• MATH 7 ACCELERATED

     CURRICULUM 

     

    THE NUMBER SYSTEM

     

    REVIEW

     

    ADD, SUBTRACT, MULTIPLY AND DIVIDE FRACTIONS.

    PROPERTIES OF OPERATIONS:  COMMUTATIVE, ASSOCIATIVE, IDENTITY, INVERSE, DISTRIBUTIVE

     

    INTRODUCTION

     

    REAL NUMBERS – RATIONAL OR IRRATIONAL, INTEGERS, WHOLE , NATURAL…

    Ø DISTINGUISH BETWEEN THE VARIOUS SUBSETS OF REAL NUMBERS

    Ø PLACE RATIONAL AND IRRATIONAL NUMBERS (APPROXIMATIONS) ON A NUMBER LINE AND JUSTIFY THE PLACEMENT OF NUMBERS.

    Ø CLASSIFY IRRATIONAL NUMBERS AS NON-REPEATING/NON-TERMINATING DECIMALS.

     

    APPLY AND EXTEND PREVIOUS UNDERSTANDINGS OF OPERATIONS WITH FRACTIONS TO ADD, SUBTRACT, MULTIPLY AND DIVIDE RATIONAL NUMBERS.

     

    ADD AND SUBTRACT INTEGERS.

     

    ADD AND SUBTRACT RATIONAL NUMBERS.  (+/- fractions & decimals)

    Ø REPRESENT ADDITION AND SUBTRACTION ON A HORIZONTAL OR VERTICAL NUMBER LINE DIAGRAM.

    Ø DESCRIBE SITUATIONS IN WHICH OPPOSITE QUANTITIES COMBINE TO MAKE 0.  (for example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.)

    Ø  INTERPRET SUMS OF RATIONAL NUMBERS BY DESCRIBING REAL-WORLD CONTEXTS.

    Ø SHOW THAT A NUMBER AND ITS OPPOSITE HAVE A SUM OF 0                  (ADDITIVE INVERSES).

    Ø UNDERSTAND p + q AS THE NUMBER LOCATED A DISTANCE |q| FROM p, IN THE POSITIVE OR NEGATIVE DIRECTION, DEPENDING ON WHETHER q IS POSITIVE OR NEGATIVE.

    Ø UNDERSTAND SUBTRACTION OF RATIONAL NUMBERS AS ADDING THE ADDITIVE INVERSE: p – q = p + (-q).

    Ø SHOW THAT THE DISTANCE BETWEEN TWO RATIONAL NUMBERS ON A NUMBER LINE IS THE ABSOLUTE VALUE OF THEIR DIFFERENCE AND APPLY THIS IN REAL-WORLD CONTEXTS.

    Ø APPLY PROPERTIES OF OPERATIONS AS STRATEGIES TO ADD AND SUBTRACT RATIONAL NUMBERS.

     

    MULTIPLY AND DIVIDE INTEGERS.

     

     

    MULTIPLY AND DIVIDE RATIONAL NUMBERS. (+/- fractions & decimals)

    Ø UNDERSTAND THAT MULTIPLICATION IS EXTENDED FROM FRACTIONS TO RATIONAL NUMBERS BY REQUIRING THAT OPERATIONS CONTINUE TO SATISFY THE PROPERTIES OF OPERATIONS, PARTICULARLY THE DISTRIBUTIVE PROPERTY, LEADING TO PRODUCTS SUCH AS (-1)(-1) = 1 AND THE RULES FOR MULTIPLYING SIGNED NUMBERS.

    Ø INTERPRET PRODUCTS OF RATIONAL NUMBERS BY DESCRIBING REAL-WORLD CONTEXTS.

    Ø UNDERSTAND THAT INTEGERS CAN BE DIVIDED, PROVIDED THAT THE DIVISOR IS NOT ZERO, AND EVERY QUOTIENT OF INTEGERS IS A RATIONAL NUMBER. 

    Ø IF p AND q ARE INTEGERS, THEN –(p/q) = (-p)/q = p/(-q).

    Ø INTERPRET QUOTIENTS OF RATIONAL NUMBERS BY DESCRIBING REAL-WORLD CONTEXTS.

    Ø APPLY PROPERTIES OF OPERATIONS AS STRATEGIES TO MULTIPLY AND DIVIDE RATIONAL NUMBERS.

    Ø CONVERT A RATIONAL NUMBER TO A DECIMAL USING LONG DIVISION; KNOW THAT THE DECIMAL FORM OF A RATIONAL NUMBER TERMINATES OR EVENTUALLY REPEATS.

    Ø IDENTIFY WHICH FRACTIONS WILL TERMINATE (THE DENOMINATOR OF THE SIMPLIFIED FRACTION IS ONLY DIVISIBLE BY 2 OR 5) AND WHICH WILL REPEAT.

     

    SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS INVOLVING THE FOUR OPERATIONS WITH RATIONAL NUMBERS. (INCLUDE COMPLEX FRACTIONS.)

     

    KNOW THAT THERE ARE NUMBERS THAT ARE NOT RATIONAL, AND APPROXIMATE THEM BY RATIONAL NUMBERS.

     

    KNOW THAT NUMBERS THAT ARE NOT RATIONAL ARE CALLED IRRATIONAL.  UNDERSTAND INFORMALLY THAT EVERY NUMBER HAS A DECIMAL EXPANSION.

     

    FOR RATIONAL NUMBERS SHOW THAT THE DECIMAL EXPANSION REPEATS EVENTUALLY.

     

    CONVERT A DECIMAL EXPANSION WHICH REPEATS EVENTUALLY INTO A RATIONAL NUMBER.

              STUDENTS UNDERSTAND THAT REAL NUMBERS ARE EITHER RATIONAL OR IRRATIONAL AND DISTINGUISH

              BETWEEN THEM, RECOGNIZING THAT ANY NUMBER THAT CAN BE EXPRESSED AS A FRACTION IS A RATIONAL

              NUMBER. 

     

              STUDENTS RECOGNIZE THAT THE DECIMAL EQUIVALENT OF A FRACTION WILL EITHER TERMINATE OR

              REPEAT.  FRACTIONS THAT TERMINATE WILL HAVE DENOMINATORS WHOSE ONLY PRIME FACTORS ARE 2

              AND/OR 5. 

     

              STUDENTS CONVERT REPEATING DECIMALS INTO THEIR FRACTION EQUIVALENT USING PATTERNS OR

              ALGEBRAIC REASONING. 

    ex.  Change  to a fraction in lowest terms.

              Let x = 0.444…        Then 10x = 4.444…

                                                    - x   - 0.444…

                                                    9x = 4

                                                    x =

     

         or, students can investigate repeating patterns that occur when fractions have denominators of 9, 99 or 11.

              is equivalent to , is equivalent to , etc.

     

     

    USE RATIONAL APPROXIMATIONS OF IRRATIONAL NUMBERS TO COMPARE THE SIZE OF IRRATIONAL NUMBERS, LOCATE THEM APPROXIMATELY ON A NUMBER LINE DIAGRAM, AND ESTIMATE THE VALUE OF EXPRESSIONS (e.g. 2).  For example, by truncating the decimal expansion of , show that  is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

     

              STUDENTS LOCATE RATIONAL AND IRRATIONAL NUMBERS ON THE NUMBER LINE.  STUDENTS COMPARE AND

              ORDER RATIONAL AND IRRATIONAL NUMBERS.  STUDENTS ALSO RECOGNIZE THAT SQUARE ROOTS MAY BE

              NEGATIVE AND WRITTEN AS -.

     

                       ex. Compare  and .

                                 Solutions  could include:   and  are between the whole numbers 1 and 2.

                                                                        is between 1.7 and 1.8.

                                                                        is less than .

     

              ADDITIONALLY, STUDENTS UNDERSTAND THAT THE VALUE OF A SQUARE ROOT CAN BE APPROXIMATED

              BETWEEN INTEGERS AND THAT NON-PERFECT SQUARE ROOTS ARE IRRATIONAL.

                       ex.  Find a rational approximation of .

    « Determine the perfect squares 28 is between, which would be 25 and 36.

    « The square roots of 25 and 36 are 5 and 6, respectively, so we know that  is between 5 and 6.

    « Since 28 is closer to 25, an estimate of the square root would be closer to 5.  One method to get an estimate is to divide 3 (the distance between 25 and 28) by 11 (the distance between 25 and 36) to get 0.27.

    « The estimate of  would be 5.27 (the actual is 5.29).

     

     

     

     

     

     

    RATIOS AND PROPORTIONAL RELATIONSHIPS

     

    REVIEW

     

    RATIO VOCABULARY; EQUIVALENCES; PROPORTIONALITY TEST

    SOLVING PROPORTIONS BY CROSS-MULTIPLYING

     

    ANALYZE PROPORTIONAL RELATIONSHIPS AND USE THEM TO SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS.

     

    COMPUTE UNIT RATES ASSOCIATED WITH RATIOS OF FRACTIONS, INCLUDING RATIOS OF LENGTHS, AREAS AND OTHER QUANTITIES MEASURED IN LIKE OR DIFFERENT UNITS.  (For example, if a person walks ½ mile in each ¼ of an hour, compute the unit rate as the complex fraction ½ / ¼ miles per hour, equivalently 2 mph.)

     

    RECOGNIZE AND REPRESENT PROPORTIONAL RELATIONSHIPS BETWEEN QUANTITIES.

    Ø DECIDE WHETHER TWO QUANTITIES ARE IN A PROPORTIONAL RELATIONSHIP, E.G., BY TESTING FOR EQUIVALENT RATIOS IN A TABLE OR GRAPHING ON A COORDINATE PLANE AND OBSERVING WHETHER THE GRAPH IS A STRAIGHT LINE THROUGH THE ORIGIN.

    Ø IDENTIFY THE CONSTANT OF PROPORTIONALITY (UNIT RATE) IN TABLES, GRAPHS, EQUATIONS, DIAGRAMS AND VERBAL DESCRIPTIONS OF PROPORTIONAL RELATIONSHIPS.

    Ø REPRESENT PROPORTIONAL RELATIONSHIPS BY EQUATIONS.  (For example, if total cost, t, is proportional to the number, n, of items purchased at a constant price, p, the relationship between the total cost and the number of items can be expressed as     t = pn.)

    Ø EXPLAIN WHAT A POINT (x,y) ON THE GRAPH OF A PROPORTIONAL RELATIONSHIP MEANS IN TERMS OF THE SITUATION, WITH SPECIAL ATTENTION TO THE POINTS (0,0) AND (1,r), WHERE r IS THE UNIT RATE.

     

     

    USE PROPORTIONAL RELATIONSHIPS TO SOLVE MULTISTEP RATIO AND PERCENT PROBLEMS.


    Ø SIMPLE INTEREST

    Ø TAX

    Ø MARKUPS AND MARKDOWNS

    Ø GRATUTITIES AND COMMISSIONS

    Ø FEES

    Ø PERCENT INCREASE AND DECREASE

    Ø PERCENT ERROR


     

    ANALYZE PROPORTIONAL RELATIONSHIPS AND USE THEM TO SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS.

     

    COMPUTE UNIT RATES ASSOCIATED WITH RATIOS OF FRACTIONS, INCLUDING RATIOS OF LENGTHS, AREAS AND OTHER QUANTITIES MEASURED IN LIKE OR DIFFERENT UNITS.  (For example, if a person walks ½ mile in each ¼ of an hour, compute the unit rate as the complex fraction ½ / ¼ miles per hour, equivalently 2 mph.)

     

    RECOGNIZE AND REPRESENT PROPORTIONAL RELATIONSHIPS BETWEEN QUANTITIES.

    Ø DECIDE WHETHER TWO QUANTITIES ARE IN A PROPORTIONAL RELATIONSHIP, E.G., BY TESTING FOR EQUIVALENT RATIOS IN A TABLE OR GRAPHING ON A COORDINATE PLANE AND OBSERVING WHETHER THE GRAPH IS A STRAIGHT LINE THROUGH THE ORIGIN.

    Ø IDENTIFY THE CONSTANT OF PROPORTIONALITY (UNIT RATE) IN TABLES, GRAPHS, EQUATIONS, DIAGRAMS AND VERBAL DESCRIPTIONS OF PROPORTIONAL RELATIONSHIPS.

    Ø REPRESENT PROPORTIONAL RELATIONSHIPS BY EQUATIONS.  (For example, if total cost, t, is proportional to the number, n, of items purchased at a constant price, p, the relationship between the total cost and the number of items can be expressed as     t = pn.)

    Ø EXPLAIN WHAT A POINT (x,y) ON THE GRAPH OF A PROPORTIONAL RELATIONSHIP MEANS IN TERMS OF THE SITUATION, WITH SPECIAL ATTENTION TO THE POINTS (0,0) AND (1,r), WHERE r IS THE UNIT RATE.

     

    USE PROPORTIONAL RELATIONSHIPS TO SOLVE MULTISTEP RATIO AND PERCENT PROBLEMS.


    Ø SIMPLE INTEREST

    Ø TAX

    Ø MARKUPS AND MARKDOWNS

    Ø GRATUITIES AND COMMISSIONS

    Ø FEES

    Ø PERCENT INCREASE AND DECREASE

    Ø PERCENT ERROR

     


    EXPRESSIONS AND EQUATIONS

     

    REVIEW

     

    EVALUATE EXPRESSIONS

    SOLVE TWO-STEP EQUATIONS & INEQUALITIES

    GCF

     

    INTRODUCTION

     

    TRANSLATE TWO-STEP VERBAL EXPRESSIONS INTO ALGEBRAIC EXPRESSIONS.

     

    USE PROPERTIES OF OPERATIONS TO GENERATE EQUIVALENT EXPRESSIONS.

     

    APPLY PROPERTIES OF OPERATIONS AS STRATEGIES TO ADD, SUBTRACT, FACTOR AND EXPAND LINEAR EXPRESSIONS WITH RATIONAL COEFFICIENTS.  (DISTRIBUTIVE PROPERTY, COMBINING LIKE TERMS)

     

    UNDERSTAND THAT REWRITING AN EXPRESSION IN DIFFERENT FORMS IN A PROBLEM CONTEXT CAN SHED LIGHT ON THE PROBLEM AND HOW THE QUANTITIES IN IT ARE RELATED.  (For example, a + 0.05a = 1.05a means that “increase by 5%”  is the same as “multiply by 1.05”.)

    SOLVE REAL-LIFE AND MATHEMATICAL PROBLEMS USING NUMERICAL AND ALGEBRAIC EXPRESSIONS AND EQUATIONS.

     

    SOLVE MULTISTEP REAL-LIFE AND MATHEMATICAL PROBLEMS POSED WITH POSITIVE AND NEGATIVE RATIONAL NUMBERS, USING TOOLS STRATEGICALLY.

    Ø APPLY PROPERTIES OF OPERATIONS TO CALCULATE WITH NUMBERS IN ANY FORM.

    Ø CONVERT BETWEEN FORMS WHEN APPROPRIATE.

    Ø ASSESS THE REASONABLENESS OF ANSWERS USING MENTAL COMPUTATION AND ESTIMATION STRATEGIES.  (For example, If a woman making $25 an hour gets a 10% raise, she will make and additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50…  If you want to place a towel bar 9 ¾ inches long in the center of a door that is 27 ½ inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

     

    USE VARIABLES TO REPRESENT QUANTITIES IN A REAL-WORLD OR MATHEMATICAL PROBLEM, AND CONSTRUCT SIMPLE EQUATIONS AND INEQUALITIES TO SOLVE PROBLEMS BY REASONING ABOUT THE QUANTITIES.

    Ø SOLVE WORD PROBLEMS LEADING TO EQUATIONS OF THE FORM px + q = r AND p(x + q) = r, WHERE p, q AND r ARE SPECIFIC RATIONAL NUMBERS.  SOLVE EQUATIONS OF THESE FORMS FLUENTLY.

    Ø COMPARE AN ALGEBRAIC SOLUTION TO AN ARITHMETIC SOLUTION, IDENTIFYING THE SEQUENCE OF THE OPERATIONS USED IN EACH APPROACH.  For example, the perimeter of a rectangle is 54 cm.  Its length is 6 cm.  What is its width?)

    Ø SOLVE WORD PROBLEMS LEADING TO INEQUALITIES OF THE FORM         px + q > r OR px + q < r, WHERE p, q AND r ARE SPECIFIC RATIONAL NUMBERS.

    Ø GRAPH THE SOLUTION SET OF THE INEQUALITY AND INTERPRET IT IN THE CONTEXT OF THE PROBLEM.  (For example, As a salesperson, you are paid $50 per week plus $3 per sale.  This week you want your pay to be at least $100.  Write an inequality for the number of sales you need to make, and describe the solutions.)  ***NEGATIVE COEFFICIENTS POSSIBLE!

    Ø FIND THE MINIMUM OR MAXIMUM VALUE OF A SOLUTION SET.

     

    INTRODUCTION

     

    TRANSLATE TWO-STEP VERBAL EXPRESSIONS INTO ALGEBRAIC EXPRESSIONS.

     

    USE PROPERTIES OF OPERATIONS TO GENERATE EQUIVALENT EXPRESSIONS.

     

    APPLY PROPERTIES OF OPERATIONS AS STRATEGIES TO ADD, SUBTRACT, FACTOR AND EXPAND LINEAR EXPRESSIONS WITH RATIONAL COEFFICIENTS.  (DISTRIBUTIVE PROPERTY, COMBINING LIKE TERMS)

     

    UNDERSTAND THAT REWRITING AN EXPRESSION IN DIFFERENT FORMS IN A PROBLEM CONTEXT CAN SHED LIGHT ON THE PROBLEM AND HOW THE QUANTITIES IN IT ARE RELATED.  (For example, a + 0.05a = 1.05a means that “increase by 5%”  is the same as “multiply by 1.05”.)

    SOLVE REAL-LIFE AND MATHEMATICAL PROBLEMS USING NUMERICAL AND ALGEBRAIC EXPRESSIONS AND EQUATIONS.

    SOLVE MULTISTEP REAL-LIFE AND MATHEMATICAL PROBLEMS POSED WITH POSITIVE AND NEGATIVE RATIONAL NUMBERS, USING TOOLS STRATEGICALLY.

    Ø APPLY PROPERTIES OF OPERATIONS TO CALCULATE WITH NUMBERS IN ANY FORM.

    Ø CONVERT BETWEEN FORMS WHEN APPROPRIATE.

    Ø ASSESS THE REASONABLENESS OF ANSWERS USING MENTAL COMPUTATION AND ESTIMATION STRATEGIES.  (For example, If a woman making $25 an hour gets a 10% raise, she will make and additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50…  If you want to place a towel bar 9 ¾ inches long in the center of a door that is 27 ½ inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

     

    USE VARIABLES TO REPRESENT QUANTITIES IN A REAL-WORLD OR MATHEMATICAL PROBLEM, AND CONSTRUCT SIMPLE EQUATIONS AND INEQUALITIES TO SOLVE PROBLEMS BY REASONING ABOUT THE QUANTITIES.

    Ø SOLVE WORD PROBLEMS LEADING TO EQUATIONS OF THE FORM px + q = r AND p(x + q) = r, WHERE p, q AND r ARE SPECIFIC RATIONAL NUMBERS.  SOLVE EQUATIONS OF THESE FORMS FLUENTLY.

    Ø COMPARE AN ALGEBRAIC SOLUTION TO AN ARITHMETIC SOLUTION, IDENTIFYING THE SEQUENCE OF THE OPERATIONS USED IN EACH APPROACH.  For example, the perimeter of a rectangle is 54 cm.  Its length is 6 cm.  What is its width?)

    Ø SOLVE WORD PROBLEMS LEADING TO INEQUALITIES OF THE FORM         px + q > r OR px + q < r, WHERE p, q AND r ARE SPECIFIC RATIONAL NUMBERS.

    Ø GRAPH THE SOLUTION SET OF THE INEQUALITY AND INTERPRET IT IN THE CONTEXT OF THE PROBLEM.  (For example, As a salesperson, you are paid $50 per week plus $3 per sale.  This week you want your pay to be at least $100.  Write an inequality for the number of sales you need to make, and describe the solutions.)  ***NEGATIVE COEFFICIENTS POSSIBLE!

    Ø FIND THE MINIMUM OR MAXIMUM VALUE OF A SOLUTION SET.

     

     

     

     

     

     

     

    GEOMETRY

     

    REVIEW

     

    SIMILAR FIGURES (PROPORTIONS)

    CIRCUMFERENCE, AREA, VOLUME, SURFACE AREA

     

    INTRODUCTION

     

    COMPUTER PROGRAM/WEBSITE TO DRAW GEOMETRIC SHAPES AND EXAMINE CHARACTERISTICS?

     

    TRIANGLE RELATIONSHIPS:  ANGLE SUM, TRIANGLE INEQUALITY

     

    DRAW, CONSTRUCT AND DESCRIBE GEOMETRICAL FIGURES AND DESCRIBE THE RELATIONSHIPS BETWEEN THEM.

     

    SOLVE PROBLEMS INVOLVING SCALE DRAWINGS OF GEOMETRIC FIGURES, INCLUDING COMPUTING ACTUAL LENGTHS AND AREAS FROM A SCALE DRAWING AND REPRODUCING A SCALE DRAWING AT A DIFFERENT SCALE.

    Ø IDENTIFY THE IMPACT OF A SCALE ON ACTUAL LENGTH AND AREA.

    Ø IDENTIFY THE SCALE FACTOR GIVEN TWO FIGURES.

    Ø USING A GIVEN SCALE DRAWING, REPRODUCE THE DRAWING AT A DIFFERENT SCALE.  UNDERSTAND THAT THE LENGTHS WILL CHANGE BY A FACTOR EQUAL TO THE PRODUCT OF THE MAGNITUDE OF THE TWO SIZE TRANSFORMATIONS.

     

    DRAW (FREEHAND, WITH RULER AND PROTRACTOR, AND WITH TECHNOLOGY) GEOMETRIC SHAPES WITH GIVEN CONDITIONS.  FOCUS ON CONSTRUCTING TRIANGLES FROM THREE MEASURES OF ANGLES OR SIDES.

    Ø NOTICE WHEN THE CONDITIONS DETERMINE A UNIQUE TRIANGLE, MORE THAN ONE TRIANGLE OR NO TRIANGLE.

    o   TRIANGLE ANGLE SUM = 180o.

    o   CAN THE TRIANGLE HAVE MORE THAN ONE OBTUSE ANGLE?

    o   CAN THE SIDES OF ANY LENGTH CREATE A TRIANGLE?  STUDENTS RECOGNIZE THAT THE SUM OF THE TWO SMALLER SIDES MUST BE LARGER THAN THE THIRD SIDE. (TRIANGLE INEQUALITY)

     

    DESCRIBE THE 2-DIMENSIONAL FIGURES THAT RESULT FROM SLICING 3-D FIGURES, AS IN THE PLANE SECTIONS OF RIGHT RECTANGULAR PRISMS AND RIGHT RECTANGULAR PYRAMIDS.

    Ø CUTS MADE PARALLEL AND PERPENDICULAR TO THE BASE.

    o   PARALLEL:  SHAPE OF THE BASE

    o   PERPENDICULAR: SHAPE OF THE LATERAL (SIDE) FACE

    Ø CUTS MADE AT AN ANGLE WILL PRODUCE A PARALLELOGRAM.

     

    SOLVE REAL-LIFE AND MATHEMATICAL PROBLEMS INVOLVING ANGLE MEASURE, AREA, SURFACE AREA AND VOLUME.

     

    KNOW (UNDERSTAND) THE FORMULAS FOR THE AREA AND CIRCUMFERENCE OF A CIRCLE AND USE THEM TO SOLVE PROBLEMS.

    Ø FIND AREA OF LEFT-OVER MATERIALS WHEN CIRCLES ARE CUT FROM SQUARES OR TRIANGLES, OR VICE VERSA.

     

    GIVE AN INFORMAL DERIVATION OF THE RELATIONSHIP BETWEEN THE CIRCUMFERENCE AND AREA OF A CIRCLE.

    Ø UNDERSTAND THE RELATIONSHIP BETWEEN RADIUS AND DIAMETER.

    Ø UNDERSTAND THE RATIO OF CIRCUMFERENCE TO DIAMETER IS PI (p).

    Ø http://mathworld.wolfram.com/Circle.html

     

    USE FACTS ABOUT SUPPLEMENTARY, COMPLEMENTARY, VERTICAL AND ADJACENT ANGLES IN A MULTISTEP PROBLEM TO WRITE AND SOLVE SIMPLE EQUATIONS FOR AN UNKNOWN ANGLE IN A FIGURE.

     

    SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS INVOLVING AREA, VOLUME AND SURFACE AREA OF TWO- AND THREE-DIMENSIONAL OBJECTS COMPOSED OF TRIANGLES, QUADRILATERALS, POLYGONS, CUBES AND RIGHT PRISMS.

    Ø EXPECTATION IS UNDERSTANDING, NOT USING FORMULAS.  STUDENTS SHOULD RECOGNIZE THAT FINDING THE AREA OF EACH FACE OF A 3-D FIGURE AND ADDING THE AREAS WILL GIVE THE SURFACE AREA. (NO NETS WILL BE GIVEN.)

    INTRODUCTION

     

    COMPUTER PROGRAM/WEBSITE TO DRAW GEOMETRIC SHAPES AND EXAMINE CHARACTERISTICS?

     

    TRIANGLE RELATIONSHIPS:  ANGLE SUM, TRIANGLE INEQUALITY

     

    DRAW, CONSTRUCT AND DESCRIBE GEOMETRICAL FIGURES AND DESCRIBE THE RELATIONSHIPS BETWEEN THEM.

     

    SOLVE PROBLEMS INVOLVING SCALE DRAWINGS OF GEOMETRIC FIGURES, INCLUDING COMPUTING ACTUAL LENGTHS AND AREAS FROM A SCALE DRAWING AND REPRODUCING A SCALE DRAWING AT A DIFFERENT SCALE.

    Ø IDENTIFY THE IMPACT OF A SCALE ON ACTUAL LENGTH AND AREA.

    Ø IDENTIFY THE SCALE FACTOR GIVEN TWO FIGURES.

    Ø USING A GIVEN SCALE DRAWING, REPRODUCE THE DRAWING AT A DIFFERENT SCALE.  UNDERSTAND THAT THE LENGTHS WILL CHANGE BY A FACTOR EQUAL TO THE PRODUCT OF THE MAGNITUDE OF THE TWO SIZE TRANSFORMATIONS.

     

    DRAW (FREEHAND, WITH RULER AND PROTRACTOR, AND WITH TECHNOLOGY) GEOMETRIC SHAPES WITH GIVEN CONDITIONS.  FOCUS ON CONSTRUCTING TRIANGLES FROM THREE MEASURES OF ANGLES OR SIDES.

    Ø NOTICE WHEN THE CONDITIONS DETERMINE A UNIQUE TRIANGLE, MORE THAN ONE TRIANGLE OR NO TRIANGLE.

    o   TRIANGLE ANGLE SUM = 180o.

    o   CAN THE TRIANGLE HAVE MORE THAN ONE OBTUSE ANGLE?

    o   CAN THE SIDES OF ANY LENGTH CREATE A TRIANGLE?  STUDENTS RECOGNIZE THAT THE SUM OF THE TWO SMALLER SIDES MUST BE LARGER THAN THE THIRD SIDE. (TRIANGLE INEQUALITY)

     

    DESCRIBE THE 2-DIMENSIONAL FIGURES THAT RESULT FROM SLICING 3-D FIGURES, AS IN THE PLANE SECTIONS OF RIGHT RECTANGULAR PRISMS AND RIGHT RECTANGULAR PYRAMIDS.

    Ø CUTS MADE PARALLEL AND PERPENDICULAR TO THE BASE.

    o   PARALLEL:  SHAPE OF THE BASE

    o   PERPENDICULAR: SHAPE OF THE LATERAL (SIDE) FACE

    Ø CUTS MADE AT AN ANGLE WILL PRODUCE A PARALLELOGRAM.

    SOLVE REAL-LIFE AND MATHEMATICAL PROBLEMS INVOLVING ANGLE MEASURE, AREA, SURFACE AREA AND VOLUME.

     

    KNOW (UNDERSTAND) THE FORMULAS FOR THE AREA AND CIRCUMFERENCE OF A CIRCLE AND USE THEM TO SOLVE PROBLEMS.

    Ø FIND AREA OF LEFT-OVER MATERIALS WHEN CIRCLES ARE CUT FROM SQUARES OR TRIANGLES, OR VICE VERSA.

     

    GIVE AN INFORMAL DERIVATION OF THE RELATIONSHIP BETWEEN THE CIRCUMFERENCE AND AREA OF A CIRCLE.

    Ø UNDERSTAND THE RELATIONSHIP BETWEEN RADIUS AND DIAMETER.

    Ø UNDERSTAND THE RATIO OF CIRCUMFERENCE TO DIAMETER IS PI (p).

    Ø http://mathworld.wolfram.com/Circle.html

     

    USE FACTS ABOUT SUPPLEMENTARY, COMPLEMENTARY, VERTICAL AND ADJACENT ANGLES IN A MULTISTEP PROBLEM TO WRITE AND SOLVE SIMPLE EQUATIONS FOR AN UNKNOWN ANGLE IN A FIGURE.

     

    SOLVE REAL-WORLD AND MATHEMATICAL PROBLEMS INVOLVING AREA, VOLUME AND SURFACE AREA OF TWO- AND THREE-DIMENSIONAL OBJECTS COMPOSED OF TRIANGLES, QUADRILATERALS, POLYGONS, CUBES AND RIGHT PRISMS.

    Ø EXPECTATION IS UNDERSTANDING, NOT USING FORMULAS.  STUDENTS SHOULD RECOGNIZE THAT FINDING THE AREA OF EACH FACE OF A 3-D FIGURE AND ADDING THE AREAS WILL GIVE THE SURFACE AREA. (NO NETS WILL BE GIVEN.)

     

     

     

     

     

     

     

     

    STATISTICS AND PROBABILITY

     

    REVIEW

     

    SIMPLE PROBABILITY, SAMPLE SPACE, COUNTING PRINCIPLE

    MEAN, MEDIAN, MEAN ABSOLUTE DEVIATION (M.A.D.), INTERQUARTILE RANGE

     

    USE RANDOM SAMPLING TO DRAW INFERENCES ABOUT A POPULATION.

     

    UNDERSTAND THAT STATISTICS CAN BE USED TO GAIN INFORMATION ABOUT A POPULATION BY EXAMINING A SAMPLE OF THE POPULATION;  GENERALIZATIONS ARE VALID ONLY IF THE SAMPLE IS REPRESENTATIVE OF THE POPULATION.  RANDOM SAMPLING TENDS TO PRODUCE REPRESENTATIVE SAMPLES AND SUPPORT VALID INFERENCES.

     

    USE DATA FROM A RANDOM SAMPLE TO DRAW INFERENCES ABOUT A POPULATION WITH AN UNKNOWN CHARACTERISTIC OF INTEREST

    Ø GENERATE MULTIPLE SAMPLES (OR SIMULATED SAMPLES) OF THE SAME SIZE TO GAUGE THE VARIATION IN ESTIMATES OR PREDICTIONS.  (For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data.  Gauge how far off the estimate or prediction might be.)

    Ø ISSUES OF VARIATION IN THE SAMPLES SHOULD BE ADDRESSED.

    DRAW INFORMAL COMPARATIVE INFERENCES ABOUT TWO POPULATIONS.

     

    INFORMALLY ASSESS THE DEGREE OF VISUAL OVERLAP OF TWO NUMERICAL DATA DISTRIBUTIONS WITH SIMILAR VARIABILITIES, MEASURING THE DIFFERENCE BETWEEN THE CENTERS BY EXPRESSING IT AS A MULTIPLE MEASURE OF VARIABILITY.  (For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team;  on a dot plot, the separation between the two distributions of heights is noticeable.)

    Ø BUILD ON UNDERSTANDING OF MEAN, MEDIAN, MEAN ABSOLUTE DEVIATION AND INTERQUARTILE RANGE (REVIEW) FOR FULL UNDERSTANDING.

    Ø VARIABILITY IS RESPONSIBLE FOR OVERLAP OF TWO DATA SETS; INCREASE IN VARIABILITY CAN INCREASE OVERLAP.

    Ø MEDIAN PAIRED WITH INTERQUARTILE RANGE; MEAN PAIRED WITH MEAN ABSOLUTE DEVIATION.

     

    USE MEASURES OF CENTER AND MEASURES OF VARIABILITY FOR NUMERICAL DATA FROM RANDOM SAMPLES TO DRAW INFORMAL COMPARATIVE INFERENCES ABOUT TWO POPULATIONS.  (For example, decide whether the words in a chapter of a 7th grade science book are generally longer than the words in a chapter of a 4th grade science book.)

    Ø COMPARE TWO SETS OF DATA USING MEASURES OF CENTER AND VARIABILITY.

     

    “MEASURES OF CENTER” – MEAN, MEDIAN

    “MEASURES OF VARIABILITY” – INTERQUARTILE RANGE, M.A.D.

    INVESTIGATE CHANCE PROCESSES AND DEVELOP, USE AND EVALUATE PROBABILITY MODELS.

     

    UNDERSTAND THAT THE PROBABILITY OF A CHANCE EVENT IS A NUMBER BETWEEN 0 AND 1 THAT EXPRESSES THE LIKELIHOOD OF THE EVENT OCCURRING.  LARGER NUMBERS INDICATE GREATER LIKELIHOOD. 

    Ø PROBABILITY CAN BE REPRESENTED BY A FRACTION.

    Ø A PROBABILITY NEAR 0 INDICATES AN UNLIKELY EVENT.

    Ø A PROBABILITY NEAR ½ INDICATES AN EVENT THAT IS NEITHER LIKELY NOR UNLIKELY.

    Ø A PROBABILITY NEAR 1 INDICATES A LIKELY EVENT.

    Ø THE SUM OF THE PROBABILITIES OF ALL POSSIBLE OUTCOMES IS 1.

     

    APPROXIMATE THE PROBABILITY OF A CHANCE EVENT BY COLLECTING DATA ON THE CHANCE PROCESS THAT PRODUCES IT AND OBSERVING ITS LONG-RUN RELATIVE FREQUENCY, AND PREDICT THE APPROXIMATE RELATIVE FREQUENCY GIVEN THE PROBABILITY.  (For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.)

    Ø PROBABILITY EXPERIMENT.

    Ø RECOGNIZE THAT AS THE NUMBER OF TRIALS INCREASES, THE EXPERIMENTAL PROBABILITY APPROACHES THE THEORETICAL PROBABILITY.

    Ø FOCUS: RELATIVE FREQUENCY:OBSERVED NUMBER OF SUCCESSFUL EVENTS FOR A FINITE SAMPLE OF TRIALS.

     

    DEVELOP A PROBABILITY MODEL AND USE IT TO FIND PROBABILITIES OF EVENTS.  COMPARE PROBABILITIES FROM A MODEL TO OBSERVED FREQUENCIES; IF THE AGREEMENT IS NOT GOOD, EXPLAIN POSSIBLE SOURCES OF THE DISCREPANCY.

    Ø DEVELOP A UNIFORM PROBABILITY MODEL BY ASSIGNING EQUAL PROBABILITY TO ALL OUTCOMES, AND USE THE MODEL TO DETERMINE PROBABILITIES OF EVENTS.  (For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.)

    Ø DEVELOP A PROBABILITY MODEL (WHICH MAY NOT BE UNIFORM) BY OBSERVING FREQUENCIES IN DATA GENERATED FROM A CHANCE PROCESS.  (For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down.  Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?) ~experimental probability

    Ø USING THEORETICAL PROBABILITY, PREDICT FREQUENCIES OF OUTCOMES.

    Ø RECOGNIZE AN APPROPRIATE DESIGN TO CONDUCT AN EXPERIMENT WITH SIMPLE PROBABILITY EVENTS.

    Ø UNDERSTAND THAT EXPERIMENTAL DATA GIVE REALISTIC ESTIMATES OF THE PROBABILITY OF AN EVENT BUT ARE AFFECTED BY SAMPLE SIZE.

     

    FIND PROBABILITIES OF COMPOUND EVENTS USING ORGANIZED LISTS, TABLES, TREE DIAGRAMS AND SIMULATION.

    Ø UNDERSTAND THAT, JUST AS WITH SIMPLE EVENTS, THE PROBABILITY OF A COMPOUND EVENT IS THE FRACTION OF OUTCOMES IN THE SAMPLE SPACE FOR WHICH THE COMPOUND EVENT OCCURS.

    Ø REPRESENT SAMPLE SPACES FOR COMPOUND EVENTS USING METHODS SUCH AS ORGANIZED LISTS, TABLES AND TREE DIAGRAMS. 

    Ø FOR AN EVENT DESCRIBED IN EVERYDAY LANGUAGE (E.G. “ROLLING DOUBLE SIXES”), IDENTIFY THE OUTCOMES IN THE SAMPLE SPACE WHICH COMPOSE THE EVENT.

    Ø USING THEORETICAL PROBABILITY, PREDICT FREQUENCIES OF OUTCOMES.

    Ø RECOGNIZE AN APPROPRIATE DESIGN TO CONDUCT AN EXPERIMENT WITH SIMPLE PROBABILITY EVENTS.

    Ø UNDERSTAND THAT EXPERIMENTAL DATA GIVE REALISTIC ESTIMATES OF THE PROBABILITY OF AN EVENT BUT ARE AFFECTED BY SAMPLE SIZE.

     

    DESIGN AND USE A SIMULATION TO GENERATE FREQUENCIES FOR COMPOUND EVENTS.  (For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?)        

     

    USE RANDOM SAMPLING TO DRAW INFERENCES ABOUT A POPULATION.

     

    UNDERSTAND THAT STATISTICS CAN BE USED TO GAIN INFORMATION ABOUT A POPULATION BY EXAMINING A SAMPLE OF THE POPULATION;  GENERALIZATIONS ARE VALID ONLY IF THE SAMPLE IS REPRESENTATIVE OF THE POPULATION.  RANDOM SAMPLING TENDS TO PRODUCE REPRESENTATIVE SAMPLES AND SUPPORT VALID INFERENCES.

     

    USE DATA FROM A RANDOM SAMPLE TO DRAW INFERENCES ABOUT A POPULATION WITH AN UNKNOWN CHARACTERISTIC OF INTEREST

    Ø GENERATE MULTIPLE SAMPLES (OR SIMULATED SAMPLES) OF THE SAME SIZE TO GAUGE THE VARIATION IN ESTIMATES OR PREDICTIONS.  (For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data.  Gauge how far off the estimate or prediction might be.)

    Ø ISSUES OF VARIATION IN THE SAMPLES SHOULD BE ADDRESSED.

    DRAW INFORMAL COMPARATIVE INFERENCES ABOUT TWO POPULATIONS.

     

    INFORMALLY ASSESS THE DEGREE OF VISUAL OVERLAP OF TWO NUMERICAL DATA DISTRIBUTIONS WITH SIMILAR VARIABILITIES, MEASURING THE DIFFERENCE BETWEEN THE CENTERS BY EXPRESSING IT AS A MULTIPLE MEASURE OF VARIABILITY.  (For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team;  on a dot plot, the separation between the two distributions of heights is noticeable.)

    Ø BUILD ON UNDERSTANDING OF MEAN, MEDIAN, MEAN ABSOLUTE DEVIATION AND INTERQUARTILE RANGE (REVIEW) FOR FULL UNDERSTANDING.

    Ø VARIABILITY IS RESPONSIBLE FOR OVERLAP OF TWO DATA SETS; INCREASE IN VARIABILITY CAN INCREASE OVERLAP.

    Ø MEDIAN PAIRED WITH INTERQUARTILE RANGE; MEAN PAIRED WITH MEAN ABSOLUTE DEVIATION.

     

    USE MEASURES OF CENTER AND MEASURES OF VARIABILITY FOR NUMERICAL DATA FROM RANDOM SAMPLES TO DRAW INFORMAL COMPARATIVE INFERENCES ABOUT TWO POPULATIONS.  (For example, decide whether the words in a chapter of a 7th grade science book are generally longer than the words in a chapter of a 4th grade science book.)

    Ø COMPARE TWO SETS OF DATA USING MEASURES OF CENTER AND VARIABILITY.

     

    “MEASURES OF CENTER” – MEAN, MEDIAN

    “MEASURES OF VARIABILITY” – INTERQUARTILE RANGE, M.A.D.

    INVESTIGATE CHANCE PROCESSES AND DEVELOP, USE AND EVALUATE PROBABILITY MODELS.

     

    UNDERSTAND THAT THE PROBABILITY OF A CHANCE EVENT IS A NUMBER BETWEEN 0 AND 1 THAT EXPRESSES THE LIKELIHOOD OF THE EVENT OCCURRING.  LARGER NUMBERS INDICATE GREATER LIKELIHOOD. 

    Ø PROBABILITY CAN BE REPRESENTED BY A FRACTION.

    Ø A PROBABILITY NEAR 0 INDICATES AN UNLIKELY EVENT.

    Ø A PROBABILITY NEAR ½ INDICATES AN EVENT THAT IS NEITHER LIKELY NOR UNLIKELY.

    Ø A PROBABILITY NEAR 1 INDICATES A LIKELY EVENT.

    Ø THE SUM OF THE PROBABILITIES OF ALL POSSIBLE OUTCOMES IS 1.

     

    APPROXIMATE THE PROBABILITY OF A CHANCE EVENT BY COLLECTING DATA ON THE CHANCE PROCESS THAT PRODUCES IT AND OBSERVING ITS LONG-RUN RELATIVE FREQUENCY, AND PREDICT THE APPROXIMATE RELATIVE FREQUENCY GIVEN THE PROBABILITY.  (For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.)

    Ø PROBABILITY EXPERIMENT.

    Ø RECOGNIZE THAT AS THE NUMBER OF TRIALS INCREASES, THE EXPERIMENTAL PROBABILITY APPROACHES THE THEORETICAL PROBABILITY.

    Ø FOCUS: RELATIVE FREQUENCY:OBSERVED NUMBER OF SUCCESSFUL EVENTS FOR A FINITE SAMPLE OF TRIALS.

     

    DEVELOP A PROBABILITY MODEL AND USE IT TO FIND PROBABILITIES OF EVENTS.  COMPARE PROBABILITIES FROM A MODEL TO OBSERVED FREQUENCIES; IF THE AGREEMENT IS NOT GOOD, EXPLAIN POSSIBLE SOURCES OF THE DISCREPANCY.

    Ø DEVELOP A UNIFORM PROBABILITY MODEL BY ASSIGNING EQUAL PROBABILITY TO ALL OUTCOMES, AND USE THE MODEL TO DETERMINE PROBABILITIES OF EVENTS.  (For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.)

    Ø DEVELOP A PROBABILITY MODEL (WHICH MAY NOT BE UNIFORM) BY OBSERVING FREQUENCIES IN DATA GENERATED FROM A CHANCE PROCESS.  (For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down.  Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?) ~experimental probability

    Ø USING THEORETICAL PROBABILITY, PREDICT FREQUENCIES OF OUTCOMES.

    Ø RECOGNIZE AN APPROPRIATE DESIGN TO CONDUCT AN EXPERIMENT WITH SIMPLE PROBABILITY EVENTS.

    Ø UNDERSTAND THAT EXPERIMENTAL DATA GIVE REALISTIC ESTIMATES OF THE PROBABILITY OF AN EVENT BUT ARE AFFECTED BY SAMPLE SIZE.

     

    FIND PROBABILITIES OF COMPOUND EVENTS USING ORGANIZED LISTS, TABLES, TREE DIAGRAMS AND SIMULATION.

    Ø UNDERSTAND THAT, JUST AS WITH SIMPLE EVENTS, THE PROBABILITY OF A COMPOUND EVENT IS THE FRACTION OF OUTCOMES IN THE SAMPLE SPACE FOR WHICH THE COMPOUND EVENT OCCURS.

    Ø REPRESENT SAMPLE SPACES FOR COMPOUND EVENTS USING METHODS SUCH AS ORGANIZED LISTS, TABLES AND TREE DIAGRAMS. 

    Ø FOR AN EVENT DESCRIBED IN EVERYDAY LANGUAGE (E.G. “ROLLING DOUBLE SIXES”), IDENTIFY THE OUTCOMES IN THE SAMPLE SPACE WHICH COMPOSE THE EVENT.

    Ø USING THEORETICAL PROBABILITY, PREDICT FREQUENCIES OF OUTCOMES.

    Ø RECOGNIZE AN APPROPRIATE DESIGN TO CONDUCT AN EXPERIMENT WITH SIMPLE PROBABILITY EVENTS.

    Ø UNDERSTAND THAT EXPERIMENTAL DATA GIVE REALISTIC ESTIMATES OF THE PROBABILITY OF AN EVENT BUT ARE AFFECTED BY SAMPLE SIZE.

     

    DESIGN AND USE A SIMULATION TO GENERATE FREQUENCIES FOR COMPOUND EVENTS.  (For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?)