Mathville Middle School Ontario Correlations

Printed on recycled paper

04-163

ISBN 0-7794-8122-4

© Queen’s Printer for Ontario, 2005

The Ontario CurriculumGrades 1-8

Ministry of Education

2 0 0 5

Mathematics

R E V I S E D

Evannah Sakamoto

Mathville Middle School Curriculum Correlation

Evannah Sakamoto

This extract of the Ontario Mathematics Curriculum Guideline has been annotated to show the correlations between Specific Expectations for Grades 6, 7, and 8 and activities in Mathville Middle School v. 3.0.

Janis Gershman

Text Box

Mathville Middle School Curriculum Correlation

Janis Gershman

Text Box

This extract of the Ontario Mathematics Curriculum Guideline has been annotated to show the correlations between Specific Expectations for Grades 5, 6, 7, and 8 and activities in Mathville Middle School v. 3.0.

78 T H E O N T A R I O C U R R I C U L U M , G R A D E S 1 – 8 : M A T H E M A T I C S

Grade 5: Number Sense and Numeration

Specific Expectations

Quantity Relationships

By the end of Grade 5, students will:

– represent, compare, and order whole numbers and decimal numbers from 0.01to 100 000, using a variety of tools (e.g.,number lines with appropriate increments,base ten materials for decimals);

– demonstrate an understanding of placevalue in whole numbers and decimalnumbers from 0.01 to 100 000, using avariety of tools and strategies (e.g., usenumbers to represent 23 011 as 20 000 + 3000 + 0 + 10 + 1; use base ten materials to represent the relationshipbetween 1, 0.1, and 0.01) (Sample problem: How many thousands cubeswould be needed to make a base ten blockfor 100 000?);

– read and print in words whole numbers toten thousand, using meaningful contexts(e.g., newspapers, magazines);

– round decimal numbers to the nearesttenth, in problems arising from real-lifesituations;

– represent, compare, and order fractionalamounts with like denominators, includ-ing proper and improper fractions and

mixed numbers, using a variety of tools(e.g., fraction circles, Cuisenaire rods,number lines) and using standard frac-tional notation;

– demonstrate and explain the concept ofequivalent fractions, using concrete mate-rials (e.g., use fraction strips to show that

– demonstrate and explain equivalent repre-sentations of a decimal number, usingconcrete materials and drawings (e.g., usebase ten materials to show that threetenths [0.3] is equal to thirty hundredths[0.30]);

– read and write money amounts to $1000(e.g., $455.35 is 455 dollars and 35 cents,or four hundred fifty-five dollars andthirty-five cents);

– solve problems that arise from real-life situations and that relate to the magnitude of whole numbers up to 100 000 (Sampleproblem: How many boxes hold 100 000sheets of paper, if one box holds 8 pack-ages of paper, and one package of papercontains 500 sheets of paper?).

Overall Expectations


• read, represent, compare, and order whole numbers to 100 000, decimal numbers to hundredths, proper and improper fractions, and mixed numbers;

• demonstrate an understanding of magnitude by counting forward and backwards by 0.01;

• solve problems involving the multiplication and division of multi-digit whole numbers, andinvolving the addition and subtraction of decimal numbers to hundredths, using a variety ofstrategies;

• demonstrate an understanding of proportional reasoning by investigating whole-numberrates.

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Throughout Mathville

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79G R A D E 5 : N U M B E R S E N S E A N D N U M E R A T I O N

Counting


– count forward by hundredths from anydecimal number expressed to two decimalplaces, using concrete materials and num-ber lines (e.g., use base ten materials torepresent 2.96 and count forward by hun-dredths: 2.97, 2.98, 2.99, 3.00, 3.01, …;“Two and ninety-six hundredths, two and ninety-seven hundredths, two andninety-eight hundredths, two and ninety-nine hundredths, three, three andone hundredth, …”) (Sample problem:What connections can you make betweencounting by hundredths and measuringlengths in centimetres and metres?).

Operational Sense


– solve problems involving the addition,subtraction, and multiplication of wholenumbers, using a variety of mental strate-gies (e.g., use the commutative property:5 x 18 x 2 = 5 x 2 x 18, which gives10 x 18 = 180);

– add and subtract decimal numbers to hundredths, including money amounts,using concrete materials, estimation, andalgorithms (e.g., use 10 x 10 grids to add2.45 and 3.25);

– multiply two-digit whole numbers bytwo-digit whole numbers, using estima-tion, student-generated algorithms, andstandard algorithms;

– divide three-digit whole numbers by one-digit whole numbers, using concretematerials, estimation, student-generatedalgorithms, and standard algorithms;

– multiply decimal numbers by 10, 100,1000, and 10 000, and divide decimalnumbers by 10 and 100, using mentalstrategies (e.g., use a calculator to look forpatterns and generalize to develop a rule);

– use estimation when solving problemsinvolving the addition, subtraction, multi-plication, and division of whole numbers,to help judge the reasonableness of a solu-tion (Sample problem: Mori used a calcu-lator to add 7.45 and 2.39. The calculatordisplay showed 31.35. Explain why thisresult is not reasonable, and suggest whereyou think Mori made his mistake.).

Proportional Relationships


– describe multiplicative relationshipsbetween quantities by using simple frac-tions and decimals (e.g.,“If you have 4plums and I have 6 plums, I can say that I

– determine and explain, through investiga-tion using concrete materials, drawings,and calculators, the relationship betweenfractions (i.e., with denominators of 2, 4,5, 10, 20, 25, 50, and 100) and their equiv-alent decimal forms (e.g., use a 10 x 10

– demonstrate an understanding of simplemultiplicative relationships involvingwhole-number rates, through investigationusing concrete materials and drawings(Sample problem: If 2 books cost $6, howwould you calculate the cost of 8 books?).

have 1 or 1.5 times as many plums as12

you have.”);

grid to show that , which can also25 = 40

100be represented as 0.4);

Evannah Sakamoto

Clothing Store, Grocery Store; Work - Financial Records

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Games - Ice Cream Catch

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Grocery Store Work - Packaging, Photography

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Clothing Store Grocery Store Hardware Store

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Grade 5: Measurement


Attributes, Units, and Measurement Sense


– estimate, measure (i.e., using an analogueclock), and represent time intervals to thenearest second;

– estimate and determine elapsed time, withand without using a time line, given thedurations of events expressed in minutes,hours, days, weeks, months, or years(Sample problem:You are travelling fromToronto to Montreal by train. If the traindeparts Toronto at 11:30 a.m. and arrivesin Montreal at 4:56 p.m., how long willyou be on the train?);

– measure and record temperatures to deter-mine and represent temperature changesover time (e.g., record temperaturechanges in an experiment or over a sea-son) (Sample problem: Investigate the rela-tionship between weather, climate, andtemperature changes over time in differentlocations.);

– estimate and measure the perimeter andarea of regular and irregular polygons,using a variety of tools (e.g., grid paper,geoboard, dynamic geometry software)and strategies.

Measurement Relationships


– select and justify the most appropriatestandard unit (i.e., millimetre, centimetre,decimetre, metre, kilometre) to measure

length, height, width, and distance, and tomeasure the perimeter of various polygons;

– solve problems requiring conversion frommetres to centimetres and from kilometresto metres (Sample problem: Describe themultiplicative relationship between thenumber of centimetres and the number of metres that represent a length. Use this relationship to convert 5.1 m to centimetres.);

– solve problems involving the relationshipbetween a 12-hour clock and a 24-hourclock (e.g., 15:00 is 3 hours after 12 noon,so 15:00 is the same as 3:00 p.m.);

– create, through investigation using a varietyof tools (e.g., pattern blocks, geoboard, gridpaper) and strategies, two-dimensionalshapes with the same perimeter or thesame area (e.g., rectangles and parallelo-grams with the same base and the sameheight) (Sample problem: Using dot paper,how many different rectangles can youdraw with a perimeter of 12 units? withan area of 12 square units?);

– determine, through investigation using avariety of tools (e.g., concrete materials,dynamic geometry software, grid paper)and strategies (e.g., building arrays), therelationships between the length andwidth of a rectangle and its area andperimeter, and generalize to develop the formulas [i.e., Area = length x width;Perimeter = (2 x length) + (2 x width)];



• estimate, measure, and record perimeter, area, temperature change, and elapsed time, using avariety of strategies;

• determine the relationships among units and measurable attributes, including the area of a rectangle and the volume of a rectangular prism.

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81G R A D E 5

– solve problems requiring the estimation andcalculation of perimeters and areas of rec-tangles (Sample problem:You are helpingto fold towels, and you want them to stacknicely. By folding across the length and/orthe width, you fold each towel a total ofthree times.You want the shape of eachfolded towel to be as close to a square aspossible. Does it matter how you fold thetowels?);

– determine, through investigation, the relationship between capacity (i.e., theamount a container can hold) and volume(i.e., the amount of space taken up by anobject), by comparing the volume of anobject with the amount of liquid it cancontain or displace (e.g., a bottle has a volume, the space it takes up, and a capac-ity, the amount of liquid it can hold)(Sample problem: Compare the volumeand capacity of a thin-walled container in the shape of a rectangular prism todetermine the relationship between unitsfor measuring capacity [e.g., millilitres]and units for measuring volume [e.g.,cubic centimetres].);

– determine, through investigation usingstacked congruent rectangular layers ofconcrete materials, the relationshipbetween the height, the area of the base,and the volume of a rectangular prism,and generalize to develop the formula(i.e., Volume = area of base x height) (Sampleproblem: Create a variety of rectangularprisms using connecting cubes. For eachrectangular prism, record the area of thebase, the height, and the volume on achart. Identify relationships.);

– select and justify the most appropriatestandard unit to measure mass (i.e., mil-ligram, gram, kilogram, tonne).

Evannah Sakamoto

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Grade 5: Geometry and Spatial Sense


Geometric Properties


– distinguish among polygons, regular poly-gons, and other two-dimensional shapes;

– distinguish among prisms, right prisms,pyramids, and other three-dimensional figures;

– identify and classify acute, right, obtuse,and straight angles;

– measure and construct angles up to 90º,using a protractor;

– identify triangles (i.e., acute, right, obtuse,scalene, isosceles, equilateral), and classifythem according to angle and side properties;

– construct triangles, using a variety of tools(e.g., protractor, compass, dynamic geome-try software), given acute or right anglesand side measurements (Sample problem:Use a protractor, ruler, and pencil to con-struct a scalene triangle with a 30° angleand a side measuring 12 cm.).

Geometric Relationships


– identify prisms and pyramids from theirnets;

– construct nets of prisms and pyramids,using a variety of tools (e.g., grid paper,isometric dot paper, Polydrons, computerapplication).

Location and Movement


– locate an object using the cardinal direc-tions (i.e., north, south, east, west) and acoordinate system (e.g.,“If I walk 5 stepsnorth and 3 steps east, I will arrive at theapple tree.”);

– compare grid systems commonly used onmaps (i.e., the use of numbers and lettersto identify an area; the use of a coordinatesystem based on the cardinal directions todescribe a specific location);

– identify, perform, and describe translations,using a variety of tools (e.g., geoboard, dotpaper, computer program);

– create and analyse designs by translatingand/or reflecting a shape, or shapes, usinga variety of tools (e.g., geoboard, gridpaper, computer program) (Sample prob-lem: Identify translations and/or reflec-tions that map congruent shapes ontoeach other in a given design.).



• identify and classify two-dimensional shapes by side and angle properties, and compare andsort three-dimensional figures;

• identify and construct nets of prisms and pyramids;

• identify and describe the location of an object, using the cardinal directions, and translatetwo-dimensional shapes.

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Games - Slide Puzzle

83G R A D E 5

Grade 5: Patterning and Algebra


Patterns and Relationships


– create, identify, and extend numeric andgeometric patterns, using a variety of tools(e.g., concrete materials, paper and pencil,calculators, spreadsheets);

– build a model to represent a number pat-tern presented in a table of values thatshows the term number and the term;

– make a table of values for a pattern that isgenerated by adding or subtracting a num-ber (i.e., a constant) to get the next term,or by multiplying or dividing by a con-stant to get the next term, given either thesequence (e.g., 12, 17, 22, 27, 32, …) orthe pattern rule in words (e.g., start with12 and add 5 to each term to get the nextterm);

– make predictions related to growing andshrinking geometric and numeric patterns(Sample problem: Create growing L’s usingtiles. The first L has 3 tiles, the second Lhas 5 tiles, the third L has 7 tiles, and soon. Predict the number of tiles you wouldneed to build the 10th L in the pattern.);

– extend and create repeating patterns thatresult from translations, through investiga-tion using a variety of tools (e.g., patternblocks, dynamic geometry software, dotpaper).

Variables, Expressions, and Equations


– demonstrate, through investigation, anunderstanding of variables as changingquantities, given equations with letters orother symbols that describe relationshipsinvolving simple rates (e.g., the equationsC = 3 x n and 3 x n = C both representthe relationship between the total cost(C ), in dollars, and the number of sand-wiches purchased (n), when each sand-wich costs $3);

– demonstrate, through investigation, anunderstanding of variables as unknownquantities represented by a letter or othersymbol (e.g., 12 = 5 + � or 12 = 5 + scan be used to represent the following sit-uation: “I have 12 stamps altogether and 5of them are from Canada. How many arefrom other countries?”);

– determine the missing number in equa-tions involving addition, subtraction, mul-tiplication, or division and one- or two-digit numbers, using a variety of tools andstrategies (e.g., modelling with concretematerials, using guess and check with andwithout the aid of a calculator) (Sampleproblem: What is the missing number inthe equation 8 = 88 ÷ �?).



• determine, through investigation using a table of values, relationships in growing and shrinking patterns, and investigate repeating patterns involving translations;

• demonstrate, through investigation, an understanding of the use of variables in equations.

Figure 1 Figure 2 Figure 3

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Grade 5: Data Management and Probability


Collection and Organization of Data


– distinguish between discrete data (i.e., dataorganized using numbers that have gapsbetween them, such as whole numbers,and often used to represent a count, suchas the number of times a word is used)and continuous data (i.e., data organizedusing all numbers on a number line thatfall within the range of the data, and usedto represent measurements such as heightsor ages of trees);

– collect data by conducting a survey or anexperiment (e.g., gather and record airtemperature over a two-week period) todo with themselves, their environment,issues in their school or community, orcontent from another subject, and recordobservations or measurements;

– collect and organize discrete or continuousprimary data and secondary data and dis-play the data in charts, tables, and graphs(including broken-line graphs) that haveappropriate titles, labels (e.g., appropriateunits marked on the axes), and scales thatsuit the range and distribution of the data(e.g., to represent precipitation amountsranging from 0 mm to 50 mm over theschool year, use a scale of 5 mm for eachunit on the vertical axis and show months

on the horizontal axis), using a variety oftools (e.g., graph paper, simple spread-sheets, dynamic statistical software);

– demonstrate an understanding that sets ofdata can be samples of larger populations(e.g., to determine the most commonshoe size in your class, you would includeevery member of the class in the data; todetermine the most common shoe size inOntario for your age group, you mightcollect a large sample from classes acrossthe province);

– describe, through investigation, how a set of data is collected (e.g., by survey,measurement, observation) and explainwhether the collection method is appropriate.

Data Relationships


– read, interpret, and draw conclusions fromprimary data (e.g., survey results, measure-ments, observations) and from secondarydata (e.g., precipitation or temperaturedata in the newspaper, data from theInternet about heights of buildings andother structures), presented in charts,tables, and graphs (including broken-linegraphs);



• collect and organize discrete or continuous primary data and secondary data and display thedata using charts and graphs, including broken-line graphs;

• read, describe, and interpret primary data and secondary data presented in charts and graphs,including broken-line graphs;

• represent as a fraction the probability that a specific outcome will occur in a simple proba-bility experiment, using systematic lists and area models.

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85G R A D E 5

– calculate the mean for a small set of dataand use it to describe the shape of the dataset across its range of values, using charts,tables, and graphs (e.g.,“The data valuesfall mainly into two groups on both sidesof the mean.”; “The set of data is notspread out evenly around the mean.”);

– compare similarities and differencesbetween two related sets of data, using avariety of strategies (e.g., by representingthe data using tally charts, stem-and-leafplots, double bar graphs, or broken-linegraphs; by determining measures of cen-tral tendency [i.e., mean, median, andmode]; by describing the shape of a dataset across its range of values).

Probability


– determine and represent all the possibleoutcomes in a simple probability experi-ment (e.g., when tossing a coin, the possible

outcomes are heads and tails; when rollinga number cube, the possible outcomes are1, 2, 3, 4, 5, and 6), using systematic listsand area models (e.g., a rectangle is dividedinto two equal areas to represent the out-comes of a coin toss experiment);

– represent, using a common fraction, theprobability that an event will occur insimple games and probability experiments(e.g.,“My spinner has four equal sectionsand one of those sections is coloured red.The probability that I will land on red

– pose and solve simple probability problems,and solve them by conducting probabilityexperiments and selecting appropriatemethods of recording the results (e.g., tallychart, line plot, bar graph).

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– represent, compare, and order whole numbers and decimal numbers from 0.001 to 1 000 000, using a variety oftools (e.g., number lines with appropriateincrements, base ten materials for decimals);

– demonstrate an understanding of placevalue in whole numbers and decimalnumbers from 0.001 to 1 000 000, using avariety of tools and strategies (e.g. use baseten materials to represent the relationshipbetween 1, 0.1, 0.01, and 0.001) (Sampleproblem: How many thousands cubeswould be needed to make a base ten blockfor 1 000 000?);

– read and print in words whole numbers toone hundred thousand, using meaningfulcontexts (e.g., the Internet, referencebooks);

– represent, compare, and order fractionalamounts with unlike denominators,including proper and improper fractionsand mixed numbers, using a variety oftools (e.g., fraction circles, Cuisenaire rods,drawings, number lines, calculators) andusing standard fractional notation (Sampleproblem: Use fraction strips to show that

– estimate quantities using benchmarks of10%, 25%, 50%, 75%, and 100% (e.g., thecontainer is about 75% full; approximately50% of our students walk to school);

– solve problems that arise from real-life situations and that relate to the magnitude of whole numbers up to 1 000 000(Sample problem: How would you determine if a person could live to be1 000 000 hours old? Show your work.);

– identify composite numbers and primenumbers, and explain the relationshipbetween them (i.e., any composite num-ber can be factored into prime factors)(e.g., 42 = 2 x 3 x 7).

Operational Sense


– use a variety of mental strategies to solveaddition, subtraction, multiplication, anddivision problems involving whole num-bers (e.g., use the commutative property:4 x 16 x 5 = 4 x 5 x 16, which gives20 x 16 = 320; use the distributive prop-erty: (500 + 15) ÷ 5 = 500 ÷ 5 + 15 ÷ 5,which gives 100 + 3 = 103);

– solve problems involving the multiplica-tion and division of whole numbers (four-digit by two-digit), using a variety of tools(e.g., concrete materials, drawings, calcula-tors) and strategies (e.g., estimation,algorithms);



• read, represent, compare, and order whole numbers to 1 000 000, decimal numbers to thousandths, proper and improper fractions, and mixed numbers;

• solve problems involving the multiplication and division of whole numbers, and the additionand subtraction of decimal numbers to thousandths, using a variety of strategies;

• demonstrate an understanding of relationships involving percent, ratio, and unit rate.

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89G R A D E 6

– add and subtract decimal numbers tothousandths, using concrete materials,estimation, algorithms, and calculators;

– multiply and divide decimal numbers totenths by whole numbers, using concretematerials, estimation, algorithms, and calculators (e.g., calculate 4 x 1.4 usingbase ten materials; calculate 5.6 ÷ 4 usingbase ten materials);

– multiply whole numbers by 0.1, 0.01, and0.001 using mental strategies (e.g., use acalculator to look for patterns and gener-alize to develop a rule);

– multiply and divide decimal numbers by10, 100, 1000, and 10 000 using mentalstrategies (e.g.,“To convert 0.6 m2 tosquare centimetres, I calculated in myhead 0.6 x 10 000 and got 6000 cm2.”)(Sample problem: Use a calculator to helpyou generalize a rule for multiplyingnumbers by 10 000.);

– use estimation when solving problemsinvolving the addition and subtraction ofwhole numbers and decimals, to helpjudge the reasonableness of a solution;

– explain the need for a standard order forperforming operations, by investigatingthe impact that changing the order haswhen performing a series of operations(Sample problem: Calculate and comparethe answers to 3 + 2 x 5 using a basic four-function calculator and using a scientificcalculator.).



– represent ratios found in real-life contexts,using concrete materials, drawings, andstandard fractional notation (Sample problem: In a classroom of 28 students, 12are female. What is the ratio of male stu-dents to female students?);

– determine and explain, through investiga-tion using concrete materials, drawings,and calculators, the relationships amongfractions (i.e., with denominators of 2, 4,5, 10, 20, 25, 50, and 100), decimal num-bers, and percents (e.g., use a 10 x 10

– represent relationships using unit rates(Sample problem: If 5 batteries cost $4.75,what is the cost of 1 battery?).

= 0.25 or 25%);grid to show that 14

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Work - Financial Records, Savings and Loans

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Clothing Store, Hardware Store

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– demonstrate an understanding of the rela-tionship between estimated and precisemeasurements, and determine and justifywhen each kind is appropriate (Sampleproblem:You are asked how long it takesyou to travel a given distance. How is the method you use to determine thetime related to the precision of the measurement?);

– estimate, measure, and record length, area,mass, capacity, and volume, using the met-ric measurement system.



– select and justify the appropriate metricunit (i.e., millimetre, centimetre, decime-tre, metre, decametre, kilometre) to mea-sure length or distance in a given real-lifesituation (Sample problem: Select and justify the unit that should be used tomeasure the perimeter of the school.);

– solve problems requiring conversion fromlarger to smaller metric units (e.g., metresto centimetres, kilograms to grams, litresto millilitres) (Sample problem: Howmany grams are in one serving if 1.5 kgwill serve six people?);

– construct a rectangle, a square, a triangle,and a parallelogram, using a variety oftools (e.g., concrete materials, geoboard,

dynamic geometry software, grid paper),given the area and/or perimeter (Sampleproblem: Create two different triangleswith an area of 12 square units, using ageoboard.);

– determine, through investigation using avariety of tools (e.g., pattern blocks, PowerPolygons, dynamic geometry software,grid paper) and strategies (e.g., paper fold-ing, cutting, and rearranging), the relation-ship between the area of a rectangle andthe areas of parallelograms and triangles,by decomposing (e.g., cutting up a paral-lelogram into a rectangle and two congru-ent triangles) and composing (e.g., com-bining two congruent triangles to form aparallelogram) (Sample problem:Decompose a rectangle and rearrange theparts to compose a parallelogram with thesame area. Decompose a parallelograminto two congruent triangles, and comparethe area of one of the triangles with thearea of the parallelogram.);

– develop the formulas for the area of a par-allelogram (i.e., Area of parallelogram =base x height) and the area of a triangle [i.e.,Area of triangle = (base x height) ÷ 2], usingthe area relationships among rectangles,parallelograms, and triangles (Sample problem: Use dynamic geometry softwareto show that parallelograms with the sameheight and the same base all have the samearea.);



• estimate, measure, and record quantities, using the metric measurement system;

• determine the relationships among units and measurable attributes, including the area of a parallelogram, the area of a triangle, and the volume of a triangular prism.

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91G R A D E 6

– solve problems involving the estimationand calculation of the areas of trianglesand the areas of parallelograms (Sampleproblem: Calculate the areas of parallelo-grams that share the same base and thesame height, including the special casewhere the parallelogram is a rectangle.);

– determine, using concrete materials, therelationship between units used to mea-sure area (i.e., square centimetre, squaremetre), and apply the relationship to solveproblems that involve conversions fromsquare metres to square centimetres(Sample problem: Describe the multiplica-tive relationship between the number ofsquare centimetres and the number ofsquare metres that represent an area. Usethis relationship to determine how manysquare centimetres fit into half a squaremetre.);

– determine, through investigation using avariety of tools and strategies (e.g.,decomposing rectangular prisms into tri-angular prisms; stacking congruent trian-gular layers of concrete materials to form atriangular prism), the relationship between

the height, the area of the base, and thevolume of a triangular prism, and general-ize to develop the formula (i.e., Volume =area of base x height) (Sample problem:Create triangular prisms by splitting rec-tangular prisms in half. For each prism,record the area of the base, the height,and the volume on a chart. Identify relationships.);

– determine, through investigation using avariety of tools (e.g., nets, concrete mate-rials, dynamic geometry software,Polydrons) and strategies, the surface areaof rectangular and triangular prisms;

– solve problems involving the estimationand calculation of the surface area andvolume of triangular and rectangularprisms (Sample problem: How manysquare centimetres of wrapping paper arerequired to wrap a box that is 10 cm long,8 cm wide, and 12 cm high?).

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– sort and classify quadrilaterals by geome-tric properties related to symmetry, angles,and sides, through investigation using avariety of tools (e.g., geoboard, dynamicgeometry software) and strategies (e.g.,using charts, using Venn diagrams);

– sort polygons according to the number oflines of symmetry and the order of rota-tional symmetry, through investigationusing a variety of tools (e.g., tracing paper,dynamic geometry software, Mira);

– measure and construct angles up to 180°using a protractor, and classify them asacute, right, obtuse, or straight angles;

– construct polygons using a variety oftools, given angle and side measurements(Sample problem: Use dynamic geometrysoftware to construct trapezoids with a45° angle and a side measuring 11 cm.).



– build three-dimensional models usingconnecting cubes, given isometric sketchesor different views (i.e., top, side, front) ofthe structure (Sample problem: Given thetop, side, and front views of a structure,build it using the smallest number ofcubes possible.);

– sketch, using a variety of tools (e.g., isomet-ric dot paper, dynamic geometry software),isometric perspectives and different views(i.e., top, side, front) of three-dimensionalfigures built with interlocking cubes.



– explain how a coordinate system repre-sents location, and plot points in the firstquadrant of a Cartesian coordinate plane;

– identify, perform, and describe, throughinvestigation using a variety of tools (e.g.,grid paper, tissue paper, protractor, com-puter technology), rotations of 180º andclockwise and counterclockwise rotationsof 90°, with the centre of rotation insideor outside the shape;

– create and analyse designs made by reflect-ing, translating, and/or rotating a shape, orshapes, by 90º or 180º (Sample problem:Identify rotations of 90° or 180° that mapcongruent shapes, in a given design, ontoeach other.).



• classify and construct polygons and angles;

• sketch three-dimensional figures, and construct three-dimensional figures from drawings;

• describe location in the first quadrant of a coordinate system, and rotate two-dimensionalshapes.

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Sports - Golf

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93G R A D E 6





– identify geometric patterns, through inves-tigation using concrete materials or draw-ings, and represent them numerically;

– make tables of values, for growing patternsgiven pattern rules, in words (e.g., startwith 3, then double each term and add 1to get the next term), then list the orderedpairs (with the first coordinate represent-ing the term number and the secondcoordinate representing the term) and plotthe points in the first quadrant, using avariety of tools (e.g., graph paper, calcula-tors, dynamic statistical software);

– determine the term number of a giventerm in a growing pattern that is repre-sented by a pattern rule in words, a tableof values, or a graph (Sample problem:For the pattern rule “start with 1 and add3 to each term to get the next term”, usegraphing to find the term number whenthe term is 19.);

– describe pattern rules (in words) that gen-erate patterns by adding or subtracting aconstant, or multiplying or dividing by aconstant, to get the next term (e.g., for 1,3, 5, 7, 9, …, the pattern rule is “start with1 and add 2 to each term to get the nextterm”), then distinguish such pattern rulesfrom pattern rules, given in words, thatdescribe the general term by referring tothe term number (e.g., for 2, 4, 6, 8, …,the pattern rule for the general term is“double the term number”);



• describe and represent relationships in growing and shrinking patterns (where the terms are whole numbers), and investigate repeating patterns involving rotations;

• use variables in simple algebraic expressions and equations to describe relationships.

– determine a term, given its term number,by extending growing and shrinking pat-terns that are generated by adding or sub-tracting a constant, or multiplying ordividing by a constant, to get the nextterm (Sample problem: For the pattern5000, 4750, 4500, 4250, 4000, 3750, …,find the 15th term. Explain your reasoning.);

– extend and create repeating patterns thatresult from rotations, through investigationusing a variety of tools (e.g., pattern blocks,dynamic geometry software, geoboards,dot paper).



– demonstrate an understanding of differentways in which variables are used (e.g.,variable as an unknown quantity; variableas a changing quantity);

– identify, through investigation, the quanti-ties in an equation that vary and those thatremain constant (e.g., in the formula for

ber 2 is a constant, whereas b and h canvary and may change the value of A);

– solve problems that use two or three sym-bols or letters as variables to represent dif-ferent unknown quantities (Sample prob-lem: If n + l = 15 and n + l + s = 19,what value does the s represent?);

the area of a triangle, A = , the num-b x h2

Evannah Sakamoto


Evannah Sakamoto


Evannah Sakamoto



– determine the solution to a simple equa-tion with one variable, through investiga-tion using a variety of tools and strategies(e.g., modelling with concrete materials,using guess and check with and withoutthe aid of a calculator) (Sample problem:Use the method of your choice to deter-mine the value of the variable in theequation 2 x n + 3 = 11. Is there morethan one possible solution? Explain yourreasoning.).

95G R A D E 6





– collect data by conducting a survey (e.g.,use an Internet survey tool) or an experi-ment to do with themselves, their envi-ronment, issues in their school or commu-nity, or content from another subject, andrecord observations or measurements;

– collect and organize discrete or continuousprimary data and secondary data (e.g.,electronic data from websites such as E-Stat or Census At Schools) and displaythe data in charts, tables, and graphs(including continuous line graphs) thathave appropriate titles, labels (e.g., appro-priate units marked on the axes), andscales (e.g., with appropriate increments)that suit the range and distribution of thedata, using a variety of tools (e.g., graphpaper, spreadsheets, dynamic statisticalsoftware);

– select an appropriate type of graph to rep-resent a set of data, graph the data usingtechnology, and justify the choice of graph(i.e., from types of graphs already studied,such as pictographs, horizontal or verticalbar graphs, stem-and-leaf plots, double bargraphs, broken-line graphs, and continuousline graphs);

– determine, through investigation, howwell a set of data represents a population,

on the basis of the method that was usedto collect the data (Sample problem: Wouldthe results of a survey of primary studentsabout their favourite television shows rep-resent the favourite shows of students inthe entire school? Why or why not?).

Data Relationships


– read, interpret, and draw conclusions fromprimary data (e.g., survey results, measure-ments, observations) and from secondarydata (e.g., sports data in the newspaper,data from the Internet about movies),presented in charts, tables, and graphs(including continuous line graphs);

– compare, through investigation, differentgraphical representations of the same data(Sample problem: Use technology to helpyou compare the different types of graphsthat can be created to represent a set ofdata about the number of runs or goalsscored against each team in a tournament.Describe the similarities and differencesthat you observe.);

– explain how different scales used ongraphs can influence conclusions drawnfrom the data;

– demonstrate an understanding of mean(e.g., mean differs from median and modebecause it is a value that “balances” a set ofdata – like the centre point or fulcrum in



• collect and organize discrete or continuous primary data and secondary data and display thedata using charts and graphs, including continuous line graphs;

• read, describe, and interpret data, and explain relationships between sets of data;

• determine the theoretical probability of an outcome in a probability experiment, and use itto predict the frequency of the outcome.

Evannah Sakamoto


Evannah Sakamoto



a lever), and use the mean to compare two sets of related data, with and withoutthe use of technology (Sample problem:Use the mean to compare the masses ofbackpacks of students from two or moreGrade 6 classes.);

– demonstrate, through investigation, anunderstanding of how data from charts,tables, and graphs can be used to makeinferences and convincing arguments (e.g.,describe examples found in newspapersand magazines).

Probability


– express theoretical probability as a ratio of the number of favourable outcomes tothe total number of possible outcomes,where all outcomes are equally likely (e.g.,the theoretical probability of rolling anodd number on a six-sided number cube

because, of six equally likely outcomes,36

only three are favourable – that is, the oddnumbers 1, 3, 5);

is

– represent the probability of an event (i.e.,the likelihood that the event will occur),using a value from the range of 0 (neverhappens or impossible) to 1 (always hap-pens or certain);

– predict the frequency of an outcome of asimple probability experiment or game, bycalculating and using the theoretical prob-ability of that outcome (e.g.,“The theo-

retical probability of spinning red is 14

since there are four different-colouredareas that are equal. If I spin my spinner 100 times, I predict that red should comeup about 25 times.”). (Sample problem:Create a spinner that has rotational sym-metry. Predict how often the spinner willland on the same sector after 25 spins.Perform the experiment and compare theprediction to the results.).

Evannah Sakamoto

Sports - Soccer

Evannah Sakamoto

Sports - Soccer

99G R A D E 7




• represent, compare, and order numbers, including integers;

• demonstrate an understanding of addition and subtraction of fractions and integers, andapply a variety of computational strategies to solve problems involving whole numbers anddecimal numbers;

• demonstrate an understanding of proportional relationships using percent, ratio, and rate.




– represent, compare, and order decimals tohundredths and fractions, using a varietyof tools (e.g., number lines, Cuisenairerods, base ten materials, calculators);

– generate multiples and factors, using avariety of tools and strategies (e.g., identifymultiples on a hundreds chart; create rec-tangles on a geoboard) (Sample problem:List all the rectangles that have an area of 36 cm2 and have whole-numberdimensions.);

– identify and compare integers found inreal-life contexts (e.g., –10°C is muchcolder than +5°C);

– represent and order integers, using a vari-ety of tools (e.g., two-colour counters,virtual manipulatives, number lines);

– select and justify the most appropriaterepresentation of a quantity (i.e., fraction,decimal, percent) for a given context (e.g.,“I would use a decimal for recording thelength or mass of an object, and a fractionfor part of an hour.”);

– represent perfect squares and square roots,using a variety of tools (e.g., geoboards,connecting cubes, grid paper);

– explain the relationship between expo-nential notation and the measurement of area and volume (Sample problem:Explain why area is expressed in squareunits [units2] and volume is expressed incubic units [units3].).

Operational Sense


– divide whole numbers by simple fractionsand by decimal numbers to hundredths,using concrete materials (e.g., divide 3 by

– use a variety of mental strategies to solveproblems involving the addition and sub-traction of fractions and decimals (e.g., use

the commutative property:

use the distributive property: 16.8 ÷ 0.2can be thought of as (16 + 0.8) ÷ 0.2 =16 ÷ 0.2 + 0.8 ÷ 0.2, which gives80 + 4 = 84);

– solve problems involving the multiplica-tion and division of decimal numbers tothousandths by one-digit whole numbers,using a variety of tools (e.g., concretematerials, drawings, calculators) and strategies (e.g., estimation, algorithms);

using fraction strips; divide 4 by 0.8 12using base ten materials and estimation);

x25

133 x =

;x 253 x , which gives = 2

51 x13

25

Evannah Sakamoto

Sports - Baseball

Evannah Sakamoto

Work - Climatology

Evannah Sakamoto

Work - Scheduling

Evannah Sakamoto

Sports - Golf

Evannah Sakamoto

Clothing Store, Grocery Store, Hardware Store Work - Chef, Financial Records, Savings and Loans, Snack Services

Evannah Sakamoto



– solve multi-step problems arising fromreal-life contexts and involving wholenumbers and decimals, using a variety oftools (e.g., concrete materials, drawings,calculators) and strategies (e.g., estimation,algorithms);

– use estimation when solving problemsinvolving operations with whole numbers,decimals, and percents, to help judge thereasonableness of a solution (Sample problem: A book costs $18.49. The sales-person tells you that the total price,including taxes, is $22.37. How can youtell if the total price is reasonable withoutusing a calculator?);

– evaluate expressions that involve wholenumbers and decimals, including expres-sions that contain brackets, using order ofoperations;

– add and subtract fractions with simple likeand unlike denominators, using a varietyof tools (e.g., fraction circles, Cuisenairerods, drawings, calculators) and algorithms;

– demonstrate, using concrete materials, therelationship between the repeated additionof fractions and the multiplication of thatfraction by a whole number (e.g.,



– determine, through investigation, the rela-tionships among fractions, decimals, per-cents, and ratios;

– solve problems that involve determiningwhole number percents, using a variety oftools (e.g., base ten materials, paper andpencil, calculators) (Sample problem: Ifthere are 5 blue marbles in a bag of 20marbles, what percent of the marbles arenot blue?);

– demonstrate an understanding of rate as acomparison, or ratio, of two measurementswith different units (e.g., speed is a ratethat compares distance to time and thatcan be expressed as kilometres per hour);

– solve problems involving the calculation ofunit rates (Sample problem:You go shop-ping and notice that 25 kg of Ryan’sFamous Potatoes cost $12.95, and 10 kg ofGillian’s Potatoes cost $5.78. Which is thebetter deal? Justify your answer.).

);+ 12 = 3 x+

– add and subtract integers, using a varietyof tools (e.g., two-colour counters, virtualmanipulatives, number lines).

12

12

12

Evannah Sakamoto


Evannah Sakamoto


Evannah Sakamoto

Games - F(x) machine, Ice Cream Catch

Evannah Sakamoto

Work - Chef, Savings and Loans

Evannah Sakamoto

Games - F(x) machine, Ice Cream Catch Work - Climatology

Evannah Sakamoto

Clothing Store, Grocery Store, Sports - Baseball, Work - Savings and Loans, Snack Services

Evannah Sakamoto

Grocery Store, Hardware Store

Evannah Sakamoto

Sports - Baseball, Work - Internet Webmaster, Photography

101G R A D E 7




• report on research into real-life applications of area measurements;

• determine the relationships among units and measurable attributes, including the area of a trapezoid and the volume of a right prism.




– research and report on real-life applica-tions of area measurements (e.g., buildinga skateboard; painting a room).



– sketch different polygonal prisms thatshare the same volume (Sample problem:The Neuman Company is designing anew container for its marbles. The con-tainer must have a volume of 200 cm3.Sketch three possible containers, andexplain which one you would recommend.);

– solve problems that require conversionbetween metric units of measure (e.g.,millimetres and centimetres, grams andkilograms, millilitres and litres) (Sampleproblem: At Andrew’s Deli, cheese is onsale for $11.50 for one kilogram. Howmuch would it cost to purchase 150 g ofcheese?);

– solve problems that require conversionbetween metric units of area (i.e., squarecentimetres, square metres) (Sample prob-lem: What is the ratio of the number ofsquare metres to the number of squarecentimetres for a given area? Use this ratioto convert 6.25 m2 to square centimetres.);

– determine, through investigation using avariety of tools (e.g., concrete materials,dynamic geometry software) and strategies,

the relationship for calculating the area ofa trapezoid, and generalize to develop theformula [i.e., Area = (sum of lengths of parallelsides x height) ÷ 2] (Sample problem: Deter-mine the relationship between the area ofa parallelogram and the area of a trapezoidby composing a parallelogram from con-gruent trapezoids.);

– solve problems involving the estimationand calculation of the area of a trapezoid;

– estimate and calculate the area of compos-ite two-dimensional shapes by decompos-ing into shapes with known area relation-ships (e.g., rectangle, parallelogram, triangle)(Sample problem: Decompose a pentagoninto shapes with known area relationshipsto find the area of the pentagon.);

– determine, through investigation using a variety of tools and strategies (e.g.,decomposing right prisms; stacking con-gruent layers of concrete materials to forma right prism), the relationship betweenthe height, the area of the base, and thevolume of right prisms with simple poly-gonal bases (e.g., parallelograms, trape-zoids), and generalize to develop the for-mula (i.e., Volume = area of base x height)(Sample problem: Decompose right prismswith simple polygonal bases into triangu-lar prisms and rectangular prisms. For eachprism, record the area of the base, theheight, and the volume on a chart.Identify relationships.);


– determine, through investigation using a variety of tools (e.g., nets, concretematerials, dynamic geometry software,Polydrons), the surface area of right prisms;

– solve problems that involve the surfacearea and volume of right prisms and thatrequire conversion between metric mea-sures of capacity and volume (i.e., milli-litres and cubic centimetres) (Sample problem: An aquarium has a base in theshape of a trapezoid. The aquarium is75 cm high. The base is 50 cm long at the front, 75 cm long at the back, and 25 cm wide. Find the capacity of theaquarium.).

Evannah Sakamoto

Work - Packaging

103G R A D E 7




• construct related lines, and classify triangles, quadrilaterals, and prisms;

• develop an understanding of similarity, and distinguish similarity and congruence;

• describe location in the four quadrants of a coordinate system, dilatate two-dimensionalshapes, and apply transformations to create and analyse designs.




– construct related lines (i.e., parallel; per-pendicular; intersecting at 30º, 45º, and60º), using angle properties and a varietyof tools (e.g., compass and straight edge,protractor, dynamic geometry software)and strategies (e.g., paper folding);

– sort and classify triangles and quadrilateralsby geometric properties related to sym-metry, angles, and sides, through investiga-tion using a variety of tools (e.g., geo-board, dynamic geometry software) andstrategies (e.g., using charts, using Venn diagrams) (Sample problem: Investigatewhether dilatations change the geometricproperties of triangles and quadrilaterals.);

– construct angle bisectors and perpendicu-lar bisectors, using a variety of tools (e.g.,Mira, dynamic geometry software, com-pass) and strategies (e.g., paper folding),and represent equal angles and equallengths using mathematical notation;

– investigate, using concrete materials, theangles between the faces of a prism, andidentify right prisms (Sample problem:Identify the perpendicular faces in a set of right prisms.).



– identify, through investigation, the mini-mum side and angle information (i.e.,

side-side-side; side-angle-side; angle-side-angle) needed to describe a unique trian-gle (e.g.,“I can draw many triangles if I’monly told the length of one side, butthere’s only one triangle I can draw if youtell me the lengths of all three sides.”);

– determine, through investigation using avariety of tools (e.g., dynamic geometrysoftware, concrete materials, geoboard),relationships among area, perimeter, corre-sponding side lengths, and correspondingangles of congruent shapes (Sample prob-lem: Do you agree with the conjecturethat triangles with the same area must becongruent? Justify your reasoning.);

– demonstrate an understanding that enlarg-ing or reducing two-dimensional shapescreates similar shapes;

– distinguish between and compare similarshapes and congruent shapes, using a vari-ety of tools (e.g., pattern blocks, gridpaper, dynamic geometry software) andstrategies (e.g., by showing that dilatationscreate similar shapes and that translations,rotations, and reflections generate congru-ent shapes) (Sample problem: A largersquare can be composed from four con-gruent square pattern blocks. Identifyanother pattern block you can use tocompose a larger shape that is similar tothe shape of the block.).




– plot points using all four quadrants of theCartesian coordinate plane;

– identify, perform, and describe dilatations(i.e., enlargements and reductions),through investigation using a variety oftools (e.g., dynamic geometry software,geoboard, pattern blocks, grid paper);

– create and analyse designs involving translations, reflections, dilatations, and/or simple rotations of two-dimensionalshapes, using a variety of tools (e.g.,concrete materials, Mira, drawings,dynamic geometry software) and strategies(e.g., paper folding) (Sample problem:Identify transformations that may beobserved in architecture or in artwork[e.g., in the art of M.C. Escher].);

– determine, through investigation using avariety of tools (e.g., pattern blocks,Polydrons, grid paper, tiling software,dynamic geometry software, concretematerials), polygons or combinations ofpolygons that tile a plane, and describe thetransformation(s) involved.

105G R A D E 7




• represent linear growing patterns (where the terms are whole numbers) using concretematerials, graphs, and algebraic expressions;

• model real-life linear relationships graphically and algebraically, and solve simple algebraicequations using a variety of strategies, including inspection and guess and check.




– represent linear growing patterns, using avariety of tools (e.g., concrete materials,paper and pencil, calculators, spreadsheets)and strategies (e.g., make a table of valuesusing the term number and the term; plotthe coordinates on a graph; write a pat-tern rule using words);

– make predictions about linear growingpatterns, through investigation with con-crete materials (Sample problem: Investigatethe surface area of towers made from asingle column of connecting cubes, andpredict the surface area of a tower that is50 cubes high. Explain your reasoning.);

– develop and represent the general term ofa linear growing pattern, using algebraicexpressions involving one operation (e.g., the general term for the sequence 4, 5, 6, 7, … can be written algebraically as n + 3, where n represents the term num-ber; the general term for the sequence 5,10, 15, 20, … can be written algebraicallyas 5n, where n represents the term number);

– compare pattern rules that generate a pat-tern by adding or subtracting a constant, ormultiplying or dividing by a constant, to getthe next term (e.g., for 1, 3, 5, 7, 9, …, thepattern rule is “start at 1 and add 2 to eachterm to get the next term”) with pattern

rules that use the term number to describethe general term (e.g., for 1, 3, 5, 7, 9, …,the pattern rule is “double the term num-ber and subtract 1”, which can be writtenalgebraically as 2 x n – 1) (Sample problem:For the pattern 1, 3, 5, 7, 9, …, investigateand compare different ways of finding the50th term.).



– model real-life relationships involvingconstant rates where the initial conditionstarts at 0 (e.g., speed, heart rate, billingrate), through investigation using tables of values and graphs (Sample problem:Create a table of values and graph therelationship between distance and time for a car travelling at a constant speed of40 km/h. At that speed, how far wouldthe car travel in 3.5 h? How many hourswould it take to travel 220 km?);

– model real-life relationships involvingconstant rates (e.g., speed, heart rate,billing rate), using algebraic equationswith variables to represent the changingquantities in the relationship (e.g., theequation p = 4t represents the relationshipbetween the total number of people thatcan be seated (p) and the number of tables(t), given that each table can seat 4 people[4 people per table is the constant rate]);

Evannah Sakamoto



– translate phrases describing simple mathe-matical relationships into algebraic expres-sions (e.g., one more than three times anumber can be written algebraically as 1 + 3x or 3x + 1), using concrete mate-rials (e.g., algebra tiles, pattern blocks,counters);

– evaluate algebraic expressions by substi-tuting natural numbers for the variables;

– make connections between evaluatingalgebraic expressions and determining theterm in a pattern using the general term(e.g., for 3, 5, 7, 9, …, the general term isthe algebraic expression 2n + 1; evaluatingthis expression when n = 12 tells you thatthe 12th term is 2(12) + 1, which equals25);

– solve linear equations of the form ax = cor c = ax and ax + b = c or variations suchas b + ax = c and c = bx + a (where a, b,and c are natural numbers) by modellingwith concrete materials, by inspection, orby guess and check, with and without theaid of a calculator (e.g.,“I solved x + 7 =15 by using guess and check. First I tried6 for x. Since I knew that 6 plus 7 equals13 and 13, is less than 15, then I knew thatx must be greater than 6.”).

Evannah Sakamoto

Work - Payroll

Evannah Sakamoto

Work - Payroll

107G R A D E 7





– collect data by conducting a survey or an experiment to do with themselves,their environment, issues in their school or community, or content from anothersubject and record observations or measurements;

– collect and organize categorical, discrete,or continuous primary data and secondarydata (e.g., electronic data from websitessuch as E-Stat or Census At Schools) anddisplay the data in charts, tables, and graphs(including relative frequency tables andcircle graphs) that have appropriate titles,labels (e.g., appropriate units marked onthe axes), and scales (e.g., with appropriateincrements) that suit the range and distrib-ution of the data, using a variety of tools(e.g., graph paper, spreadsheets, dynamicstatistical software);

– select an appropriate type of graph to rep-resent a set of data, graph the data usingtechnology, and justify the choice of graph(i.e., from types of graphs already studied);

– distinguish between a census and a samplefrom a population;

– identify bias in data collection methods(Sample problem: How reliable are yourresults if you only sample girls to deter-mine the favourite type of book read bystudents in your grade?).

Data Relationships


– read, interpret, and draw conclusions fromprimary data (e.g., survey results, measure-ments, observations) and from secondarydata (e.g., temperature data or communitydata in the newspaper, data from theInternet about populations) presented in charts, tables, and graphs (including relative frequency tables and circle graphs);

– identify, through investigation, graphs thatpresent data in misleading ways (e.g., linegraphs that exaggerate change by startingthe vertical axis at a point greater thanzero);

– determine, through investigation, theeffect on a measure of central tendency(i.e., mean, median, and mode) of addingor removing a value or values (e.g., chang-ing the value of an outlier may have a sig-nificant effect on the mean but no effecton the median) (Sample problem: Use aset of data whose distribution across itsrange looks symmetrical, and change someof the values so that the distribution nolonger looks symmetrical. Does thechange affect the median more than themean? Explain your thinking.);

– identify and describe trends, based on thedistribution of the data presented in tablesand graphs, using informal language;



• collect and organize categorical, discrete, or continuous primary data and secondary data and display the data using charts and graphs, including relative frequency tables and circlegraphs;

• make and evaluate convincing arguments, based on the analysis of data;

• compare experimental probabilities with the theoretical probability of an outcome involvingtwo independent events.

Evannah Sakamoto

Work - Data Analyst

Evannah Sakamoto



– make inferences and convincing argumentsthat are based on the analysis of charts,tables, and graphs (Sample problem: Usecensus information to predict whetherCanada’s population is likely to increase.).

Probability


– research and report on real-world applica-tions of probabilities expressed in fraction,decimal, and percent form (e.g., lotteries,batting averages, weather forecasts,elections);

– make predictions about a populationwhen given a probability (Sample problem: The probability that a fish caught in Lake Goodfish is a bass is 29%.Predict how many bass will be caught in a fishing derby there, if 500 fish arecaught.);

– represent in a variety of ways (e.g., treediagrams, tables, models, systematic lists) all the possible outcomes of a probabilityexperiment involving two independentevents (i.e., one event does not affect the

other event), and determine the theoreti-cal probability of a specific outcomeinvolving two independent events (Sampleproblem: What is the probability of rollinga 4 and spinning red, when you roll anumber cube and spin a spinner that isequally divided into four differentcolours?);

– perform a simple probability experimentinvolving two independent events, andcompare the experimental probabilitywith the theoretical probability of a spe-cific outcome (Sample problem: Place1 red counter and 1 blue counter in anopaque bag. Draw a counter, replace it,shake the bag, and draw again. Comparethe theoretical and experimental probabil-ities of drawing a red counter 2 times in arow.).

111G R A D E 8




• represent, compare, and order equivalent representations of numbers, including those involving positive exponents;

• solve problems involving whole numbers, decimal numbers, fractions, and integers, using avariety of computational strategies;

• solve problems by using proportional reasoning in a variety of meaningful contexts.




– express repeated multiplication usingexponential notation (e.g., 2 x 2 x 2 x 2 = 24);

– represent whole numbers in expandedform using powers of ten(e.g., 347 = 3 x 102 + 4 x 101 + 7);

– represent, compare, and order rationalnumbers (i.e., positive and negative fractions and decimals to thousandths);

– translate between equivalent forms of anumber (i.e., decimals, fractions, percents)

– determine common factors and commonmultiples using the prime factorization ofnumbers (e.g., the prime factorization of 12 is 2 x 2 x 3; the prime factoriza-tion of 18 is 2 x 3 x 3; the greatest com-mon factor of 12 and 18 is 2 x 3 or 6; the least common multiple of 12 and 18 is 2 x 2 x 3 x 3 or 36).

Operational Sense


– solve multi-step problems arising fromreal-life contexts and involving wholenumbers and decimals, using a variety of tools (e.g., graphs, calculators) andstrategies (e.g., estimation, algorithms);

– solve problems involving percentsexpressed to one decimal place (e.g.,12.5%) and whole-number percentsgreater than 100 (e.g., 115%) (Sampleproblem: The total cost of an item withtax included [115%] is $23.00. Use baseten materials to determine the pricebefore tax.);

– use estimation when solving problemsinvolving operations with whole numbers,decimals, percents, integers, and fractions,to help judge the reasonableness of a solution;

– represent the multiplication and divisionof fractions, using a variety of tools andstrategies (e.g., use an area model to

– solve problems involving addition, sub-traction, multiplication, and division withsimple fractions;

– represent the multiplication and divisionof integers, using a variety of tools [e.g., ifblack counters represent positive amountsand red counters represent negativeamounts, you can model 3 x (–2) as threegroups of two red counters];

– solve problems involving operations withintegers, using a variety of tools (e.g., two-colour counters, virtual manipulatives,number lines);

= 0.75);34

(e.g.,

multiplied by14

represent );13

Evannah Sakamoto


Evannah Sakamoto

Work - Scheduling

Evannah Sakamoto

Sports - Basketball

Evannah Sakamoto


Evannah Sakamoto

Clothing Store, Grocery Store Sports - Baseball Work - Savings and Loans, Snack Services

Evannah Sakamoto

Clothing Store, Hardware Store, Sports - Baseball, Work - Chef, Savings and Loans, Snack Services


– evaluate expressions that involve integers,including expressions that contain bracketsand exponents, using order of operations;

– multiply and divide decimal numbers byvarious powers of ten (e.g.,“To convert230 000 cm3 to cubic metres, I calculatedin my head 230 000 ÷ 106 to get 0.23 m3.”)(Sample problem: Use a calculator to helpyou generalize a rule for dividing numbersby 1 000 000.);

– estimate, and verify using a calculator, thepositive square roots of whole numbers,and distinguish between whole numbersthat have whole-number square roots (i.e.,perfect square numbers) and those that donot (Sample problem: Explain why asquare with an area of 20 cm2 does nothave a whole-number side length.).



– identify and describe real-life situationsinvolving two quantities that are directlyproportional (e.g., the number of servingsand the quantities in a recipe, mass andvolume of a substance, circumference anddiameter of a circle);

– solve problems involving proportions,using concrete materials, drawings, andvariables (Sample problem: The ratio ofstone to sand in HardFast Concrete is 2 to 3. How much stone is needed if 15 bags of sand are used?);

– solve problems involving percent that arisefrom real-life contexts (e.g., discount, salestax, simple interest) (Sample problem: InOntario, people often pay a provincialsales tax [PST] of 8% and a federal salestax [GST] of 7% when they make a pur-chase. Does it matter which tax is calcu-lated first? Explain your reasoning.);

– solve problems involving rates (Sampleproblem: A pack of 24 CDs costs $7.99.A pack of 50 CDs costs $10.45. What isthe most economical way to purchase 130 CDs?).

Evannah Sakamoto


Evannah Sakamoto

Sports - Baseball Work - Chef, Internet Webmaster, Photography

Evannah Sakamoto

Sports - Baseball Work - Chef, Internet Webmaster, Photography

Evannah Sakamoto

Clothing Store, Grocery Store Sports - Baseball Work - Savings and Loans, Snack Services

Evannah Sakamoto

Clothing Store, Hardware Store

113G R A D E 8




• research, describe, and report on applications of volume and capacity measurement;

• determine the relationships among units and measurable attributes, including the area of a circle and the volume of a cylinder.




– research, describe, and report on applica-tions of volume and capacity measurement(e.g., cooking, closet space, aquarium size)(Sample problem: Describe situationswhere volume and capacity are used inyour home.).



– solve problems that require conversionsinvolving metric units of area, volume, andcapacity (i.e., square centimetres andsquare metres; cubic centimetres andcubic metres; millilitres and cubic cen-timetres) (Sample problem: What is thecapacity of a cylindrical beaker with aradius of 5 cm and a height of 15 cm?);

– measure the circumference, radius, anddiameter of circular objects, using con-crete materials (Sample Problem: Usestring to measure the circumferences ofdifferent circular objects.);

– determine, through investigation using avariety of tools (e.g., cans and string,dynamic geometry software) and strate-gies, the relationships for calculating thecircumference and the area of a circle, andgeneralize to develop the formulas [i.e.,Circumference of a circle = πx diameter ;Area of a circle = πx (radius)2] (Sample problem: Use string to measure the circum-ferences and the diameters of a variety of

cylindrical cans, and investigate the ratioof the circumference to the diameter.);

– solve problems involving the estimationand calculation of the circumference andthe area of a circle;

– determine, through investigation using avariety of tools and strategies (e.g., gener-alizing from the volume relationship forright prisms, and verifying using thecapacity of thin-walled cylindrical con-tainers), the relationship between the areaof the base and height and the volume ofa cylinder, and generalize to develop theformula (i.e., Volume = area of base x height);

– determine, through investigation usingconcrete materials, the surface area of acylinder (Sample problem: Use the labeland the plastic lid from a cylindrical con-tainer to help determine its surface area.);

– solve problems involving the surface areaand the volume of cylinders, using a vari-ety of strategies (Sample problem:Compare the volumes of the two cylin-ders that can be created by taping the topand bottom, or the other two sides, of astandard sheet of paper.).





• demonstrate an understanding of the geometric properties of quadrilaterals and circles andthe applications of geometric properties in the real world;

• develop geometric relationships involving lines, triangles, and polyhedra, and solve problemsinvolving lines and triangles;

• represent transformations using the Cartesian coordinate plane, and make connectionsbetween transformations and the real world.




– sort and classify quadrilaterals by geome-tric properties, including those based ondiagonals, through investigation using avariety of tools (e.g., concrete materials,dynamic geometry software) (Sampleproblem: Which quadrilaterals have diagonals that bisect each other perpendicularly?);

– construct a circle, given its centre andradius, or its centre and a point on the circle, or three points on the circle;

– investigate and describe applications ofgeometric properties (e.g., properties oftriangles, quadrilaterals, and circles) in thereal world.



– determine, through investigation using avariety of tools (e.g., dynamic geometrysoftware, concrete materials, geoboard),relationships among area, perimeter, corre-sponding side lengths, and correspondingangles of similar shapes (Sample problem:Construct three similar rectangles, usinggrid paper or a geoboard, and compare theperimeters and areas of the rectangles.);

– determine, through investigation using avariety of tools (e.g., dynamic geometrysoftware, concrete materials, protractor)and strategies (e.g., paper folding), theangle relationships for intersecting linesand for parallel lines and transversals, andthe sum of the angles of a triangle;

– solve angle-relationship problems involv-ing triangles (e.g., finding interior anglesor complementary angles), intersectinglines (e.g., finding supplementary angles oropposite angles), and parallel lines andtransversals (e.g., finding alternate anglesor corresponding angles);

– determine the Pythagorean relationship,through investigation using a variety oftools (e.g., dynamic geometry software;paper and scissors; geoboard) and strategies;

– solve problems involving right trianglesgeometrically, using the Pythagorean relationship;

– determine, through investigation usingconcrete materials, the relationshipbetween the numbers of faces, edges,and vertices of a polyhedron (i.e.,number of faces + number of vertices =number of edges + 2) (Sample problem:Use Polydrons and/or paper nets to

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115G R A D E 8

construct the five Platonic solids [i.e.,tetrahedron, cube, octahedron, dodecahe-dron, icosahedron], and compare the sumof the numbers of faces and vertices to thenumber of edges for each solid.).



– graph the image of a point, or set ofpoints, on the Cartesian coordinate planeafter applying a transformation to theoriginal point(s) (i.e., translation; reflec-tion in the x-axis, the y-axis, or the anglebisector of the axes that passes through thefirst and third quadrants; rotation of 90°,180°, or 270° about the origin);

– identify, through investigation, real-worldmovements that are translations, reflec-tions, and rotations.

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• represent linear growing patterns (where the terms are whole numbers) using graphs,algebraic expressions, and equations;

• model linear relationships graphically and algebraically, and solve and verify algebraic equations, using a variety of strategies, including inspection, guess and check, and using a “balance” model.




– represent, through investigation with concrete materials, the general term of alinear pattern, using one or more algebraicexpressions (e.g.,“Using toothpicks, Inoticed that 1 square needs 4 toothpicks,2 connected squares need 7 toothpicks,and 3 connected squares need 10 tooth-picks. I think that for n connected squaresI will need 4 + 3(n – 1) toothpicks, becausethe number of toothpicks keeps going upby 3 and I started with 4 toothpicks. Or, ifI think of starting with 1 toothpick andadding 3 toothpicks at a time, the patterncan be represented as 1 + 3n.”);

– represent linear patterns graphically (i.e.,make a table of values that shows the termnumber and the term, and plot the coor-dinates on a graph), using a variety oftools (e.g., graph paper, calculators,dynamic statistical software);

– determine a term, given its term number,in a linear pattern that is represented by agraph or an algebraic equation (Sampleproblem: Given the graph that representsthe pattern 1, 3, 5, 7, …, find the 10thterm. Given the algebraic equation thatrepresents the pattern, t = 2n – 1, find the100th term.).



– describe different ways in which algebracan be used in real-life situations (e.g., thevalue of $5 bills and toonies placed in aenvelope for fund raising can be repre-sented by the equation v = 5f + 2t);

– model linear relationships using tables ofvalues, graphs, and equations (e.g., thesequence 2, 3, 4, 5, 6, … can be representedby the equation t = n + 1, where n repre-sents the term number and t represents theterm), through investigation using a varietyof tools (e.g., algebra tiles, pattern blocks,connecting cubes, base ten materials)(Sample problem: Leah put $350 in a bankcertificate that pays 4% simple interesteach year. Make a table of values to showhow much the bank certificate is worthafter five years, using base ten materials tohelp you. Represent the relationship usingan equation.);

– translate statements describing mathemati-cal relationships into algebraic expressionsand equations (e.g., for a collection of triangles, the total number of sides is equalto three times the number of triangles or s = 3n);

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117G R A D E 8

– evaluate algebraic expressions with up tothree terms, by substituting fractions, deci-mals, and integers for the variables (e.g.,

y = 0.6, and z = –1);

– make connections between solving equa-tions and determining the term number ina pattern, using the general term (e.g., forthe pattern with the general term 2n + 1,solving the equation 2n + 1 = 17 tells youthe term number when the term is 17);

– solve and verify linear equations involvinga one-variable term and having solutionsthat are integers, by using inspection, guessand check, and a “balance” model (Sampleproblem: What is the value of the variablein the equation 30x – 5 = 10?).

,12

evaluate 3x + 4y = 2z, where x =

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– collect data by conducting a survey or an experiment to do with themselves,their environment, issues in their school or community, or content from anothersubject, and record observations or measurements;

– organize into intervals a set of data that isspread over a broad range (e.g., the age ofrespondents to a survey may range over 80years and may be organized into ten-yearintervals);

– collect and organize categorical, discrete,or continuous primary data and secondarydata (e.g., electronic data from websitessuch as E-Stat or Census At Schools), anddisplay the data in charts, tables, and graphs(including histograms and scatter plots)that have appropriate titles, labels (e.g.,appropriate units marked on the axes), andscales (e.g., with appropriate increments)that suit the range and distribution of thedata, using a variety of tools (e.g., graphpaper, spreadsheets, dynamic statistical software);

– select an appropriate type of graph to rep-resent a set of data, graph the data usingtechnology, and justify the choice of graph(i.e., from types of graphs already studied,including histograms and scatter plots);

– explain the relationship between a census,a representative sample, sample size, and apopulation (e.g.,“I think that in mostcases a larger sample size will be morerepresentative of the entire population.”).

Data Relationships


– read, interpret, and draw conclusions fromprimary data (e.g., survey results, measure-ments, observations) and from secondarydata (e.g., election data or temperaturedata from the newspaper, data from theInternet about lifestyles), presented incharts, tables, and graphs (including fre-quency tables with intervals, histograms,and scatter plots);

– determine, through investigation, theappropriate measure of central tendency(i.e., mean, median, or mode) needed tocompare sets of data (e.g., in hockey,compare heights or masses of players on defence with that of forwards);

– demonstrate an understanding of theappropriate uses of bar graphs and histo-grams by comparing their characteristics(Sample problem: How is a histogram sim-ilar to and different from a bar graph? Useexamples to support your answer.);

– compare two attributes or characteristics(e.g., height versus arm span), using a scat-ter plot, and determine whether or not



• collect and organize categorical, discrete, or continuous primary data and secondary data and display the data using charts and graphs, including frequency tables with intervals,histograms, and scatter plots;

• apply a variety of data management tools and strategies to make convincing argumentsabout data;

• use probability models to make predictions about real-life events.

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the scatter plot suggests a relationship(Sample problem: Create a scatter plot tocompare the lengths of the bases of severalsimilar triangles with their areas.);

– identify and describe trends, based on therate of change of data from tables andgraphs, using informal language (e.g.,“The steep line going upward on thisgraph represents rapid growth. The steepline going downward on this other graphrepresents rapid decline.”);

– make inferences and convincing argu-ments that are based on the analysis ofcharts, tables, and graphs (Sample problem:Use data to make a convincing argumentthat the environment is becoming increas-ingly polluted.);

– compare two attributes or characteristics,using a variety of data management toolsand strategies (i.e., pose a relevant ques-tion, then design an experiment or survey,collect and analyse the data, and drawconclusions) (Sample problem: Comparethe length and width of different-sizedleaves from a maple tree to determine ifmaple leaves grow proportionally. Whatgeneralizations can you make?).

Probability


– compare, through investigation, the theo-retical probability of an event (i.e., theratio of the number of ways a favourableoutcome can occur compared to the totalnumber of possible outcomes) with exper-imental probability, and explain why theymight differ (Sample problem: Toss a faircoin 10 times, record the results, andexplain why you might not get the pre-dicted result of 5 heads and 5 tails.);

– determine, through investigation, the tendency of experimental probability toapproach theoretical probability as thenumber of trials in an experimentincreases, using class-generated data andtechnology-based simulation models(Sample problem: Compare the theoreticalprobability of getting a 6 when tossing anumber cube with the experimental prob-abilities obtained after tossing a numbercube once, 10 times, 100 times, and 1000 times.);

– identify the complementary event for agiven event, and calculate the theoreticalprobability that a given event will notoccur (Sample problem: Bingo uses thenumbers from 1 to 75. If the numbers arepulled at random, what is the probabilitythat the first number is a multiple of 5? isnot a multiple of 5?).

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Documents

Mathville Middle School Ontario Correlations