LEARNING DISTRICT TRAINING SESSION REVISED

Learning

The revised curriculum supports students learning mathematics with understanding and actively building new knowledge from experience and prior knowledge.

Conceptual Understanding Conceptual understanding refers to an integrated and functional grasp of mathematics. It is more than knowing isolated facts and procedures.

Conceptual Understanding Conceptual understanding supports retention. When facts and procedures are learned in a connected way, they are easier to remember and use and can be reconstructed when forgotten. Hiebert and Wearne 1996; Bruner 1960, Katona 1940

Conceptual Understanding Knowledge that has been learned with understanding provides the basis for generating new knowledge and for solving new and unfamiliar problems. Bransford, Brown and Cocking 1998

Activate Your Memory Try This!

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Making a Connection! ABC DEF GHI

We use the ideas we already have (blue dots) to construct a new idea (red dot). The more ideas used and the more connections made, the better we understand. Developing Understanding John Van de Walle

Making Connections 1999 Curriculum -determine, from examination of patterns, the exponent rules for multiplying and dividing monomials and the exponent rule for the power of a power, and apply…. -determine the meaning of negative exponents and of zero as an exponent from activities involving graphing, using technology, and from activities involving patterning COURSE: GRADE 9 Applied and Academic Revision -describe the relationship between the algebraic and geometric representations of a single variable term up to degree three ( i.e., length, which is one dimensional, can be represented by x; area, which is two dimensional can be represented by x^2, and volume, which is three dimensional can be ….

“Education that consists in learning things and not the meaning of them is like feeding upon husks and not the corn.” Mark Twain

1999 Curriculum Gr 9 Applied: -solve simple problems, using the formulas for the surface area of prisms and cylinders and for the volume of prisms, cylinders, cones and spheres; COURSE: GRADE 9 Applied to GRADE 10 Applied Revision Grade 10 Applied -determine, through investigation, the relationship for finding the surface area of a pyramid (e.g., use the net of a square based pyramid to determine that the surface area is the area of the square base, plus the areas of the four congruent triangles) Developmental Continuum

1999 Curriculum Grade 9 Applied: Linear relationships are generalised as analytic geometry (linear modelling) COURSE: GRADE 9 Applied to 10 Applied Revision Grade 9 Applied: Linear relationships (understanding of, and applications of “real life” examples) Grade 10 Applied: Linear relationships are generalized as analytic geometry, spreading this concept over 2 years Developmental Continuum

Continuum of Learning : Support Resource for Draft Revision Spring 2005 Number Sense and Numeration GRADE 1: read, represent, order, and compare whole numbers to 50, and investigate money amounts and fractions; GRADE 5: read, represent, order, and compare whole numbers to , decimal numbers to hundredths, proper and improper fractions, and mixed numbers; GRADE 2: read, represent, order, and compare whole numbers to 100, and represent money amounts and fractions using concrete materials GRADE 6: read, represent, order, and compare whole numbers to , decimal numbers to thousandths, proper and improper fractions, and mixed numbers GRADE 3: read, represent, order, and compare whole numbers to 1000, and demonstrate their understandings about money and fractions GRADE 7: represent, order, and compare numbers, including integers GRADE 4: read, represent, order, and compare whole numbers to , decimal numbers to tenths, and simple fractions, and expand their understandings about money GRADE 8:represent, order, and compare equivalent representations of numbers including those involving exponents

Developing Concepts Across the Grades 1997 Curriculum Proportional Reasoning Draft Revision Spring 2005 Proportional Reasoning Grade 7: No Specific ReferenceGrade 7: Number Sense and Numeration Proportional Relationships Grade 8: Under ApplicationsGrade 8: Number Sense and Numeration Proportional Relationships Grade 9 Applied: Number Sense and Algebra Solving Numerical Problems Grade 9 Applied: Number Sense and Algebra: Proportional Reasoning Grade 10 Applied: Proportional Reasoning Grade 10 Applied: Measurement and Trigonometry: Solving Problems Involving Similar Triangles

How the Revised Curriculum fits together… It fits like a jigsaw puzzle…

Understanding Exponents Gr 7 Gr 8 Gr 9 Gr 10 Explain the relationship between exponent notation and the measurement of area and volume Express repeated multiplication using exponential notation 2x2x2x2=2 4 Represent whole numbers in expanded form using powers 347 = 3x10²+4x10+7 2x2x2x2=2 4 Substitute into and evaluate algebraic expressions involving exponents Derive through investigation the exponent rules for multiplying and dividing monomials Extend the multiplication rule to derive and understand the power of a power rule f a x f b = f a+b f a /f b = f a-b (f a ) b = f axb Determine the meaning of zero and negative exponents

Developing Concepts - Volume Grade 4 Measure Volume Grade 5Grade 6 Grade 7 Gr 8 Gr 9 Grade 9 Grade 9 Grade 10 Academic Applied Applied Solve problems Develop formulas Solve problems Involving optimal for the volumes of involving volumes volume. Solve the pyramids, cones of prisms, pyramids max and min vol and spheres. cylinders, cones, pyramids, cones spheres, and a and spheres. combination.

Ratio, Rate and Proportion Grade 9 – the seven specific expectations are: 1)Perform operations with rational numbers, as necessary to support other topics of this course (e.g., rate of change, proportionality, measurement, percent) 2)Illustrate equivalent ratios using a variety of tools 3)Represent directly proportional relationships with equaivalent ratios and proportions, arising from realistic situations (Sample problem: You are building a skateboard ramp whose ratio of height to base must be 2:3. Write a proportion that could be used to find the base if the height is 4.5 m) 4)Solve for the unknown value in a proportion 5)Make comparisons using unit rate 6)Solve problems involving ratios, rates, and directly proportional relationships in various contexts 7)Solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms (e.g., calculating simple interest and sales tax, analysing data (Sample problem: Of the 29 students in a Grade 9 Math class, 13 are taking science this semester. If this class is representative of all the Grade 9 students in the school, what percent of the 236 Grade 9 students are taking science this semester? How many grade 9 students does this percent represent?)

Ratio, Rate and Proportion Gr 7 Gr 8 Gr 9 Gr 10 There are 7 specific expectations under the Overall expec: solve problems involving proportional reasoning Use their knowledge of ratio and proportion … and solve problems. Determine the lengths of sides of similar triangles … Your turn to find the expectations for gr 7 and 8

Ratio, Rate and Proportion Grade 8 1)Solve problems involving rates (Sample problem: A pack of 24 CD’s costs $7.99. A pack of 50 CD’s costs $ What is the most economical way to purchase 130 CD’s? 2)Recognize and describe real-life situations involving two quantities that are directly proportional (e.g., number of servings and quantities in a recipe, mass and volume, circumferences and diameters of circles) 3) Solve problems involving percent arising from real-life contexts (e.g., discounts, sales tax, simple interest) 4) Solve problems involving proportions using concrete materials, drawings, and variables (Sample Problem. The ratio of stone to sand in HardFast Concrete is 2 to 3. How much stone is needed if 15 bags of sand are used?) Grade 7 1)Solve problems that involve determining whole number percents (Sample problem: If there are 5 blue marbles in a bag of 20 marbles, what percentage of the marbles are not blue?) 2)Define rate as a comparison of two quantities with different units (e.g., speed is a rate that compares distance to time) 3)Determine through investigation, the relationship amoung fractions, decimals, percent and ratios 4)Solve problems involving the calculation of unit rates (Sample Problem: You go shopping and notice that 25 kg of Carol’s Famous Potaotoes costs $ And 10 kg of Gillian’s Potatoes costs $5.78. Which is the better deal?

Your turn to put the puzzle together Gr 7 Gr 8 Gr 9 Gr 10 Your turn to investigate expectations from grade 7, 8, 9 and 10 that build on one another.

Concept Development Expectations that introduce and develop a concept often include the phrase “through investigation”. Expectations that involve concepts that give rise to procedural learning and require some level of proficiency often include the phrase “solve problems”.

1999 Curriculum -solve simple problems, using the formulas for the surface area of prisms and cylinders and for the volume of prisms, cylinders, cones and spheres COURSE: GRADE 9 Applied and Academic Revision -develop through investigation (e.g. using concrete materials) the formula for the volume of a pyramid, a cone, and a sphere (e.g., use 3 dimensional figures to show that the volume of a pyramid (cone) is one third the volume of a prism (or cylinder) with the same base and height -solve problems… Through Investigation

1997 CURRICULUM Grade 7 SPRING 2005 DRAFT Grade 7 describe data using measures of central tendency (mean, median and mode); compare, through investigation, how the data values affect the median and the mean (e.g., changing the value of an outlier can have a significant effect on the mean and no effect on the median); Grades 1 - 8

1999 Curriculum -define the formulas for the sine, the cosine, and the tangent of angles, using the ratios of sides in right triangles COURSE: GRADE 10 Applied and Academic Revision -determine through investigation (e.g., using DGS, concrete materials), the relationship between the ratio of two sides in a right triangle and the ratio of the two corresponding sides in a similar right triangle, and define the sine, cosine, and tangent ratios (e.g., sinA=opposite/hypontenuse) Through Investigation

Culminating With Solving Problems Learning should culminate with the application of knowledge and skills to solve problems.

Culminating with Solving Problems 1997 CURRICULUM Grade 8 FEBRUARY 2005 DRAFT Grade 8 multiply and divide integers; solve problems involving operations with integers, using a variety of tools;

1999 Curriculum -solve quadratic equations using the quadratic formula COURSE: GRADE 10 Academic Revision -explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numerical example; follow a demonstration of the algebraic development (student reproduction of the development of the general case is not required) -solve quadratic equations… Culminating With Solving Problems

Procedural Fluency Procedural Fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately and efficiently. Kilpatrick et al, 2001

Yours is not to reason why, just invert and multiply!

Looking back and asking why! = 6  1 

Balancing Conceptual Understanding and Procedural Fluency Pitting procedural fluency against conceptual understanding creates a false dichotomy. Understanding makes learning skills easier, less susceptible to common errors and less prone to forgetting. Also, a certain level of skill is required to learn many mathematical concepts with understanding Hiebert and Carpenter 1992

Procedural Fluency A good conceptual understanding of place value supports the development of fluency in multidigit computation. Heibert, Carpenter et al 1997; Resnick and Omanson 1987

1999 Curriculum -determine, through investigation, the relationships between the angles and sides in acute triangles (e.g., the largest angle is opposite the longest side; the ratio of the sines of the opposite angles), using DGS COURSE: GRADE 10 Academic Revision -explore the development of the cosine law within acute triangles (e.g., use DGS to verify the cosine law; follow the algebraic development of the cosine law and identify its relationship to the Pythagorean theorem and the cosine ratio (student reproduction of the development of the formula is not required) -solve problems … Procedural / Conceptual

Connecting to the problem What concepts are incorporated into this problem? Are there aspects of the problem that begin at the conceptual stage and move towards the procedural?