Math

Math Art & The Right Side of Normal

I’m currently purging the schoolroom, starting to prepare for our move this summer. While cleaning out, I found these pictures I created long ago with my oldest daughter.

3x5-math-art

Ten years ago she was almost 7 years old.  To say she was struggling with math is an understatement; no matter how much we worked on math, especially facts, the next day all would be forgotten.  Zero retention.  I began doubting my ability to teach her, especially since she was also struggling with reading.

Fortunately for both of us, around this time I also stumbled upon a yahoo group run by Cindy Gaddis at The Right Side of Normal.  I learned so much from her and the other moms on the group.  (The yahoo group has pretty much died out, but the Facebook page is still active, and her book and website contain wonderful information.)  My child was absolutely a “resistant learner,” and thanks to Cindy this was the beginning of learning to work with my daughter, rather than against her.

4x4-math-art

We dropped all formal math curriculum and began to play.  These pictures are the very first thing we created, a treasure for me as they mark the beginning of my journey as a teacher of (and learner with) my children, not just an instructor.

Math Art Week 4 – Division Facts

This “division tree” is similar to the multiplication leaves from last week. The first step in this project is to cut circles of different sizes from varying types of paper.  If the student struggles to cut circles, circle punches can be used, or they can be cut ahead of time by the instructor.  (Some of my students wanted to make their shapes more like leaves than circles.) Next, arrange the circles in stacks; use the term “divide” as much as possible: “Divide your circles into 3 stacks?  What about 4?  How many circles are in each stack? Can you divide them evenly without any leftover?”  Once the student grasps the concept of dividing, proceed with the project.

week4pic1

Materials

  • various colors and types of paper*
  • scissors and/or circle punches
  • glue or mod podge

*We used scrapbook paper for the background, colored paper for the tree and circles, and some origami paper for the decorative bits. In the picture above, origami paper is used for the top 2 layers, which makes it difficult to distinguish that there are 2 layers!

Directions

  1. Cut and divide circles as describe above.
  2. Decide on a permanent arrangement / number of stacks.
  3. Cut a tree to match the number of stacks.
  4. Glue all parts to a heavy piece of paper.
  5. Create a label to show which division fact is being illustrated.

Math Art Week 3 – Multiplication Facts

This week in Math Art we created an art panel to illustrate a multiplication fact. Our plants had leaves broken into 2 sections; each section was doodled differently.  The students multiplied the 2 sections by the number of leaves they created to figure out their math fact.

week3pic1


Materials

  • cardboard pieces from boxes
  • gesso
  • various paintbrushes
  • acrylic paint
  • plain colored paper for leaves and stem
  • decorative paper
  • mod podge
  • fine tip permanent ink art marker, such as a Sharpie Fine Point Pen*

*It is important to use a permanent ink pen, or the doodles will smear when mod podge is applied.


Directions

  1. Paint cardboard with gesso to prime it.  Let dry.  (Use hairdryer to speed up drying time).
  2. Paint a background color on cardboard with acrylic paint.  Let dry.
  3. On colored paper, draw leaf shapes.
    • Divide each leaf shape into sections.  You may choose 2, 3, 4 or more sections.  Each section will be doodled differently.
    • If you are working on a specific math fact, instruct the students to divide their leaves into that number of sections. For example, if you are creating, 6×4, you may have 6 leaves each with 4 sections, or 4 leaves each with 6 sections.
  4. Decorate the leaves by doodling each section with zentangles.
  5. Cut out a slightly larger leaf from the decorative paper to mount under the doodled leaf.
  6. Cut out a stem shape.
  7. Use mod podge to apply all shapes to the dried cardboard.
  8. Write out the math fact and mod podge it on the cardboard.

 

Math Art Week 2 – Fractions

Our project this week was based on the book “Picture Pie” by Ed Emberley.  Picture Pie helps the reader create simple pieces of art based on the fractions of a circle.   While these look simple, the execution can be quite tedious.  Below are the details for a simple bookmark.

FullSizeRender

“Fraction Fish Bookmark”

week2pic11


Materials

  • black construction paper, cut in 2″ strips for bookmarks
  • colored or patterned paper for circles
  • 1″ circle punch
  • scissors
  • glue
  • laminator (optional)

Directions

Teach a short lesson on the fractions of a circle.  Have kids use the circle punch and scissors to cut up and sort fractions.

 

Choose pieces for the Fraction Fish bookmark. This project requires 12 quarters and 2 eighths.


week2pic9

Arrange pieces in desired configuration.  Glue onto bookmark.

week2pic10

Let dry.  Laminate if desired.  (If you choose not to laminate, press the dried bookmark until all pieces are nicely flattened).


Other Projects

Math Art Week 1 – Number Bonds

This summer we are hosting a small “Math Art” class for a few of our friends.  Each week we will create a math-themed art project (or an art-themed math project!).

week1pic3

Ways to Make “10”

For Week 1 we created a mixed media board that illustrated a Number Bond.  Each student picked a Number, then cut numbers out of newspaper and magazines to show at least 4 “ways to make” their chosen Number.   The ages of these kids were from 7 to 11 years old.


Materials

  • cardboard
  • various paintbrushes
  • Mod Podge
  • papers to create a background (scrapbook paper, newspaper, old worksheets)
  • acrylic paint
  • small container to mix paint with water
  • wax paper
  • old magazines and newspapers
  • scissors
  • pencils

Directions

  1. Cut pieces of cardboard from old boxes.
  2. Use a paintbrush & Mod Podge to paste on a background.  We used old math worksheets & tests. Tear paper to fit. (If you haven’t used Mod Podge before, just paint the cardboard with a thin coat, place your piece of decorative paper, then paint another thin coat on top.  Continue layering.)
  3. Let dry, about 15 minutes if the Mod Podge is a thin coat.  Trim edges.
  4. Paint the background with watered down acrylic paint.  If the paint is too thick to see the background, wipe some off with a towel.
  5. Let dry.
  6. Have each student choose a Number.  On a piece of paper, have them list at least 4 “ways to make” that Number.  Provide magazines, newspapers, and scissors to hunt for numbers.  (The weekly ads are a great source!)  They can keep track of the numbers they have found by laying them on the paper where they wrote their answers.
  7. When the background paint is dry, use a pencil to sketch the body of a dragonfly.
  8. Paint the dragonfly body with acrylic paint (NOT watered down!).  I place a dab of red, yellow, blue, and white paint on a piece of wax paper for each child, and let them mix colors as they wish.
  9. Let dry.
  10. Have each student look through magazines to find a pretty pattern for wings.  Cut wings and set aside.
  11. Once the paint is mostly dry, use Mod Podge to attach the numbers and wings.
  12. After all paint is completely dry, you can Mod Podge over the entire piece as a finish coat.

Learning Fractions Through Art

One of my areas of frustration with early math learning is fractions.  There’s only so many times we can break a candy bar into pieces, or cut a pan of brownies, or slice a pizza.  Why are all the fraction activities based around food?

The Idea

Explore fractions through art projects.

The Execution

Last week we completed two art projects where we incorporated fractions as a natural part of what we were creating.  The first project was based on Ed Emberley’s Picture Pie book.  I used a large 1-inch circle punch and punched a pile of circles in different colors.  The kids helped me cut some of the circles into halves, quarters, and eighths.  First, we made some of the animals in Picture Pie.  I made sure to use “fraction speak” when asking my kids for pieces: “can you hand me three eighths in orange for the bird’s wings and beak?”

Fractions 1

Bird from “Picture Pie.” Age 5

A few days later my daughter asked to do some more Picture Pie.  We looked at some of the fancy mosaic pictures, and she decided to create her own “Christmas Picture.”  Note that after she glued her mosaic, she embellished her picture with some fraction-esque drawings.

Fractions 2

Age 7

Later in the week we incorporated more of fourths and eighths into our daily Doodles.  I taught the kids a new pattern we called “Flying Saucers.”

Fractions 3

Flying Saucers is in the bottom right quadrant.

We practiced Flying Saucers on scrap paper before starting our Doodles.  Even my 5 year was able to “draw an oval, then draw one line down, draw one line across, and draw four lines from the center out,” breaking the oval into eighths.  Before I even mentioned it, the kids had a collective light bulb moment where they shouted out that we were drawing eighths!  None of them chose to use the new pattern in their daily Doodles, but I later discovered this picture that had been created during free time:

Fractions 4

 

The Extension

We will be returning to these projects later in the year to learn twelfths.  We will also add in fractions from squares and rectangles.

I would like to create some mixed media pieces (cut and paste, markers, pencils, etc) using the ideas we learn from Picture Pie and our Doodles.  Kandinsky is a good inspiration for these pieces.

Composition 8 by Kandinsky

 

 

 

 

Trading Pennies 2

After the student is comfortable with Trading Pennies 1, and can do some simple regrouping addition on paper, you can move on to more complicated problems.

The Idea

Use pennies and dimes to practice more challenging addition problems with regrouping.

The Execution

Foundational Skills

  • fluent in Trading Pennies 1
  • fluent in counting by 10’s

Concurrent Skills

  • learning to count out money

Step 1 

Set up an addition problem.  In the example below, we are adding 25 cents and 16 cents.  Have your child count out the dimes and pennies.

Trade Pennies for Dimes 2

Step 2

Have your student rearrange the dimes so that they are all grouped together.

Next, have her rearrange the pennies so that there is one group of ten, and another group of the leftovers.

Step 3

Make a big show of trading the ten pennies for one dime.  Add the dime to the dime pile.

The Extension

Once your child is extremely comfortable with this stage of adding, start to moving increasingly to paper math.  If they enjoy counting with money, start to play with concepts like making change for a dollar.

Resources

PDF of this problem.
Word DOC of this problem.  Feel free to change and use.  (The formatting looks terrible online, but it should download and look right).
Clip art of Pennies and Dimes to create your own worksheets.

Trading Pennies 1

One of the foundational skills of early math is the ability to understand regrouping (also known as ‘carrying’ in addition and ‘borrowing’ in subtraction).

The Idea

Use pennies and dimes to expose your child to the interchange of ones and tens in place value.

The Execution

Foundational Skills

  • fluent in counting to 10

Concurrent Skills

  • 1 dime = 10 pennies
  • 10 + 1 = 11, 10 + 2 = 12, 10 + 3 = 13, etc.

Step 1 

Set up an addition problem.  In the example below, we are adding 7 pennies and 5 pennies.  Have your child place actual pennies on the illustrations.

Trade Pennies for DimesStep 2

Have him rearrange the pennies so that there is one group of ten, and another group of the leftovers.

Step 3

Make a big show of trading the ten pennies for one dime.

The Extension

Do the activity just to this point for several days.  You may need to park here for quite a while.  However, if your child quickly picks up the idea, take the next step of showing how the dime (10’s) and pennies (1’s) make the written answer to the addition problem.  Next, start adding bigger numbers using dimes and pennies (future post).

Resources

PDF of this problem.
Word DOC of this problem.  Feel free to change and use.  (The formatting looks terrible online, but it should download and look right).
Clip art of Pennies and Dimes to create your own worksheets.

Build-Your-Own Math Curriculum

I’m going to start this post out by saying that Math on the Level (MOTL) is, hands-down, my favorite math curriculum for K-5. We still use it every week in our house. However, the price can be a challenge for some families, especially those beginning their home ed journey who aren’t exactly sure of their child’s learning style or their personal teaching style.

The Idea

Build your own math scope-and-sequence using internet resources.

The Execution

Scope and Sequence

A few years ago I was curious as to exactly what the US Common Core math standards were teaching.  I dug through them and created a chart to illustrate the progression of topics from year to year.  I don’t necessarily agree with their placement of certain standards, but I do find it a useful resource to see how a skill progresses over the years.

Common Core Math Standards Chart (K-5)

page 1

If this chart seems confusing, you can find the full text many places.  One of my favorite resources is Mr. Nussbaum’s page.  He has the Common Core text, plus links to online games and printable resources on this page.  (Scroll to the bottom of the page).

This is just one example of a possible scope and sequence.   Another good one is this visual chart of math topics.  Math on the Level also has a wonderful list of concepts with charts and ideas on how to progress through the sequence.

Use your outline as a rough estimate.  If other things pop up that you want to learn, write them in!

Teaching

There are as many ways to teach as there are families; here I will outline how we do it.

First, I choose 2-3 concepts for each child.  Over the course of a month, I introduce those concepts to each child via games, manipulatives, and discussion. Sometimes we find a good book to read on the subject.  If the child seems to be grasping the concept quite easily, I will show them how that concept translates to paper work.  If they absolutely 100% understand it, I will add a page of those problems to their Math Binder (more on that below) and add in a new concept to the rotation.

My children seem to learn in spurts.  If they are primed and ready with a learning spurt, we may blow through several concepts in the month.  If not, we may tread water and do the same games and manipulatives over and over.  As long as they are enjoying it, we continue.  If I start to meet resistance, we put that concept on the back burner and try something new.  Don’t feel boxed in by grade levels … if you have a child that is loving addition, let them add 1, 2, 3, or 10 digit numbers!  Let them start multiplication in kindergarten.  Conversely, let them stay with the easier concepts longer if they need the time.

Among my children, I have all sorts of math learners.  My oldest daughter resisted math for many years.  However, around the age of 9/10, it suddenly began to click.  When it clicked, she caught completely up to grade level in 6 months.  I struggled with doubts, but looking back I realize that her brain development just had to catch up to the point where she was ready.  We continued many games, books, and activities during that time, trying to keep those math brain cells firing!  My second child learned math instantly.  He is the child that would do great with a workbook program, but I have tried to pull in some of the more creative things to stretch that part of his brain.  I want him to understand and play with math, not just do his pages and be done.  My third child tolerates math.  She usually understand the concepts, but doesn’t have the energy to do a lot of work.  She enjoys a lot of our math art projects.  My youngest enjoys math the most.  He watches big brother, and is determined to work at the same pace.  At 5 years old he wants to learn multiplication, so I make him worksheets with x0, x1, and a few x2 math facts.

Review – The Math Binder

As your child starts to become proficient in certain math concepts, it is important to continue review.  Repeated review over a long period will move the math knowledge from short term to long term memory.  Don’t move any concept to review until you are certain your child knows it.  These are problems she should never get wrong!

From Math on the Level I learned a concept called the “5-A-Day” review.  It’s very simple: start each day with 5 review problems.  There are a few different ways to do this: some parents like to handwrite a sheet of paper with 5 unique problems on it.  This method got to be a bit unwieldy for me with 4 kids, so I started what we call the “Math Binder.”  Details for creating the Math Binder are on my Math Skills page.

Once you have started a binder, you need a way to schedule which problems to review, and how often.  As your child grows the list of concepts can get quite long.  MOTL has a wonderful spreadsheet that we often use.  It allows you to input how often you want to review a concept (daily, every 2 days, weekly, etc), and spits out a schedule for you.  In lieu of that, you could simply keep a checklist By Week or By Day to track it.  For very new concepts, it is good to review daily or every other day.  As time goes on, you can move to once a week, every other week, or once a month review.

 

Adding Doubles

This is one of the first math activities I do with my children, usually starting around age 4 or 5.  It is simple enough for little ones, but is easily customized for older kids as well.  (See The Extension below).

The Idea

Use things found in the natural world (or the child’s world!) to expose him to early addition.  We memorize all the “doubles” math facts early on.  Later they can move to “doubles +1,” “doubles-1,” etc.

5+5

The Execution

Print pictures that show doubles addition.   If you are artistic, draw a picture.  If your child is artistic, let him draw a picture.  (To find pictures on the internet, search Google > Images.  Then choose Search Tools > Type > Line Drawing.) You can use paint, handprints, stamps, or any other medium your child enjoys. Label and write the math fact on the paper.  (See picture above).  In the early years I will usually label and write for my kids.  As they get a little older and better at numerals they can do it if desired. Display the picture for reference, or slide it into plastic protectors and make a “math fact book.”

Examples:

2+2=4

Any 4 legged animal.  Label the legs “1, 2, 3, 4.”  Others ideas: car.

3+3=6

Insects, like this Bumblebee.  Label the legs “1, 2, 3, 4, 5, 6.”
Other ideas: construction vehicle, train, flowers.

4+4=8

Spiders, like this one.  Label the legs, “1, 2, 3, 4, 5, 6, 7, 8.”
Other ideas: trains, flowers with 2 layers of 4 petals.

5+5=10

Trace 2 hands or 2 feet. Label the fingers/toes “1, 2, 3, 4, 5, 6, 7, 8, 9, 10.”
Other ideas: flowers with 2 layers of 5 petals.

6+6=12

A dozen eggs.  Label the eggs, “1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.”
Other ideas: flowers with 2 layers of 6 petals.

10+10=20

Trace 10 fingers and 10 toes. Label.

3+3

The Extension

Teaching strategies using “doubles.”

Once the child knows the “doubles” facts well, she can practice them on worksheets using doubles.

This same method can be used for older children to investigate multiplication facts.