Sacramento Homeschool Math By Hand

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 270

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful.

Focusing on the basics (the 4 processes up through long multiplication and division, multiplication tables’ fluency, factoring, fractions, and decimals) before embarking on the complexities of the number line makes sense.  Zero, positive and negative numbers, and using the 4 processes with the number line is best reserved until there’s sufficient capacity for understanding.

The Math By Hand Grade 4 / Kit 4 / Integers and Algebra presents these concepts with clear learning aids and tools that are oversized, colorful, and incremental.  Beginning with the history of numbers is an excellent lead-in to our Arabic numeral system, most specifically the role of zero in relation to positive and negative numbers or integers.  A brief history of numbers is presented in booklet form, along with basic rules for number line calculation.

Here are two suggested captioned drawings, “The Story of Zero” and “Tokens, Pictographs, and Cuneiform.” that can be taken from a detailed but anecdotal and biographical history of our modern system of numbers.

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for the next in a series of posts on the Math By Hand number line lessons.

The post The Number Line: Why Wait Until Grade 4? (#270) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 269

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Understand decimal notation for fractions and compare decimal fractions.
7. Compare two decimals to hundredths by reasoning about their size.  Recognize that comparisons are valid only when the two decimals refer to the same whole.  Record the results of the comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using the number line or another visual model.

Distinguishing between tenths and hundredths happens easily, with practice.  All the methods in the previous fractions standards posts: coins and decimals, the decimal strip, the laminated decimal-fraction conversion chart, etc., along with lots of dedicated practice, will accomplish this.

Any fourth grader can easily grasp the difference between three cents, thirty cents, and three dollars!  Coins and bills are much more effective and clear as models than abstract ones.  Math By Hand teaches the greater-less-than symbols in mid Grade 2, and they are then used consistently with whole numbers, fractions, and decimals.  Once their function is learned they can be readily applied to any set of numbers. The next several posts will address the number line as it’s taught in the Math By Hand system, including fractions and decimals.

Now on to another of Nasrudin’s wise, witty, and wonderful Middle Eastern folk tales.  This one concerns the over emphasis on appearance, featuring a beautiful, richly embroidered coat that Nasrudin wore to a banquet at a wealthy friend’s house.  His coat may have been made with fabric similar to this intricate Persian design.

And here is the story, The Hungry Coat, as it appeared in Mary Harris Todd’s Mustard Seed Journal.

A wealthy friend invites Nasrudin to a banquet. Wearing his old patchwork coat, Nasrudin sets out for the banquet. Along the way he stops to help capture a runaway goat. When Nasrudin arrives at his friend’s house, the friend, the servants, and all the other guests ignore him. He realizes that it is because his coat is now dirty and smelly as well as worn-out. Nasrudin hurries home, bathes, puts on a magnificent new coat, and returns to the banquet. Now everyone is glad to welcome him. Delicious food is set before him, which he proceeds to feed to his coat. “Eat, coat! Eat!” he says. The host and guests are aghast. “Why surely you wanted my coat to eat,” Nasrudin responds. “When I first arrived in my old coat, there was no food for me. Yet when I came back in this new coat, there was every kind of food for me. This shows that it was the coat–not me–that you invited to your banquet.”

Teach fractions and decimals, but be light-hearted about it and please do integrate the arts!  Falling victim to test-prep-itis, happening too frequently in our schools to the detriment of so-called “frills” like PE, Art, Drama, Music, to name just a few, is beyond foolish and short-sighted.

Because knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in starting tomorrow for a special series of posts on the Math By Hand number line lessons.

The post 4.NF 7: The Last Fraction/Decimal Standard & More Nasrudin! (#269) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 268

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Understand decimal notation for fractions and compare decimal fractions.
6. Use decimal notation for fractions with denominators 10 or 100.  For example, rewrite 0.62 as 62/100, describe a length as 0.62 meters, locate 0.62 on a number line diagram.

The coin exercise from post #267 would fulfill this standard nicely since the circled coin amounts are labeled as both fractions and decimals, and are also expanded or reduced from or to tenths to hundredths and vice versa.  The Math By Hand / Grade 4 / Kit 3 / Decimals includes a reversible write-on/wipe-off conversion chart.  On side 1: fractions to decimals, and on side 2: decimals to fractions.

Oversized fraction lines and a division sign are featured on both work surfaces.  A wipe-off crayon and eraser is supplied with the laminated charts, with the conversions later copied into workbooks.  The decimal version in measurement could be integrated with a project (building or gardening, etc.) so it’s less abstract and more user-friendly.  Math By Hand covers the number line with an extensive introduction and tools for working with the 4 processes and positive/negative numbers.  After the CCSS 4.NF 7 post, several posts will be devoted to the Math By Hand number line materials.

On to the glorious Grade 4 trickster tales!  One of Math By Hand’s multi-cultural tales features Nasrudin, a Middle Eastern folklore figure whose humorous escapades often involve his hapless donkey.  These tales, both wry and wise, have earned Nasrudin a beloved place in many hearts, young and old.  And they most certainly quench the fourth grader’s thirst for rambunctious adventure!

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.NF 6: Tenths & Hundredths & Decimals & Fractions (#268) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 267

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Understand decimal notation for fractions and compare decimal fractions.
5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.  For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

Math By Hand covers this standard nicely with two modalities:
1) Coin rubbings or copies that show decimal/fraction conversion and equivalency, as well as reductions of both fractions and decimals from 100ths to 10ths.  For coin rubbings: scatter quarters, dimes, nickels, and pennies on a tabletop, cover with newsprint, lightly tape the corners, and rub with the broad end of a black crayon.  For coin copies: scatter the same configuration on a copy machine glass and copy.  Proceed as shown below.

2) The Decimal Strip: A felt strip and numbered index cards represents both coin and dollar amounts, and translates these into decimal 10s and 100s.  The top image is an excerpt from the Grade 4 binder and the bottom image is the felt strip itself with numbered index cards and quarters.

With these and other tools and plenty of skills practice, decimals are a cinch.  Having the basics of fractions and decimals on hand builds the kind of confidence needed to follow that Grade 4 need for adventure and exploring: today the neighborhood, tomorrow the world!  Here is a Grade 5 chalkboard drawing from Catie Johnson (not meant to be a globe but rather a depiction of the North American continent).  See her gallery of chalkboard drawings here.

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.NF 5: Coins & Dollars & Decimals, O My! (#267) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 266

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.  For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed.  And between what two whole numbers does your answer lie?

Since this standard is so similar to 4.NF 3d below, I will repeat portions of that post here.  The same operation, multiplying a fraction by a whole number, is required for the last three standards.  This repetitiveness seems counterproductive, roast beef or no.  And the sorts of practical applications posited by word problems can be readily accomplished when skills are in place.

d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to solve the problem.

Today’s standard requires multiplication rather than addition.  Here is the multiplication page from Math By Hand’s little reference guide, Fraction Rules & Keys.  Examples are given on the facing page of the booklet.

This is a no-nonsense guide to anything that might need to happen involving multiplication with a fraction or mixed number.  Note that the two elements that are most relevant are shortened to RCF (reduce/cancel first) and GCF (greatest common factor).  A picture is worth 1,000 words of course, so the examples make all of this so much clearer.  The rest of this post duplicates post #263, because the essence of both standards are so similar.  (With the exception of the art, because there are so many outstanding Grade 4 images to choose from!

Along with many other basic tools and strategies for understanding the fundamentals of fractions and how they relate to each other through the 4 processes, a comprehensive overview enables confidence and the ability to apply all rules to any situation, including multi level word problems.

Though the Math By Hand materials do not include word problems per se, a plethora of such problems and worksheets can be found online or in textbooks. However, before implementing them, it’s a good idea to research the ins and outs of word problem construction, to circumvent any comprehension problems that may arise. Building the foundation for understanding by using clear, pictorial, graphic, hands-on materials comes first and then continues to be used as support while applying it to any purpose.

And as always, there’s the artful side. Here’s a sketch for a Celtic Knot cross-stitch project by a fourth grader from fiveofus.ca on Pinterest.  The joyful process of creating something from scratch looms large for all of us, but especially for children, because there is no more effective way of finding one’s true path in the world.  And no boring worksheet or standardized test prep should get in the way of that journey.  Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.NF 4c: Word Problems Redux. Roast Beef? (#266) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 265

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful.  The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
4.  Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.  For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5), recognizing this product as 6/5.  (In general n x (a/b) = (n x a)/b.)

Isn’t 4b the same as 4a?  It seems the addition of parenthesis is the only difference, since 4b’s example: 3 x 2/5 = 6/5 = 1 1/5, is the same as 4a’s example: 5 x 1/4 = 5/4 = 1 1/4.  The addition of parenthesis introduces an algebraic concept that may be best reserved until later, therefore teaching only one concept at a time.  Math By Hand slowly and carefully introduces the basics of beginning algebra at the end of Grade 4, with stories and hands-on tools.

Having students participate in group discussion is a staple of the Common Core classroom, in both math and language arts.  I conducted the usual lesson plans search for this standard and found many, many relevant resources and websites, with most of them advocating student group learning and discussion.

At first glance, the goal of student-centered learning is a valid one.  But on closer inspection, questions arise.  Such as, is too much time devoted to repetitious discussion?  Or, is it a stretch to expect children under the age of reason (12-14 years old) to engage in logical discussion and debate?

Here is a list of questions focused on suggestions for student feedback, from the website CPALMS

Feedback to Students
1. Monitor students during individual work time by circulating and seeing what is being recorded as they work the problem. Encourage students working out a correct solution strategy and challenge them to find another way to confirm their solution. Prompt students who are struggling or not addressing the task with the following possible questions:

What do we know about the situation?

Can a drawing help you see what is happening?

Is there another way to prove your answer is correct?

What does that amount tell you?

Does this look like any problems we have done before? Why or why not?

How did you determine that amount?

What do the numbers used in the problem represent?

What does the 4 tell us? The 2/3?

How is the number of ribbons related to 2/3 yards?

What operations might we use to find a solution?

How did you decide in this task that you needed to use…?

Could we have used another operation or property to solve this task? Why or why not?

2. During the group sharing and discussion encourage students to share and listen to others. Acknowledge those asking questions of each other that promote deeper understanding. (This addresses the Math Practice Standard: Construct viable arguments and critique the reasoning of others.)

3. In preparation for the instruction after this lesson, sort the summative assessments according to understanding. Form small groups from this work. Once small groups are assembled according to understanding, point out misconceptions and use fraction bars, the area model, or number line to illustrate why the solution was incorrect.

Summative Assessment
1.At the conclusion of this lesson, students will be given an incorrect solution and asked to determine whether or not it is correct and to justify their thinking. 

I am especially troubled by the last sentence.  Why give a student an incorrect solution?  It sounds like trickery, and to what end?  Imposing rigor and deep thinking in this manner is a mistaken notion.  It’s like watching a pot boil.  Students need facts, they need to be given tools for lifelong learning, then be allowed to go to it.  Rather than having them toil endlessly over small, detailed tasks, shouldn’t we be giving them a true renaissance education, one that’s broad and motivating?

This sort of endless questioning promotes doubt and insecurity.  In the Waldorf classroom, the teacher stands before the students as a loved authority figure, one who represents the world to those in his or her care as a living font of wisdom and knowledge.  This sort of teaching is not pedantic or overbearing.  It draws forth innate knowledge from the student, but (and this is most important) it does so indirectly.  Student and teacher together bear witness to the wonder of the world, with every concept or idea taught as an element of that wonder.

In this light, fractions are wonder-ful.  They shine with their own glory and are downright, interesting fun.  When this is seen, when student interest is kindled, learning occurs naturally.  The content of the above standard is simple.  It can be taught briefly, economically, and hands-on.  With modeling, but without the repetition and inherent distrust that’s endemic in Common Core style probing and questioning.

Caroline Myss, medical intuitive, teacher and author, once relayed a story about intuitive knowing.  Years ago as a novice gardener, she found herself continually pulling carrots out of the ground to see how they were doing and track their progress.  Excellent analogy.  (See below for recipegirl.com’s wonderful portrait of multicolored carrots.)

As parents and teachers, we must let the learning process happen in a nurturing environment that is not intrusively counterproductive.  Because knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.NF 4b: How Much Discussion is Too Much Discussion? (#265) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 264

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful.  The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
4.  Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b.  For example, use a visual fraction model to represent 5/4 as the product of 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4).

In my usual search for Common Core examples, I found the usual scenarios, with repetition and worksheets as a mode of instruction.  Conversely, if concepts are taught clearly and economically, there’s no need to bore and browbeat students with demeaning tasks.  One such task I found was: to find the product of 7 x 2/5, use repetitive addition by drawing objects.  In this case, 14 objects would need to be drawn.  This sort of activity is demeaning to a fourth grader.

Math By Hand uses beans to teach fraction concepts, with kidney beans representing whole numbers and black beans representing fractions’ numerators and denominators.  Similarly to how beans are used to represent place values in Grade 2 (10 black beans/ones are cashed in for 1 kidney bean/ten), mixed numbers are generated by cashing in fractions (with remainders).  This example taken from the Math By Hand Grade 4 binder shows an addition problem, but the same method is used for the other 3 processes.

Students, using a graph paper workbook to facilitate correct placement, learn the basics of the 4 processes with fractions in a concentrated way.  The trust that’s implicit in this sort of teaching is empowering and and builds self-confidence, because once the method is learned there’s no need for repetition and boring, demeaning worksheets.  The knowledge gained can be applied where needed.

Less time on repetitive tasks leaves more time for joyful learning, like deconstructing the woven pattern in this Longobardian Knot, from waldorftoday,com.  This knot is #4 in complexity, see the page for increasingly complex knots up to #8.  Just another example of how knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.NF 4a: Bean-o Fractions & Mixed Numbers (#264) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 263

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful.  The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3. Understand a fraction a/b with a> 1 as a sum of fractions 1/b.
d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to solve the problem.

The Math By Hand / Grade 4 / Kit 2 / Fractions is a thorough, seamless approach to teaching and learning fractions.  As in Grade 1 when all 4 processes were learned at once so relationships and correspondences between them were clear, so are all 4 processes with fractions introduced and learned at the same time.  A basic, clear “rules” booklet is used as a reference as long as it’s needed.  

Here’s a page from the addition and subtraction section of the booklet.  The choice here was either slightly out of focus or too small to see, but note that the originals are perfectly clear and sharp.  (The booklet is a compact pocket guide that is only 8 pages long and measures 4 1/4″ x 5 1/2″ very portable!)

Along with many other basic tools and strategies for understanding the fundamentals of fractions and how they relate to each other through the 4 processes, a comprehensive overview enables confidence and the ability to apply all rules to any situation, including multi level word problems.

Though the Math By Hand materials do not include word problems per se, a plethora of such problems and worksheets can be found online or in textbooks.  However, before implementing them, it’s a good idea to research the ins and outs of word problem construction, to circumvent any comprehension problems that may arise.  Building the foundation for understanding by using clear, pictorial, graphic, hands-on materials comes first and then continues to be used as support while applying it to any purpose.

And as always, there’s the artful side.  Here’s another wonderful image from the waldorftoday.com gallery of main lesson book drawings: a study of the eagle from the human and animal block.  Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

The post 4.NF 3d: Beyond Adding & Subtracting Fractions (#263) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 262

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful. 

Grade 4: the year of the Trickster!  Here is a wonderful tale wherein Raven tricks Crow into giving away his entire winter’s cache of food, and of the surprising consequences that result in the raucous crow call we know so well today.  The trickster tale speaks to the fourth grader on many levels.  As s/he experiences a surge in confidence and bravado, stepping further into the world-at-large, the rambunctiousness that often accompanies this is well-met with tales of moral consequences resulting from unjustly taking power.  Or even if, as in this tale, moral consequences are not immediately apparent.

Knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

Raven and Crow’s Potlatch
A Skagit Raven Tale as told by Eldrbarry

Raven used to live high up in the upper Skagit River country. He was very lazy. In the summer when the other animals were busy gathering food for winter, he would be flying from rock to stump and stump to rock making fun of them. Raven just laughed when Crow (his cousin) urged him to follow squirrel’s example – but Raven never prepared for the cold months, when the snow would drift over the ground and cover all the remaining food.  But now Raven was in trouble. Winter had come and the snows were deep. He was hungry – and Raven loved to eat. He had to find someone who would share their food with him.  Raven went to see Squirrel. He had a huge supply of pine nuts and seeds and other food hidden all over the place. Raven poked his head in squirrel’s nest in a old fir tree. Squirrel had lots to eat. Raven politely begged for some food. Squirrel scolded him – that was always Squirrel’s way – “You refused to work and save for winter – and you poked much fun at me – you deserve to starve!”  Raven went looking for Bear. But Bear was sound asleep in his cave and could not be wakened. Raven looked around for some food, but it was all in Bear’s belly – Bear had already eaten it all and was sleeping till spring.  Raven was now very hungry. He thought: “Who can give me something to eat? Everyone is either stingy like Squirrel or sleeping like Bear and Marmot, or they have gone South for winter like the snowbirds.” Then he thought of Crow – he would be easy to fool!  Raven flew to Crow’s nest. “Cousin Crow, we must talk about your coming potlatch!” Crow answered. “I have not planned a potlatch”  Raven ignored his response. “Crow, everyone is talking about your potlatch – will you sing at it?” “Sing?” Crow had not known that anybody really cared for his singing voice – though in those days, Crow’s song was much more like that of Wood Thrush than it is today.  Raven continued to talk of Crow’s potlatch. “You are very talented and possess a beautiful voice – everyone will be so disappointed if you don’t sing at your potlatch!”  “What potlatch? . . . You really like my singing?”  “We love your singing, Crow,” Raven answered. “The Winter’s cold has chilled the forest and we’re cold and hungry and singing will help us forget our cold feet and empty stomachs. Now you get started fixing the food – looks like you have plenty here – and I will go invite the guests to your potlatch. You can practice your songs as you cook!”  Crow’s hesitation now overcome, he began to prepare all the food he had collected for winter, and as he prepared it, he practiced his songs. The more he thought the feast and how everyone wanted to hear him sing, the more excited he got about it.  Meanwhile Raven was offering invitations to all the animals of the forest. (Of course Marmot and Beaver were sleeping like Bear, and Robin and Goose were gone South) To each he said the same thing: “Come to My potlatch! I have worked hard to prepare it. There will be much food at Raven’s potlatch and Crow is helping and will sing for us. There will be fern roots and wild potatoes, dried berries, fish and meat. Come to My potlatch! It will be a great occaision.” Raven did not invite Squirrel however since he had refused to share his food with Raven. But all the rest of the animals were invited to Raven’s Potlatch.  When he returned to Crow – he was busy singing and cooking. Raven told him – “Everyone is coming – be sure and fix all your food – they will be hungry after their journey. And your songs are sounding so good! Crow’s potlatch will be a great feast!”  As the guest arrived, Raven welcomed each one to his potlatch. There was deer and mountain goat and mouse, rabbit, ptarmigan and jay. The guests were seated and the food was brought out. Crow started to sit and eat – but Raven asked him for a song first. “It’s not good to sing on a full stomach, Crow”. So crow began to sing. Every time he would stop to eat – Raven would insist he sing another song. “You can’t sing with your mouth full, Crow!” Encouraged again and again by the guests – who were busy stuffing themselves with Crow’s food – Crow sang song after song after song – all day until night – and Crow’s voice became hoarser and hoarser until all he could do was “Caw – caw.”  As was the custom – the left over food was collected by the guests and taken by them for their homeward journey. Even Raven had taken his share and left as Crow was cleaning up. Crow had nothing left to eat. ” At least,” Crow thought, “I won’t go hungry – I will be invited to their feasts.” For it was the custom that having been entertained, each guest was now obliged to return the favor and invite the host for a return potlatch.  But the invitations never came. Since all the guests thought it was Raven who hosted the feast, Raven was invited to enough dinners to keep his stomach full for several winters – and he never went hungry.  But Crow, who had been fooled, had been reduced to starving, and never regained his singing voice either. He was destined to spend his winters begging in the camps of men for scraps of food. And that’s where we find him today – squabbling over scraps in grocery store parking lots – Caw! Caw! Caw!”

The post A Winter Trickster Tale With Raven & Crow (#262) appeared first on Math By Hand.

A Year in the Life: Ambient Math Wins the Race to the Top!
Day 261

For one year, 365 days, this blog will address the Common Core Standards from the perspective of creating an alternate, ambient learning environment for math. Ambient is defined as “existing or present on all sides, an all-encompassing atmosphere.”

And ambient music is defined as: “Quiet and relaxing with melodies that repeat many times.” Why ambient? A math teaching style that’s whole and all encompassing, with themes that repeat many times through the years, is most likely to be effective and successful.  The CCSS math standards are listed here in blue followed by their ambient counterparts.

Number and Operations – Fractions 4.NF
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3. Understand a fraction a/b with a> 1 as a sum of fractions 1/b.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

The Math By Hand / Fractions / Kit 2 includes dice and a set of wooden cubes ( 0-6 and 7-12) for creating fractions and mixed numbers.  Using the Fractions Rules and Keys booklet and a graph paper workbook, addition and subtraction is carried out with fractions and mixed numbers.  Here are several images: the Math By Hand cubes and dice, examples of using them to create proper and improper fractions and mixed numbers, and the instruction page from the dice half of the workbook.

Fractions, equivalency, mixed numbers, and calculating with all 4 processes can be learned fairly quickly.  Time is then freed up for the real creative work of engaging the right brain through challenging tasks.  This frees up every child’s genius, allowing that sometimes latent ability to grow and thrive.  Form drawing, for example serves many excellent purposes, and deep interest is always piqued when the intricate, woven or braided Celtic forms are introduced to the fourth grader, so that the challenge that these complex drawings present is well met.  (The beauty below comes from Artopotamus’ blog.)

Have your fourth grader write his or her name in Celtic letters!  Again, the challenge will be well met.  As always, knowledge ensues in an environment dedicated to imaginative, creative knowing, where student and teacher alike surrender to the ensuing of knowledge as a worthy goal. Tune in tomorrow for more Grade 4 math CCSS and their ambient counterparts.

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