Math Games

Determine the pattern rule to make predictions about subsequent elements.
• Achievement Indicators
- Extend a given pattern with and without concrete materials, and explain how each element differs from the proceeding 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
- Express a given problem in context 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.
- Create a problem in context for a given equation.

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
- 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
- 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.
- 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.

• Achievement Indicators
- 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.

Design and construct different rectangles given either perimeter or area, or both (whole numbers) and draw conclusions.
• Achievement Indicators
- 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
- 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
- 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
- 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
• horizontal.

• Achievement Indicators
- 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
- 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
- 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
- Provide an example of a translation, a rotation and a reflection.
- Identify a given single transformation as a translation, rotation or reflection.
- Describe a given rotation by the direction of the turn (clockwise or counterclockwise).

Represent and describe whole numbers to 1 000 000.
• Achievement Indicators
- 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.

• Achievement Indicators
- 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
- 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
- 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
- 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 81and related division facts.

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

• Achievement Indicators
- Determine the products when one factor is a multiple of 10, 100 or 1000 by annexing zero or adding zeros, 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 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
- 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
- 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
- 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
- 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
- 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 .