Math Crafts – Tangram Puzzles

Monday, September 21st, 2015

tangram pint2 20150730_172329_resized

Tangrams are ancient Chinese puzzles.  Each tangram is made with the same 7 pieces, called “tans.”

This math activity involves making your own tans, and using geometry vocabulary: triangles, right triangles, right angles, parallelogram, square, trapezoid, congruent, similar.  Then students develop spatial sense by moving the pieces to make pre-determined pictures, or to design tangram pictures of their own.

For younger students, precut the tangram shapes and introduce them to the shapes one at a time, finding the shapes that are exactly alike (the congruent triangles), and the ones that are bigger and smaller versions of the same right triangle (similar triangles).  Then discuss the two quadrilaterals (the pieces with four sides, the square and the parallelogram).  How are they the same? How are they different?

For older students who can use scissors to cut precisely, have them fold and cut the pieces as pictured below, or by using the directions found in this downloadable PDF.

Note that, in the pictures below, we are using an 8.5”x11” piece of paper, so the beginning square is 8.5”x8.5” and the resulting tangram pieces are larger than the standard size tangrams that can be purchased in a teacher store or online from a math catalog.  (Standard tangrams are cut from a square that is 4”x4”.)

MATERIALS

To trace tangrams (as shown above):

  • Sheets of colored foam
  • One packet of tangrams to use for tracing.
  • Pencil and scissors.

To cut your own tangrams (as described below):

  • Sheets of colored cardstock.

(For large tangrams use 8.5”x11” and start with a square that has 8.5” sides.)

(Standard tangrams start with a square that has 4” sides.)

  • Scissors

HOW TO FOLD AND CUT YOUR OWN TANGRAM SHAPES

1. Start with a piece of paper that is 8.5”x11”.

Fold it as shown, then cut off the extra piece so you are left with a square folded in half diagonally.  (Throw out the “extra piece.”)

tanstep1

2. Cut along the diagonal, making two big right triangles.

Notice that they are exactly the same shape, so they are congruent to each other.  (Technically, they are isosceles right triangles, because the two legs of the triangle are the same length!)

tanstep2

3. Take ONE of the big triangles and cut it in half to make two other right triangles, as shown.

These two triangles are also congruent to each other, but they are smaller than the big right triangle you started with.  They are “similar” to the big one, but congruent to each other.

tanstep3

4. Take the OTHER big triangle and fold the right angle down.

Make sure that the top is parallel to the bottom.  Then cut it off.  Notice that it, too, is a right triangle.   It is smaller than the other right triangles, but the same shape. We say it is similar to the larger triangles, but not congruent.

tanstep4

5. Now look at the long trapezoid (a quadrilateral with just two parallel sides).

Fold it in half and cut along the fold.  You are left with two smaller trapezoids, called “right” trapezoids because they have a right triangle in them.  The two smaller trapezoids are congruent to each other.

tanstep5

6. Take ONE right trapezoid.

Fold and cut so that you have a small right triangle and a little square.  Notice that two of the small right triangles would make a square!  Also notice that the small right triangle is similar to the other right triangles that you have made, but not congruent.

tanstep6

7. Now use the OTHER right trapezoid.

The final fold is the hardest!  Fold up the right angle to make a little triangle and a parallelogram.  Check with someone to make sure that you really have a right triangle and a parallelogram. Then cut.  Notice that the small triangle is congruent to the other small triangle.  Also notice that you can make a parallelogram using two of the small triangles, so the parallelogram has the same area as the square!

tanstep7

8. Move the pieces around to make other geometry discoveries.

For an extra challenge, see if you can make a square using just two pieces, then three pieces, or four, five, six seven. [No one has found a square with 6 pieces, but maybe YOU can! We know that all the others are possible!]  Here is the solution for all seven pieces:

tanstep8

9. If you followed the directions, you should have seven tangram pieces (as shown above).

  • Two large right triangles
  • One medium right triangle
  • Two small right triangles
  • One square
  • One parallelogram

These seven “tans” are used to make all kinds of pictures.  Find some in books, or online.

GENERAL LESSON DIRECTIONS 

1. Make and discuss the characteristics of the seven tans.

Hold up and manipulate the pieces as you discuss the shapes, to demonstrate the new vocabulary.   (It doesn’t hurt for little kids to learn big words!)

Here are some examples of what you can discuss:

-All the triangles are right triangles because they all have right angles.

-The two large right triangles are congruent to each other.  (The angles and sides are the same size.)

-The two small right triangles are congruent to each other.

-The two small triangles are similar to the large triangles.  (The angles are the same, but the sides of the large triangle are longer.)

-The medium triangle is similar to both the large and small triangles. (All are right triangles.)

-The hypotenuse (long side) of the medium triangle = the short side (leg) of the large triangle.

-The leg of the medium triangle = the hypotenuse of the small triangle.

-It takes two small triangles to make the little square.

-It takes two small triangles to make the parallelogram.

-The parallelogram has two sets of parallel sides, and so does the square.

-Ask kids what other discoveries they can make about the pieces.

2. Move the pieces around to make one of the tangram puzzles shown.

 3. Find books of other tangram pictures, or do tangram puzzles online.

Note: If you buy a workbook with tangram puzzles, it probably uses tangram pieces in the standard size (made from a 4-inch square). You may want to purchase a set of plastic or foam tangram pieces, especially if you elect to trace them instead of cutting and folding.

 

GET A PACKET

tan books 20150730_173007_resized_1

There are two Let’s Read Math lesson packets about tangrams, featuring these two books.  Both books are tales told with tangram pictures.  Lesson packets can be purchased on the Let’s Read Math website.

The Pigs Packet, is related to an easy reader called Three Pigs, One Wolf, and Seven Magic Shapes by Grace Maccarone. This book is a spoof on the story of the Three Little Pigs.

The Grandfather Tang Packet is related to Grandfather Tang’s Story, by Ann Tompert.  This story is about two authentic Chinese fairy tale characters called fox fairies, who turn themselves into other animals as the story unfolds.

Math Crafts – Graphing Sticker Pictures

Monday, September 21st, 2015

rooster farm pic,JPG

rooster farm graph

This is a Let’s Read Math activity about beginning graphing.  It was inspired by the book Rooster’s Off to See the World, by Eric Carle.  Ultimately, this activity involves sorting, classifying, making pictures and graphs, and interpreting graphs made by other children.

In the story book, Rooster gets bored of the barnyard and decides to leave the farm and go see the world.  He takes some animal friends with him, and each page of the book has a small picto-graph, showing each kind of animal lined up in a row, so you can see how many frogs, cats, and other animals are currently part of the adventure.  It was natural to follow up with a lesson about graphing.

The lesson involves making a sticker picture, then making a bar graph to represent the data in the picture.  Since the book was about farm animals, we made pictures and graphs using animal stickers on a barnyard scene.  Then we counted the animals in the picture to make a graph representing how many animals were on the farm.  Click below for downloads:

If you had fun with this activity, go through your sticker collection and think of other kinds of graphs you can make!  One idea would be to make a zoo picture or a jungle picture and put different kinds of jungle animals in the picture.  To make your graph interesting, though, try to have different numbers of each kind of sticker, not the same number.  (For example, it will be a boring graph if your zoo picture has exactly 6 lions, 6 tigers, 6 monkeys, 6 alligators, 6 ostriches, and 6 giraffes!)

OTHER IDEAS FOR STICKER PICTURES

  • Draw roadways and use stickers of different kinds of vehicles.  Then count cars by type, or by color.
  • Draw a garden and use stickers of different kinds of flowers.
  • Draw a sky and use stickers of different kinds of aircraft.
  • Draw a desolate scene and use stickers of different kinds of dinosaurs.
  • Draw a living room and use stickers of different kinds of cats.

rooster cat pic

rooster cat graph

Another idea would be to sort some stickers by making a picto-graph.  Then turn the picto-graph into a bar graph, so you can see that the same number of items are represented in both kinds of graphs.

rooster horn picgraph

rooster horn bargraph

ENRICH THE LESSON BY ASING GOOD QUESTIONS, LIKE…

How many instruments in all?

How many French horns? Drums? Saxophones? Trumpets?

Were there more drums or trumpets?  How many more?

Were there fewer saxophones or French horns?  How many fewer?

How many brass instruments were there in all?  (French horns + trumpets + saxophones)

MATERIALS  

  • Downloadables of the blank farm picture, the little farm animals, and the blank graph
  • Blank paper for making your own sticker pictures
  •  Stickers to make into a sticker picture or picto-graph of your own
  •  One-inch graph paper.
  •  Blank paper to write about your picture and graph.
  •  Crayons, pencils or markers for drawing or writing.

pwa-07 rooster (1)

GET A PACKET

Purchase a Rooster Lesson Packet on the Let’s Read Math website.  It includes the book, lesson outline, stickers, and worksheets  (including one-inch graph paper for making your own graphs).

Math Crafts – Crab Counting

Sunday, September 20th, 2015

crab make

This Let’s Read Math activity was inspired by the book One is a Snail, Ten is a Crab by April Pulley Sayre and Jeff Sayre.

pwa-06 one is snail (1)

The book starts by explaining that a snail has just one foot.  Then the snail is used to “add one.”  For example.  If “One is a snail,” and “Two is a person,” then “Three is a person and a snail.”  We extended the activity so that children solve math problems by counting the number of feet on other animals that they know.   We ask them. “How many feet do other animals have?”  Then have them solve problems like: “If one ladybug has 6 feet, how many feet are on 2 ladybugs?”

Another critical piece of information is that all crabs have ten feet – two claws, 2 back feet for swimming, and 6 feet for walking!  This leads to the concept of counting by tens.  If one crab is 10, then 2 crabs are 20, 3 crabs are 30, etc.   Make paper plate crab “puppets” with ten legs, then have children use their crab puppets to practice counting by tens!

The next part of the lesson involves using pictures of animals with different numbers of feet to practice addition, multiplication and beginning ALGEBRA!

MATERIALS FOR MAKING THE CRAB:

  • One red paper plate cut in half. (Need scissors.)
  • Red cardstock or colored paper to cut out legs and head. (Click here to get a .pdf of the crab parts.)
  • Scotch tape to attach legs.
  • Two brass fasteners to attach the claws.
  • Two googly eyes and a glue stick.

DIRECTIONS FOR THE CRAB:

1. Cut the plate in half.

2. Cut out two claws, six side legs, the two back legs, and a heart-shaped head.

3. Tape down the side legs and back legs on one of the half-plates.

Then tape down the head on the flat upper side of the same half-plate.

4. Put the second half-plate on top of the bottom half-plate, to hide the scotch tape.

5. Insert the claws between the two plates.

Attach them by using a brass fastener that goes through the top plate, the claw, and the bottom plate. This allows the claws to move. (To make it easier to insert the brass fastener, you may want to use a hole puncher.)

Finally, use the glue stick to glue down the googly eyes.

6. Put your hand between the plates to make a puppet.

Have your puppet count by tens!

CRAB ALGEBRA

westchestr crabs08 (1) crab algebra

DIRECTIONS FOR DOING ADDITION, MULTIPLIICATION, AND ALGEBRA

1. Have fun thinking about the number of feet on different animals.

Make up a set of animal pictures and tell how many feet on each animal.

2. Practice Addition.

What if you had a cat and a dog?  (4+4=8)

Or a spider and a bird? (8+2=10)

Or a crab and a fish? (10+0=10)

3. Practice Multiplication.

What if you had 3 cats? (3×4=12)

Or 5 ladybugs? (5×6=30)

Or 8 spiders? (8×8=64)

Or 6 dogs and 10 snails? (6×4)+(10×1) = 34

4. Practice Algebra.

Use variables to record your discoveries.

If C=cat and D=dog, C+D=8,   or  3C=12.

If Sn=Snail, and Cr=Crab, 5Cr + 1Sn = 51.

GET A PACKET

Purchase a Crab Lesson Packet on the Let’s Read Math website.  It includes:

  •      one copy of the book
  •      the black line master for making the crab
  •      a lesson outline
  •      a collection of animal pictures for practicing addition, multiplication, or algebra
  •      two worksheets for practicing “Animal Algebra.”

 

Math Crafts – Paper Chains and Place Value

Sunday, September 20th, 2015

chains

In math, it’s important for students in beginning grades to understand place value so they can do computation with regrouping.   Make paper chains, with ten loops of a color.  Have each child make ten loops, then staple ten tens to make a hundred.  If adults are present, they can contribute to the effort as well!  Estimate how far 1000 loops will reach (across the room? down the hall? across the playground?)    Staple ten hundreds to make 1000.  Then see how close you came with your estimate.

This activity helps kids understand that:  ten ones = 10; ten tens = 100; and ten hundreds = 1000.

Then help them understand things like:   20 ones = 20     30 tens= 300    50 hundreds = 5000

MATERIALS:

  •      1” paper strips in several different colors. (I used 100 sheets of  8.5”x11”  photocopy paper cut into 11 strips each)
  •      Small rolls of scotch tape to make each loop.
  •      Stapler to staple the tens together.  (Note: Mini-staplers are not a good idea; they tend to jam.)

DIRECTIONS:

Note: 1000 loops is a lot of loops for one kid to make.  Get ten or more kids to work together, and enlist the help of any adults in the vicinity.

1. Use scotch tape to make a string of ten loops, all the same color.

(Having tens in the same color facilitates counting by tens to 100, rather than counting by ones.)

2. Put an adult in charge of stapling the tens together to make a hundred.

Each time you reach 100, let the group know.  Have the group “check” that the chain has 100 by counting the tens:

10,20,20,40,50,60,70,80,90,100!

chains parent

3. After you reach the first 100, ask kids to estimate how far 1000 will be.

Then put the hundred aside and start a new hundred.

4. Each time you reach another hundred, celebrate.

“Check” that there are 100 links by counting tens, then put the hundred aside to start a new hundred.

5. When you reach ten hundreds, count the hundreds:

100,200,300,400,500,600,700,800,900….1000!  

Some kids will say “ten hundred.” Great! That is your opportunity to talk about the fact that ten hundred is indeed the same as one thousand.   Ask: What are two names for 1100? (Correct answers: “11 hundred,” or “one thousand one hundred”).  Watch out for your genius kids. They maybe be very clever and say “110 tens”!  Smarty pants.

6. Test out their estimate about how far 1000 loops will reach.

chains kids

READ ABOUT 1000

To build number sense, read the book How Much, How Many, How Far, How Heavy, How Long, How Tall is 1000?  by Helen Nolan.

chains how big

This book is one of 16 books featured in the Let’s Read Math Funbook 2 for elementary grades.

Math Crafts – Alphabet Symmetry

Tuesday, September 15th, 2015

mirrors alphabet

Attending to visual details is an important skill in math and science education: Make a set of letters and use a small mirror to look for horizontal and vertical symmetry.

Next, use mirrors to experiment with other properties of reflection. Look for interesting images in books, newspapers and magazines and see if you can make things happen by using your mirror.

If you have two mirrors, put them together to make different angles of reflection and see rotated images! (Mirrors and reflective properties are interesting for students of all ages, even adults!)

MATERIALS:

  • 3×5 index cards.
  • Packet of alphabet letters (I got mine at the dollar store.)
  • Glue stick
  • 1-2 small mirrors (plastic mirrors for kids)
  • Small items like earrings, beads, macaroni; or magazine pictures. (These are for studying angles of reflection.)

DIRECTIONS FOR ACTIVITIES WITH ONE MIRROR:

1. Glue letters to index cards.  Use one mirror to look for lines of symmetry in the letter.

For example:     B has one line of symmetry (horizontal).

H has two lines of symmetry (horizontal and vertical).

A perfectly round circle has an infinite number of lines of symmetry!

2. Think of reflection challenges. For example:

Start with the letter R.  Can you make a B out of an R?

Start with an R.  Can you make a K out of an R?

Can you make a W out of an M?

Can you make an M out of an N?

3. Look for the book M is for Mirror: Find the Hidden Pictures, by Duncan Birmingham.

mirror book

4. Look for magazine pictures that have symmetry.

For example, find the line of symmetry in a butterfly, a light bulb, a person’s face (though faces are not perfectly symmetrical!).  Play with your mirror(s) and make other reflection discoveries.

mirror mag pix

DIRECTIONS FOR ACTIVITIES WITH TWO MIRRORS:

If you have two mirrors, you can study angles of reflection.

1. Little kids can make big angles and little angles.

Put small objects between the two mirrors and look inside.  What do you see?  (Lots of images!)  Make the angle get bigger and smaller to see a different number of reflections!

2. Older students can think about why this happens, by considering the exact angle of reflection.

A complete rotation is 360◦.

Put two mirrors together to make a right angle (90◦).

Select a small item to put between the mirrors.  If you make an exact right angle, you will see four images: the original item and three reflections, because 4×90=360.

If you make a 60◦ angle you will see 6 images: the original and 5 copies, because 6×60=360.

Use a protractor and some small objects (like earrings, paperclips, or stickers).  Deliberately experiment with other angles.

Try 45◦ (for 8 images); 120◦ (for 3 images); 40◦ (for 9 images).

Here’s an example using 72◦ (for 5 images):

mirror watch

QUILT DESIGNS:

Another fun activity is to use mirrors to see reflected images in quilt designs.  Many quilt designs are made by sliding, rotating, or reflecting a basic design.  These two books are featured in the Let’s Read Math Quilts packet:

Eight Hands Round, by Ann Whitford Paul, and

Sam Johnson and the Blue Ribbon Quilt, by Lisa Campbell Ernst.

pw1-07 eight-hands

pw1-08 samjohnson