Wednesday Nov. 4th 2009Emidio DiAntonio

AgendaWelcome & PrayerThe who, what, where and why of manipulatives.Activity #1: Scavenger HuntActivity #2: Algebra TilesActivity #3: Using the Trig TrainerActivity #4: Using GeoboardsActivity #5: Lets Make a Mess!Activity #6: Virtual Manipulatives LibraryActivity #7: Exploring GizmosLunch Activity },71|{ Zttt

A Prayer for Math Teachers:Lord, as we gather here today, we pray for your

guidance…You were there when Jesus multiplied the

loaves and fishes, and when Moses divided the Red Sea

We ask that you be with us nowHelp us trust that your highest power will be

our guideHelp us to be a positive influence on our

studentsHelp us to be a fraction of the teacher Jesus

wasHelp us to recognize the infinite possibilities

that are born of faithAmen

Why use manipulatives to teach Mathematics?A focus on deep learning of particular mathematics topics –

through a variety of strategies, including working with concrete materials – leads to greater conceptual depth (Ben-Chaim, Fey, Fitzgerald, Benedetto, & Miller, 1998; Fletcher, Hope, & Wagner, 2001; Siemon et al., 2001).

Manipulatives allow students to concretely explore mathematical relationships that will later be translated into symbolic form. The key to the successful use of manipulatives lies in the bridge – which must be built by the teacher – between the artifact and the underlying mathematical concepts (D’Ambrosio et al., 1993). The mathematics is in the connections, not the objects (Kilpatrick & Swafford, 2002).

Leading Math Success, 2004

Why use manipulatives to teach Mathematics? Continued …“Students need to develop the ability to select

the appropriate electronic tools, manipulatives, and computational strategies to perform particular mathematical tasks, to investigate mathematical ideas, and to solve problems.”

Students should be encouraged to select and use concrete learning tools to make models of mathematical ideas. Students need to understand that making their own models is a powerful means of building understanding and explaining their thinking to others.

Introduction Curriculum Document Grades 9 – 10 Mathematics, 2005

Why use manipulatives to teach Mathematics? Continued … Using manipulatives to construct representations helps students

to: see patterns and relationships; make connections between the concrete and the abstract; test, revise, and confirm their reasoning; remember how they solved a problem; communicate their reasoning to others.

Even at the secondary level, manipulatives are necessary tools for supporting the effective learning of mathematics. These concrete learning tools invite students to explore and represent abstract mathematical ideas in varied, concrete, tactile, and visually rich ways. Manipulatives are also a valuable aid to teachers. By analyzing students’ concrete representations of mathematical concepts and listening carefully to their reasoning, teachers can gain useful insights into students’ thinking and provide supports to help enhance their thinking.

Introduction Curriculum Document Grades 9 – 10 Mathematics, 2005

Why use manipulatives to teach Mathematics? Continued ….

Studies on the use of manipulatives by students described as low achievers, at risk, having behaviour problems, or with limited English proficiency have found positive effects on achievement (Ruzic & O’Connell, 2004).

Manipulatives are necessary tools for supporting the effective learning of mathematics by all students. These learning materials invite teachers and students to explore and represent abstract mathematical ideas in varied, concrete, tactile, and visually rich ways.

Manipulatives support the conceptual development of important mathematical ideas for tactile and visual learners. Manipulatives allow teachers to provide alternative ways for students to see and think about mathematical concepts. Paper-and-pencil drill does not lead to conceptual learning for at-risk students, but effective use of manipulatives can.

Leading Math Success, 2004

A list of Essential ManipulativesAlgebra tiles (2 colour) – class set, clear plastic organizer trays,

overhead or magnetic setBase-ten materials (clearview with interlocking pieces)Circular fraction set and frames (decimal, degree, percentage,

time, fraction, and compass points), translucent pieces and overhead set

Coloured tiles and overhead setColoured relational rods and overhead setConnecting cubes (1 cm, 2 cm)Connecting plastic shapes to build 2-D shapes and nets for 3-D

solidsFull circle protractorsGeoboards (minimum of 15 cm by 15 cm dimensions) clear 11 x

11, 5 x 5, circular, elastics

Leading Math Success, 2004

A list of Essential Manipulatives Continued …GeolegsGraphing calculators, preferably with projection capabilitiesMeasuring tapes (minimum 150 cm wind-up tape in

protective case calibrated in centimetres and millimetres)Number cubes: 6-sided in two colours; 10-, 12-, and 30-sidedOverhead graphing calculator with projection unitPlastic transparent toolsRelational geometric 3-D solids and large demonstration setSpinners (number, colour)Trundle wheelsTwo-colour counters and overhead set

Leading Math Success, 2004

Starting Points for TeachersBegin by selecting one major mathematical idea

(e.g., fractions) and exploring that idea with students from many different perspectives, employing a variety of manipulatives.

Plan how the mathematics concept will be developed from the experience with the manipulatives.

Plan the assessment of students’ mathematics knowledge with and without the presence of the manipulatives.

Leading Math Success, 2004

Activity #1: Scavenger HuntIn your assigned course groups, make a

note of the specific expectations that include the language:

“aided by a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil) and strategies (e.g. patterning)”

(e.g., using concrete materials)

Activity #2: Algebra TilesAlgebra tiles allow students a way to represent

algebraic terms using concrete materials. Students are able to solve algebraic problems using concrete materials rather then the abstract concepts they represent.

A typical set of algebra tiles consists of a number of small squares, large squares and rectangles of different colour and size.

All secondary schools should have at least 1 set of Red/Blue algebra tiles.

Algebra Tiles: Lets use a set of Algebra Tiles to explore

each of the following concepts:Adding & Subtracting IntegersSolving Linear EquationsExpanding Monomials & BinomialsFactoring BinominalsFactoring TrinomialsCompleting the square

Activity #3: Using the Trig TrainerCan be used to explore the primary trigonometric ratios.Prior Knowledge: Students should have already

completed an activity where they compare the ratios of the sides of different right triangles with a fixed acute angle.

On a Trig Trainer, the length of the hypotenuse is set to one unit. The angles of rotation are about the unit circle.

I have found that the calculator buttons sine, cosine and tangent become a “black box” where students don’t always know the why and when to press the particular buttons and more importantly that they are ratios of sides.

A short history lesson:The two legs of the trig trainer are called the

sine and cosine legs.The significance of those names comes from

the historical origin of the words sine and cosine.

In the 5th century AD, the sine leg was called “ardha-jya” which means “half-chord”. Eventually shortened to “jya”, and later translated to “jiba” by Arab scolars and then “jaib”.

Fast forward a few years …In 1150, the Italian Gerardo of Cremona translated

the Arab works into Latin replacing the word “jaib” with the word “sinus”, from which the English word sine is derived. The word “sinus” means “bend” or “curve” – picture the graph of the sine function.

The term cosine comes from the shortened phrase “complementary sine”. Edmund Gunter in 1620 later abbreviated this to “co-sinus” which became “cosine”.

Over 1000 years of history in less than 5 minutes .

Trig Trainer: ActivitiesIntroduction to the parts of the trig trainer.Finding ratios of anglesUsing the trig trainer to find angles.Exploring the following relationships:

Sin (90°-A)=cos(A)Cos(90°-A)=sin(A)How sine increases and then decreases as

angles increase.How cosine decreases then increases as angles

increase.

Activity #4: Using Geoboards Geoboards are grids of pegs that can hold rubber bands in position.

Geoboards are available in a variety of sizes, styles and colours. The preferred model is the transparent Geoboard as it can be placed on

an overhead projector. The best size to use is an 11x11 peg Geoboard. There are three types of Geoboards that are available in varying sizes.

Some Geoboards can be “connected” together to form larger work areas.

Each type of Geoboard has different applications – you don’t need to have each type – virtual Geoboards can be used as well.

Square Isometric Circular

Geoboards: Where can we use them?Exploring area and perimeter (including composite

figures)Exploring fractions and operations with fractionsEuclidean Geometry – lines, symmetries, congruence,

similarityPythagorean TheoremAnalytic Geometry – plotting ordered pairs, slopes, linesOptimization ProblemsBuild 3D objects using 2D materialsOther suggestions?Frisbee

Geoboards: Tricks and HintsBe sure that you have the right type of Geoboard

before you go to class! There are three types of Geoboards – Square, Isometric and Circular.

Pass out the Geoboards with a set number of elastics already on the pegs. This makes it easier to ensure that all the elastics have been handed in at the end and should cut down on misuse.

Longer elastics are preferable to shorter ones since they can be doubled around the shape if necessary.

Geoboards: Tricks and HintsUse transparent Geoboards whenever possible – at a

minimum the teacher should use a transparent one. Transparent Geoboards can be used on the overhead or can be stacked to check congruence.

Bring a stack of overhead acetates and markers so that students can more easily count areas.

Count spaces NOT pegs. When creating a 4 by 4 square, elastics should go around 5 pegs in each direction.

The distance between diagonal pegs is not the same as vertical/horizontal pegs. Create a 4x4 square using vertical pegs, and then using horizontal pegs. Compare for congruence.

Geoboards – Working With FractionsGeoboards are great to use when introducing the

concept of fractions or providing remedial assistance to students having difficulties with fractions.

Lets explore the following concepts:Equivalent fractionsLowest Common DenominatorAdding fractionsSubtracting fractionsMultiplying fractionsDividing fractions

Geoboards: Model Areas with Same PerimeterProblem:

You have 12m of rope to fence off a rectangular play area at a summer day camp. (MHR: Principles 9).

Use a Geoboard to explore the different rectangles that can be formed with a perimeter of 12m.

Geoboards: Maximizing AreaProblem:

Carina wants to build a rectangular play area for her new puppy. What dimensions will maximize the enclosed area if she is to use 24m of fencing?

Rectangle

Number

Width (m)

Length (m)

Perimeter (m)

Area (m2)

1 1 11 24 11

2 2 10 24 20

And so on

Geoboards: Maximizing Area - ExtensionProblem:

In hopes of creating a larger play area for her puppy, Carina decides to use the side of the house as one of the sides of her puppy’s play area. What dimensions will maximize the rectangular play area using only 24m of fencing? Rectangl

e Number

Width (m)

Length (m)

Perimeter (m)

Area (m2)

1 1 22 24 22

2 2 20 24 40

3 3 18 24 54

And so on

Geoboards: Analytic GeometryWith the aid of an Overhead marker, a

Geoboard can be used to investigate slopes and equations of lines.

Draw a Cartesian Grid on the back of a transparent Geoboard with overhead marker.

When using a Geoboard for slopes and lines, count pegs.

Geoboard ResourcesVirtual Geoboard:

Rectangular Geoboard: http://nlvm.usu.edu/en/nav/frames_asid_279_g_4_t_3.html?open=acti

vities&from=category_g_4_t_3.html

Isometric Geoboard: http://nlvm.usu.edu/en/nav/frames_asid_129_g_4_t_3.html?

open=activities&from=category_g_4_t_3.html Circular:

http://nlvm.usu.edu/en/nav/frames_asid_285_g_4_t_3.html?open=activities&from=category_g_4_t_3.html

Co-ordinates:. http://nlvm.usu.edu/en/nav/frames_asid_303_g_4_t_3.html?

open=activities&from=category_g_4_t_3.html

Some Classroom Examples: http://mathforum.org/trscavo/geoboards/contents.html

Activity #5: Lets Make A Mess! Using Relational Solids Investigation A:

Using a rectangular prism and pyramid with the same height and base area, compare the relative volumes of the two containers by filling the pyramid repeatedly and dumping it into the rectangular prism.

Investigation B: Using a cone with same base radius and height as a cylinder,

compare the relative volumes of the two containers by filling the cone repeatedly and dumping it into the cylinder.

Investigation C: Using a sphere with the same radius as a cylinder, and whose

diameter is equal to the height of the cylinder, can we find a relationship between these two volumes?

Fill the sphere with sand or water, and dump it into the cylinder. What fraction of the cylinder is filled with sand/water?

Activity #6: Virtual Manipulatives LibraryAmazing website for both students and

teachers to use manipulatives. http://nlvm.usu.edu/en/nav/vlibrary.html

Virtual Manipulative Library is also available for download for non-internet connected machines.

Activity #7: Exploring GizmosEach school was allotted a bank of user ids and

passwords to the website: http://www.explorelearning.com/

Lets explore what is there!Maximize Area with given perimeter:

http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=73

Let’s not make a mess! http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=193

More fun … http://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=349

Finished ….Where do we go from here?