Math Models: Arithmetic Concepts

 

 

Topic

Math Idea

Model

Vocabulary

Manipulatives

Addition

Joining Sets

Put on quantities, group together

·         Put on

·         Group

·         Equal Trades

Base 10 Blocks

Pattern Blocks

C-Rods

Algebra Tiles

Subtraction

Difference

Make fist quantity; take away second

·         Put on

·         Take Away

·         Equal Trades

Base 10 Blocks

Pattern Blocks

C-Rods

Algebra Tiles

Adding Opposites

Put on opposites (negative) quantity—idea of multiplying by  is turning piece over.

·         Opposites

·         Negatives

·         Flipping

Algebra Tiles

2 Color Chips

Trades & Exchanges

Regrouping

Trade 10 units for a rod (addition), or a rod for 10 units (subtraction)

·         Equal Trades

·         Place Value

Base 10 Blocks

Simplifying

Common Denominators

Trade for all one color/one shape in both fractional quantities.  For C-Rods, choose unit to be LCM (CD).

·         Equal Trades

Pattern Blocks

C-Rods

Equivalent Fractions or Lowest Terms (Reducing—avoid using this term)

Make simplest mixed number:

·         “Race” to units

·         Use all one color/one shape for fractional quantities

·         Use fewest blocks/rods possible

·         Unit

·         Whole

 

Pattern. Blocks

C-Rods

Multiplication

Repeated Addition

Repeatedly add equal quantities

Ex:3x5—Put on 5 groups of 3

·         Equal Groups

·         Repeated “Putting on”

Pattern Blocks

C-Rods

Algebra Tiles

Area

Rectangular Arrays with factors as length/width and product as area

Ex: 3x5—Make a 3 by 5 rectangular array

·         Length/Width

·         Factors

·         Area/Array

·         Rectangle

·         Square (102)

Base 10 Blocks

Algebra Tiles

Division

Equal Groups

Divide total into equal groups of a given size

Ex:: Make 3 equal piles of 5 OR ask “How many groups of 3 ‘fill up’ 15?” (esp. pat. blocks)

·         Equal Groups

·         Remainder (B10)

·         Fractional Part (PB)

·         Unit/Whole (PB)

Base 10 Blocks

Pattern Blocks

Area or Reverse Multiplication

Make rectangular array of total (dividend) and find factor(s).

·         Factors

·         Remainder

·         Equal Groups

Base 10 Blocks

Algebra Tiles

 

Math Models: Algebraic Concepts

 

Topic

Math Idea

Model

Vocabulary

Manipulatives

Distributing

Distributing

Do operation outside parentheses to each term inside parentheses.

·         Terms vs. Factors

·         Multiplication

·         Distribute the negative

Algebra Tiles

Opposites

Negative vs. Positive

The opposite side of the algebra tile (or chip) is the mathematical opposite.  The opposite of -7 is 7, and vice versa.

·         Opposite (neg)

·         Neg/Pos

Algebra Tiles

2 Color Chips

Like Terms

Like Terms

Only pieces that both same size and same shape can be grouped or combined (x-terms cannot be added to x2-terms)

·         No trades!

·         Same size/same shape

Algebra Tiles

2 Color Chips

Simplifying

Adding Zero

Adding “nothing” changes nothing (same for subtraction)

We can add/take away as many zeros as we wish at any time

1 positive piece and 1 red (same size & shape) make zero

·         Convenient Zeros

·         Adding “nothing” changes nothing

Algebra Tiles

2 Color Chips

Subtracting Zero

Multiplying

Polynomial Multiplication

Make a rectangular array with factors as length/width.  Ties in precisely to B-10 blocks with one exception: multiplying opposites (pos x neg, etc) is negative while multiplying “sames” is positive.

·         Binomials

·         Trinomials

·         Polynomials

Algebra Tiles

Factoring

Polynomial Division

Make a rectangular array, add convenient zeros until an array is possible.  Ties in precisely to B-10 Blocks division as reverse multiplication (see above) and should be connected to it explicitly during discussion.

·         Factor = Divide

·         BW Multiplication

·         Convenient Zeros

Algebra Tiles

Perfect Squares

Algebraic Squares

If the rectangular array is actually square, then the factors are the same.  Square here is geometric.  The word “squared” has meaning.  Write (x-3)(x-3) as (x-3) 2.  The words match the concept!

·         Rectangles

·         Area/Array

·         Squares

Algebra Tiles

Completing Square

Quadratics

With only x2- and x-terms, place half of x’s vertically and half horizontally beside/beneath x2 ’s.  Fill in square “hole” with units.

·         SQUARES!!

Algebra Tiles

Equation Solving Concepts

Equation Solving Steps

Equivalent Equations

With an equation mat, any action performed on one side must also be done on the other.  Any action with pieces is legitimate: flipping, dividing into equal groups, and adding or subtracting any pieces.

·         Must do same thing to both sides

Algebra Tiles

Simplifying

Add Like Terms

Make or use convenient zeros to “transform” equation.

·         Like Terms

Algebra Tiles

Equation Solving Strategies

X By Itself

Solving for Variable

Idea: get only x’s on one side with all numeric quantities on other.

·         Strategy

·         Goal

Algebra Tiles

Completing Square

Solving Quadratics

After separating x2- and x-terms from numeric terms, complete square.  Factors on left are equal to factors on right.  Remember (-x) (-x) also equals x2.  Use above steps twice to find both solutions.

·         Completing Square

·         Side lengths same

Algebra Tiles