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16 Cards in this Set

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IMPORTANT TWO WORDS
INDUCTIVE
DEDUCTIVE
DESC 1
**USES TECHNIQUES OF FORMAL AND INFORMAL REASONING TO REACH VALID CONCLUSIONS, EVALUATE VALIDITY AND JUSTIFY MATHMATICAL IDEAS
DESC. 2
USE A VARIETY OF APPPROOPREATE STRAEGIES TO SOLVE PROBLEMS
TYPES OF REASSONING-INDUCTIVE (INFORMAL)
REASONING INDUCTIVBIELY IS THE PROCESS OF USING DATE AND PATTERNS TO REACH A GENAERALIZZED CONCLUSION
INTUTION
EXAMPLE OF INDCUTIVE REASINON
IF YOUR BEST FRIEND PICKS U UP EVERY MORNING AND A RED SPORTS CAR. YOU WOULD ALWAYS EXPECT TO HAVE YOUR FREIND PICK YOU UP IN ONE. nOT A VAN, TRUCK OR SUV.
DEDUCTIVE (FORMAL): REASONING DEDUCTIVILY IS THE PROCESS OF REACHING CONCLUSIONS, BASED ON ACCEEPTED TRUTHS ADN LOGICAL REASONING-EXAMPLES
Gravity makes things fall. The apple that hit my head was due to gravity.
They are all like that -- just look at him!
Toyota make wonderful cars. Let me show you this one. There is a law against smoking. Stop it now.
MATHMATICAL MODELS:
REPRESENTS A SITUATION OR RELATIONSHIP USING TABLES, GRAPHS, CHARTS, EQUATIONS, AND FUNCTIONS IS THE PROCESS OF MODELLING THE APPLICATION MATHMATICALLY
EXAMPLE OF MM: THE RELATIONSHIP BETWEEN TEMPERATURE WRTTEN, IN CELSIUS VS FAHRENHEIT CAN BE USED FOR MULTIPLE REPRESENTATION
EQUATION/FUNDCTION F=9/C +32
USE OF A DIAGRAM
USE OF A GRAPH OR CHART
TYPE OF MATHMAITICLA MODELS USED BY 4TH GRADERS:
know, explain, and use mathematical models such as process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate planes/grids) to model computational procedures, mathematical relationships, and equations; place value models (place value mats, hundred charts, base ten blocks, or unifix cubes) to compare, order, and represent numerical quantities and to model computational procedures; fraction and mixed number models (fraction strips or pattern blocks) and decimal models (base ten blocks or coins) to compare, order, and represent numerical quantities; money models (base ten blocks or coins) to compare, order, and represent numerical quantities; function tables (input/output machines, T-tables) to model numerical and algebraic relationships; two-dimensional geometric models (geoboards, dot paper, pattern blocks, or tangrams) to model perimeter, area, and properties of geometric shapes and three-dimensional geometric models (solids) and real-world objects to compare size and to model properties of geometric shapes; two-dimensional geometric models (spinners), three-dimensional models (number cubes), and process models (concrete objects) to model probability; graphs using concrete objects, pictographs, frequency tables, horizontal and vertical bar graphs, Venn diagrams or other pictorial displays, line plots, charts, tables, line graphs, and circle graphs to organize and display data; Venn diagrams to sort data and show relationships to represent mathematical concepts, procedures, and relationships
create a mathematical model to show the relationship between two or more things
recognize that various mathematical models such as process models (concrete objects, pictures, diagrams, number lines, hundred charts, measurement tools, multiplication arrays, division sets, or coordinate planes/grids) to model computational procedures and mathematical relationships; place value models (place value mats, hundred charts, base ten blocks, or unfix cubes) to model mathematical relationships; fraction and mixed number models (fraction strips or pattern blocks) and decimal models (base ten blocks or coins) to compare, order, and represent numerical quantities; money models (base ten blocks or coins) to compare, order, and represent numerical quantities; function tables (input/output machines, T-tables) to model numerical and algebraic relationships; two-dimensional geometric models (geoboards, dot paper, pattern blocks, or tangrams) to model perimeter, area, and properties of geometric shapes and three-dimensional geometric models (solids) and real-world objects to compare size and to model properties of geometric shapes; two-dimensional geometric models (spinners), three-dimensional geometric models (number cubes), and process models (concrete objects) to model probability; graphs using concrete objects, pictographs, frequency tables, horizontal and vertical bar graphs, Venn diagrams or other pictorial displays, line plots, charts, tables, and line graphs to organize, display, explain, and interpret data; Venn diagrams to sort data and show relationships can be used to represent the same problem situation
select a mathematical model and explains why some mathematical models are more useful than other mathematical models in certain situations
Geometric Figures and Their Properties
TYPE OF MATHMAITICLA MODELS USED BY 5THThe segment strips and grid can be used to represent dimensions of rectangles whose areas depend on the unit of measurement chosen. Suppose we say that three blue segments represent one linear unit:

GRADERS:
Fraction Segment Strips

Fraction Concepts, More Fraction Concepts, Fraction Operations, Decimals and Fractions

In previous fraction lessons, we used egg cartons and base ten mats as area concept models for fractions. In this lesson, we use a linear model, represented by segment strips. These are long, narrow strips of paper, which are subdivided into congruent segment lengths. In this lesson, they serve as a means for comparing fractions, given a relative unit, and as a model for combining various fractions. There are six segment strips:




The actual strips are about 11"

long - these are short versions
USED BY 5THThe segment strips and grid can be used to represent dimensions of rectangles whose areas depend on the unit of measurement chosen. Suppose we say that three blue segments represent one linear unit:

GRADERS:In a previous lesson, we learned about measuring dimension and area in base five units. In this lesson, we apply the same concepts to base ten measurement, which lead to models for the basic operations in base ten. We begin by filling in a rectangular region with base ten pieces, then looking at what multiplication and division equations might be represented by the model:



The measured area of the rectangle depends on the value of the various pieces. If the small square represents one unit, then we can regroup to find the area is 192 square units. If the mat is the unit, then the area of the rectangle is 1.92 units:



Drawing on our experience with base five area and dimension and multiplication and division models, we can see the following multiplication and division equations modeled by the above rectangle area and dimensions:







After some exploration with other rectangles and collections of base ten pieces, the students can build or sketch models of situations like these, and try to find out as much as they can about them:

A square with an area of 1.44 square units
A rectangle with dimensions of 1.5 linear units by 2.4 linear units
A square whose area is 361 square units
A rectangle whose area is 437 square units, and one dimension is 23 linear units
A rectangle with an area of 5.13 square units, and one dimension of 1.9 linear units
By working with examples like these, students build concepts about linear and area measurement relationships, and intuitions about multiplying and dividing in base ten. We'll do some more work with these ideas in the next couple of lessons.'Graphing' often is taken to mean lines on a coordinate grid, but there are many ways of displaying data in a visual format. In this lesson, students use their understanding of angle measure, area, fractions and probability to create a variety of graphs to represent various situations. Here is some sample data, in the form of a puzzle, and graphs similar to the ones the students produce in class.

72 students voted in an election:

Julie received 1/6 of the votes
Jackie and Pete each received 1/18 of the votes
Vondra received 4 times as many votes as Jackie
Barbara received 1 vote more than 1/12 of the votes
Sam received the rest of the votes
In a circle graph, the 360 degrees must be divided evenly among the 72 votes, so each vote would be represented by 5 degrees.

We draw on students experiences with base ten numeration to dig into percentage concepts and applications. Base ten pieces make conceptualizing percent concepts meaningful to students and allow them to explore percentage in a familiar setting.

The 10 X 10 square mat is the basic model for percent quantities:



We draw on students experiences with base ten numeration to dig into percentage concepts and applications. Base ten pieces make conceptualizing percent concepts meaningful to students and allow them to explore percentage in a familiar setting.

The 10 X 10 square mat is the basic model for percent quantities:



We draw on students experiences with base ten numeration to dig into percentage concepts and applications. Base ten pieces make conceptualizing percent concepts meaningful to students and allow them to explore percentage in a familiar setting.

The 10 X 10 square mat is the basic model for percent quantities:

describe the relationship between sets of data in graphic organizers such as lists, tables, charts, and diagrams; and
TYPE OF MATHMAITICLA MODELS USED BY 6H GRADERS:
A) model addition and subtraction situations
involving fractions with objects, pictures,
words, and numbers;
identify prime and composite numbers using concrete objects, pictorial models, and patterns in factor pairs.
identify prime and composite numbers using concrete objects, pictorial models, and patterns in factor pairs.
represent ratios and percents with concrete
models, fractions, and decimals; and
(A) construct sample spaces using lists, tree
diagrams, and combinations; and
TYPE OF MATHMAITICLA MODELS USED BY 7H GRADERS:
represent squares and square roots using
geometric models.
represent multiplication and division
situations involving fractions and decimals
with concrete models, pictures, words, and
numbers;
use concrete models to solve equations and use
symbols to record the actions; and
The student is expected to:

(A) sketch a solid when given the top, side, and
front views;

(B) make a net (two-dimensional model) of the
surface area of a solid; and

(C) use geometric concepts and properties to solve
problems in fields such as art and
architecture.
WHAT ARE THE MATHMATICAL MODELS THAT 8TH GRADERS USE?
C) approximate (mentally and with calculators)
the value of irrational numbers as they arise
from problem situations (p, Ö2); and

(D) express numbers in scientific notation,
including negative exponents, in appropriate
problem situations using a calculator.
8.5)Patterns, relationships, and algebraic thinking.
The student uses graphs, tables, and algebraic
representations to make predictions and solve
problems. The student is expected to:
A) estimate, find, and justify solutions to
application problems using appropriate tables,
graphs, and algebraic equations; and

(B) graph dilations, reflections, and translations
on a coordinate plane.
(8.7)Geometry and spatial reasoning. The student uses
geometry to model and describe the physical world.
The student is expected to:

(A) draw solids from different perspectives;

(B) use geometric concepts and properties to solve
problems in fields such as art and
architecture;

(C) use pictures or models to demonstrate the
Pythagorean Theorem; and

(D) locate and name points on a coordinate plane
using ordered pairs of rational numbers.
find surface area of prisms and cylinders
using concrete models and nets (two-
dimensional models);

(B) connect models to formulas for volume of
prisms, cylinders, pyramids, and cones\
(C) construct circle graphs, bar graphs, and
histograms, with and without technology.
OTHER 8TH MATHMATICAL MODELS
Grade 8 mathematics
through informal and mathematical language,
representations, and models. The student is
expected to:

(A) communicate mathematical ideas using language,
efficient tools, appropriate units, and
graphical, numerical, physical, or algebraic
mathematical models; and

(B) evaluate the effectiveness of different
representations to communicate ideas.Grade 8 mathematics
through informal and mathematical language,
representations, and models. The student is
expected to:

(A) communicate mathematical ideas using language,
efficient tools, appropriate units, and
graphical, numerical, physical, or algebraic
mathematical models; and

(B) evaluate the effectiveness of different
representations to communicate ideas.
DO EXAMPLES 1 AND 2 ON PAGE 101
DID YOU COMPLETE PROBLEMS
(5.12) Probability and statistics. The student
describes and predicts the results of a probability
experiment. The student is expected to:

(A) use fractions to describe the results of an
experiment; and

(B) use experimental results to make predictions.
(5.12) Probability and statistics. The student
describes and predicts the results of a probability
experiment. The student is expected to:

(A) use fractions to describe the results of an
experiment; and

(B) use experimental results to make predictions.