Math Games

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.

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.