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!


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.

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Tessellating squares is easy!

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

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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.

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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.

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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.

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Hexagons made of rhombuses

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.

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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.

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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.

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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.

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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.

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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!

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Look at how fun these are!!

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

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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.

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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.



When the student is ready, the manipulatives will appear.

L continues to demonstrate to me just how much she will gravitate toward a particular material/concept when she’s ready to have mastered it. We’ve had the Melissa and Doug pattern blocks for awhile now, and beyond having quickly matched up the blocks with the pre-printed patterns and engaging in one free-play session at the kitchen table, she hasn’t shown any interest in them. I, on the other hand, have materials at the ready! In particular, I downloaded The Math Learning Center‘s sets of pattern block lessons to meet the Common Core State Standards, both grades K-2 and 3-5. I downloaded both sets because a) L likes to whip through materials quite quickly, and b) I like to see concepts in developmental sequence so I know what types of prompts and questions to use. Anyway, they’d been printed, three-hole-punched, and were sitting in a binder. Just waiting for when L wanted to do pattern blocks again…

… which came today! We use the Ikea Trofast system for many of our learning materials. I use address labels on the drawers to identify the materials within a given drawer. I was guiding her to choose a puzzle when she noticed the pattern blocks drawer and asked if she could do that instead. The drawer, by the way, has been in the EXACT SAME SPOT for months. I have no idea if she’s ever noticed it before. In any case, you want to do pattern blocks? Absolutely!

Pattern block drawer in use!

Pattern block drawer in use!

She had already completed the templates that came with the blocks in the first place and handily enough, there is a rocket ship template in the K-2 lessons packet (pages 9-13 of the pdf). I grabbed that and started her on her way. She quickly built the other four templates (the baby is shown in the picture).

I then wanted to capitalize on her interest with the blocks to push her thinking a bit. I used page 35 of the K-2 packet. I skipped the “game play” versions of the lesson because she’s already drawn towards being competitive, and given her creativity and quick understanding, she’s often better than us at lots of things! I don’t want to feed into that any more than is naturally going to occur. However, I did like the idea of trying to find the minimum number of pieces required to fill in a given space. I used the top of the page without any direction for minimizing the number of pieces used. Instead, we started by looking at if we could fill the entire space with just one shape of block (sorted by color on the right-hand side). We found that the only block you could use exclusively to fill the shape was green (triangle). We then experimented with different ways to use multiple colors simultaneously to fill the space. We noticed that it took 5, 6, or 7, depending on the blocks you used in combination.

(Please forgive my upside-down writing!)

(Please forgive my upside-down writing!)

I then used the bottom half of the sheet to introduce the idea of minimizing the number of pieces used. We used a similar process, but noticed something about the greens. We noticed that there were some places where you could substitute one blue (rhombus) in for two greens (triangle). By stacking the two greens on top of the blue, L was able to see that each green was a half of a blue. Two halves equals one whole. We repeated the process with yellows (hexagons) and reds (trapezoids). In the end, L found two solutions that used six blocks. She chose one to replicate and capture by coloring the spaces underneath each of the blocks the appropriate color.

Capturing her solution

Capturing her solution

So, while we haven’t dealt with the idea of symmetry yet, and I also want to move to an outlined shape with no guide lines in it, she was able to apply some excellent critical thinking and reasoning to about an hour’s worth of work. As is my theme these days, I could have continued worrying about when she’d do pattern blocks – and what holes am I leaving in her education?¬† – and what will happen when, when, when… – or I could just breathe deeply and trust that I’ve created a learning environment for her that’s hands-on and engaging and she will lead the way when it’s time.

Gosh, if only I could remember that overnight. =)

Math is fun!

We’ve had a blast today working on math!

No, really, we have!

We get lots of funny looks when we talk with others about how excited we all get about learning together, but I swear to you, this is a snark-free blog entry. As much as I can make one, I suppose. =)

We started off our morning working with place value. L still inverts the ones and tens places when reading and writing numbers (15 is likely to be read 51, etc). We have worked hard on visualizing the numbers themselves (What does 15 look like? What does 51 look like?) and I decided today to link it explicitly to written notation.

Rolling place value, writing it out, and stamping the base ten blocks, too

Rolling place value, decomposing, and stamping the base ten blocks

We have place value dice (the dark green ones say “hundreds” under the numerals, the blue “tens”, and the purples “ones”). We have thousands and ten-thousands, too, but I didn’t want to be overwhelming. I think she has a pretty solid grasp of 10 tens being 1 hundred, so I decided to stop there. I also grabbed our one, ten, and hundred base ten stamps and our decomposition templates (100, 200, 300, etc through 900 on green tagboard; 10, 20, 30 through 90 on blue; the numerals 1-9 on purple).

She rolled the dice and worked on putting the hundreds on the far left, the tens in the middle, and the ones on the far right. She then read the number aloud (six hundred fifteen in this picture) and I transcribed it for her. I asked if her she wanted to stamp it or decompose it first. She chose decompose, so she separated this number into 600+10+5. She then stamped out 6 hundreds, 1 ten, and 5 ones. We repeated this for a few numbers and I could tell she was growing bored…

So I pulled out another concept! We took the two numbers she’d rolled most recently and I told her about the hungry alligator. We agreed that given the choice, the alligator would always eat the bigger snack. We giggled about how the alligator would eat me, not L, and the dog, not the cat. Silly! Anyway, I showed her the alligator sign with its open mouth toward the bigger number:

Introducing greater than/less than signs

Introducing greater than/less than signs

I did not introduce the idea that one sign was called “greater than” and the other was called “less than” because it seemed to me that if she grasped the idea that the mouth was eating the bigger number, we were good to go at this point. We played with this for awhile (you can see that at the top of the paper, we inverted our example to show how the mouth opened the opposite way).

Whew! Math is fun!

We then decided to play with our new magnet kit. After we both built a few shapes for fun, she noted that I’d built a hexagon. I promptly decided that a geometry construction lesson was in order! I pulled off the “names of the polygons” chart we’d made a few weeks ago and challenged her to build a triangle. Easy! A quadrilateral? Surprisingly challenging.

Construction of polygons with magnets

Construction of polygons with magnets

She spent some time struggling with adding line segments onto a vertex but it never transformed into a quadrilateral. We stopped for a minute and looked at the hexagon to notice how many line segments and vertices there were, and how many line segments touched each vertex. We then pulled apart the attempt at a quadrilateral and started from scratch. Slowly, narrating her way through it, she built it! I then challenged her to transform her quadrilateral into a pentagon. She grinned, opened a vertex/line segment, and added another line segment and vertex. Easy peasy! So fun.

Mindware's pattern play

Fun with patterns

We ended our “school time” with¬†Pattern Play (which we LOVE!) She found an error on card 19 – if you look closely at the card in the picture, you can see it. In the triangle facing us, there are two yellow segments and there are none on the triangle against the edge of the picture, while there ought to be one yellow per triangle. She found the error but decided to replicate it (see her example) since she was creating the pattern on the card. Funny girl!

In addition to this, we’ve spent some quality time over the past few days (too much time!) watching They Might Be Giants videos on YouTube – her current favorites include The Sun is a Mass of Incandescent Gas, I am a Paleontologist, The Bloodmobile, and How Many Planets? (especially when it says “Jupiter” in the deep, deep voice!).

We may be creating a monster. =)