Magic Mirror Tricks: Exploring Symmetry

Two oldie-but-goodie books give children some fun exposure to symmetry. The Magic Mirror Book and Magic Mirror Tricks show incomplete illustrations--sometimes halves of objects, especially in the tricks book, hence the link to symmetry--that children can complete by holding a small mirror up next to the picture. The books, though out-of-print, can be found very cheaply.

To continue the symmetry lesson, I drew a light pencil line down the center of a piece of paper and asked my son to make a pattern block pattern on one side of the line only. He traced the blocks to make a new symmetry puzzle which he then completed by finishing the design on the other side of the line. He quickly discovered it was easier to trace blocks if he used the magnetic blocks on our new cookie sheet math board so the pieces didn't move.

I then asked him to make a riddle to give a hint as to the picture he'd made. He wrote one about a lion, the second about a hyena.

It would be fun to further extend the lesson by making his own Magic Mirror book. Sounds like a good summer project!

365 Penguins...and Problem Solving

I first read 365 Penguins while preparing for my math and literature class. (*See note at bottom of post about the related Advent Calendar.)

I love it for it's problem solving possibilities. Since it also starts on New Year's Day, it's timely...

Summary: When a box containing a penguin arrives anonymously on New Year's Day, a family of four is puzzled, but as they continue to receive one penguin each day their problems--and food budget, and storage issues--are multiplied.

While I do not think problem solving should be exploited every time you see a number in a book (an inherent danger with linking lit and math) the scenarios in this book are too good to pass up. Here are just a few of the possibilities. Quotes are from the book. The ensuing problems are mine.

• “At the end of January, there were thirty-one penguins in the house.”

Problem Solve: If penguins continue arriving at this rate, how many penguins will there be in another week? Two weeks? At the end of February?

• The family organizes the penguins in triangular stacks to discover that “Four times fifteen equals sixty!”

Problem Solve: What other equal groups could you put the penguins in? (Exploring factors of 60.)

• On the following page, yet another penguin arrives, making it sixty “Plus one!"

Problem Solve: Can you put the 61 penguins into equal groups? Why or why not? (Exploring prime and odd/even numbers)

• The penguins must be fed. "Each penguin ate 2.5 pounds of fish per day. A pound costs 3 dollars..."

Problem Solve: By this time, there are 100 penguins. How much does it cost to feed them? How much more will it cost in a month if they continue arriving at this rate? How much will food cost for a year? 

• Penguins must be organized. Again! The family makes space for 12 boxes of 12 penguins. 

Problem Solve: How many total penguins will fit?

• Penguins are organized into a cube of 6 x 6 x 6 penguins for storage.

Problem Solve: Now how many penguins?

On December 31st, the 365 guests gather. Uncle Victor arrives to take them away. Life begins to return to normal. Until the doorbell rings with ANOTHER DELIVERY! (I'll let you read the book to find out what happens! It's funny.)

This would make a great living book for math workboxes! It's also going to solve one of my Christmas problems...what to give my 8yo? Shhhhh...he's getting one for Christmas. ;)

*If you order QUICK you could also get the 24 Penguins Before Christmas: A 365 Penguins Advent Calendar. I haven't seen it but it got stunning reviews.

More Polygon Play! (Geometry Class #4, Fall 2010)

Time for some more polygon play...wanna play along?

I made a pdf (print it, free!) to review the terms for the number of sides of various polygons.

1. Print the sheet on white cardstock, one per person.

2. Have each child take a red crayon or colored pencil. As you say (or draw) the number of sides, they lightly color the rectangle with that term. Example: "Color in the word that means a 4 sided polygon." [Color in "quadrilateral."] After you've gotten through the first 8 terms on the sheet...

3. Now do the same thing in reverse. Have each child take a blue crayon or colored pencil. As you say one of the terms from the top of the page (already colored in red), they color in the card with that number of sides using blue. Example: "Color in the number of sides on an octagon." [Color in "8 sides."] Now everything should be colored in except the last two terms.

4. Do students know what "regular" vs. "irregular" polygons are? Review. If they don't, look here. Color the regular/irregular cards green.

5. Cut out cards.

6. Mix up cards. Match corresponding red and blue cards.

As you read the book, The Greedy Triangle, ask students to lay out the red and blue cards that match the polygon on each page.

Give children some popsicle sticks. Tell them to lay out the "triangle"  and "regular" cards. Ask them to make a polygon with those criteria. What do students notice about their polygons? Are they alike? Different?

 Ask them to lay out the "quadrilateral" and "irregular" cards. Make a polygon. Are their figures the same?

Ask students to set aside all cards except the following: triangle, quadrilateral, pentagon, hexagon, regular and irregular. Each child should select one card of each color (red and green) and lay it out in front of him. (Use the blue cards as needed if kids need reinforcement to recall how many sides are on each polygon.)

This is one awesome irregular pentagon. Dude!

Now, have kids switch places and build a polygon based on the cards at his new spot. Rotate around the table, placing cards, building, and checking polygons.

See our first Polygon Play session here. 

Look, Ma! Another irregular pentagon!

Captain, we are ready for irregular triangle takeoff!

Whoa. Irregular quadrilaterals are too cool for words.

P.S. The content in this post has been heavily influenced by Bridges lessons. Love them! :)

Pumpkin Math

One of my favorite books, How Many Seeds in a Pumpkin?, is perfect for this time of year. Children estimate how many pumpkin seeds are in pumpkins of varying sizes. In order to find the number of seeds, children count by 2s, 5s, and 10s, and learn that the smallest pumpkin actually has the most.

Maybe you'll want to read it and do some of the "Pumpkin Problem Solving" or "Fall Math Activities" over at Mathwire...one of my FAVORITE math idea sites! ;)

Polygon Play (Geometry Class #2 Part I, Fall 2010)

Math class today. We did so many things that I'm going to break it into a series of posts. :)

Started with a brief review of tangrams by reading Grandfather Tang's Story. I provided tangrams so the kids could construct the animals as the story went along, but the story became too captivating...they slowly abandoned the pieces to fully concentrate on what was happening. Great book!

We continued with a review of polygons. Last week we discussed the characteristics of a polygon. I drew a variety of figures on the board and they told me which were polygons and which were not. Since we'd read The Greedy Triangle last week, we quickly flipped through the book to see the polygons used there.

We then made a flapbook...the word "polygon" on the outside of the flap and the characteristics listed inside. (This is not pictured.)

And now...to make a quiz for their parents...to see if they know about polygons. They LOVED that idea! We made a series of 6 flaps (see right photo...one flap is open.) I had them cut off the top flap to make a "title" area..."Is this a polygon?" They then drew 5 figures of their choice on top of the 5 remaining flaps. Under each, they wrote "yes" or "no" to indicate whether the figure is a polygon. They used rulers to make sure the straight lines were actually straight.

In the process of assessing their parents, this became a quick assessment to see if they had a solid understanding of polygons. With fun in the process!

Nice addition to polygon lessons:


Geometric Stick Lessons

Tangrams (Geometry Class #1, Fall 2010)

[Note: Most of what we did today comes from Bridges materials.]

We began by reading a Marilyn Burns book, The Greedy Triangle, in which a shape takes on one additional side after another, transforming from a triangle to a quadrilateral to a pentagon, etc.  I then asked the kids to consider what they already knew about geometry. They recorded their thoughts (words and/or pictures) in a 2-flap book. (We made our own flap books--photo left--but you can see a template under "Flip Flap with 2 flaps.") They wrote what they knew under the left flap and questions or things they wonder about under the right flap.

We then compiled our information on a class chart.  It was exciting to see lightbulbs begin to go on; several kids had little to nothing on their individual flaps as they thought "geometry" was something totally new. As they heard ideas from other kids, our list became longer and longer as they began to make connections.

We began thinking about Tangrams by reading Three Pigs, One Wolf, and Seven Magic Shapes. After the book, I slowly took them through the process of cutting their own Tangrams from 6x6" pieces of wallpaper. Here are instructions if you'd like to make your own. As they worked, we used a lot of mathematical vocabulary in context. Two words we especially highlighted: congruent and similar. A debate arose as to whether all triangles are "right triangles" or not. Some children were certain that they are; others were positive that there are exceptions. I told them to think about that for a time.

When the Tangrams were complete, we used them to create a variety of shapes using just 2 pieces: square, rectangle, parallelogram, trapezoid, etc. They were then challenged to create the same shapes with 3 pieces. They have homework (optional) in which they'll try to construct shapes out of 4 or 5 pieces.

Another challenge? To put the seven pieces back into their original square. This will be made easier by the fact that we used wallpaper with a pattern on one side. We finished with a reading of Agatha's Feather Bed.

During the week, students may try Tangram puzzles on an iPad/Touch/Phone or select from the following:

Tangram Puzzles

Cyberchase Tangram Game

Sagwa Tangrams

NCTM Tangram Puzzles

NCTM Illuminations Tangram Lesson

Tangram Puzzles to Print

Online Tangram Puzzles

We also have several tangram puzzle kits and a game called Classic Tangoes. Manipulating the puzzles by hand definitely uses a different set of skills than doing them on the computer. If you'd like to read some more books with Tangrams, pick up Grandfather Tang's Story or The Tangram Magician.

What an enjoyable class you are! See you next week! :)

P.S. We talked quite a bit about the properties of a trapezoid. Please have them look at this site.  It allows you to manipulate a figure so that it remains a trapezoid but allows you to see the wide range of possible figures.

Square Numbers

I LOVE this lesson at Childplay about square numbers. What fun! It's on my to-do list. When we do the lesson, I'll add the book, Sea Squares, in which "Rhyming text and illustrations of such sea animals as whales, gulls, clown fish, and seals provide opportunities to practice counting and squaring numbers from one to ten." (from book summary)

What lessons have you done with square numbers? Post your link! ;)

Place Value Structures, Class #4

Few things are more thrilling that seeing a child with an "ah ha" moment. Times that by several children and you've made my day!

Today was our last class on place value structures using material from Bridges. For the last three weeks we've been considering place value in base five, looking at units (1), strips (5), mats (25), and stripmats (125). Today I asked students to imagine what the pieces might look like if we were working in base four. They began by building the pieces with tile. They debated over whether the mat should have 16 pieces or 24 pieces. (I think that a few jumped from the logic that if base 5 had 25 pieces, then base 4 should have 24 pieces.) We looked at the base five chart we'd made. Students recalled that with base five, each place value was x5, so decided that with base four, each would be x4 with a total of 16 for a mat. No easy task.

They cut a unit, strip and mat from graph paper. I then asked them to cut out a strip mat. When they couldn't agree on the area, I asked them to build mats with tile. Ah-ha! Then they remembered how to follow the place value pattern and had no trouble building a stripmat of 64.

Since the base five building didn't come easy, I wondered how the next question (and one of my big goals for the whole class!) would go...  "Close your eyes. Picture base ten. What would a unit look like? A strip? A mat?" The quietest child in the class couldn't hold back..."100!" They all immediately told me that base ten would have a unit of 1, a strip of 10, and a mat of 100!!!  I gave them base ten pieces and asked them to create a strip mat. With shining eyes and wide grins they asked me for TEN MATS and told me that a strip mat would have 1000 units. They built a strip mat and then proceeded to tell me what the next piece would look like and that it would have 10,000 units. "And then 100,000 and then 1,000,000...!!!!!!!!!!!!!" They helped me to complete a venn diagram comparing base five and base ten.

We read Sir Cumference and All the King's Tens and talked about place value in base ten. I also read The 329th Friend and asked them to build 329 (and a few additional numbers) with the base ten pieces.

It isn't unusual for teachers to ask, "Why do we teach other bases to children?" It's elementary, my dear! When we teach other bases, we give children an opportunity to develop conceptual understanding of place value. That deep understanding transfers to work in "our base," base ten. The lightbulbs in my classroom were going off so fast today that the electric meter must have been smokin'! Base ten place value means something new and exciting.

It's elementary. :)

You've been a wonderful class! Can't wait to see you again!

Who Sank the Boat?

Yesterday, my 4yo ds received a book at church; this was his reward for reading for 10 hours this summer. (Well, I actually did the reading. That could be a whole 'nother blog entry. Do you know how many hours my older son read on his own compared to how many hours I read to his younger brother? Shameful, I tell you! About 2.5:1 ratio!)

The book, Who Sank the Boat?, is one that I've wanted to buy.

Summary: The reader is invited to guess who causes the boat to sink when five animals of varying sizes decide to go for a row.

Spoiler Alert (as if you can't see this coming!): four huge animals (cow, donkey, sheep, pig) get in, but it's the final passenger, the mouse, that causes the boat to sink.

We did some math and science activities based on the story.

First, collect a variety of objects from around the house. Set out two objects at a time and ask child to predict which is heavier. For some comparisons, I use large, light objects (fluffy pillow) vs. small, heavy objects (rock). Place one object in each of the child's hands and ask him which one is heavier. If you have a balance scale, you can do a little informal work there as well.

We did some float/sink lessons. (pdfs available--I used the picture drawing one for the 4yo, the writing one for my 8yo) The site also links a BBC interactive video where kids get to predict whether various items will sink or float.

Although this lesson is primarily for my 4yo, my 8yo enjoyed participating in the sink/float scientific inquiry. Challenge question...find an object around the house that will trick mom...one that I think will float when it will actually sink or vice versa. (My son did it with Playmobile animals.)

Check out more Math Monday links at Joyful Learner or click on the Math Box below to find more math activities suitable for Workboxes.