Number Bases in Montessori: Binary Mat & Octal Mat

My oldest student started the year with an exploration of Number Bases (Jamie York suggests doing this in Waldorf in grade 8; I didn't learn it until my math teacher training at Smith). We focused on Binary and Octal.

The Number Bases page on my website gives a number of resources; here, I want to focus on the new Montessori-inspired Not-Golden Mats we made for the classroom! The original Golden Mat material looks like this:

It is base-10, and it uses a set of Colored Bead Material that goes up to 9 (the highest possible digit in any place value).

With base-2, the mat would be green (the Montessori color coding for two). The only beads you would use would be the 1-bars. This is because 1 is the highest possible digit in any place value in the binary system. The place value headings across the top of the mat would also be different:

BINARY

n⁰ = 1
n¹ = 2
n² = 4
n³ = 8
n⁴ = 16
n⁵ = 32
n⁶ = 64

and so on, multiplying the value of the column by 2 each time

Figuring these out is a wonderful exercise for the childen, and we were very pleased with the resulting felt mat.

In a traditional classroom setting, it would be very easy for each student to make their own Binary Mat using a sheet of large green construction paper. Simply place a bead in a section to show a 1; an empty section is a 0.

It's fun to see all the zeroes and ones change as you count up from 1 to 20!

We also made an octal mat, which was super helpful for Jamie's Number Bases Practice Sheet #2 (PDF sample from Jamie's grade 8 workbook ).

Note: It works out really well for base-8 to read Weslandia by Paul Fleischman first!

For base-8, the mat would then logically be made of a dark brown felt, and we removed the extra bead bars from my Decanomial Bead Bar Box so that we didn't have anything higher than 7. Here are the place value headings:

OCTAL

n⁰ = 1
n¹ = 8
n² = 64
n³ = 512
n⁴ = 4,096
n⁵ = 32,768
n⁶ = 262,144

In my experience, it works best to use the Multi-Base Bead Frame when converting from decimal to the other number base (base-2 to base-16) and the Not-Golden Mats when converting from another number base to decimal.

For example, 37_oct would be represented by a 3-bar in the 8s place and a 7-bar in the 1s place. 3 x 8 is 24, and 24 + 7 is 31_dec.

Oops, One More... Tree Bingo

Yesterday I finished my Botany Books post and was feeling all proud of myself. Then, walking down the hall at the end of the day to close the door of the Board Games closet, my eye fell upon Laurence King's Tree Bingo.
I pulled out the box, and what do you think was on the front?? Date Palm!

AARGH, I thought. We did that plant last week! But, of course, we can play the game at any time to help review the trees we've learned. Here are all the trees in the box (as with their other games, a fact booklet is included):

Alder (Alnus glutinosa)

Ash (Fraxinus excelsior)

Banyan (Ficus macrophylla)

Basswood (Tilia americana)

Beech (Fagus sylvatica)

Blue Quandong (Elaeocarpus angustifolius)

Bois Dentelle (Elaeocarpus bojeri)

Bottle (Pachypodium lealii)

Candelabra (Euphorbia ingens)

Cedar (Cedrus libani)

Chestnut (Aesculus hippocastanum)

Coco-de-Mer (Lodoicea maldivica)

Cypress (Cupressus sempervirens)

Date Palm (Phoenix dactylifera)

Dragon's Blood (Dracaena cinnabari)

Durian (Durio zibethinus)

Field Maple (Acer campestre)

Frankincense (Bowsellia sacra)

Giant Redwood (Sequoiadendron giganteum)

Grandidier's Baobab (Adansonia grandidieri)

Hawthorn (Crataegus monogyna)

Hazel (Corylus avellana)

Holly (Ilex aquifolium)

Jacaranda (Jacaranda mimosifolia)

Juniper (Juniperus communis)

Kapok (Ceiba pentandra)

Kauri (Agathis australis)

Larch (Larix decidua)

London Plane (Platanus x hispanica)

Madrone (Arbutus menziesii)

Maidenhair (Ginkgo biloba)

Monkey Puzzle (Araucaria araucana)

Nordmann Fir (Abies nordmanniana)

Oak (Quercus robur)

Olive (Olea europaea)

Orange (Citrus x sinensis)

Paperbark (Melaleuca quinquenervia)

Pecan (Carya illinoinensis)

Quaking Aspen (Populus tremuloides)

Quiver (Aloidendron dichotomum)

Rainbow Eucalyptus (Eucalyptus deglupta)

Rowan (Sorbus aucuparia)

Sandbox (Hura cerpitans)

Sassafras (Sassafras albidum)

Saucer Magnolia (Magnolia x soulangeana)

Scots Pine (Pinus sylvestris)

Shagbark Hickory (Carya ovata)

Silver Birch (Betula pendula)

Spruce (Picea abies)

Sweetgum (Liquidambar styraciflua)

Sycamore (Acer pseudoplatanus)

Traveller's Palm (Ravenala madagascariensis)

Tulip (Liriodendron tulipifera)

Tupelo (Nyssa sylvatica)

Upas (Antiaris toxicaria)

Walking Palm (Socratea exorrhiza)

Walnut (Juglans regia)

Whistling Thorn (Vachellia drepanolobium)

Wild Apple (Malus sieversii)

Wisteria (Wisteria sinensis)

Wollemi Pine (Wollemia nobilis)

Wych Elm (Ulmus glabra)

Yew (Taxus baccata)

Yoshino Cherry (Prunus x yedoensis)

What's in My Botany Books

First, I would like to update everyone that I'm no longer an Amazon Affiliate.

I cancelled my entire Amazon account last night after seeing news items about their invasive "About You" page and the new glasses their delivery drivers will be wearing to take constant photos of the path to your front door (that you cannot opt out of) which they can then use to train their humanoid delivery robots. This means that while in the past I spent a lot of time tracking down Amazon links to the specific books I was recommending, I will no longer be supplying links to purchase materials. I get most of my curriculum things used on eBay anyway, but Amazon was so pervasive that I felt those links were actually helpful to people. Now that so many people are boycotting Amazon, there's no reason for me to take the time.

End rant.

Instead of using a prepared Botany curriculum with a specifically laid out scope & sequence, we are just doing plants as they come onto our radar. They drop nuts, bear fruit, change colors, open flowers, give us a rash, etc. And we want to learn about them!

This means I need to look at my Botany books in a new light, and simply note which plants are covered in each. When we choose a plant, I can go to this blog post and hit "Find" to see if I have any resources for it.

The Burgess Book of Nature Lore
by Thornton W. Burgess

chapter 3
"Tommy's Insect Soup" - pitcher plant

I also wrote up a 2023 blog post detailing the Bloodthirsty Plants series.

Animal and Plant Anatomy
11 volume encyclopedia

vol. 1 - apple tree

vol. 2 - diatom, fern

vol. 4 - hawkmoth, honeybee

vol. 6 - marsh grass, mushroom, orchid

vol. 7 - potato

vol. 8 - saguaro cactus, seaweed, sequoia

vol. 10 - Venus flytrap, weevil

Diatoms and mushrooms are not plants, but we may end up talking about them and why they are not plants, so I want to remember that they're included. The hawkmoth and honeybee are pollinators; the weevil is a pest.

Drawing from the Book of Nature
by Dennis Klocek

chapter 11, "The Leaf"

11.1 - pinnate & lanceolate (willow), cordate (violet, morning glory)
11.2 - round or lobe-shaped (mallow), dentate (nettle)
11.3 - white oak, vetch, clover, cranesbill, carrot

chapter 12 - "Algae, Fungi, Lichen, Ferns"

12.1 - algae (bull kelp, fucus "rockweed")
12.3 - fern, horsetail

chapter 13 - "Higher Plants"

13.1 - bean, pussy willow
13.2 - plum - a perfect flower
13.3 - conifer (pine)
13.4 - tree growth
13.5 - venation (sycamore, maple, willow, poplar, birch)
13.6 - foliage shading
13.7 - live oak
13.8 - cottonwood
13.9 - birch
13.10 - fir
13.11 - apple

chapter 14 - "The Flowering Plants"

14.1 - shepherd's purse, cranesbill, storksbill
14.2 - crucifer (radish, burnip, mustard, cabbage, broccoli)
14.3 - bulbs (narcissus, jonquil, daffodil, crocus, amaryllis)
14.4 - rose
14.6, 14.7, 14.8, 14.9 - grasses (oat)
14.10 - legumes (peas, lentils, peanuts, beans)
14.11 - composites (sunflower, zinnia)

New Eyes for Plants: A Workbook for Observing and Drawing Plants
Margaret Colquhoun & Axel Ewald

an absolutely lovely book of seasonal art lessons!

chapter 1 - Late Autumn
potato, beech, groundsel, rose

chapter 2 - Winter
iceflowers, hogweed, oak

chapter 3 - Early Spring
snowdrop, horse chestnut, beech, tulip, daffodil, groundsel

chapter 4 - Late Spring / Early Summer
broad bean, sunflower, groundsel, cuckoo flower, musk mallow, charlock

chapter 5 - Midsummer
chamomile, peony, sow thistle, dog-rose, creeping buttercup, columbine

chapter 6 - Early Autumn
apple, beech, horse chestnut, wild cherry, vetch, buttercup, rose

chapter 7

yarrow, chrysanthemum, bramble, hawthorn

metamorphosis within the Brassica family:
Brussels sprout, cabbage, kale, cauliflower, kohlrabi

metamorphosis within the Rose familiy:
apple, rosehip, strawberry, blackberry, cherry

For my notes doing this block using Thomas Elpel's books, see Resources & Presentation Order for Plant Families. In that 2021 block I focused on Shanleya's Quest & Shanleya's Quest 2: Botany Adventure at the Fallen Tree

Matching Memory Games - Botany Edition

While working on my Botany planning for October, I found an old note about a Horse Chestnut card in Match a Leaf: A Tree Memory Game.

That inspired me to inventory all the plants included in all three Botany memory games!

First up we have Gardener's Match, which is largely a matching game between the whole plant and the edible food it produces. In an interesting twist, there are also two special cards (poison ivy & stinging nettle). If you turn over one, you lose your turn. If you turn over both, you lose the game!

Next up we have Double Take: Plants & Flowers. To the best of my knowledge, this game is out of print. An old copy was donated years ago.

In this matching game you pair the image of a flower with its close up.

The set includes Wonder Flower, Amaryllis, Hibiscus, Rose, Mum, Aster, Painted Daisy, Transvaal Daisy, Bougainvillea, Blue Hydrangea, Wax Plant, Loosestrife, Iris, Carnation, Lantana, Friesia, Hypericum, Blue Tit, Cyclamen, Flamingo Plant, Snap Dragon, Star Lily, Day Lily, Peruvian Lily, Moth Orchid, Sunflower, Scarlet Trumpet, Marigold.

Finally we have Match a Leaf: A Tree Memory Game. Like Gardener's Match, this game is made by Laurence King! Here you match the entire tree -- helpful for practicing identifying trees at a distance -- with a leaf picture.

The set includes

Red Maple (Acer rubrum)

Grandidier's Baobab (Adansonia grandidieri)

Horse Chestnut (Aesculus hippocastanum)

Monkey Puzzle (Araucaria araucana)

European White Birch (Betula pendula)

Shagbark Hickory (Carya ovata)

European Chestnut (Castanea sativa)

Swamp Gum (Eucalyptus regnans)

European Beech (Fagus sylvatica)

European Ash (Fraxinus excelsior)

Ginkgo (Ginkgo biloba)

English Holly (Ilex aquifolium)

English Walnut (Juglans regia)

Sweetgum (Liquidambar styraciflua)

Tulip Tree (Liriodendron tulipifera)

Olive (Olea europaea)

Scotch Pine (Pinus sylvestris)

London Plane (Platanus x hispanica)

English Oak (Quercus robur)

Northern Red Oak (Quercus rubra)

Sassafras Tree (Sassafras albidum)

Coastal Redwood (Sequoia sempervirens)

Giant Sequoia (Sequoiadendron giganteum)

European Mountain Ash (Sorbus aucuparia)

Scotch Elm (Ulmus glabra)

Now when I search my blog for the name of a particular plant, this post will pop up and I will remember that I have these resources!

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

Practical Botany in October

It's easiest for me to write up my notes in pieces, updating as we go along. Here, if you missed it, is my topic overview post: Notes on Practical Botany.

Now, on to the plants!

Milkweed

Common Milkweed (Asclepias syriaca)
Swamp Milkweed (Asclepias incarnata)

read Monarch and Milkweed by Helen Frost

look at dried milkweed pods & milkweed seeds

make Abstract Textile Seedpods from Making Zen workshop

nature sketch

add to MLB under Botany Basics (Seed Dispersal)




for more on Seed Dispersal

Seeds Move! by Robin Page

Bur Oak

Bur Oak (Quercus macrocarpa)

field trip to Superblock to pick up bur oak acorns & acorn caps

nature sketch

add to MLB under Tree Identification
note: oak wood is traditionally used in barrel making aka cooperage

The Mighty Bur Oak: Sentinel Tree of the Prairie Peninsula
Agricultural Electronic Bulletin Board, University of Missouri Extension

Another Gift from Oaks
International Oak Society

Judean Date Palm

Date Palm (Phoenix dactylifera)

read A Seed Is Sleepy by Dianna Hutts Aston

read The Miracle Seed by Martin Lemelman

open a fresh date to find its seed

nature sketch

add to MLB under Botany Basics (Germination)



Six New Ancient Date Trees
The Arava Institute for Environmental Studies


for more on Germination

Morning Glories by Sylvia Johnson (pp.14-15, 36-41)
available FREE at Internet Archive



terminology:
cotyledon, monocot, dicot, seed coat, embryo, endosperm

Osage Orange

Osage Orange (Maclura pomifera)

look at osage orange fruit and thorny branches

learn interesting facts about this tree - three links below

learn the term "dioecious" and cut open a fruit to see if the female tree it came from was pollinated (does it contain seeds?)

experiment with using osage orange as a natural dye

nature sketch

add to MLB under Tree Identification

note: due to its nasty spines, osage orange was widely planted as fencing to define property lines until the introduction of barbed wire (thus the name "hedge-apple")

Pruning the Osage Orange Trees
Martha Stewart

Natural Dyes - Harvesting Osage Orange
Folk Fibers

Osage Oranges in Peñasquitos Canyon
Friends of Los Peñasquitos Canyon Preserve

Persimmon

American Persimmon (Diospyros virginiana)

look at persimmon tree, leaf, fruit, and seeds

recipe?
here is my 2024 post of Persimmon Recipes

nature sketch

add to MLB under Special Plants - Edible

How Do Persimmons Breed?
Lady Bird Johnson Wildflower Center

Flower Dissection with Emily
Lowell Land Trust

terminology:
perfect flower, monoecious, dioecious

Chestnut

Chinese Chestnut (Castanea mollissima)
American Chestnut (Castanea dentata)
European Chestnut (Castanea sativa)
Horse Chestnut (Aesculus hippocastanum)

look at Chinese chestnut burr and nuts

recipe?
here is my 2023 post of Chestnut Recipes

look at European Chestnut and Horse Chestnut trees in Match a Leaf

look at Chestnut illustration in Tree Bingo

check fact booklet - what kind of chestnut is it?

scientific names matter because they prevent confusion!

nature sketch

add to MLB under Botany Basics (Scientific Name)


horse chestnuts are extremely poisonous and MUST NOT be confused with Chinese chestnuts; note that they share the first part of their scientific name with the also-poisonous red buckeye (Aesculus pavia)

sketch Chinese/American chestnut leaves (edible nut) compared to horse chestnut/red buckeye leaves (poisonous nut)

add to MLB under Botany Basics (Simple vs. Compound Leaves)


Match a Leaf: A Tree Memory Game

Cattail

Common Cattail (Typha latifolia)

look at cattails

read From Seed to Cattail by Lisa Owings

put students in pairs, and have them use the index of foraging books to find and share facts about foraging cattails

Field Guide to Edible Wild Plants by Bradford Angier

Midwest Foraging: 115 Wild and Flavorful Edibles from Burdock to Wild Peach by Lisa M. Rose

Stalking the Wild Asparagus by Euell Gibbons

nature sketch

add to MLB under Special Plants - Edible

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

Notes on Practical Botany

We are beginning our study of Botany! In Waldorf, this is a major grade 5 topic. I have a LOT of resources in my stash already, which can be really overwhelming. So I've decided to just think about one plant at a time.

Since we are focusing -- if at all possible -- on plants that are located near us and doing something interesting at the present moment, I'm going to organize these notes by month. (I'm in the temperate forest in the Northern Hemisphere, so it's early Autumn right now.) We have two Botany MLBs which we will keep out and available all year, in case a plant near us unexpectedly starts doing something interesting. They are tricky that way!

Our two MLBs are divided into two parts each, so four chapters in all:

MLB #1

Botany Basics

Tree Identification

MLB #2

Special Plants - Medicinal

Special Plants - Edible

I even got different MLBs (one with onion skin and one without... both green of course) so that it's easier for the children to tell them apart while we are adding in our information. I'm excited to see how the books come together!

Many thanks to Ms. Megan for the Botany books she's lending us this year:

Field Guide to Edible Wild Plants
by Bradford Angier

The Lost Book of Herbal Remedies: The Healing Power of Plant Medicine
by Nicole Apelian & Claude Davis

Manual of Woody Landscape Plants: Their Identification, Ornamental Characteristics, Culture, Propogation and Uses
by Micahel Dirr

Midwest Foraging: 115 Wild and Flavorful Edibles from Burdock to Wild Peach
by Lisa M. Rose

Special Forces Survival Guide: Wilderness Survival Skills from the World's Most Elite Military Units
by Chris McNab

Stalking the Wild Asparagus
by Euell Gibbons

A Weaver's Garden: Growing Plants for Natural Dyes and Fibers
by Rita Buchanan

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

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

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