Math Games

Represent and describe whole numbers to 1 000 000.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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.
- Read a given numeral without using the word “and,” e.g., 574 321 is five hundred seventy-four thousand three hundred twenty-one, NOT five hundred AND seventy-four thousand three hundred AND twenty one. Note: The word “and” is reserved for reading decimal numbers.

Compare and order decimals (to thousandths) by using:
• Benchmarks
• place value
• equivalent decimals.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Order a given set of decimals by placing them on a number line that contains benchmarks, 0.0, 0.5, 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.

Demonstrate an understanding of addition and subtraction of decimals (limited to thousandths).
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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 including:
• front-end rounding
• compensation
• compatible numbers
in problem-solving contexts.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Provide a context for when estimation is used to:
• | make predictions
• | check 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, such as:
• skip counting from a known fact
• using doubling or halving
• using patterns in the 9s facts
• using repeated doubling or halving
to determine answers for basic multiplication facts to 81 and related division facts.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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.
- Explain why division by zero is not possible or undefined, e.g., 8 ÷ 0.
- Recall multiplication facts to 81 and related division facts.

Apply mental mathematics strategies for multiplication, such as:
• annexing then adding zero
• halving and doubling
• using the distributive property.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Determine the products when one factor is a multiple of 10, 100 or 1000 by annexing zero or tacking on zeros, e.g., for 3 × 200 think 3 × 2 hundreds which equals six hundreds (600).
- 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 involving multiplying factors that are close to multiples of 10, e.g., 98 × 7 = (100 × 7) – (2 × 7)

Demonstrate an understanding of multiplication (2-digit by 2-digit) to solve problems.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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.

Demonstrate, with and without concrete materials, an understanding of division (3- digit by 1-digit) and interpret remainders to solve problems.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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.

Demonstrate an understanding of fractions by using concrete and pictorial representations to:
• create sets of equivalent fractions
• compare fractions with like and unlike denominators.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Create a set of equivalent fractions and explain why there are many equivalent fractions for any given fraction using concrete materials.
- 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.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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 (to thousandths).
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Write a given decimal in fractional 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.

Determine the pattern rule to make predictions about subsequent elements.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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.

Solve problems involving single-variable, one-step equations with whole number coefficients and whole number solutions.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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.
- Express a given problem as an equation where the unknown is represented by a letter variable.
- Create a problem for a given equation with one unknown.
- 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.

Design and construct different rectangles given either perimeter or area, or both (whole numbers) and draw conclusions.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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.
- Illustrate that for any given perimeter, the square or shape closest to a square will result in the greatest area.
- Illustrate that for any given perimeter, the rectangle with the smallest possible width will result in the least area.
- 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.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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., ruler.
- Show that 1000 millimetres is equivalent to 1 metre using concrete materials, e.g., 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 by using referents for cm³ or m³
• measuring and recording volume (cm³ or m³)
• constructing rectangular prisms for a given volume.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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 rectangular prism for a given volume.
- Explain that many rectangular prisms are possible for a given volume by constructing more than one 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 by using referents for mL or L
• measuring and recording capacity (mL or L).

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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 which capacity unit is represented by 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 or horizontal.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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 or 3-D objects that have edges, faces and sides 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; squares
• trapezoids
• parallelograms
• rhombuses
according to their attributes.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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.

Perform a single transformation (translation, rotation, or reflection) of a 2-D shape (with and without technology) and draw and describe the image.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Translate a given 2-D shape horizontally, vertically or diagonally, and describe the position and orientation of the image.
- Rotate a given 2-D shape about a point, and describe the position and orientation of the image.
- Reflect a given 2-D shape in a line of reflection, and describe the position and orientation of the image.
- Perform a transformation of a given 2-D shape by following instructions.
- 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 and describe the direction of the turn (clockwise or counterclockwise), the fraction of the turn and point of rotation.
- 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.

Identify a single transformation, including a translation, rotation, and reflection of 2-D shapes.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Provide an example of a translation, a rotation and a reflection.
- Identify a given transformation as a translation, rotation or reflection.
- Describe a given rotation by the direction of the turn (clockwise or counter-clockwise).

Differentiate between first-hand and second-hand data.
• Achievement Indicators
- 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.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- 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.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Provide examples of events that are impossible, possible or certain from personal contexts.
- Classify the likelihood of a single outcome occurring in a probability experiment as impossible, possible or certain.
- Plot the likelihood of a single outcome occurring in a probability experiment along a continuum.
- 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.
- Identify outcomes from a given probability experiment which 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 as 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.

Compare the likelihood of two possible outcomes occurring using words, such as:
• less likely
• equally likely
• more likely.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Provide examples of events that are impossible, possible or certain from personal contexts.
- Classify the likelihood of a single outcome occurring in a probability experiment as impossible, possible or certain.
- Plot the likelihood of a single outcome occurring in a probability experiment along a continuum.
- 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.
- Identify outcomes from a given probability experiment which 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 as 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.