Math Games

Represent and describe whole numbers to 1 000 000.
• Write a numeral using proper spacing without commas (e.g., 934 567 and not 934,567).
• Describe the pattern of adjacent place positions moving from right to left.
• Describe the meaning of each digit in a 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 in expanded notation.

Compare and order decimals (tenths, hundredths, thousandths) by using
• benchmarks
• place value
• equivalent decimals

• Order a set of decimals by placing them on a number line (vertical or horizontal) that contains benchmarks, 0.0, 0.5, 1.0.
• Order a set of decimals including only tenths using place value.
• Order a set of decimals including only hundredths using place value.
• Order a 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 set of decimals including tenths, hundredths, and thousandths using equivalent decimals.

Demonstrate an understanding of addition and subtraction of decimals (to thousandths), concretely, pictorially, and symbolically, by
• using personal strategies
• using the standard algorithms
• using estimation
• solving problems

• Estimate 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) and place the decimal in the appropriate place.
• 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 problem that involves addition and subtraction of decimals, to thousandths.
• Model and explain the relationship that exists between an algorithm, place value, and number properties.
• Determine the sum and difference using the standard algorithms of vertical addition and subtraction. (Numbers are arranged vertically with corresponding place value digits aligned.)
• Refine personal strategies, such as mental math, to increase efficiency when appropriate (e.g., 3.36 + 9.65 think, 0.35 + 0.65 = 1.00, therefore, 0.36 + 0.65 = 1.01 and 3 + 9 = 12 for a total of 13.01)

Apply estimation strategies, including
• front-end rounding
• compensation
• compatible numbers
in problem-solving contexts.

• 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 problem not requiring an exact answer.
• Estimate a sum or product using compatible numbers.
• Estimate the solution to a problem using compensation, and explain the reason for compensation.
• Select and use an estimation strategy to solve a 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 or 400 and less than 25 × 25 or 625)
- quotients (e.g., the quotient of 831 ÷ 4 is greater than 800 ÷ 4 or 200)

Apply mental math strategies to determine multiplication and related division facts to 81 (9 x 9).
• Describe the mental mathematics strategy used to determine a 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)
- halving/doubling (e.g., for 8 × 3 think 4 × 6 = 24)
- use patterns when multiplying by 9 (e.g., for 9 × 6, think 10 × 6 = 60, then 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)
- relating multiplication to division facts (e.g., for 7 x 8, think 56 ÷ 7 = )
- use multiplication facts that are squares (1 x 1, 2 x 2, up to 9 x 9)
• Refine personal strategies to increase efficiency (e.g., for 7 x 6, use known square 6 x 6 + 6 instead of repeated addition 6 + 6 + 6 + 6 + 6 + 6 + 6).

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

• 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 product (e.g., 32 × 5 is the same as 16 × 10).
• Apply the distributive property to determine a product involving multiplying factors that are close to multiples of 10 [e.g., 98 × 7 = (100 × 7) – (2 × 7)].

Demonstrate an understanding of multiplication (1- and 2-digit multipliers and up to 4-digit multiplicands), concretely, pictorially, and symbolically, by
• using personal strategies
• using the standard algorithm
• estimating products
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-10 blocks, and record the process symbolically.
• Describe a solution procedure for determining the product of two 2-digit factors using a pictorial representation, such as an area model.
• Model and explain the relationship that exists between an algorithm, place value, and number properties.
• Determine products using the standard algorithm of vertical multiplication. (Numbers arranged vertically and multiplied using single digits which are added to form a final product.)
• Solve a multiplication problem in context using personal strategies, and record the process.
• Refine personal strategies such as mental math strategies to increase efficiency when appropriate [e.g., 16 x 25 think 4 x (4 x 25) = 400].

Demonstrate an understanding of division (1- and 2-digit divisors and up to 4-digit dividends), concretely, pictorially, and symbolically, and interpret remainders by
• using personal strategies
• using the standard algorithm
• estimating quotients
to solve problems.

• Model the division process as equal sharing using base-10 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 or money)
• Model and explain the relationship that exists between algorithm, place value, and number properties.
• Determine quotients using the standard algorithm of long division. (The multiples of the divisor are subtracted from the dividend.)
• Solve a division problem in context using personal strategies, and record the process.
• Refine personal strategies such as mental math strategies to increase efficiency when appropriate (e.g., 860 ÷ 2 think 86 ÷ 2 = 43 then 860 ÷ 2 is 430).

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

• Create a set of equivalent fractions and explain why there are many equivalent fractions for any fraction using concrete materials.
• Model and explain that equivalent fractions represent the same quantity.
• Determine if two 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 fraction.
• Compare two fractions with unlike denominators by creating equivalent fractions.
• Position a set of fractions with like and unlike denominators on a number line (vertical or horizontal), and explain strategies used to determine the order

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

Relate decimals to fractions (tenths, hundredths, thousandths).
• Write a decimal in fractional form.
• Write a fraction with a denominator of 10, 100, or 1000 as a decimal.
• Express a 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 ).

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 set of double bar graphs.
• Represent a set of data by creating a double bar graph, label the title and axes, and create a legend with or without the use of technology.
• Draw conclusions from a 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 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 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 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 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.
• 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 perimeter, the square or shape closest to a square will result in the
• greatest area.
• Illustrate that for any 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

• 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

• 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 3-D object using personal referents.
• Determine the volume of a 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 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)

• 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 (mL or L) is represented by a given referent.
• Estimate the capacity of a container using personal referents.
• Determine the capacity of a 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 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 3-D object using terms such as parallel, intersecting, perpendicular, vertical, or horizontal.
• Describe the sides of a 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.

• Identify and describe the characteristics of a pre-sorted set of quadrilaterals.
• Sort a set of quadrilaterals and explain the sorting rule.
• Sort a set of quadrilaterals according to the lengths of the sides.
• Sort a 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, and draw and describe the image.
• Translate a 2-D shape horizontally, vertically, or diagonally, and describe the position and orientation of the image.
• Rotate a 2-D shape about a point, and describe the position and orientation of the image.
• Reflect a 2-D shape in a line of reflection, and describe the position and orientation of the image.
• Perform a transformation of a 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 (e.g., the circle moved 3 cm to the left).
• Draw a 2-D shape, rotate the shape, and describe the direction of the turn (clockwise or counter-clockwise), 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 (translation, rotation, or reflection) of 2-D shapes.
• Provide an example of a translation, a rotation, and a reflection.
• Identify a single transformation as a translation, rotation, or reflection.
• Describe a rotation by the direction of the turn (clockwise or counter-clockwise).

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

Solve problems involving single-variable (expressed as symbols or letters), one-step equations with whole-number coefficients, and whole-number solutions.
• Express a problem in context as an equation where the unknown is represented by a letter variable.
• Solve a 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 an equation.