Math Games

Determine the pattern rule to make predictions about subsequent elements.
• Extend a given pattern with and without concrete materials, and explain how each element differs from the preceding one.
• Describe, orally or in writing, a given pattern, using mathematical language such as one more, one less, five more.
• Write a mathematical expression to represent a given pattern, such as r + 1, r – 1, r + 5.
• Describe the relationship in a given table or chart, using a mathematical expression.
• Determine and explain why a given number is or is not the next element in a pattern.
• Predict subsequent elements in a given pattern.
• Solve a given problem by using a pattern rule to determine subsequent elements.
• Represent a given pattern visually to verify predictions.

Express a given problem as an equation in which a letter variable is used to represent an unknown number (limited to whole numbers).
• Explain the purpose of the letter variable in a given addition, subtraction, multiplication or division equation with one unknown; e.g., 36 ÷ n = 6.
• Express a given pictorial or concrete representation of an equation in symbolic form.
• Identify the unknown in a problem, and represent the problem with an equation.
• Create a problem for a given equation with one unknown.

Solve problems involving single-variable, one-step equations with whole number coefficients and whole number solutions.
• Express a given problem as an equation where the unknown is represented by a letter variable.
• Solve a given single-variable equation with the unknown in any of the terms; e.g., n + 2 = 5, 4 + a = 7, 6 = r – 2, 10 = 2c.
• Identify the unknown in a problem; represent the problem with an equation; and solve the problem concretely, pictorially or symbolically.
• Create a problem for a given equation.

Differentiate between first-hand and second-hand data.
• Explain the difference between first-hand and second-hand data.
• Formulate a question that can best be answered using first-hand data, and explain why.
• Formulate a question that can best be answered using second-hand data, and explain why.
• Find examples of second-hand data in print and electronic media, such as newspapers, magazines and the Internet.

Construct and interpret double bar graphs to draw conclusions.
• Determine the attributes (title, axes, intervals and legend) of double bar graphs by comparing a given set of double bar graphs.
• Represent a given set of data by creating a double bar graph, label the title and axes, and create a legend without the use of technology.
• Draw conclusions from a given double bar graph to answer questions.
• Provide examples of double bar graphs used in a variety of print and electronic media, such as newspapers, magazines and the Internet.
• Solve a given problem by constructing and interpreting a double bar graph.

Describe the likelihood of a single outcome occurring, using words such as:
• impossible.
• possible.
• certain.
• Provide examples of events from personal contexts that are impossible, possible or certain.
• Classify the likelihood of a single outcome occurring in a probability experiment as impossible, possible or certain.
• Design and conduct a probability experiment in which the likelihood of a single outcome occurring is impossible, possible or certain.
• Conduct a given probability experiment a number of times, record the outcomes, and explain the results.

Compare the likelihood of two possible outcomes occurring, using words such as:
• less likely.
• equally likely.
• more likely.
• Identify outcomes from a given probability experiment that are less likely, equally likely or more likely to occur than other outcomes.
• Design and conduct a probability experiment in which one outcome is less likely to occur than the other outcome.
• Design and conduct a probability experiment in which one outcome is equally likely to occur as the other outcome.
• Design and conduct a probability experiment in which one outcome is more likely to occur than the other outcome.

Identify 90º angles.
• Provide examples of 90º angles in the environment.
• Sketch 90º angles without the use of a protractor.
• Label a 90º angle, using a symbol.

Design and construct different rectangles, given either perimeter or area, or both (whole numbers), and make generalizations.
• Construct or draw two or more rectangles for a given perimeter in a problem-solving context.
• Construct or draw two or more rectangles for a given area in a problem-solving context.
• Determine the shape that will result in the greatest area for any given perimeter.
• Determine the shape that will result in the least area for any given perimeter.
• Provide a real-life context for when it is important to consider the relationship between area and perimeter.

Demonstrate an understanding of measuring length (mm) by:
• selecting and justifying referents for the unit mm.
• modelling and describing the relationship between mm and cm units, and between mm and m units.
• Provide a referent for one millimetre, and explain the choice.
• Provide a referent for one centimetre, and explain the choice.
• Provide a referent for one metre, and explain the choice.
• Show that 10 millimetres is equivalent to 1 centimetre, using concrete materials; e.g., a ruler.
• Show that 1000 millimetres is equivalent to 1 metre, using concrete materials; e.g., a metre stick.
• Provide examples of when millimetres are used as the unit of measure.

Demonstrate an understanding of volume by:
• selecting and justifying referents for cm³ or m³ units.
• estimating volume, using referents for cm³ or m³.
• measuring and recording volume (cm³ or m³).
• constructing right rectangular prisms for a given volume.
• Identify the cube as the most efficient unit for measuring volume, and explain why.
• Provide a referent for a cubic centimetre, and explain the choice.
• Provide a referent for a cubic metre, and explain the choice.
• Determine which standard cubic unit is represented by a given referent.
• Estimate the volume of a given 3-D object, using personal referents.
• Determine the volume of a given 3-D object, using manipulatives, and explain the strategy.
• Construct a right rectangular prism for a given volume.
• Construct more than one right rectangular prism for the same given volume.

Demonstrate an understanding of capacity by:
• describing the relationship between mL and L.
• selecting and justifying referents for mL or L units.
• estimating capacity, using referents for mL or L.
• measuring and recording capacity (mL or L).
• Demonstrate that 1000 millilitres is equivalent to 1 litre by filling a 1 litre container using a combination of smaller containers.
• Provide a referent for a litre, and explain the choice.
• Provide a referent for a millilitre, and explain the choice.
• Determine the capacity unit of a given referent.
• Estimate the capacity of a given container, using personal referents.
• Determine the capacity of a given container, using materials that take the shape of the inside of the container (e.g., a liquid, rice, sand, beads), and explain the strategy.

Describe and provide examples of edges and faces of 3-D objects, and sides of 2-D shapes that are:
• parallel.
• intersecting.
• perpendicular.
• vertical.
• horizontal.
• Identify parallel, intersecting, perpendicular, vertical and horizontal edges and faces on 3-D objects.
• Identify parallel, intersecting, perpendicular, vertical and horizontal sides on 2-D shapes.
• Provide examples from the environment that show parallel, intersecting, perpendicular, vertical and horizontal line segments.
• Find examples of edges, faces and sides that are parallel, intersecting, perpendicular, vertical and horizontal in print and electronic media, such as newspapers, magazines and the Internet.
• Draw 2-D shapes that have sides that are parallel, intersecting, perpendicular, vertical or horizontal.
• Draw 3-D objects that have edges and faces that are parallel, intersecting, perpendicular, vertical or horizontal.
• Describe the faces and edges of a given 3-D object, using terms such as parallel, intersecting, perpendicular, vertical or horizontal.
• Describe the sides of a given 2-D shape, using terms such as parallel, intersecting, perpendicular, vertical or horizontal.

Identify and sort quadrilaterals, including:
• rectangles according to their attributes.
• squares according to their attributes.
• trapezoids according to their attributes.
• parallelograms according to their attributes.
• rhombuses according to their attributes.
• Identify and describe the characteristics of a pre-sorted set of quadrilaterals.
• Sort a given set of quadrilaterals, and explain the sorting rule.
• Sort a given set of quadrilaterals according to the lengths of the sides.
• Sort a given set of quadrilaterals according to whether or not opposite sides are parallel.

Identify and describe a single transformation, including a translation, rotation and reflection of 2-D shapes.
• Provide an example of a translation, rotation and reflection.
• Identify a given single transformation as a translation, rotation or reflection.
• Describe a given rotation about a vertex by the direction of the turn (clockwise or counterclockwise).
• Describe a given reflection by identifying the line of reflection and the distance of the image from the line of reflection.
• Describe a given translation by identifying the direction and magnitude of the movement.

Perform, concretely, a single transformation (translation, rotation or reflection) of a 2-D shape, and draw the image.
• Translate a given 2-D shape horizontally, vertically or diagonally, and draw the resultant image.
• Rotate a given 2-D shape about a vertex, and describe the direction of rotation (clockwise or counterclockwise) and the fraction of the turn (limited to ¼, ½, ¾ or full turn).
• Reflect a given 2-D shape across a line of reflection, and draw the resultant image.
• Draw a 2-D shape, translate the shape, and record the translation by describing the direction and magnitude of the movement.
• Draw a 2-D shape, rotate the shape about a vertex, and describe the direction of the turn (clockwise or counterclockwise) and the fraction of the turn (limited to ¼, ½, ¾ or full turn).
• Draw a 2-D shape, reflect the shape, and identify the line of reflection and the distance of the image from the line of reflection.
• Predict the result of a single transformation of a 2-D shape, and verify the prediction.

Represent and describe whole numbers to 1 000 000.
• Write a given numeral, using proper spacing without commas; e.g., 934 567.
• Describe the pattern of adjacent place positions moving from right to left.
• Describe the meaning of each digit in a given numeral.
• Provide examples of large numbers used in print or electronic media.
• Express a given numeral in expanded notation; e.g., 45 321 = (4 × 10 000) + (5 × 1000) + (3 × 100) + (2 × 10) + (1 × 1) or 40 000 + 5000 + 300 + 20 + 1.
• Write the numeral represented by a given expanded notation.

Compare and order decimals (to thousandths) by using:
• benchmarks.
• place value.
• equivalent decimals.
• Order a given set of decimals by placing them on a number line that contains the benchmarks 0.0, 0.5 and 1.0.
• Order a given set of decimals including only tenths, using place value.
• Order a given set of decimals including only hundredths, using place value.
• Order a given set of decimals including only thousandths, using place value.
• Explain what is the same and what is different about 0.2, 0.20 and 0.200.
• Order a given set of decimals including tenths, hundredths and thousandths, using equivalent decimals; e.g., 0.92, 0.7, 0.9, 0.876, 0.925 in order is: 0.700, 0.876, 0.900, 0.920, 0.925.

Demonstrate an understanding of addition and subtraction of decimals (limited to thousandths).
• Place the decimal point in a sum or difference, using front-end estimation; e.g., for 6.3 + 0.25 + 306.158, think 6 + 306, so the sum is greater than 312.
• Correct errors of decimal point placements in sums and differences without using paper and pencil.
• Explain why keeping track of place value positions is important when adding and subtracting decimals.
• Predict sums and differences of decimals, using estimation strategies.
• Solve a given problem that involves addition and subtraction of decimals, limited to thousandths.

Use estimation strategies in problem-solving contexts.
• Provide a context for when estimation is used to:
- make predictions.
- check the reasonableness of an answer.
- determine approximate answers.
• Describe contexts in which overestimating is important.
• Determine the approximate solution to a given problem not requiring an exact answer.
• Estimate a sum or product, using compatible numbers.
• Estimate the solution to a given problem, using compensation, and explain the reason for compensation.
• Select and use an estimation strategy for a given problem.
• Apply front-end rounding to estimate:
- sums; e.g., 253 + 615 is more than 200 + 600 = 800.
- differences; e.g., 974 – 250 is close to 900 – 200 = 700.
- products; e.g., the product of 23 × 24 is greater than 20 × 20 (400) and less than 25 × 25 (625).
- quotients; e.g., the quotient of 831 ÷ 4 is greater than 800 ÷ 4 (200).

Apply mental mathematics strategies and number properties in order to understand and recall basic multiplication facts (multiplication tables) to 81 and related division facts.
• Describe the mental mathematics strategy used to determine a given basic fact, such as:
- skip count up by one or two groups from a known fact; e.g., if 5 × 7 = 35, then 6 × 7 is equal to 35 + 7 and 7 × 7 is equal to 35 + 7 + 7.
- skip count down by one or two groups from a known fact; e.g., if 8 × 8 = 64, then 7 × 8 is equal to 64 – 8 and 6 × 8 is equal to 64 – 8 – 8.
- doubling; e.g., for 8 × 3 think 4 × 3 = 12, and 8 × 3 = 12 + 12.
- patterns when multiplying by 9; e.g., for 9 × 6, think 10 × 6 = 60, and 60 – 6 = 54; for 7 × 9, think 7 × 10 = 70, and 70 – 7 = 63.
- repeated doubling; e.g., if 2 × 6 is equal to 12, then 4 × 6 is equal to 24 and 8 × 6 is equal to 48.
- repeated halving; e.g., for 60 ÷ 4, think 60 ÷ 2 = 30 and 30 ÷ 2 = 15.
• Explain why multiplying by zero produces a product of zero (zero property of multiplication).
• Explain why division by zero is not possible or is undefined; e.g., 8 ÷ 0.
• Determine, with confidence, answers to multiplication facts to 81 and related division facts.
• Demonstrate understanding, recall/memorization and application of multiplication and related division facts to 9 × 9.

Apply mental mathematics strategies for multiplication.
• Determine the products when one factor is a multiple of 10, 100 or 1000 by annexing and adding zero; e.g., for 3 × 200 think 3 × 2 and then add two zeros.
• Apply halving and doubling when determining a given product; e.g., 32 × 5 is the same as 16 × 10.
• Apply the distributive property to determine a given product that involves multiplying factors that are close to multiples of 10; e.g., 98 × 7 = (100 × 7) – (2 × 7).

Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems.
• Illustrate partial products in expanded notation for both factors; e.g., for 36 × 42, determine the partial products for (30 + 6) × (40 + 2).
• Represent both 2-digit factors in expanded notation to illustrate the distributive property; e.g., to determine the partial products of 36 × 42, (30 + 6) × (40 + 2) = 30 × 40 + 30 × 2 + 6 × 40 + 6 × 2 = 1200 + 60 + 240 + 12 = 1512.
• Model the steps for multiplying 2-digit factors, using an array and base ten blocks, and record the process symbolically.
• Describe a solution procedure for determining the product of two given 2-digit factors, using a pictorial representation such as an area model.
• Solve a given multiplication problem in context, using personal strategies, and record the process.
• Refine personal strategies to increase their efficiency.
• Create and solve a multiplication problem, and record the process.
• Solve a given problem using the standard/traditional multiplication algorithm.

Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems.
• Model the division process as equal sharing, using base ten blocks, and record it symbolically.
• Explain that the interpretation of a remainder depends on the context:
- ignore the remainder; e.g., making teams of 4 from 22 people.
- round up the quotient; e.g., the number of five passenger cars required to transport 13 people.
- express remainders as fractions; e.g., five apples shared by two people.
- express remainders as decimals; e.g., measurement and money.
• Solve a given division problem in context, using personal strategies, and record the process.
• Refine personal strategies to increase their efficiency.
• Create and solve a division problem, and record the process.
• Solve a given problem using the standard/traditional division algorithm.

Demonstrate an understanding of fractions by using concrete, pictorial and symbolic representations to:
• create sets of equivalent fractions.
• compare fractions with like and unlike denominators.
• Create a set of equivalent fractions; and explain, using concrete materials, why there are many equivalent fractions for any given fraction.
• Model and explain that equivalent fractions represent the same quantity.
• Determine if two given fractions are equivalent, using concrete materials or pictorial representations.
• Formulate and verify a rule for developing a set of equivalent fractions.
• Identify equivalent fractions for a given fraction.
• Compare two given fractions with unlike denominators by creating equivalent fractions.
• Position a given set of fractions with like and unlike denominators on a number line, and explain strategies used to determine the order.

Describe and represent decimals (tenths, hundredths, thousandths), concretely, pictorially and symbolically.
• Write the decimal for a given concrete or pictorial representation of part of a set, part of a region or part of a unit of measure.
• Represent a given decimal, using concrete materials or a pictorial representation.
• Represent an equivalent tenth, hundredth or thousandth for a given decimal, using a grid.
• Express a given tenth as an equivalent hundredth and thousandth.
• Express a given hundredth as an equivalent thousandth.
• Describe the value of each digit in a given decimal.

Relate decimals to fractions and fractions to decimals (to thousandths).
• Write a given decimal in fraction form.
• Write a given fraction with a denominator of 10, 100 or 1000 as a decimal.
• Express a given pictorial or concrete representation as a fraction or decimal; e.g., 250 shaded squares on a thousandth grid can be expressed as 0.250 or 250/1000.