Math Games

Represent and describe whole numbers to 10 000, concretely, pictorially and symbolically.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Write a given numeral 0 – 10 000 in words.
- Represent a given numeral using a place value chart or diagrams.
- Describe the meaning of each digit in a given numeral.
- Express a given numeral in expanded notation, e.g., 4301 = 4000 + 300 + 1.
- Write the numeral represented by a given expanded notation. e.g. 2000 + 400 + 60 = 2460.
- Explain and show the meaning of each digit in a given 4-digit numeral with all digits the same, e.g., for the numeral 2222, the first digit represents two thousands, the second digit two hundreds, the third digit two tens and the fourth digit two ones.
- Write a given numeral using proper spacing without commas, e.g., 4567 or 4 567, 10 000.
- Read numerals up to four-digits without using the word “and,” e.g., 365 is read as “three hundred, sixty-five; 5321 is read as “five thousand, three hundred, twenty one.” The word “and” is reserved for reading decimal numbers, e.g., 3.8 is “three and eight tenths”.

Relate decimals to fractions (to hundredths).
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Read decimals as fractions, e.g., 0.5 is zero and five tenths.
- Express orally and in written form a given decimal in fractional form.
- Express orally and in written form a given fraction with a denominator of 10 or 100 as a decimal.
- Express a given pictorial or concrete representation as a fraction or decimal, e.g., 15 shaded squares on a hundred grid can be expressed as 0.15 or 15/100 .
- Express orally and in written form the decimal equivalent for a given fraction, e.g., 50/100 can be expressed as 0.5

Demonstrate an understanding of addition and subtraction of decimals (limited to hundredths) by:
• using compatible numbers
• estimating sums and differences
• using mental math strategies to solve problems.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Predict sums and differences of decimals using estimation strategies.
- Solve problems, including money and measurement, which involve addition and subtraction of decimals, limited to hundredths.
- Ask students to determine which problems do not require an exact solution.
- Determine the approximate solution of a given problem using compatible numbers.
- Determine an exact solution using mental computation strategies.
- Count back change for a given purchase.

Compare and order numbers to 10 000.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Order a given set of numbers in ascending or descending order and explain the order by making references to place value.
- Create and order three different 4-digit numerals.
- Identify the missing numbers in an ordered sequence or on a number line.
- Identify incorrectly placed numbers in an ordered sequence or on a number line.

Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3 and 4-digit numerals) by:
• using personal strategies for adding and subtracting
• estimating sums and differences
• solving problems involving addition and subtraction.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Explain how to keep track of digits that have the same place value when adding numbers, limited to 3- and 4-digit numerals.
- Explain how to keep track of digits that have the same place value when subtracting numbers, limited to 3- and 4-digit numerals.
- Represent concretely, pictorially, symbolically the addition and subtraction of whole numbers up 4-digit by 4-digit.
- Describe a situation in which an estimate rather than an exact answer is sufficient.
- Estimate sums and differences using different strategies, i.e., front-end estimation and compensation.
- Solve problems that involve addition and subtraction of whole numbers (one or more steps/where some numbers may be irrelevant). Explain solutions to problems.
- Create a problem given a number sentence for addition or subtraction.
- Solve problems that involve addition and subtraction in more than one way, limited to 3- and 4- digit numerals. For example, 385 +  = 500 or 500 – 385 = .

Explain the properties of 0 and 1 for multiplication and the property of 1 for division.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Explain the property for determining the answer when multiplying numbers by one.
- Explain the property for determining the answer when multiplying numbers by zero.
- Explain the property for determining the answer when dividing numbers by one.

Describe and apply mental mathematics strategies, such as:
• skip counting from a known fact
• using doubling or halving
• using doubling or halving and adding or subtracting one more group
• using patterns in the 9s facts
to determine basic multiplication facts to 9 × 9 and related division facts.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Provide examples for applying mental mathematics strategies:
• | doubling (for 4  3, think 2  3 = 6, so 4  3 = 6 + 6)
• | doubling and adding one more group (for 3  7, think 2  7 = 14, and 14 + 7 = 21
• | using known facts (for example, when multiplying 9  6, think 10  6 = 60, and 60 – 6 = 54)
• | halving (if 4  6 is equal to 24, then 2 × 6 is equal to 12)
• | think division for multiplication facts (for 64 ÷ 8, think 8   = 64)

Demonstrate an understanding of multiplication (2- or 3-digit by 1-digit) to solve problems by:
• using personal strategies for multiplication with and without concrete materials
• using arrays to represent multiplication
• connecting concrete representations to symbolic representations
• estimating products.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Model a given multiplication problem using the distributive property, e.g., 8 × 365 = (8 × 300) + (8 × 60) + (8 × 5).
- Use concrete materials, such as base ten blocks or their pictorial representations, to represent multiplication and record the process symbolically.
- Create and solve a multiplication problem that is limited to 2- or 3-digits by 1-digit.
- Estimate a product using a personal strategy, e.g., 2 × 243 is close to or a little more than 2 × 200, or close to or a little less than 2 × 250.
- Model and solve a given multiplication problem with and without an array and record the process.

Demonstrate an understanding of division (1-digit divisor and up to 2-digit dividend) to solve problems by:
• using personal strategies for dividing with and without concrete materials
• estimating quotients
• relating division to multiplication

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Solve a given division problem without a remainder using arrays or base ten materials.
- Solve a given division problem with a remainder using arrays or base ten materials.
- Solve a given division problem using a personal strategy and record the process.
- Create and solve a word problem involving a 1- or 2-digit dividend.
- Estimate a quotient using a personal strategy, e.g., 86 ÷ 4 is close to 80 ÷ 4 or close to 80 ÷ 5.
(It is not intended that remainders be expressed as decimals or fractions.)

Demonstrate an understanding of fractions less than or equal to one by using concrete and pictorial representations to:
• name and record fractions for the parts of a whole or a set
• compare and order fractions
• model and explain that for different wholes, two identical fractions may not represent the same quantity
• provide examples of where fractions are used.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Represent a given fraction using concrete materials.
- Identify a fraction from its given concrete representation.
- Name and record the shaded and non-shaded parts of a given set.
- Name and record the shaded and non-shaded parts of a given whole.
- Represent a given fraction pictorially by shading parts of a given set.
- Represent a given fraction pictorially by shading parts of a given whole.
- Explain how denominators can be used to compare two given unit fractions with numerator 1.
- Order a given set of fractions that have the same numerator and explain the ordering.
- Order a given set of fractions that have the same denominator and explain the ordering.
- Identify which of the benchmarks: 0, or 1 is closer to a given fraction.
- Name fractions between two given benchmarks on a number line.
- Order a given set of fractions by placing them on a number line with given benchmarks.
- Provide examples of when two identical fractions may not represent the same quantity, e.g., half of a large apple is not equivalent to half of a small apple; half of ten oranges is not equivalent to half of sixteen oranges.
- Provide an example of a fraction that represents part of a set and, a fraction that represents part of a whole from everyday contexts.

Describe and represent decimals (tenths and hundredths) 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.
- Explain the meaning of each digit in a given decimal with all digits the same.
- Represent a given decimal using money values (dimes and pennies).
- Record a given money value using decimals.
- Provide examples of everyday contexts in which tenths and hundredths are used.
- Model, using manipulatives or pictures, which a given tenth can be expressed as hundredths, e.g., 0.9 is equivalent to 0.90 or 9 dimes is equivalent to 90 pennies.

Read and record time using digital and analog clocks, including 24‐hour clocks.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- State the number of hours in a day.
- Express the time orally and numerically from a 12-hour analog clock.
- Express the time orally and numerically from a 24-hour analog clock.
- Express the time orally and numerically from a 12-hour digital clock.
- Describe time orally and numerically from a 24-hour digital clock.
- Describe time orally as “minutes to” or “minutes after” the hour.
- Explain the meaning of AM and PM, and provide an example of an activity that occurs during the AM and another that occurs during the PM.

Read and record calendar dates in a variety of formats.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Write dates in a variety of formats, e.g., yyyy/mm/dd, dd/mm/yyyy, March 21, 2006, dd/mm/yy.
- Relate dates written in the format yyyy/mm/dd to dates on a calendar.
- Identify possible interpretations of a given date, e.g., 06/03/04.

Demonstrate an understanding of area of regular and irregular 2-D shapes by:
• recognizing that area is measured in square units
• selecting and justifying referents for the units cm² or m²
• estimating area by using referents for cm² or m²
• determining and recording area (cm² or m²)
• constructing different rectangles for a given area (cm2 or m2) in order to demonstrate that many different rectangles may have the same area.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Describe area as the measure of surface recorded in square units.
- Identify and explain why the square is the most efficient unit for measuring area.
- Provide a referent for a square centimetre and explain the choice.
- Provide a referent for a square metre and explain the choice.
- Determine which standard square unit is represented by a given referent.
- Estimate the area of a given 2-D shape using personal referents.
- Determine the area of a regular 2-D shape and explain the strategy.
- Determine the area of an irregular 2-D shape and explain the strategy.
- Construct a rectangle for a given area.
- Demonstrate that many rectangles are possible for a given area by drawing at least two different rectangles for the same given area.

Describe and construct rectangular and triangular prisms.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Identify and name common attributes of rectangular prisms from given sets of rectangular prisms.
- Identify and name common attributes of triangular prisms from given sets of triangular prisms.
- Sort a given set of rectangular and triangular prisms using the shape of the base.
- Construct and describe a model of rectangular and triangular prisms using materials, such as pattern blocks or modeling clay.
- Construct rectangular prisms from their nets.
- Construct triangular prisms from their nets.
- Identify examples of rectangular and triangular prisms found in the environment.

Demonstrate an understanding of line symmetry by:
• identifying symmetrical 2-D shapes
• creating symmetrical 2-D shapes
• drawing one or more lines of symmetry in a 2-D shape.

• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Identify the characteristics of given symmetrical and nonsymmetrical 2-D shapes.
- Sort a given set of 2-D shapes as symmetrical and nonsymmetrical and explain the process.
- Complete a symmetrical 2-D shape given half the shape and its line of symmetry and explain the process.
- Identify lines of symmetry of a given set of 2-D shapes and explain why each shape is symmetrical.
- Determine whether or not a given 2-D shape is symmetrical by using a Mira or by folding and superimposing.
- Create a symmetrical shape with and without manipulatives and explain the process.
- Provide examples of symmetrical shapes found in the environment and identify the line(s) of symmetry.
- Sort a given set of 2-D shapes as those that have no lines of symmetry, one line of symmetry or more than one line of symmetry.

Demonstrate an understanding of congruency, concretely and pictorially.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Determine if two given 2-D shapes are congruent and explain the strategy used.
- Create a shape that is congruent to a given 2-D shape and explain why the two shapes are congruent.
- Identify congruent 2-D shapes from a given set of shapes shown in different orientations.
- Identify corresponding vertices and sides of two given congruent shapes.
- Explain the connections between congruence and symmetry using 2-D shapes.

Identify and describe patterns found in tables and charts, including a multiplication chart.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Identify and describe a variety of patterns in a multiplication chart.
- Determine the missing element(s) in a given table or chart.
- Identify error(s) in a given table or chart.
- Describe the pattern found in a given table or chart

Reproduce a pattern shown in a table or chart using concrete materials.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Create a concrete representation of a given pattern displayed in a table or chart.
- Explain why the same relationship exists between the pattern in a table and its concrete representation.

Represent and describe patterns and relationships using charts and tables to solve problems.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Extend patterns found in a table or chart to solve a given problem.
- Translate the information provided in a given problem into a table or chart.
- Identify and extend the patterns in a table or chart to solve a given problem

Identify and explain mathematical relationships using charts and diagrams to solve problems.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Complete a Carroll diagram by entering given data into correct squares to solve a given problem.
- Determine where new elements belong in a given Carroll diagram.
- Solve a given problem using a Carroll diagram.
- Identify a sorting rule for a given Venn diagram.
- Describe the relationship shown in a given Venn diagram when the circles intersect, when one circle is contained in the other and when the circles are separate.
- Determine where new elements belong in a given Venn diagram.
- Solve a given problem by using a chart or diagram to identify mathematical relationships

Express a given problem as an equation in which a symbol is used to represent an unknown number.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Explain the purpose of the symbol, such as a triangle or circle, in a given addition, subtraction, multiplication or division equation with one unknown, e.g. 36   = 6
- Express a given pictorial or concrete representation of an equation in symbolic form.
- Identify the unknown in a story problem, represent the problem with an equation and solve the problem concretely, pictorially or symbolically.
- Create a problem in context for a given equation with one unknown.

Solve one-step equations involving a symbol to represent an unknown number.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Solve a given one-step equation using manipulatives.
- Solve a given one-step equation using guess and test.
- Describe, orally, the meaning of a given one-step equation with one unknown.
- Solve a given equation when the unknown is on the left or right side of the equation.
- Represent and solve a given addition or subtraction problem involving a “part-part-whole” or comparison context using a symbol to represent the unknown.
- Represent and solve a given multiplication or division problem involving equal grouping or partitioning (equal sharing) using symbols to represent the unknown.

Demonstrate an understanding of many-to-one correspondence.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Compare graphs in which different intervals or correspondences are used and explain why the interval or correspondence was used.
- Compare graphs in which the same data has been displayed using one-to-one and many-to-one correspondences, and explain how they are the same and different.
- Explain why many-to-one correspondence is sometimes used rather than one-to-one correspondence.
- Find examples of graphs in which many-to-one correspondence is used in print and electronic media, such as newspapers, magazines and the Internet, and describe the correspondence used.

Construct and interpret pictographs and bar graphs involving many-to-one correspondence to draw conclusions.
• Achievement Indicators
• Students who have achieved this outcome(s) should be able to:
- Identify an interval and correspondence for displaying a given set of data in a graph and justify the choice.
- Create and label (with categories, title and legend) a pictograph to display a given set of data using many-to-one correspondence, and justify the choice of correspondence used.
- Create and label (with axes and title) a bar graph to display a given set of data using many-to-one correspondence, and justify the choice of interval used.
- Answer a given question using a given graph in which data is displayed using many-to-one correspondence.