Course description
This course is designed to prepare students for careers as elementary school teachers. It develops mathematical reasoning, problem solving, and communication skills, with an emphasis on a deep understanding of concepts in the elementary school mathematics curriculum. Topics include problem solving, set theory, the real number system and its subsets, arithmetic, elementary number theory, basic geometric concepts, measurement geometry, transformations and coordinate geometry, congruence, and similarity. Students should have solid arithmetic, numeracy, and algebra skills before registering in this course.
Course content
1. Problem solving processes and strategies
- Polya’s four-step method
- tables and pictures
- guess, test, and revise method
- algebraic models
- reasoning skills
- working with patterns
- number sequences
- arithmetic and geometric sequences
- pattern recognition and solving a simplified version
- working backward and problem restatement
2. Sets
- elements, subsets, power sets, and cardinal numbers
- complement, relative complement, intersection, and union
- Cartesian product, relations, and equivalent sets
- associative, commutative, identity, inverse, closure properties, and De Morgan's laws
- word problems
- Venn diagrams
3. Whole numbers
- operations with whole numbers
- addition and subtraction models: set model and measurement model
- subtraction models: take-away, missing addend, and comparison
- multiplication models: the set model and repeated addition
- division models: partition, grouping, measurement, and missing factor
- properties: associativity, commutativity, distributivity, and closure
- order of operations
- solution set for equations and inequalities
4. The set of integers and elementary number theory
- integers: properties, operations
- the coloured chips (square) model
- order of integers
- absolute value
- primes, relatively prime numbers, composite numbers, and test for divisibility
- the Fundamental Theorem of Arithmetic
- divisibility a|b, and its properties
- prime factorization and standard prime factorization: factor tree method, and division (ladder) model
- the greatest common factor (GCF) and least common multiple (LCM) of two and three positive integers: list method, prime factorization, division method, and Euclidean algorithm
- word problems involving applications of GCFand LCM
5. The set of rational numbers
- rational numbers and fractions
- equivalent fractions, reduced fractions, and the Fundamental Law of Fractions
- proper fractions, improper fractions, and mixed numbers
- operations and properties
- mental arithmetic and estimation with rational numbers
- order in fractions
- equations and inequalities involving rational numbers
- ratios and proportions, and applications
6. Real numbers
- decimals
- rational numbers versus irrational numbers
- exponentials, and exponent rules
- terminating and non-terminating decimal representations
- decimal operations models
- order in decimals
- square root, other roots, and rational exponents
- Pythagorean theorem and its applications
- mental arithmetic and estimation for decimals and radicals
- ratios, decimals, and percents
- word problems involving applications of decimals and percentages
7. Geometry
- informal geometry
- points, lines, rays, line segments, and planes in space with their subsets
- relations among lines and planes
- terms related to lines: collinearity, parallel, perpendicular, skew, and transversal
- terms related to angles, including supplementary, complementary, adjacent, vertical, alternate, acute, obtuse, and reflex
- terms related to triangles, including equilateral, isosceles, scalene, right, similarity and congruence
- figures in plane
- simple and closed curves
- the Jordan Curve Theorem
- convex and concave figures
- polygons, circles, and their combinations
- special polygons: equilateral, equiangular, and regular
- classification of triangles: equilateral, isosceles, scalene, right, similar and congruent
- quadrilaterals
- angles of polygons
- figures in space
- simple closed surfaces and polyhedra
- prisms, pyramids, cones, cylinders, spheres, and their combinations
- Euler’s formula for polyhedra
- regular polyhedra and Platonic solids
8. Length, area, volume, and measurement in the metric and imperial systems
- fundamental measurement properties, including covering, congruence, additive, and comparison, to determine length, area, surface area, and volume
- perimeter and area of polygons, circles, triangles, parallelograms, trapezoids, rhombuses, kites, squares, circles, and their combinations
- volume and surface area of prisms, pyramids, cylinders, spheres, cones, and their combinations
- SI units for capacity, volume, mass, weight, and temperature
9. Transformations and coordinate geometry
- coordinate geometry: Cartesian coordinates, midpoint formula, distance formula and its properties
- motion geometry:
- rigid transformations: translations, reflections, rotations, and glide reflections
- combining transformations
- scaling: enlargements and reductions
10. Congruence and similarity
- congruent shapes
- similarity and scale factor
Learning activities
Lectures, discussions, problem solving, group work, assignments
Means of assessment
Assessment will be in accordance with the Douglas College Evaluation Policy. The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester. Evaluation will be based on the following:
| Weekly written assignments |
15-30% |
| In-class assignments/group work |
5-10% |
| Participation/attendance |
0-5% |
| Term test(s) |
20-50% |
| Term project |
0-10% |
| Final exam |
30-40% |
| Total |
100% |
A minimum grade of 40% on the final exam is required to receive a grade of D or higher in the course.
Note that calculator use is prohibited for all graded work, including tests and the final exam.
Learning outcomes
Upon successful completion of the course, students will be able to:
- use various processes and strategies for problem solving;
- use Venn diagrams to solve set theory word problems;
- apply and work with the symbols and notations of set theory;
- define and work with subsets, power sets, set complements, relative complements, intersections, unions, and Cartesian products of sets;
- apply properties of set operations including the associative, commutative, and distributive laws, De Morgan’s laws, closure, inverse, and identity properties, and set cardinality;
- state and apply set-theoretic definitions of numbers and their operations;
- demonstrate addition, subtraction, multiplication, and division of whole numbers and integers using models such as sets, the real number line, and arrays;
- distinguish between different interpretations of subtraction and division;
- define unit, prime, and composite numbers;
- use tests for divisibility by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13;
- find the prime factorization of an integer using a factor tree and a division (ladder) model;
- find the greatest common factor (GCF) and least common multiple (LCM) using prime factorization and the Euclidean Algorithm;
- use the relationship between the GCF and LCM to find the LCM given the GCF and vice versa;
- use the Sieve of Eratosthenes algorithm to find primes up to at least 100;
- state and apply the Fundamental Theorem of Arithmetic;
- represent and work with a rational number in different ways, including parts of a whole, relative amount, division of whole numbers, and as a point on the number line;
- convert between decimals and fractions;
- extend the place value system to expand decimals in terms of integer powers of 10;
- demonstrate equivalence and perform addition, subtraction, multiplication, and division of rational numbers and decimals using appropriate models;
- order fractions and decimals in terms of increasing or decreasing size;
- distinguish between terminating, repeating, and non-repeating, non-terminating decimals, and explain how they relate to rational and irrational numbers;
- state and use the Pythagorean Theorem in different contexts;
- state and apply the definition of square roots;
- use the Pythagorean theorem to locate square roots on a number line;
- represent a decimal in scientific notation and vice versa;
- evaluate expressions involving integer exponents using exponential rules;
- solve word problems involving applications of percent, ratio, and fractions;
- use mental arithmetic to estimate the value of calculations involving various number types;
- define and solve problems using terms from informal geometry, such as point, line, ray, line segment, plane, and space;
- state the Jordan Curve Theorem and apply it to analyze the properties of simple and closed curves in the plane;
- define terms related to lines, including collinearity, parallel, perpendicular, skew, and transversal;
- define terms related to angles, including supplementary, complementary, adjacent, vertical, alternate, acute, obtuse, and reflex;
- define terms related to triangles, including equilateral, isosceles, scalene, and right triangles;
- solve problems involving lines, angles, and triangles;
- describe various components and properties of two-dimensional shapes, including sides, vertices, angles, perimeter, and area;
- describe various components and properties of three-dimensional shapes, including faces, edges, vertices, angles, face shapes, surface area, and volume;
- determine similarity and congruence in triangles;
- define and apply formulas for vertex (interior), exterior, and central angles of polygons;
- calculate perimeter and area of polygons, circles, and composite figures;
- calculate surface area and volume of polyhedra, spheres, cones, cylinders, and composite solids;
- apply Euler's Formula when solving problems related to polyhedra;
- state and apply conversions within and between metric and imperial measurements for length, area, volume, and temperature;
- plot points on a Cartesian plane, calculate distances, and midpoints;
- solve problems involving symmetry, reflection, translation, rotation, and combinations of up to two rigid transformations.
Textbook materials
Consult the Douglas College Bookstore for the latest required textbooks and materials. Example textbooks and materials may include:
Wheeler and Wheeler. (Current Edition). Modern Mathematics for Elementary Educators. Kendall-Hunt Publishing.
Musser, Peterson, and Burger. (Current Edition). Mathematics for Elementary Teachers: A Contemporary Approach. Wiley Publishing.
Your instructor will inform you if additional materials such as a ruler, straightedge, or adjustable compass are required.