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Form Drawing: A Journey Toward Geometry (#280)

February 1st, 2015 · Common Core

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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 279

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.

Both the Waldorf and Math By Hand curriculums begin form drawing at the very beginning, by focusing on it in the first Grade 1 main lesson block.  Each week thereafter begins with a Monday form drawing main lesson.  Form drawing is primarily just that: formative, so its beneficial qualities and effects translate to all other subjects and lessons.  Here are excerpts from the Math By Hand Form Drawing book.

Mirrored or symmetrical forms continue in Grade 3. The midline, beyond being merely a separator between the forms, is now crossed. New intricacy appears in the sequential forms, as well as in the now more challenging transformational ones.  Freehand geometry continues, with a greater complexity and accuracy. A decorative element promotes an artistic and motivating creativity. You’ll want to frame and/or display many of these beautiful drawings!

 

G3 FD2

G3 FD

In Grade 4, fraction forms combine math and form, though you’ll find that math is often a component of form drawing, as the forms do lend themselves naturally to a mathematical function. Much more complex forms are introduced, and weaving is combined with both the new and the more familiar forms. Weaving is challenging and rewarding, while helping with concentration and manual dexterity. New design elements are added, reflecting ancient folk cultures like the Norse and Celtic. And freehand geometry continues to develop, building toward formal geometry.

 

G4 FD2
G4 FD

 

Take your time with the whole process of form drawing, as it’s an investment in so many aspects of learning: concentration, discipline, experiences with story telling and art, mastering challenging tasks, and seeing, applying, and creating basic forms. The rewards are many, not least of which is the message that each of us can be creative, using just our own inner resources.

Save the form drawings in a simple folder like the one shown below. Fold a piece of poster board in half (score the fold first with a ruler and pencil or ballpoint pen). Punch holes in the top as shown, then tie it closed with yarn. Collect all of the form drawings, as real works of art!

FD FOLDER

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 ambient math alternatives to the CC.  

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Form Drawing: A Journey Toward Geometry (#279)

January 30th, 2015 · Common Core

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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 279

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.

Both the Waldorf and Math By Hand curriculums begin form drawing at the very beginning, by focusing on it in the first Grade 1 main lesson block.  Each week thereafter begins with a Monday form drawing main lesson.  Form drawing is primarily just that: formative, so its beneficial qualities and effects translate to all other subjects and lessons.  Here are excerpts from the Math By Hand Form Drawing book.

Form drawing promotes an awareness of archetypal forms while enhancing eye-hand coordination and creativity. Forms are first introduced through images and stories, translated to movement, traced in the air, and finally drawn on paper. The story or image provides a picture of the form, thereby making it more accessible. Using large motor movements to introduce new concepts incorporates a spatial orientation that effectively transitions to working with fine motor skills later on.

The simpler forms can be run very large, on the floor or ground, indoors or out. The more complex forms worked with in Grades 3 and 4 tend to go beyond verbal description or a translation into movement, and so need to be presented differently. Many forms can be contained within a circle or other visual parameters, which can be drawn on the floor or the pavement with chalk. Running the forms is an effective technique, and is also helpful with letter and number formation.  For the mirrored/symmetrical forms, the teacher moves the form on one side while the student watches, mirroring the movement on the other side.

The beginning drawings in Grade 1 illustrate the idea that all things are made up of straight or curved lines or some combination of the two. Vertical symmetrical or mirrored forms are drawn on one side, and mirrored on the other. They improve balance, spatial orientation, and eye-hand coordination, while forms with invisible lines help develop control and mastery, along with fluidity and flexibility. The horizontal sequencing forms that contain repeating or contrasting elements sharpen and develop visual judgment and acuity.

G1 FD

G1 FD2

 

In Grade 2, mirrored or symmetrical forms become horizontal and more complex, increasing fluidity and accuracy. Variations on straight and curved lines follow; transformative forms are drawn side by side on the page. Freehand geometric forms are introduced, with many variations, and continue to be a focus until formal geometry with instruments in Grade 6. The spiral is a recurring theme, later appearing in many different areas like life science, astronomy, and geometry.

 

G2 FD
     G2 FD2
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 on form drawing in Waldorf and Math By Hand.

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4.MD 5-7, 4.G 1-3: CCSS Geometry. Too Much, Too Soon? (#278)

January 27th, 2015 · Uncategorized

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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 278

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.

Measurement and Data 4.MD
Geometric measurement: understand concepts of angle and measure angles.
5. Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
a) An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
b) An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

Geometry 4.G
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.
2. Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.

In a word (or two): developmentally inappropriate.  Rather, it should be “sufficient unto the day” for your fourth grader to make reasonable strides in math within a sensible and methodical framework that includes mastery of the following concepts and skills: times tables, factoring, prime numbers, as well as fractions, decimals and the number line with positive and negative integers (including whole numbers, fractions, and decimals) applying 4 processes’ rules to all of the above, and a first look into beginning algebra.  All of this is included in both the Waldorf and Math By Hand Grade 4 curriculums.

The Common Core begins its harmful, abusive overreach in Pre-K and Kindergarden, and it continues to do so up through all of the lower school grades.  Waldorf covers geometry indirectly through form drawing until Grade 6 when it is formally studied, with instruments.  Children cannot be rushed into logic, reason, and abstract thinking before they are ready to make that leap.  This does not preclude building a solid foundation, however.  Which is exactly what form drawing does for geometry.  The posts over the next few days will be focused on form drawing.

Here is a youtube video of a brave New York State teacher who stood up for her sixth grade students against the Common Core’s developmentally inappropriate testing and teaching to the tests.  See below for another spectacular Grade 4 chalkboard form drawing from Catie Johnson’s page.  

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 on form drawing in Waldorf and Math By Hand.

 

FD

 

 

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4.MD 4: Line It Up With Coins & Fractions & Push Pins (#277)

January 22nd, 2015 · Uncategorized

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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 277

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.

Measurement and Data 4.MD
Represent and interpret data.
4. Make a line plot to display a data set of measurements in fractions of a unit 1/2, 1/4, 1/8).  Solve problems involving addition and subtraction of fractions by using information presented in line plots.  For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

The problem here, as with so much of the Common Core, is relevance and integration (or the lack thereof).  One of the online examples for this standard involved students measuring paper clips, erasers, and pencils, to create factions for placement on a line plot.  Such work is deadening, certainly not creative or enlivening.  As Sir Ken Robinson said in one of his wonderful TED talks on education, we have our schoolchildren doing tasks comparable to low-paid office clerks.

If the point is to teach a concept, then it should be taught in a clear straightforward way, so that once is enough.  No need for endless drill and testing.  You could revisit the fraction-coins activity found in post #267, setting it up for one or more student(s).  Use lots of coins so the student(s) can circle them, creating many fractions and repeating each one several times.

Set up a large, colorful line plot on the floor (carpeted is best) using poster paper or poster board, with colored masking tape as the baseline, and push pins for the points.  Actual coins could be used under the baseline to represent fractions (one dime = 1/10, two quarters = 2/4 or 1/2, etc).  Then simply record the number of fractions circled, using the push pins, employing addition and subtraction as needed.  Translate to a pencil and paper line plot when finished.

 

COINS

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.MD 4: Line It Up With Coins & Fractions & Push Pins (#277) appeared first on Math By Hand.

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4.MD 3: The Inside Scoop On Getting Around, Area & Perimeter (#276)

January 20th, 2015 · Common Core

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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 276

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.

Measurement and Data 4.MD
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
3. Apply the area and perimeter formulas for rectangles in real-world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

The Grade 4 Math By Hand does not cover area and perimeter, nor does Waldorf Ed introduce it until Grade 5.  It is a simple, straightforward concept however, that can easily be covered informally.  Real-world problems are key here, with project-based learning being the most practical for both introduction and skills practice.

The local geography lesson block may be an ideal fit.  Often, this study begins at homeschool or school with the kitchen table or desk as a starting point.  Either of these could be measured and the concept of perimeter explained something like this.  “When we bought this table, we had to make sure it was a good fit for our kitchen.  So we had to measure the room and then measure the table at the store before deciding to buy it, to be sure it would be a good fit.”  Then proceed to do both while explaining how perimeter works.

Teaching the concept of area might sound like this.  “We needed a table that could comfortably fit place settings for our family.  We use place mats at meals, so we had to be sure that the mats would fit on the tabletop with room to spare.”  Many options here.  Explain how area works then take the area of one place mat and multiply it by the number of mats.  Find the area of the table and compare the two.  Or map it all out on graph paper.

Make it large and adventurous!  Find the perimeter of the local ball field by walking it, translating large steps to yards or feet.  Later, using graph paper again, plot out the placement of the bases and outfield positions, according to regulation standards.  Or plan next season’s garden using graph paper, while researching the amount of space needed for each plant.

Once the concept is learned, by grounding it in real-world situations, it can easily be applied to anything.  Putting the cart before the horse by insisting on rote, abstract learning with endless drill and testing is counter productive at best.  “Experience is the best teacher,” is a saying that’s especially true 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.

b-ball

 

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4.MD 2: Dry/Liquid Measure? Bake A Cake! (#275)

January 18th, 2015 · Common Core

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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 275

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.

Measurement and Data 4.MD
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
2. Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.  Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

Though the Math By Hand materials do not include word problems per se, a plethora of such problems and worksheets can be found online, in textbooks, or create your own!  However, before using 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 should come first and then continue to be used as support while applying them to any purpose.

That said, consider tying investigations into all of the above to other subjects as suggested yesterday because integrating math skills practice with other lessons and activities is most effective.  See blog post #267 to work with money amounts involving simple fractions and decimals.

Try a new recipe for dinner or bake a cake with your fourth grader to review/learn dry and liquid measures.  Review posts #270, 271, and 272 for working with the number line and any units of measure.  All others could be covered in the context of other subject matter such as yesterday’s post suggested: measuring the parts of a whale (or any other animal) or using a pedometer to count the number of steps to the corner store.  Steps could then be converted into feet, yards, and fractions of a mile.

Keep it real!  Bake that cake and remember that 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.

 

CHOC-CRAN CAKE

 

 

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4.MD 1: The Corner Store & A Whale’s Tail! (#274)

January 18th, 2015 · Common Core

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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 274

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.

Measurement and Data 4.MD
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
1. Know relative sizes of measurement units within one system of units including km, m, cm, kg, g; lb, oz; l, ml; min, sec.  Within a single system of measurement, express measurements of a larger unit in terms of a smaller unit.  Record measurement equivalents in a two column table.  For example, know that 1 foot is 12 times as long as 1 in.  Express the length of a 4 foot snake as 48 in.  Generate a conversion table for feet and inches, listing the number pairs (1, 12), (2, 24), (3, 36), . . .

An online search for CCSS lessons and materials turned up lots of interesting interpretations of this standard, some of them game-based, some pattern-based, and some online skills practice.  While most of these were thoughtful and/or challenging, and engaging to some extent, they still left something to be desired.  That is, the element of story and picture, making it all colorfully relevant to the fourth grader.

Grade 3 measurement lessons are intensive and complete.  All of the information listed above is covered in a 3 to 4 week comprehensive measurement block.  The facts are made compelling, relevant, and effective because anecdotal biography, history, and pictorial examples and illustrations are included.  Economy is the soul of good teaching.  Faith and trust in both student and teacher is evident when concepts are taught deeply (just once) then reviewed briefly, not extensively and not overly repetitively.  Fourth grade measurement could be integrated with all other subjects, thereby adding relevance and meaning.

For example, measurements could be taken as part of a human and animal study: how long is a whale, in yards, in feet?  Comparatively, how long are its fins or its tail?  What is the relative size of the krill eaten by a Baleen whale?  Tiny!  Larger measurements could be included in the study of local geography: how many feet measures the distance from your front door to the corner store?  What part of a mile is that distance, as a fraction, as a decimal?  Love this wonderful whale pair in a fourth grader’s main lesson book (from a jamedaris’ pinterest page).

 

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In short, math cannot/should not be extricated from its source, stood in a corner, and made to perform, on demand.  Subtlety is where math shines and lives.  Children understand this so much more than we do, consequently it violates a sacred trust to expose the beautiful nature of math (or theirs for that matter) to a harsh reality that distorts its (and their) innate beauty and integrity.

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.MD 1: The Corner Store & A Whale’s Tail! (#274) appeared first on Math By Hand.

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Beginning Algebra: Dear Aunt Sally! (#273)

January 15th, 2015 · Uncategorized

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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 273

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.

Math By Hand introduces beginning algebra at the end of Grade 4 with anecdotal, biographical, and historical stories.  As always, the story carries the content and new concepts!  Here’s an excerpt from Math By Hand’s supplemental booklet, Order of Operations.  

Algebra: the word is taken from the Arabic “Al Jabr” which means “bone setting.” Al Khwarizmi of Baghdad, a mathematician, wrote a book about algebra in the year 830 CE. Its title translates as The Science of Restoring What Is Missing, and Equating Like with Like. Think of “x” as the missing piece, and the relationship of what’s on both sides of the equals sign as the equating of like with like. Algebra, like fractions and the concept of zero, was introduced to Europe from the Middle East.

Here is an example of how the order of operations (Please Excuse My Dear Aunt Sally: Parenthesis Exponents Multiplication Division Addition Subtraction) is entered into students’ main lesson books, then continually used during skills practice.  The problem could be written on the board including all of the five steps, color coding each step with different color.  The properties of operations: commutativity, associativity, and distributivity should also be included in this lesson.

aunt sally
The gist of both the Waldorf and Math By Hand approaches is that the “Race to the Top” needs to be won by the tortoise rather than the hare.  Slow and steady wins because from Kindergarten on, sensible and incremental steps are taken, always following the lead of developmental stages and the growing needs of the child.  “To everything there is a season” and algebra’s season can wait until there’s a real readiness for it.

slow
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 as we return to the Common Core math standards for measurement and data.

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The Number Line: Add, Subtract, Multiply & Divide! (#272)

January 12th, 2015 · Uncategorized

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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 272

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.

As stated earlier, the ins and outs of the number line are quite complex. It’s really best to build a solid foundation of 4 processes skills up through long multiplication and division, multiplication tables’ fluency, and a good, working grasp of fractions and decimals before even touching on the number line.

Math By Hand devotes an extensive section to the number line at the end of the Grade 4 binder. In addition, a supplemental booklet that includes a brief history of numbers also provides a clear, handy set of rules for using the 4 processes with the number line.  This reference tool should be used as needed until all concepts are learned.

See below for two excerpts from the number line reference booklet and the binder.  The first is the multiplication section of the reference booklet, and the second is a sample of how a summary of the number line 4 processes would be reviewed and recorded in the student’s main lesson book.

X INTEGERS

# LINE RULES
And then on to your fourth grader’s love for adventure and fascination with the Norse Myth’s codes and symbols.  Here is a wonderful rendering of those from Rachel Britton’s Pinterest page.

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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.

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The Number Line: Nuts & Bolts (#271)

January 11th, 2015 · Common Core

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A Year in the Life: Ambient Math Wins the Race to the Top!
Day 271

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.

As stated earlier, the ins and outs of the number line are quite complex.  It’s really best to build a solid foundation of 4 processes skills up through long multiplication and division, multiplication tables’ fluency, and a good, working grasp of fractions and decimals before even touching on the number line.

Math By Hand devotes an extensive section to the number line at the end of the Grade 4 binder.  After a detailed introduction, instructions are provided for creating number cards to be used on a large, red and black felt version of the number line.  The zero is placed in the center, with red representing negative numbers to the left and black, positive numbers to the right.

Students create cards divided into six groups of positive and negative whole numbers, fractions, and mixed numbers, then order the cards from lowest to highest value so they are easily found as needed.  Here’s an excerpt from the binder for getting started with the number line (note that this does not reflect the quality of the binder pages, as it’s slightly out of focus).

#LINE
And then have some form drawing fun with this woven star, leprechaun, rainbow, and pot o’ gold!  (It’s from jamedaris’ weebly page.)

 

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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.

 

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