Factor Trees Are Confusing My Child

A special blog post for parents whose children find the process of creating factor trees to be completely opaque. You could also introduce this technique to a class before moving them on to factor trees. This is from Jamie York!

Without exception, I would recommend Jamie York's excellent book for Middle School Math.

Making Math Meaningful:
A Middle School Math Curriculum for Teachers and Parents

The process Jamie recommends requires rewriting the number multiple times, so just know that it will likely take your child one sheet of paper for each prime factorization problem. Let's take 270,000 as our example.

270,000

300 ⋅ 900

2 ⋅ 150 ⋅ 900

2 ⋅ 50 ⋅ 3 ⋅ 900

Get the idea?

It is a lot of rewriting, but the children can easily see that all of these rows will still multiply together to equal 270,000, which they CANNOT see with a factor tree. You just keep going until the whole thing is prime numbers.

2 ⋅ 25 ⋅ 2 ⋅ 3 ⋅ 900

2 ⋅ 5 ⋅ 5 ⋅ 2 ⋅ 3 ⋅ 900

2 ⋅ 5 ⋅ 5 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 300

2 ⋅ 5 ⋅ 5 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 ⋅ 150

While it may be tempting for us as adults to start to put the prime factors in order from smallest to largest, don't have a child do that. They need to get an idea of where these numbers are coming from! Plus, you want to be able to go back and check their work if they get the wrong answer at the end.

2 ⋅ 5 ⋅ 5 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 ⋅ 6 ⋅ 25

2 ⋅ 5 ⋅ 5 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 25

2 ⋅ 5 ⋅ 5 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 5

2⁴ ⋅ 3³ ⋅ 5⁴

When you are ready for factor trees, I love

You Can Count on Monsters: The First 100 Numbers and Their Characters
by Richard Evan Schwartz

See my past blog post on how to draw these "Monstrously" Cool Factor Trees

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John Cage - Oh, How We Love You

This year I'm teaching an Art Club for 5 & 6 year old children.

I absolutely love John Cage...

and so I chose him as the focus of our first two Art Club sessions!

Beautiful Noise: The Music of John Cage
by Lisa Jean Rogers

In the past, we've often learned about John Cage in Philosophy:

Is It Music?
Jan 1, 2024

What Is Music?
Oct 12, 2016

My favorite quote ever from doing that lesson with children:
"That music lacked music."

If you're unfamiliar with 4'33" here is an NPR article:

The Story Of '4'33"'
https://www.npr.org/2000/05/08/1073885/4-33

A quote from this article:

For the second, the provisionally entitled "Four Pieces," Tudor starts a stopwatch, sits down in front of the piano, closes the lid and begins a performance in which he never plays a note.

After 30 seconds of silence, Tudor resets the stopwatch and times another two minutes, 23 seconds of silence, then another one minute, 40 seconds of silence. But is it silence? ....

The silent composition, which became known by its duration of four minutes and 33 seconds, was influenced by Cage's encounter with the so-called "white paintings" by his friend Robert Rauschenberg — huge canvasses of undifferentiated white whose surfaces vary infinitely with particles of dust and light reflections.

Nothing: John Cage and 4'33"
by Nicholas Day

John Cage was in the news during COVID:

A 639-Year Concert, With No Intermission for Coronavirus
https://www.nytimes.com/2020/09/07/arts/music/john-cage-as-slow-as-possible-germany.html

The long, slow performance of a piece by the American composer John Cage is scheduled to run until 2640. On Saturday, an audience in Germany heard a sound change — the first since 2013.

John Cage has come up in our Art History studies of Sister Mary Corita Kent, Yves Klein, and Yoko Ono. His book with Lois Long, Mud Book: How to Make Pies and Cakes, is also featured in my list of Books in ALL CAPS.

Mud Book: How to Make Pies and Cakes
by John Cage & Lois Long

We will also probably spend some time on John Cage when we do Inventors and the Slinky! In 1959, John Cage composed an avant garde work called "Sounds of Venice" scored for (among other things) a piano, a slab of marble and Venetian broom, a birdcage of canaries, and an amplified Slinky.

But on to my notes from Art Club:

Fri Sep 25 - introduction to mark making / graphic notation

read Beautiful Noise: The Music of John Cage by Lisa Jean Rogers

pass out large sheets of paper and colored pencils

make sounds and have the children create marks which go with them (shake water bottle, stamp feet, tap pencil, sweep broom)

look at examples of birdsongs in graphic notation

have each person sing one of the birdsongs
can we figure out which one they're singing?

have each person sing a wordless song of their choice
while we draw the colors & marks that go with their voice

set out a selection of musical instruments

demonstrate the sound made by each

have each person draw a set of colorful marks
then choose instruments and perform their "piece"

take a Nature walk and listen for birdsongs



this page is from a vintage children's encyclopedia

Fri Oct 2 - tactile exploration / sensory play

read Nothing: John Cage and 4'33" by Nicholas Day

sit on a yoga mat in silence for 4 minutes and 33 seconds
what did you hear?

add drops of food coloring to water and watch it swirl around

read Mud Book: How to Make Pies and Cakes by John Cage

go outside to the mud kitchen to mix up recipes!
to the mud we added our colored water and some seashells

I did a whole blog post on Fun Things We Add to the Mud Kitchen. I find that colored water is such a simple addition and a surprisingly big hit with kids!

This post contains affiliate links to materials I truly use for homeschooling. Qualifying purchases provide me with revenue. Thank you for your support!

Botany: Art & Handwork Ideas

It's Making Zen right now and I'm having a great time. I also remembered that there have been several projects over the past 3 years (6 Making Zen conferences) that I mentally tagged as being good possibilities for Botany.

And now we are beginning Botany!

I'm just going to make myself a master list of Art & Handwork Ideas. Feel free to add to it if you found a great online tutorial, or a lesson in a book.

Speaking of books... Drawing from the Book of Nature is KEY for this block. I'll have to make another post listing all of the Botany chapters.

workshops from Making Zen

Amy Maricle

Fine Line Flowers
Oct 2025

watercolor paints: indigo, rose madder, lemon yellow




Mini Watercolor Leaves: Beautiful Botanicals Made Simple
Sep 2026
inspired by Broadhead Arrowleaf

cold press watercolor paper

watercolor paints: indigo, lemon yellow, olive green

Darcy Hunter

Hand-Stitched Red Poppies with 3D Elements
Oct 2025




Painting Fabric with Watercolour and Inktense
Sep 2026

Kristin Peterson

Abstract Textile Seedpods
Sep 2026

inspired by Milkweed

Lisa Goddard

Mixed Media Textile Moth
May 2024

POLLEN: Darwin's 130-Year Prediction by Darcy Pattison

other workshops, tutorials & ideas

Wet Felted Flower
YouTube - Spry Whimsy Fiber Arts

Wet Felted Trumpet Flower
YouTube - Esther's Place

Sycamore Keys
YouTube - Marion's World
embroidery pattern available as a FREE download

This post contains affiliate links to materials I truly use for homeschooling. Qualifying purchases provide me with revenue. Thank you for your support!

Geodesic Dome... Conquered!

HOORAY!

First of all, color coding is key.

For our dome project (see my previous post on Buckminster Fuller and Geodesic Domes), we used

25 yellow + 5 orange straws (cut to 17.5 inches in length)

35 pink straws

The 5 orange straws are the star at the top of the dome. This helps keep everyone on your team oriented spatially as you assemble the structure.

We also used

40 green paperclips
(to make ten 4-way connectors)

30 yellow paperclips
(to make six 5-way connectors)

60 white paperclips
(to make ten 6-way connectors)

These jumbo 20 inch long straws are awesome, and having all the different colors really helps!!!

STEPS IN ASSEMBLY

1. Lay out straws by color. Cut the yellow and orange straws to 17.5 inches.

2. Lay out paper clips by color. Make 4-way, 5-way, and 6-way connectors.

Use the yellow paper clips to make the 5-way connectors
(these are for the centers of the orange and yellow stars)

Use the green paper clips to make the 4-way connectors
(these are for the base; they go into the "feet" of the yellow stars)

Use half of the white paper clips to make five 6-way connectors
(these are for the top points of the yellow stars)

Use the remaining white paper clips to make ten 3-way connectors
(these are for the "hands" of the yellow stars and will join together to become 6-way connectors upon assembly of the dome)

3. Make the orange star (which is the top of the dome). Set it aside.

4. Make a yellow five point star.

5. Add the following connectors to the open ends of your yellow star

4-way connectors at the feet of the star (left and right)

3-way connectors (1/2 of six) at the arms of the star (left and right)

6-way connector on the top point of star
note: this connector will go into the orange star "cap" at the end

5. Insert the pink straws into the connectors all the way around your star to form a pentagon panel. These panels are the sides of your dome.

6. Make 4 more yellow stars / pentagons in this way. You will have five in all.

6. Assemble your pentagon panels into a circle, with the left and right arms touching. The white 3-way connectors there will need to be linked together to form a 6-way connector. At the base and at the top of the space between the panels you will place a pink straw. It will look like an hourglass shape.

On the floor, all the way around the base of the dome, you will see green connectors and pink straws.

The first row of triangles above the base will alternate between having yellow sides and having pink sides.

There will be five pentagon panels that make up the circular walls of the dome. One pentagon panel will be the cap on top.

yellow stars, highlighted

the orange star proudly takes its place at the top of the dome

In my experience, it takes A LOT of people to help hold each panel in place while the pink straws are being put in between them.

The design will be very floppy and it can be quite frustrating... until the cap is placed on top, at which point it will snap into place and become rigid and very strong. Buckminster Fuller came up with an amazing building!

This post contains affiliate links to materials I truly use for homeschooling. Qualifying purchases provide me with revenue. Thank you for your support!

Buckminster Fuller & Geodesic Domes

Science Club weeks 2 & 3 are follow up to our week 1 disaster. We are learning about the work of Buckminster Fuller and his Geodesic Domes.

The combination of curve and triangle in dome structures makes them strong and stable during an earthquake. Dome houses are also economical and environmentally friendly, requiring only a small amount of materials.

According to Buckminster Fuller: Poet of Geometry by Cole Gerst, Fuller's personal Dome Home in Carbondale IL "took only seven hours to build."

The American Institute of Architects calls the geodesic dome “the strongest, lightest and most efficient means of enclosing space known to man.” (pacificdomes.com)

Preparation:

straws - Perfect Stix 20" Extra Long Neon Straws - Pack of 300

35 straws, 20 inches long

30 straws, cut to 17.5 inches

our dome design is smaller and simpler than the illustration above, which shows 3 colors for 3 strut lengths

paper clips - enough to make the three following types of connectors

a huge thank you to the creator of DesertDomes.com!

the connectors slide inside the straws to hold them together, as shown below

Buckminster Fuller Book Display

Buckminster Fuller: Poet of Geometry
by Cole Gerst

Cosmic Fishing: An Account of Writing Synergetics With Buckminster Fuller
by E.J. Applewhite

Fuller's Earth: A Day With Buckminster Fuller and the Kids
by Richard J. Brenneman

Buckminster Fuller to Children of Earth
compiled and photographed by Cam Smith

also

Mortensen material Basic Operations Kit
(to demonstrate Richter Scale)

Earth: By the Numbers by Steve Jenkins

Activities:

review last week's disaster

explain Richter Scale and demonstrate with a math material to scale

level 1 - 1

level 2 - 10

level 3 - 100

level 4 - 1000

level 5 - 10,000

imagine level 6 - ten 10,000s

imagine level 7 - a hundred 10,000s

imagine level 8 - a thousand 10,000s

imagine level 9 - ten thousand 10,000s!!!

a level 6 would be ten of these 10,000 squares!

recall the destruction in the 1811 New Madrid Earthquake (8.1)

look at Earthquakes infographic in Earth: By the Numbers, pp.14-15

read excerpts from Buckminster Fuller: Poet of Geometry

use straws and paper clip connectors to build a geodesic dome

DOME NOTES

As part of a Housebuilding summer camp in August 2018, my three daughters successfully built a geodesic dome with these straws:

I could never have done it without Natalie and Leah!

This year we were determined to do it again. And so I am going to take VERY detailed notes as to how to be successful!

We had a lot of questions. Do you make the stars first (some with 5 spokes and some with 6)? Do you start at the top and work your way down? Do you start at the bottom and work your way up? Do you make all the connectors first or construct them as you need them?

As soon as my students and I figure this out, I will write it up!

Articles & Links:

10 Largest Earthquakes Ever Recorded
usgs.gov

Many thanks to desertdomes.com for help with the math. This is an amazing website which helps you build your own geodesic dome. Simply put in the radius of the dome you wish to build, and it will tell you how long to make your struts and how many of each length you will need, plus how many connectors of each type you will need.

Field Trip:

Buckminster Fuller's Carbondale Dome Home
407 S. Forest Avenue, Carbondale IL

This post contains affiliate links to materials I truly use for homeschooling. Qualifying purchases provide me with revenue. Thank you for your support!