Embedded Files
Overview of Math Program
The mathematical processes that support effective learning in mathematics are as follows: • problem solving • reasoning and proving • reflecting • connecting • communicating • representing • selecting tools and strategies .
Problem Solving
Problem solving is central to doing mathematics. By learning to solve problems and by learning through problem solving, students are given, and create, numerous opportunities to connect mathematical ideas and to develop conceptual understanding. Problem solving forms the basis of effective mathematics programs that place all students’ experiences and queries at the centre. Thus, problem solving should be the mainstay of mathematical instruction. It is considered an essential process through which all students are able to achieve the expectations in mathematics and is an integral part of the Ontario mathematics curriculum. Problem solving: • increases opportunities for the use of critical thinking skills (e.g., selecting appropriate tools and strategies, estimating, evaluating, classifying, assuming, recognizing relationships, conjecturing, posing questions, offering opinions with reasons, making judgements) to develop mathematical reasoning; • helps all students develop a positive math identity; • allows all students to use the rich prior mathematical knowledge they bring to school; • helps all students make connections among mathematical knowledge, concepts, and skills, and between the classroom and situations outside the classroom; • promotes the collaborative sharing of ideas and strategies and promotes talking about mathematics; • facilitates the use of creative-thinking skills when developing solutions and approaches; • helps students find enjoyment in mathematics and become more confident in their ability to do mathematics. Most importantly, when problem solving is in a mathematical context relevant to students’ experiences and derived from their own problem posing, it furthers their understanding of mathematics and develops their math agency.
Problem-Solving Strategies.
Problem-solving strategies are methods that can be used to solve problems of various types. Common problem-solving strategies include the following: simulating; making a model, picture, or diagram; looking for a pattern; guessing and checking; making an organized list; making a table or chart; solving a simpler version of the problem (e.g., with smaller numbers); working backwards; and using logical reasoning. Teachers can support all students as they develop their use of these strategies by engaging with solving various kinds of problems – instructional problems, routine problems, and non-routine problems. As students develop this repertoire over time, they become more confident in posing their own questions, more mature in their problem-solving skills, and more flexible in using appropriate strategies when faced with new problem-solving situations.
Reasoning and Proving
Reasoning and proving are a mainstay of mathematics and involves students using their understanding of mathematical knowledge, concepts, and skills to justify their thinking. Proportional reasoning, algebraic reasoning, spatial reasoning, statistical reasoning, and probabilistic reasoning are all forms of mathematical reasoning. Students also use their understanding of numbers and operations, geometric properties, and measurement relationships to reason through solutions to problems. Teachers can provide all students with learning opportunities where they must form mathematical conjectures and then test or prove them to see if they hold true. Initially, students may rely on the viewpoints of others to justify a choice or an approach to a solution. As they develop their own reasoning skills, they will begin to justify or prove their solutions by providing evidence.
Reflecting
Students reflect when they are working through a problem to monitor their thought process, to identify what is working and what is not working, and to consider whether their approach is appropriate or whether there may be a better approach. Students also reflect after they have solved a problem by considering the reasonableness of their answer and whether adjustments need to be made. Teachers can support all students as they develop their reflecting and metacognitive skills by asking questions that have them examine their thought processes, as well as questions that have them think about other students’ thought processes. Students can also reflect on how their new knowledge can be applied to past and future problems in mathematics.
Connecting
Experiences that allow all students to make connections – to see, for example, how knowledge, concepts, and skills from one strand of mathematics are related to those from another – will help them to grasp general mathematical principles. Through making connections, students learn that mathematics is more than a series of isolated skills and concepts and that they can use their learning in one area of mathematics to understand another. Seeing the relationships among procedures and concepts also helps develop mathematical understanding. The more connections students make, the deeper their understanding, and understanding, in turn, helps them to develop their sense of identity. In addition, making connections between the mathematics they learn at school and its applications in their everyday lives not only helps students understand mathematics but also allows them to understand how useful and relevant it is in the world beyond the classroom. These kinds of connections will also contribute to building students.
Communicating Communication is an essential process in learning mathematics. Students communicate for various purposes and for different audiences, such as the teacher, a peer, a group of students, the whole class, a community member, or their family. They may use oral, visual, written, or gestural communication. Communication also involves active and respectful listening. Teachers provide differentiated opportunities for all students to acquire the language of mathematics, developing their communication skills, which include expressing, understanding, and using appropriate mathematical terminology, symbols, conventions, and models. For example, teachers can ask students to: • share and clarify their ideas, understandings, and solutions; • create and defend mathematical arguments; • provide meaningful descriptive feedback to peers; and • pose and ask relevant questions. Effective classroom communication requires a supportive, safe, and respectful environment in which all members of the class feel comfortable and valued when they speak and when they question, react to, and elaborate on the statements of their peers and the teacher.
Representing
Students represent mathematical ideas and relationships and model situations using tools, pictures, diagrams, graphs, tables, numbers, words, and symbols. Teachers recognize and value the varied representations students begin learning with, as each student may have different prior access to and experiences with mathematics. While encouraging student engagement and affirming the validity of their representations, teachers help students reflect on the appropriateness of their representations and refine them. Teachers support students as they make connections among various representations that are relevant to both the student and the audience they are communicating with, so that all students can develop a deeper understanding of mathematical concepts and relationships. All students are supported as they use the different representations appropriately and as needed to model situations, solve problems, and communicate their thinking.
A: Social-Emotional Learning (SEL) Skills in Mathematics and the Mathematical Processes
1. express and manage their feelings, and show understanding of the feelings of others, as they engage positively in mathematics activities
2. work through challenging math problems, understanding that their resourcefulness in using various strategies to respond to stress is helping them build personal resilience
3. recognize that testing out different approaches to problems and learning from mistakes is an important part of the learning process, and is aided by a sense of optimism and hope
4. work collaboratively on math problems – expressing their thinking, listening to the thinking of others, and practising inclusivity – and in that way fostering healthy relationships
5. see themselves as capable math learners, and strengthen their sense of ownership of their learning, as part of their emerging sense of identity and belonging
6. make connections between math and everyday contexts to help them make informed judgements and decisions
B1. Number Sense
Whole Numbers
B1.1
read, represent, compose, and decompose whole numbers up to and including 10 000, using appropriate tools and strategies, and describe various ways they are used in everyday life
B1.2
compare and order whole numbers up to and including 10 000, in various contexts
B1.3
round whole numbers to the nearest ten, hundred, or thousand, in various contexts
Fractions and Decimals
B1.4
represent fractions from halves to tenths using drawings, tools, and standard fractional notation, and explain the meanings of the denominator and the numerator
B1.5
use drawings and models to represent, compare, and order fractions representing the individual portions that result from two different fair-share scenarios involving any combination of 2, 3, 4, 5, 6, 8, and 10 sharers
B1.6
count to 10 by halves, thirds, fourths, fifths, sixths, eighths, and tenths, with and without the use of tools
B1.7
read, represent, compare, and order decimal tenths, in various contexts
B1.8
round decimal numbers to the nearest whole number, in various contexts
B1.9
describe relationships and show equivalences among fractions and decimal tenths, in various contexts
Properties and Relationships
B2.1
use the properties of operations, and the relationships between addition, subtraction, multiplication, and division, to solve problems involving whole numbers, including those requiring more than one operation, and check calculations.
Math Facts
B2.2
recall and demonstrate multiplication facts for 1 × 1 to 10 × 10, and related division facts
Mental Math
B2.3
use mental math strategies to multiply whole numbers by 10, 100, and 1000, divide whole numbers by 10, and add and subtract decimal tenths, and explain the strategies used.
Addition and Subtraction
B2.4
represent and solve problems involving the addition and subtraction of whole numbers that add up to no more than 10 000 and of decimal tenths, using appropriate tools and strategies, including algorithms.
Multiplication and Division
B2.5
represent and solve problems involving the multiplication of two- or three-digit whole numbers by one-digit whole numbers and by 10, 100, and 1000, using appropriate tools, including arrays
B2.6
represent and solve problems involving the division of two- or three-digit whole numbers by one-digit whole numbers, expressing any remainder as a fraction when appropriate, using appropriate tools, including arrays
B2.7
represent the relationship between the repeated addition of a unit fraction and the multiplication of that unit fraction by a whole number, using tools, drawings, and standard fractional notation
B2.8
show simple multiplicative relationships involving whole-number rates, using various tools and drawings.
C. Algebra
Patterns
C1.1
identify and describe repeating and growing patterns, including patterns found in real-life contexts
C1.2
create and translate repeating and growing patterns using various representations, including tables of values and graphs
C1.3
determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in repeating and growing patterns
C1.4
create and describe patterns to illustrate relationships among whole numbers and decimal tenths,
Variables
C2.1
identify and use symbols as variables in expressions and equations
Equalities and Inequalities
C2.2
solve equations that involve whole numbers up to 50 in various contexts, and verify solutions
C2.3
solve inequalities that involve addition and subtraction of whole numbers up to 20, and verify and graph the solutions.
Coding Skills
C3.1
solve problems and create computational representations of mathematical situations by writing and executing code, including code that involves sequential, concurrent, repeating, and nested events
C3.2
read and alter existing code, including code that involves sequential, concurrent, repeating, and nested events, and describe how changes to the code affect the outcomes.
D1. Data Literacy
Data Collection and Organization
D1.1
describe the difference between qualitative and quantitative data, and describe situations where each would be used
D1.2
collect data from different primary and secondary sources to answer questions of interest that involve comparing two or more sets of data, and organize the data in frequency tables and stem-and-leaf plots.
Data Visualization
D1.3
select from among a variety of graphs, including multiple-bar graphs, the type of graph best suited to represent various sets of data; display the data in the graphs with proper sources, titles, and labels, and appropriate scales; and justify their choice of graphs
D1.4
create an infographic about a data set, representing the data in appropriate ways, including in frequency tables, stem-and-leaf plots, and multiple-bar graphs, and incorporating any other relevant information that helps to tell a story about the data .
Data Analysis
D1.5
determine the mean and the median and identify the mode(s), if any, for various data sets involving whole numbers, and explain what each of these measures indicates about the data
D1.6
analyse different sets of data presented in various ways, including in stem-and-leaf plots and multiple-bar graphs, by asking and answering questions about the data and drawing conclusions, then make convincing arguments and informed decisions.
Probability
D2.1
use mathematical language, including the terms “impossible”, “unlikely”, “equally likely”, “likely”, and “certain”, to describe the likelihood of events happening, represent this likelihood on a probability line, and use it to make predictions and informed decisions
D2.2
make and test predictions about the likelihood that the mean, median, and mode(s) of a data set will be the same for data collected from different populations.
E. Spatial Sense
Geometric Reasoning
E1.1
identify geometric properties of rectangles, including the number of right angles, parallel and perpendicular sides, and lines of symmetry
Location and Movement
E1.2
plot and read coordinates in the first quadrant of a Cartesian plane, and describe the translations that move a point from one coordinate to another
E1.3
describe and perform translations and reflections on a grid, and predict the results of these transformations.
The Metric System
E2.1
explain the relationships between grams and kilograms as metric units of mass, and between litres and millilitres as metric units of capacity, and use benchmarks for these units to estimate mass and capacity
E2.2
use metric prefixes to describe the relative size of different metric units, and choose appropriate units and tools to measure length, mass, and capacity.
Time
E2.3
solve problems involving elapsed time by applying the relationships between different units of time.
Angles
E2.4
identify angles and classify them as right, straight, acute, or obtuse.
Area
E2.5
use the row and column structure of an array to measure the areas of rectangles and to show that the area of any rectangle can be found by multiplying its side lengths
E2.6
apply the formula for the area of a rectangle to find the unknown measurement when given two of the three.
F1. Money and Finances
Money Concepts
F1.1
identify various methods of payment that can be used to purchase goods and services
F1.2
estimate and calculate the cost of transactions involving multiple items priced in whole-dollar amounts, not including sales tax, and the amount of change needed when payment is made in cash, using mental math.
Financial Management
F1.3
explain the concepts of spending, saving, earning, investing, and donating, and identify key factors to consider when making basic decisions related to each
F1.4
explain the relationship between spending and saving, and describe how spending and saving behaviours may differ from one person to another
Consumer and Civic Awareness
F1.5
describe some ways of determining whether something is reasonably priced and therefore a good purchase.
Report abuse