5?”

I still had no idea what she was talking about. Luckily, my incredible co-teacher, Sara, looked at L and asked if she meant like a cube. Yes! L’s face lit up. She explained that we had only counted the squares we could see, not all the ones we couldn’t see. She was seeing the visible squares as the front-facing side of a 3-D cube…

I didn’t want to lose the other kids, so I decided to stick a pin in that idea. I told L we would talk more about that later, erased the drawing, put up a set of triangles, and moved on with the lesson. L clarified, “We should only count the triangles we see, right?” Right!

A trickier picture with triangles – we found 48. How many can you find?

On the way home, L piped up again. “It’s not times 5! It’s times 6!” She had realized that you had to count the squares on the face you COULD see in addition to those on the faces you couldn’t. I realized that clearly, L wasn’t done with this yet.

When we got home, I printed and cut out a cube net.While I was doing that in the office, I gave her a piece of scratch paper and asked her to represent what she meant mathematically. This is what she created:

I then gave L the cube to represent her work in three dimensions. She asked me to draw the 3×3 grid on each face and then proceeded to complete her calculations.

She began by counting each small square, realizing that her total was 54. I had her check using multiplication to see that 9×6 is 54.

She then said, “But there are 14×6 squares,” remembering that we had found 14 on each side. I gave her a high five and we decided to multiply it out to find the total. We broke 14 into 10 and 4, giving us two problems: 10×6 and 4×6. She immediately knew that 10×6=60. 4×6 was harder for her. She did know that 4×2=8, so we used that fact three times to compose 4×6=24. We then joined the 60 with the 24 to learn that there were 84 squares on the six faces. There are our written notes from the process:

Because L is L, though, we weren’t done yet. She wanted to figure out how to build the cube out of smaller cubes. As she was considering the build, she gasped, “The cubes are all hidden in plain sight!” When I was done laughing, I told her she was right.

We pulled out Cuisenaire Rods to build. I wouldn’t usually use cuisenaire rods for this build, but we had recently taken our base ten blocks into the school and I could tell we needed to do this build right now. I was particularly aware of the fact that we don’t have 27 1-unit rods. I decided to let it ride, though, and see what she did with it.

L began to construct a 3×3 base out of the 1-unit cubes. I went outside to collect cherry tomatoes from the garden (yes, the garden is still crazily producing). L ran up to me and said, “I don’t have enough cubes!” I said, “Oh.” As I was thinking of what to say, she spurted out, “But it’s ok! I can get one that’s 3 long and use it in place of three ones!” She turned around and ran into the house. Ok. I thought. She’s got a plan.

I came back in to find her build complete.

As she began to figure out how many 1×1 cubes were in the whole cube, I kept quiet and watched. She sorted the white cubes from the purple rods. There were 12 white cubes and 5 rods. She counted the white cubes one by one, moving them over as she counted them. When she got to 12, she began touching the first purple rod. She touched it on the end, the middle, and the other end as she counted, “13, 14, 15.” She moved that rod over with the white cubes and repeated the same process until she got to 27 as her sum.

The look of pride on her face was palpable. I loved that this was all self-driven and to solve a question that she had been puzzling over. She’s also been talking about the math on the board and the cube daily since then. Hidden in plain sight, indeed!

# I can’t keep up with my kid!

Why am I writing at 1:00 in the morning, you may ask? Midway through the first week of teaching classes at the university? Just a little freaking out, of course.

I’ve been really worried lately about how L will respond to what’s coming her way – her involvement with a micro-school designed for a handful of kiddos to work at their own pace. That part sounds great – the part I’m worried about is that I’ll be leading instruction there two days a week and she and I still haven’t figured out how to work together on a daily basis.

L doesn’t respond the way I anticipate a child will respond. Doesn’t matter what I incentivize her work with. Doesn’t matter what I attempt to use as consequences.

She will do what she will do when she wants to do it.

For the most part, I try to breathe deeply and trust that. It’s easy for me to do with reading, since she reads voraciously. Recently, I attempted to remind myself how to do running records by practicing on her. I downloaded some grade-leveled passages and sat down with her, asking her to read them out loud so I could practice taking running records. I anticipated that the second grade passage would be too easy and the third or fourth grade would be just about right.

Grade level Words per minute Accuracy Comprehension questions
2 102 98.6% 2/2 questions
3 69 (proper nouns were a struggle) 96% 2/2 questions
4 92 97% 2/2 questions
5 73 96% 2/2 questions
6 74 96% 1/2 questions

She turned 6 years old last month. So, yeah. I don’t worry about her reading. She is constantly lugging a book around (DK Eyewitness Books: Ocean and others in the series are a huge hit these days).

Her writing is a concern, for sure. She struggles with letter formation and isn’t drawn to production of text. I am playing around with cursive for her (which has been better than printing) as well as typing, but we don’t have a magic bullet yet.

I don’t worry about science. At all. She is constantly reading and watching videos and documentaries and conducting experiments. She’s probably most advanced in science, which is ironic given that I am not drawn to natural sciences. She came by that one all on her own!

Social studies? Well, I would have argued that she’s not interested in social studies, but I think I’ve been too narrow in what I defined as social studies. She knows where many countries are and what their flags are from looking at a map on her wall. Same for the states and the state flags. She was deeply interested in the American enslavement movement and Civil Rights era for awhile and read deeply on those topics. She has onboarded much of the events covered by the musical Hamilton and has enjoyed reading the lyrics and asking questions about them. She has also followed along with NPR coverage of the presidential election season and has explored the ideologies behind the Democratic and Republican parties, even going so far as to define herself as an “environmental one-issue voter.” So maybe that’s pretty good coverage of social studies at age 6?

## But math!

Anyway, this brings us all to math. Math is the source of today’s hijinks and is why I am still up (it’s almost 2 now, for those of you following along).

L has been “fighting” about math, pretty much always. Her fight is saying she doesn’t know how to do something or she’s bored. There was a great period early on when she used Todo Math (which she blew through but enjoyed), Slice Fractions (same) and Dreambox (which was new for awhile but she became bored with it). Even Beast Academy, the “go to” for elementary g&t kiddos, was interesting to her to read (she lugged books 3A-D as her bedtime reading for awhile), but she was never interested in the practice books.

And yet, we’re going to be working together in this micro-school, so I need her to be able to sustain work in math. I grabbed a couple of story problems from a third grade problem set and gave them to her as today’s work.

Let me be clear: I was at work and she was in my office with me. My attention was clearly divided and I couldn’t reinforce to her that she should continue working. However, she essentially didn’t do anything for long enough that I gave up. I decided that the battle wasn’t worth it today, that I needed to get a bit of my work done, and that we would try again tomorrow.

However, after school, she was fighting my husband as well, and he decided that she was going to work through these 6 problems because, well, sometimes you need to listen to your parents.

So she sat down and figured out the answers to all six problems. Most of them in her head.

Sample 1: There are 2,532 students at a school. 1,312 of them are girls. How many of the students are boys?

She picked up her pencil and wrote down 1,220. Didn’t write the problem out. Didn’t use manipulatives. Nothing.

Sample 2: Mom has 11 apples. She needs 5 apples to make 1 pie. She wants to make 5 pies. How many more apples does she need?

She circled the words “5 pies” and wrote out “25-11=14”.

I posted about this on my favorite facebook group and another mom suggested that maybe she is bored. Her kiddo presented with “it’s too hard” when it was really boredom.

Bored? No! She couldn’t possibly be bored! She is just now 6 and has had almost no formal math instruction. It’s all been picked up through apps and occasional problem-based lessons. And I pulled those from a third-grade book.  When she got bored with Dreambox last year, she was 84% finished with second grade.

Seriously?!?!

## The freakout

How am I supposed to stay ahead of this kid? I feel like every time I have an idea of where she is, I turn around and she’s past it. I have felt that way for three years now.

I think I’ve made significant progress in changing how I think about education as it relates to L. Instead of thinking of myself as a teacher who sets out the path, I think of myself as an intense kid watcher. I watch her for emerging interests and skills and then scour the world for the most appropriate resources, which I place in her path in the most time-efficient manner I can manage. She is thus constantly picking up high-quality, high-interest materials and, since I know her pretty well, they’re typically in alignment with her interests. I don’t lead her down a path. I don’t even walk next to her on the path. I walk behind her on the path and slip goodies onto the path, hoping not to be seen.

But I’m at a loss here.

I don’t even know how to assess her appropriately to figure out where she is mathematically. I don’t know what curriculum to turn to. I don’t know how far off my estimate of her progress is.

I was re-reading an article about the opportunities the internet allows for gifted kiddos. The article refers to the Art of Problem Solving, a name I’d certainly heard bandied about (and the middle- and high-school wing of the Beast Academy company).

Tonight, I checked the diagnostic test to see if she’s ready for their first class – prealgebra (for students who have completed elementary math – grades 5/6). To be clear, I know she wouldn’t pass all of it right now, because we haven’t talked about division. She could nail about half of it (basic arithmetic with negative numbers, addition and subtraction of fractions with same denominators, basic fraction comparison, and some of the word problems).

Dear me. She can nail about half of the assessment for the prealgebra class “specifically designed for high-performing math students”. And she’s 6. And we don’t really do math in a sustained way.

But I’m pretty sure that if I introduced division and decimals, she could be ready in a month or two.

And then I look back at the online prealgebra class schedule and it occurs to me: she can’t take this class anyway, because it goes past her bedtime! How ridiculous is that? A class that she could be ready for in a month or two that she can’t do in a month or two because she still goes to bed at 8 BECAUSE SHE’S 6 YEARS OLD!

And it hits me again. She is atypical. She is asychronous. I am so lucky to get to be her mom. It is so terrifying to be her mom. And I’ve simply got to get some sleep.

# MC Escher and tessellations: Where math meets art

In our ongoing quest to keep L engaged with math without necessarily pushing her through more and more abstract concepts. I still harbor fantasies of her going back to school at some point, and I worry that the growing disconnect between her age and her abilities is only going to make finding a fit harder. However, I want her to continue to push past the zone where things are easy and have to persist on some difficult tasks, too. She already struggles with shutting down if things don’t come instantly to her (or if she doesn’t do them “correctly”) so one of my goals for her educationally is to grapple with that which is just out of reach.

We recently completed a lesson in Beast Academy related to using polyominos to fill defined spaces. We’ve also been using pattern blocks in relation to our study of fractions, so it occurred to me that we could use pattern blocks to begin to explore tessellations.

A tessellation is a repeating pattern that has no overlap or gaps between the pieces. You can tessellate lots of shapes, but if you want to see how cool tessellations can be, you’ve got to check out the artwork of M.C. Escher.

I found a really cool link that shows how to make your own tessellating shape, but I knew that opening with that level of open-endedness was likely to freak L out. Instead, we started with our pattern blocks.

I took a cookie sheet and used washi tape to define a small (about 4″) square on the cookie sheet. We defined this as our field. We then sorted the pattern blocks by shape. L chose a shape to begin with and we began seeing how we could cover the entire field with that single shape with no overlap and no gap.

Tessellating squares is easy!

We then moved onto hexagons, which were also simple to tessellate.

We had a nice connection to the honeycomb in nature when we did this one

We then moved onto a shape which I’m not sure they had “when I was a kid” – or if they did, I certainly didn’t know anything about it… rhombuses! L loves the shape and the word – and I love the way she says the word (a mildly trilled “r” and like rum-busses). She first arranged the small rhombuses in a non-standard pattern, which we decided also looked like nature.

Like the wing of a bald eagle!

When we moved onto the larger rhombuses, I asked her to arrange them differently than the previous set of rhombuses. One of my strategies with her is always to ask her to reflect on what she’s just done and find a slightly different take on the task. Here’s what she came up with for the larger rhombuses.

A different arrangement of rhombuses

I decided at this point that she clearly understood the basics of the task. I asked her to remove most of the blue rhombuses from the field and instead, use a few rhombuses to make a different shape. Instead of tessellating rhombuses, we would tessellate this new shape she created.

L put together three blue rhombuses to create a hexagon. She was concerned that they didn’t fit together perfectly, but I told her that we could pretend there were tiny white rhombuses filling in the gaps because the gaps themselves were regular. She then began tessellating the sets of three rhombuses and came up with quite a cool pattern.

As we were admiring the work, L decided that we could now add some of those whole yellow hexagons to the field. I asked her to think about how to add them in a pattern, like she might find on a floor or a wall. She came up with stripes.

Yellow and blue striped hexagon tessellation

And then, of course, she decided to input the red half-hexagons in sets of two to complete the stripes.

Full on hexagon stripes

Very cool!!

Building off the idea of altering patterns, we then picked up the final shape we hadn’t yet used: the humble equilateral triangle. She designed a tessellation in which the vertex of one triangle rested at the midway point of a side in each line.

Each line is the same with the triangles in the same places

She then pushed over lines two and four to line up the lengths of opposing triangles with one another to form a slightly different pattern – and in it, she found hexagons! We had a conversation about how we could re-create the three-lined hexagon tessellation above with additional green lines or how we could use three triangles in the place of any one of the red half-hexagons to complicate it further.

Look, mom! Hexagons!

I was feeling pretty good about the open-ended result we’d experienced so far on this day, and I stepped away to take part in a quick phone conversation. When I returned, she’d created this tessellation. The green triangles are the wingspan and the single triangle above them serves as the head of one bird and the tail of the next bird.

The birds in mid-flight

She also used this time to find tessellations on the floors and walls of our bathrooms. Since she was still really into it, I pulled out a recent supply I’d ordered from Nasco, anticipating both her enjoyment of this concept and her love of animals.

Animal. Tessellation. Templates.

I kid you not.

I mean, what in the what? Right?!

Anyway. They were a hit!

Look at how fun these are!!

Let me be clear: I am jealous that we didn’t have these.

She tessellated fish

The fish was the end of it for the day for her – I mean, she had been at it for a solid few hours. However, a few days later, we revisited the templates again. This time, I urged L to think about coloring in a pattern to enhance her tessellation. She picked up the dog and came up with this.

Red and black tessellated dogs.

We’ll get to the self-made templates in the coming weeks. Overall, I feel relatively certain that she engaged her pattern-making brain, build some fine-motor skills, and also had a pretty darn good time, too.

#HomeschoolWin

# A note on our curriculum choices

As a teacher by training, I like curricula. I like pacing charts and a spiraling curriculum. I like scaffolding big ideas. I like anchor charts. However, I already know that when I’ve tried to use curricular resources, we do a big lesson and then skip the next 10% because L’s already figured out and applied the pattern. Have I mentioned that she excels at patterns? It doesn’t help that she’s got massive gaps between what she can write and what she can express, or that she focuses on subjects of interest to the exclusion of everything else. These traits make it hard to find an “all in one” curriculum. Instead, we would be best described as eclectic (a word my husband hates), relaxed, thematic, and passion-based homeschoolers. Here’s what we’re using now (age 5).

Language Arts

I could care less if she can spell words right now. She’s 5. Spelling will come. It may be an area of strength for her (as it was for me) or an area of frustration. Just as she learned to read when she was ready and using her own method, so too will spelling come. I could care less if her handwriting is legible right now. A huge part of that is that she’s 5. I want her to focus on the idea that she has important things to say and recording them allows them to be shared with others who aren’t with you right now. So we jot things down all the time and I scribe for her when she requests it. Our house is littered with notes.

Th closest we come to spelling practice is pointing out word parts like prefixes, suffixes, and word roots as we decode complex texts. We also read Grammar Island  on the couch as if it’s a read-aloud. We have a moveable alphabet and stamps. We don’t use them often. We play Wordsearch and Pathwords Jr.

L reads. Beautifully. Narrowly. And far above grade level. She hates most fiction. She doesn’t like reading for strangers. And she absorbs everything she reads, so working on non-fiction graphic organizers is a waste of time right now. We just read. A lot. Vociferously, in fact. And I often curl up and read my own books while she reads hers. (The link at the beginning of this paragraph is her reading a page from Almost Gone: The World’s Rarest Animals – Lexile AD1020L, DRA 34).

We also do a ton of stuff that strengthens her hands. Sewing. Play doh. Drawing. Snap circuits. Lego. Her handwriting will come along as her hands grow. So I call all of that hand-strengthening stuff “language arts” too!

Science

We don’t go more than a few hours ever without doing science. She reads science books. Plays science apps. Watches documentaries and other science shows. Is known by name at both our zoo and natural history museum. Her pretend play with animal figures is based in actual appropriate behaviors for the species represented.

Oh, and we make stuff. Snap circuits. Building with “garbage”. Tinkercrate. Chemistry kit. Marble run. I think we’re good on science.

Math

I have really struggled with math for L. One issue we consistently have is that she gets bored (habituates) quickly. This manifests itself in a number of ways.

First, she is not interested in repetition, especially if the repetition is intended to do things like teach math facts. Exploring addition? A-ok. Practicing number pairs which compose 10? Only in context, my friend.

And how does she manifest said habituation? Well, through stubborn shutting down, of course! I will not be moved, her behavior says. One thing we work on is helping her learn to tolerate (and it is literally that – tolerate) that which she doesn’t find engaging. However, I feel pretty strongly that my 5 year olds world shouldn’t be primarily or even a lot about learning to tolerate disengagement!

Where that lands us, then, is in a land where what we find that appeals to her get used manically for some period of time and then discarded, never to be touched again.

Good times. Expensive times.

Some of you may recall how L was obsessed with dinosaurs for almost 2 years. She now won’t look at them. Literally. She keeps her eyes down at that part of the museum. Because she’s done with them.

In the past, she worked through all of Todo Math which contains PreK-2nd grade concepts aligned with several domains of the Common Core (which, to be clear, I’m not opposed to as a set of standards). She has outgrown that app. She tried the Redbird mathematics curriculum and didn’t respond well to the format. That was very quickly a struggle. She responds really well to the Dreambox Learning app (aligned pretty broadly to Common Core), but she powers through it in pretty big chunks. A few months ago, she walked through a review of kindergarten, all of first grade, and most of second grade in about 6 weeks. She then got stuck on a particular concept the app had her working through (perhaps it was at the edge of what she could do?) and started fighting about it.

We use a variety of approaches right now, mostly low-cost ones! As I mentioned in an earlier post, we have some mad love for manipulatives right now. We also continue to use Beast Academy on a very casual basis. I also surf pinterest and grab ideas that look interesting. Anytime we come across a math app that looks interesting and seems reasonably priced, we grab it. Right now, we’re playing with Slice Fractions, Quick Math, Jr., and Attributes.

We also play lots of logic games. We love Rush Hour, Blokus, Kanoodle, Pattern Play, Set, and chess.

Social Studies

L was interested in enslavement earlier this year, so we spent an intense 6 weeks learning a bit about US history and specifically those who didn’t escape on the Underground Railroad (she told me that since so few people ever escaped, it was better to really focus on those who didn’t escape). Luckily we live very close to the National Underground Railroad Freedom Center, so we were able to visit several times and obtain lots of information.

Very quickly, that interest passed.

This will shock you, but we listen to NPR. I’ll give you a second to recover from your shock. In any case, we were listening in the car and I attempted to talk with L about one of the stories. She cut me off and said, “Mom. I only care about the people I already love and nature. Not anyone else.” Ok. We’ll come back to social studies another time.

So overall…

Follow her passions. Watch closely. Be prepared with lots of high-quality, open-ended, inquiry tools. Drink a lot of coffee. These are the things that make our homeschool work.

At least, it works sometimes.

# Exploring equivalent fractions: I’m 5 1/2 now!

This is the first in a series of posts about the mad love we have in our home for math manipulatives.

I’ve spent some quality time thinking obsessing about how to deal with L’s math brain. She doesn’t deal well with repetition and likes to think “big picture” things. I have had to deal with the idea that her automaticity will come through repeated use of facts rather than memorization simply because she shuts down if we attempt to do anything resembling math facts. There’s another blog post brewing in my head right now about the approaches we’ve tried…

In any case, our idea right now is that we are exploring in-depth topics that she seems ready to absorb.We’ve been playing with fractions already, especially as we cook and bake pretty frequently so she’s used to seeing them in recipes and on cooking tools.

A previous day’s work: Find 10 different ways to make 1 using halves, thirds, quarters, and sixths

Recently, L turned 5 1/2. When I was a kid, we celebrated my summer birthday at the 6-month mark at school. L is also a summer birthday, and at the awesome play-based nature school she attends two days a week, they celebrate half birthdays.

In the car, we began discussing how L was 5 1/2 and how there were other ways to express that same idea. I didn’t delve into the conversation at that moment, but knew what the next day’s lesson would be.

I started by layout out for her the conversation we’d had the day before, telling her that there were lots of fractions that would express the same amount. I wrote out for her a variety of fractions with 5 as a whole number and 3, 4, 5, 6, 8, 10, and 12 as denominators (with no numerators). I then asked her to use her fraction circles to figure out which of those denominators could be made into a fraction that showed the same amount as 1/2. I very specifically wanted to include denominators that wouldn’t work to help her think about what fractions mean, not just how to work with them.

Playing with the fraction circles

She used a few really interesting approaches, including laying pieces on top of the 1/2 piece (so putting the 2 1/4 pieces on top of the 1/2 piece to show they were the same) as well as using the additional pieces to complete a circle along with the 1/2 piece.

Completing the circle with tenths

She was pretty quickly able to record 2/4, 3/6, 4/8, 5/10, and 6/12 as representing the same amount as 1/2. I told her she’d discovered a really cool idea called “equivalency” and asked her what word she could find in the beginning of “equivalent”. She found “equal” and we discussed how equivalent fractions represent the same amount as one another.

L showed me how the 1/3 pieces couldn’t be made to be the same amount as 1/2. She showed me how one 1/3 piece was too small to fill up the same amount of space as a 1/2 piece while two 1/3 pieces filled up too much space. We set those aside. She repeated the same reasoning with two 1/5 pieces and three 1/5 pieces. We put those in a pile we called the “crying, sad fractions” and crossed them out on the paper =)

We then looked for a pattern that emerged between which fractions could be made equivalent to 1/2 and which couldn’t. I introduced the words “numerator” and “denominator” so we could talk. We looked at the denominators and noticed that each of the denominators in fractions that could be made equivalent to 1/2 were even. We decided to test this theory by taking all of the pieces of each type of fraction and putting even amounts of them on each side of a dotted line.

Checking our theory

After getting the even fractions divided into twos, we then considered the third and fifth fraction pieces again. L put one 1/3 on one side of the line and another 1/3 on the other side of the line. She sat there holding the third 1/3.

I wrote out for her 1/2 = 6/12 = 5/10 = 4/8 = 3/6 = 2/4 and we discussed again the pattern. This time she noticed that if you added the numerator to itself, you would get the denominator. I told her again what a cool discovery this was, and then directed her to this question:

If you notice that all of the denominators in the fractions equivalent to 1/2 are even, why can’t thirds or fifths be equivalent to 1/2?

It was a big question. She sat there. She put the final 1/3 underneath the dotted line. I worried that I’d pushed too far.

“Their denominators are odd!! They aren’t equivalent to 1/2 because they are odd!”

Right, kiddo.

“And you can’t divide 3 or 5 by half and get even groups!”

Precisely.

“And you can’t add any numerator to itself and get the denominator!!”

Amen, sister.

As we were high-fiving, I decided to double check her understanding of the underlying concept, so I wrote up a quick challenge sheet: Can you make an equivalent fraction to 1/2 out of sevenths? fourteenths? ninths? twentieths? What’s another denominator that you can make an equivalent fraction to 1/2 of?

She spent a minute puzzling over the first four, figuring them out correctly. Then, I heard it in relation to that open-ended question: “Well, I’ll just start in the thirties. 30, 32, 34, 36, 38, 40… any even number would do it.” Eureka!

A hard day’s work

In true form, though, we got to discuss one more great math term. L said, “So I could say that I am 5 6/12, but I never would.” I asked her why, and she told me, “It’s too hard to understand that one. 5 1/2 is easy.” I reminded her about the word simple, which she told me meant the same thing as easy. It was a quick hop over to writing down “simplification” and defining that as a fraction in the form of the smallest numerator and denominator that represents the same amount.

All in all, a solid day’s work.

# 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!

Yes, baby girl, it sure is.

# What a cute little angle!

First of all, I’d like to thank you for the tremendous outpouring of support (on facebook, twitter, and through comments here) for my last post (on being scared). It reinforced for me how important my own vulnerability is through this process, because I think it really spoke to lots of other people’s fears. I will continue sharing the internal as well as the external.

Today, though, I’m turning towards math. L is definitely pulled most towards science and language arts. She gets math pretty intuitively, but it’s not her love. I had heard great things about Beast Academy, but knew it was aimed at third grade level and above. She’s definitely not at third grade level! I looked closely at the first book, though, and saw that it was about shapes and skip counting, both of which she’s fairly fluent with. I also thought the narrative components would appeal to her (though I anticipated the comic-book style to be an obstacle). I decided to order book 3A and the practice guide to see how things went.

Today, we opened the book. I explained to L that this was a completely different way to do math. It was a reading way and a doing way. She flipped through the opening pages and noticed that it was a story with information embedded in it. “I am interesting in this book,” she said. Ok, I’ll take that (PS – I love her babyisms in her speech. I know they’re going to disappear soon and I will mourn them!).

We read the “how to read this book” pages and while she wasn’t wild about the comic book format, she was able to persevere through it and continue. We read the four-page lesson on angles – right, acute, and obtuse. I decided to use a hands-on approach to reinforce the concepts. I grabbed a couple of toothpicks and skewers.

First, we used the toothpicks to construct right angles.

Making a right angle

I wanted to make sure she was thinking about the shape of the angle, so I made a right angle out of skewers, too. I asked L, “Which angle is bigger?” She pointed at the skewers. I asked her to show me the space of the angle. She correctly identified it. I asked her to show me the space of the toothpick angle. She identified it. I then asked her to compare the space of the toothpick angle and the space of the skewer angle, and she replied, “They are the same.” We got to have a great conversation about line segments, lines and infinity versus the intersections of lines and the angle size.

Comparing angle sizes

We continued reading and L decided to use the toothpicks to show a hidden angle in one of the illustrations. I was glad that she decided to use them for her own purposes.

Owning the manipulatives herself

She also decided to make the three types of angles with her body (the right angle is not quite there, but she kept falling when she tried to make it more accurate!)

Obtuse, right, and acute angles

Again, I loved that she took ownership over this portion of the lesson!

Finally, we turned to the pencil-and-paper component of the lesson. Three follow-up practice pages are aligned with the lesson, but we decided that one was enough for today. She used her toothpicks to show the angles in the questions, then identified the angles as acute, right, or obtuse.

Transferring the ideas to paper

I love that she’s beginning to feel more confident in recording her ideas, even when they’re not spelled or formed perfectly.

When she was finished, we decided it was time for a vegan cookie snack. I asked her (as casually as I could manage) what she thought about the books we’d just used.

“COOL!” came the reply. That’s a pretty ringing endorsement…