onap - nelson · · 2009-07-20the measurement strand of the ontario curriculum for grade 7...

Ontario Numeracy Assessment PackageONAP

8SAMPLER

Measurement

MEASUREMENT

The Measurement strand of the Ontario Curriculum for Grade 7 identifies sevenMathematical Process Expectations: problem solving, reasoning and proving,reflecting, selecting tools and computational strategies, connecting, representing, andcommunicating. Using these process expectations, students study, learn, and applyconcepts and skills organized under the big ideas/headings of Attributes, Units, andMeasurement Sense; and Measurement Relationships.

The following chart highlights key knowledge and skill development as students movefrom Grade 7 to 8.

97Ontario Numeracy AssessmentNEL

Measurem

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GRADE 7 GRADE 8

• research and report on real-lifeapplications of area measurement

• solve problems that require conversionbetween metric units of measure (e.g.,millimetres and centimetres, grams andkilograms, millilitres and litres, squarecentimetres and square metres)

• estimate and calculate the area ofcomposite two-dimensional shapes bydecomposing into shapes with knownarea relationships

• determine the relationship forcalculating the area of a trapezoid andgeneralize to develop a formula

• solve problems involving the estimationand calculation of the area of atrapezoid

• determine the surface area of rightprisms

• determine the relationship betweenthe height, the area of the base, andthe volume of right prisms with simplepolygonal bases, and generalize todevelop the formula

• sketch different polygonal prisms thatshare the same volume

• solve problems that involve the surfacearea and volume of right prisms andthat require conversion between metricmeasures of capacity and volume

• research, describe, and report on real-lifeapplications of volume and capacitymeasurement

• solve problems that require conversionsinvolving metric units of area, volume,and capacity (e.g., millimetres andcentimetres, grams and kilograms, squarecentimetres and square metres)

• measure the circumference, radius, anddiameter of circular objects, usingconcrete materials

• determine the relationships for calculatingthe circumference and the area of a circle,and generalize to develop the formula

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

• determine the surface area of a cylinder

• determine the relationship between thearea of the base and height and thevolume of a cylinder, and generalize todevelop the formula

• solve problems that involve the surfacearea and the volume of cylinders

For each strand of the curriculum, ONAP provides three types of assessment materials.Consider the following points when administering the assessments for the strand.

Part A: Activating Prior Knowledge (page 99) • The activities in Part A have been created to activate students’ knowledge before

they complete the Part B assessment or the Part C performance tasks.

• No score is assigned during this part of the assessment.

• It is recommended that you spend one or two periods working on and discussingthe activities provided in this part.

Part B: Concepts and Skills Assessment (page 107)• The assessment in Part B addresses each of the specific expectations within the

two overall expectations in the Measurement strand.

• Students will be responding to a mix of questions: short response, fill-in-the-blank,and multiple choice.

• Most students will be able to complete the entire assessment in a 60-minuteperiod. Individual students may be allowed additional time to complete theassessment if needed as long as they complete the assessment in one sitting.

Part C: Performance-Based Assessment Tasks (page 125)• The two performance tasks in Part C are designed to provide insights into how

well students are able to perform in terms of the categories of the OntarioAchievement Chart: Knowledge and Understanding, Thinking, Communicationand Application.

• All of the overall expectations for this strand have been assessed through theConcept and Skills Assessment in Part B.

• It is recommended that you select one performance task for the Measurementstrand.

• Each task is designed to be completed in a 45- to 60-minute period. If necessary,provide additional time as long as students complete the task in one sitting.

98 Ontario Numeracy Assessment NEL

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Part A: Activating Prior Knowledge

Administration

To activate students’ knowledge of the Measurement strand, choose one or both of thefollowing activities to work on prior to administering the assessments. Introductoryand culminating suggestions are provided for each. No score is assigned for theseactivities.

Timing

It is recommended that you spend one or two periods working on and discussing theactivities provided in this part.

Accommodating Students with Special Needs

Observe students as they complete the activities. While the activities in this sectionare not designed as a formal diagnostic assessment, you may want to consider whetherstudents who are having extreme difficulties with the activities are ready to participatein the full ONAP assessment for this grade level. Observations at this stage might alsoindicate students who will need special accommodations during the assessment, suchas having someone read questions to them or scribe responses.

Activity 1: The Climbing ConversionMaterials

• BLM A1: Activity 1: The Climbing Conversion—Game Cards (one per student pair)

• BLM A2: Activity 1: The Climbing Conversion—Game Board (one per student)

• scissors

Introducing the Activity

In preparation for playing this game, review conversions using metres and kilometres,and metres and millimetres. Ask students:

• How many metres are in 1 km? (1000)

• How many kilometres are in 1 m? (0.001)

• How many millimetres are in 1 m? (1000)

• How many metres are in 1 mm? (0.001)

As well, review conversions between the area units of square metres and square centimetres. Ask students:

• How many centimetres are in 1 m? (100)

• How many square centimetres are in 1 m2? (100 � 100, or 10 000)

• How many metres are in 1 cm? (0.01)

• How many square metres are in 1 cm2? (0.01 � 0.01, or 0.0001)

Record the following area conversions on the board for students’ reference:1 m2 � 10 000 cm2 1 cm2 � 0.0001 m2

Distribute BLM A1: Activity 1: The Climbing Conversion—Game Cards and BLM A2:Activity 1: The Climbing Conversion—Game Board to students to use for this activity.


Procedure

Number of players: twoGoal: to fill the spaces on the game board with game cards so that the calculatednumerical answers on the cards are placed in order from greatest to leastHow to play:

• Step 1—Players cut out the cards and lay them out face up.

• Step 2—Players take turns choosing a card, completing the conversion, andplacing the card on their game board. They will try to place the cards so that thenumerical values that they calculate are placed in order from greatest to least.Players may use a turn to change the position of a card on their game board, butthey must wait for their next turn to choose a new card.

• Step 3—Play continues until one player has covered all the spaces on his or hergame board.

Students may play subsequent games using the remaining cards.

Variation:Students choose five cards that will give them the greatest or least total. Beforestarting to play, the two students must decide whether they are aiming for the greatestor least total. They must use the conversion number that they calculate, but do notneed to include the unit of measure.

Culminating Discussion

1. What strategies helped as you played the game?When I filled in the top of the board, I started with cards that had the larger unit on theleft side because I knew that the number for the equivalent smaller unit would be large.When I filled in the bottom of the board, I looked for cards with the smaller unit on theleft side. I also noticed that cards that converted square metres to square centimetreshad big numbers in the blank.

2. Which conversions were the most difficult?I had to think more about converting area units than I did about converting units forlength. Sometimes it was a little hard to convert the units if there was a decimal in oneof the numbers.

3. Did the game become more challenging as you played a second time with fewercards?Yes, because I picked the easiest numbers to convert in the first game.

Answers

Activity cards:120 mm � 12.0 cm 0.5000 m2 � 5000 cm2

3500 L � 3.500 kL 4000 mL � 4.000 L5250 g � 5.250 kg 7 km � 7000 m30 000 cm2 � 3.000 m2 8000 m � 8.000 km74 500 m � 74.500 km 4.0000 m2 � 40 000 cm2

324.0 cm � 3240 mm 20 000 cm2 � 2.0000 m2

14.0 m2 � 140 000 cm2 5.60 L � 5600 mL19.0 g � 1900 cg 89.0 m � 8900 cm


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70 000 g � 70.000 kg 35 000 g � 35.000 kg2500 cm2 � 0.250 m2 80 000 cm2 � 8.0000 m2

Activity 2: Capacity—Least to GreatestMaterials

• BLM A3: Activity 2: Capacity—Least to Greatest (one per student)

• a metric measuring cup (for the teacher)

• 10 cm � 10 cm � 10 cm cube (for the teacher)

Introducing the Activity

Review the relationship between the base area and volume of prisms. As well, reviewthe formula for calculating the area of the trapezoid, A � (a � b) � 2 � h, where a and b are the parallel sides and h is the height.

Demonstrate the relationship between volume and capacity. Fill a 10 cm � 10 cm � 10 cm(or 1000 cm3) cube with 1000 mL of accurately measured water to demonstrate that thevolume of the cube is 1000 cm3 and its capacity is 1000 mL or 1 L.

Distribute BLM A3: Activity 2: Capacity—Least to Greatest and have students examinethe information given for the three vases. Have them predict the order from least togreatest capacity. Then have them calculate the capacity of each container and orderthem. Students may work individually or in pairs.

Culminating Discussion

1. How did your prediction compare to your calculated answer?I predicted that the order from least to greatest capacity would be A, C, B, but thecalculated order was A, B, C. I shouldn’t have thought that just because vase B wastallest it would have the greatest capacity.

2. Explain how you calculated the capacity of vase C.I imagined three separate prisms. Two were trapezoid-based prisms and one was asquare-based prism, which was in the middle.To get the volume of each, I multiplied the area of the base by the height of the prism.Then I added the volumes together. Finally, I converted the total volume to capacity.

Answers

Least to greatest capacity: A, B, C

Vase AArea of base: 10 � 6 � 60 cm2

Height: 15 cmVolume: 60 cm2 � 15 cm � 900 cm3

Capacity: 900 mL or 0.9 L

Vase BArea of base: [(8 � 6) � 2] � 5 � 35 cm2

Height � 30 cm Volume: 35 cm2 � 30 cm � 1050 cm3



Vase CBase consists of 2 trapezoids and 1 square.Area of base: (3 � 7) � 2 � 4 � (3 � 3) � (3 � 7) � 2 � 4 � 49 cm2



OR

Each trapezoid prism:Area of trapezoid: [(7 � 3) � 2] � 4 � 20 cm2


Rectangular prism in the middle:Area of square base: 3 � 3 � 9 cm2


Total volume: 500 cm3 � 500 cm3 � 225 cm3 � 1225 cm3



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Copyright © 2010 by Nelson Education Ltd.

ONAP MEA A-1

Activity 1: The Climbing Conversion—Game Cards

NEL

120 mm � cm 0.5000 m2 � cm2

3500 L � kL 4000 mL � L

5250 g � kg 7 km � m

30 000 cm2 � m2 8000 m � km

74 500 m � km 4.0000 m2 � cm2

324.0 cm � mm 20 000 cm2 � m2

14.0 m2 � cm2 5.60 L � mL

19.0 g � cg 89.0 m � cm

70 000 g � kg 35 000 g � kg

2500 cm2 � m2 80 000 cm2 � m2


Name: __________________________________________ Date: ____________________________

NEL

How to play:

Step 1 Cut out the game cards and lay them out face up.

Step 2 Take turns choosing a card, completing the conversion, andplacing the card on the game board. Try to place the cards so thatthe numerical values that you calculate are in order from greatestto least. You may use a turn to change the position of a card onthe game board, but then you must wait for your next turn tochoose a card.

Step 3 Play continues until one player has covered all spaces on his orher game board.

ONAP MEA A-2

Activity 1: The Climbing Conversion—Game Board

Copyright © 2010 by Nelson Education Ltd.NEL

ONAP MEA A-3.1

Activity 2: Capacity—Least to Greatest (page 1)

Predict the order of the three vases from least to greatest capacity.

_____ _____ _____

Calculate the capacity of each vase, and then list them in order

from least to greatest capacity: _____ _____ _____

Name: __________________________________________ Date: ____________________________

8 cm

6 cm

5 cm

h � 30 cm

Vase Bcapacity: ________ mL

6 cm

10 cm

h � 15 cm

Vase Acapacity: ________ mL

Copyright © 2010 by Nelson Education Ltd. NEL

Activity 2: Capacity—Least to Greatest (page 2)

ONAP MEA A-3.2

Name: __________________________________________ Date: ____________________________

3 cm 7 cm7 cm

3 cm

4 cm4 cm

h � 25 cm

Vase Ccapacity: ________ mL

Part B: Concepts and Skills AssessmentAdministration

This assessment addresses each of the specific expectations within the two overallexpectations in the Measurement strand. Part B includes several styles of questions:short response, fill-in-the-blank, and multiple choice.

Timing

Most students will be able to complete the entire assessment in a 60-minute period. Ifnecessary, provide students with additional time to complete the assessment as long asthey complete it in one sitting.

Materials

Introducing the Assessment

Inform students that they will be completing an assessment to help you get to knowwhat they have learned about math in earlier grades. Tell students that it is importantthat they answer the questions as fully as possible. To communicate effectively, theycan use pictures, numbers, words, and/or diagrams to represent their thinking.

Encourage students to use other material, such as centimetre cubes or centimetre gridpaper, that they think might help them to answer the questions.

Note: Calculators are not recommended during this assessment.


If individuals or groups of students have difficulties with reading, consider reading thequestions aloud as they complete the assessment.

If individual students have difficulties explaining their thinking in writing, considerproviding scribes to record for the students, or encourage students to show andexplain their thinking using concrete materials.

Some students will require additional time to complete the assessment. You may wishto note this accommodation in your anecdotal notes about the students. However,there should be no reduction of the student’s overall score in terms of the amount oftime it takes the student to complete the assessment.


FOR THE TEACHER FOR EACH STUDENT OPTIONAL MATERIALS

• Individual StudentScoring Guide: pp. 119–121

• Class Tracking Sheet: pp. 122–123

• ONAP 8 CD-ROM(optional)

• Assessment Part B: pp. 109–117

• pencil

• eraser

• ruler

• centimetre cubes

• centimetre grid paper

Scoring the Assessment

A detailed Individual Student Scoring Guide is provided on pages 119 to 121. Theguide is designed to be completed for each student. The individual scores can then beused to fill in the Class Tracking Sheet on pages 122 to 123. Alternatively, you mayrecord student results directly on the Class Tracking Sheet. The results may also berecorded electronically using the ONAP 8 CD-ROM.

While great care has been taken to consider the range of possible answers for eachquestion, there will be times when you will need to apply your professional judgmentto score an individual answer. You may use the Curriculum Correlation chart providedon page 124 to help you to determine whether the student has demonstrated theintended concept knowledge or skill based on the overall and specific expectationsbeing assessed by the particular question.

At times, a student may provide an answer that you think does not completelyrepresent his or her knowledge and skill level. You may ask probing questions to betterassess the student’s overall understanding.

Some questions are delivered in more than one part (a and b) and are given more thanone point. Should a student’s answer in one part reveal that a correct answer in theother part was arrived at for the wrong reason, a score of zero should be given forboth parts.


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Name: __________________________________________ Date: ____________________________

NEL

Measurement

1. Three flower beds were created in a rectangular area 10 m by 8 m, asshown below. Small stones were spread between the flower beds.

a) What is the total area of the flower beds? _____________________

ONAP MEA B-1

3

2

1

Scale: 1 cm represents 1 m

Show your work.


Name: __________________________________________ Date: ____________________________

NEL

ONAP MEA B-2

b) What is the total area covered by the small stones? _____________

2. The Caramel Candy Company is designing a new container for itscaramel candy. The container must have a volume of 300 cm3.

a) Sketch and label the dimensions of two possible containers thatare rectangular prisms.

b) Sketch and label the dimensions of a triangular base for atriangular prism container that also has a volume of 300 cm3.What is the height of the container?

Show your work.


Name: __________________________________________ Date: ____________________________

NEL

3. a) How many 160 cm lengths of ribbon can be cut from a ribbonthat is 8 m long?

A 500

B 20

C 50

D 5

b) A craft store sells model paint in 250 mL bottles. A litre of themodel paint costs $44.84. What is the cost for one 250 ml bottle?

A $17.93

B $4.48

C $11.21

D $112.10

4. Andy is putting a new countertop in his kitchen.

a) A countertop is 1 m � 1 m. What is the area of the countertop insquare centimetres (cm2)?

______________________________________________________________

b) What is the ratio of the number of square metres (m2) to thenumber of square centimetres (cm2) for a given area?

______________________________________________________________

c) Use the ratio from part b) to determine the number of square centimetres (cm2) in a countertop that is 5.75 m2.

______________________________________________________________

ONAP MEA B-3


Name: __________________________________________ Date: ____________________________

NEL

ONAP MEA B-4

5. Viraf wants a formula for the area of a trapezoid. He knows the areaof a parallelogram. So he made two copies of the trapezoid into aparallelogram.

a) What is the formula for the area of the parallelogram? Use thevariables in the diagram.

______________________________________________________________b) What is the formula for the area of each trapezoid? Use the

variables in the diagram.

______________________________________________________________c) How are the formulas in 5 a) and 5 b) related?

h

b

a

h

b a

a b

h


Name: __________________________________________ Date: ____________________________

NEL

6. Angelo’s class is participating in a math olympics. Each team isdesigning a team logo. This is Angelo’s team logo.

a) What is the area of the whole grid? ______ square unitsHow can you estimate the area of the trapezoid?

b) Calculate the area of the trapezoid. Was your estimate reasonable?

ONAP MEA B-5

Show your work.

Explain your thinking.

The area of the trapezoid is about ______ square units.


Name: __________________________________________ Date: ____________________________

NEL

ONAP MEA B-6

7. a) Look at the polygons. Draw lines to show how each could bebroken up into parallelograms, triangles, or trapezoids so that youcan use formulas to calculate the area of the whole polygon.

b) Calculate the area of each polygon below.

_____________ square units _____________ square units

Show your work.


Name: __________________________________________ Date: ____________________________

NEL

8. Examine the prisms.

a) Complete the chart for each shape.

b) What relationship did you use to determine the volume?

______________________________________________________________

ONAP MEA B-7

A

4 cm

4 cm8 cm

B C

4 cm

4 cm8 cm

6 cm

2 cm

4 cm8 cm

Dimensions (cm)

Prism Area of base (cm2) Height (cm) Volume (cm3)

A

B

C


Name: __________________________________________ Date: ____________________________

NEL

ONAP MEA B-8

9. a) Calculate the volume and the surface area of cubes in the table below.

b) If the pattern in the table continues, will the volume and surfacearea of a cube ever be the same numerical value?

1 cm � 1 cm � 1 cm

2 cm � 2 cm � 2 cm

Length (cm) Width (cm) Height (cm) Volume (cm3) Surface area (cm2)

1 1 1

2 2 2

3 3 3

4 4 4

Explain your thinking.


Name: __________________________________________ Date: ____________________________

NEL

10. The Dolphin Swim Club is building a new lap pool for training. The pool is 2 m deep, 25 m long, and 10 m wide.

a) What is the volume of the pool? ____________________

b) The pool is filled so that the water is 1.5 m deep. How many litresof water are in the pool?

c) What is the total surface area of the sides and bottom of the

pool? ____________________

ONAP MEA B-9

Show your work.

______________ L of water are in the pool.

Show your work.

The total surface area is ________________ m2.

Copyright © 2010 by Nelson Education Ltd. 119NEL

ONAP INDIVIDUAL STUDENT SCORING GUIDEGRADE 8: MEASUREMENT—PART B

Name: ________________________________________________________ Date: _______________________________________

Overall Expectation 7m31 (Attributes, Units, and Measurement Sense):Report on research into real-life applications of area measurements.

7m33

1. a) 1 point for calculating with one errorOR 2 points for the correct process and calculations; e.g.,

Area of shape 1: 2 � 3 � 2 � 3 m2

Area of shape 2: 2 � 4 � 8 m2

Area of shape 3: (7 � 3) � 3 � 2 � 15 m2

Total area of the gardens: 3 m2 � 8 m2 � 15 m2 � 26 m2

7m33

1. b) 1 point for calculating with one errorOR 2 points for calculating correctly; e.g.,

Rectangle: 10 m2 � 8 m2 � 80 m2

Flower beds: 26 m2

Stone: 80 m2 � 26 m2 � 54 m2

NOTE: Give points for correct process and calculations even if answer to a) is incorrect.

Total for Overall Expectation 7m31

Overall Expectation 7m32 (Measurement Relationships): Determine the relationships among units and measurable attributes, including the area of a trapezoid and thevolume of a right prism.

7m34

2. a) 1 point for two correctly labelled sketches of containers

NOTE: Possible dimensions include but are not limited to 3 cm � 10 cm � 10 cm;

or 6 cm � 5 cm � 10 cm; or 15 cm � 2 cm � 10 cm

7m34

2. b) 1 point for a correctly labelled sketch of a triangle and a recorded prism height; e.g., a sketch of a triangle with height labelled 6 cm and base labelled 10 cm, and prism height is 10 cm or a sketch of a triangle with height labelled 10 cm and base labelled 10 cm, and prism height is 6 cm.

NOTE: It is the base and height of the triangular base that are required; these are not sides unless the triangle sketched is a right triangle.

7m35

3. a) D 5

1 point

7m35

3. b) C $11.21

1 point

4

Copyright © 2010 by Nelson Education Ltd.120 NEL

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7m36

4. a) 100 � 100 � 10 000 cm2

1 point

7m36

4. b) 1:10 000; or 1 m to 10 000 cm

1 point

7m36

4. c) 5.75 � 10 000 � 57 500 cm2

1 point

7m37

5. a) A � (a � b) � h; or A � (a � h) � (b � h)

1 point

7m37

5. b) A � (a � b) � 2 � h; or A � (a + b) � h � 2; or A � �(a �

2h)

� � �(b �

2h)

�

1 point

7m37

5. c) 1 point for a brief explanation; e.g., The area of each congruent trapezoid is �12

� the area of the parallelogram.

AND 1 point for a complete explanation of why they are related; e.g., I can make a parallelogram from two trapezoids, but one needs to be upside down. I know the formula for the area of a rectangle and I know that the area of one of the trapezoids will be half. It is still base � height, but I need to divide by 2. The base is the sum of the two lengths, or the top and bottom of the trapezoid.

7m38

6. a) 1 point for a reasonable explanation; e.g., The area of the grid is 30 square units. The area of the trapezoid is about �

12

� of the grid so it's about 15 square units.

7m38

6. b) 14 square units

1 point for the correct area and relating it to estimate; e.g.,

Sample calculation:

Trapezoid: (2 � 5) � 2 � 4 � 14 square units

My calculation is close to my estimate.

7m39

7. a) Sample answers:

1 point for one correct answer

OR 2 points for two correct answers


7m39

7. b) 48 square units; 42 square units

1 point for one correct answer

OR 2 points for two correct answers with explanations; e.g., The polygon on the left is made up of 2 triangles and a square. The square has an area of 6 � 4 � 24 square units and each triangle has an area of �

12

� � 6 � 4 � 12 square units. 24 � 12 � 12 � 48 square units

The polygon on the right is made up of a triangle with area �12

� � 6 � 4 � 12 square units and a trapezoid with area (12 � 8) � 3 � 2 � 30 square units. 12 � 30 � 42 square units

7m40

8. a) A: 8, 8, 64; B: 16, 8, 128 or 32, 4, 128; C: 16, 8, 128

1 point for 6 to 8 correct answers

OR 2 points for all 9 answers correct

7m40

8. b) area of base times height

1 point

7m41

9. a) Volume: 1, 8, 27, 64

Surface area: 6, 24, 54, 96

1 point for 6 or 7 answers correct

OR 2 points for all 8 answers correct

7m41

9. b) 1 point for identifying that the numbers will be the same, accompanied by a reasonable explanation; e.g., Yes, a 6 cm � 6 cm � 6 cm cube will have a volume of 216 cm3 and a surface area of 216 cm2. This makes sense because there are six surfaces, so the surface area will be 6 � 6 cm � 6 cm.

7m42

10. a) 2 m � 25 m � 10 m � 500 m3

1 point

7m42

10. b) 150 cm � 2500 cm � 1000 cm � 375 000 000 cm3

� 375 000 000 mL

� 375 000 L

1 point for correctly calculating the volume

AND 1 point for correctly converting to litres

7m42

10. c) 2(2 � 10) � 2(2 � 25) � (10 � 25)

� 40 � 100 � 250

� 390 m2

1 point

Total for Overall Expectation 7m32

27


ONAP GRADE 8: MEASUREMENT

Date: ______________________________ Grade: _________ School: ______________________________________________

Overall Expectation 7m31 (Attributes,Units, and

Measurement Sense)Report on research

into real-life applications of area

measurements.

Specific Expectation # 7m33 7m34 7m35

Question # 1. a) 1. b) 2. a) 2. b) 3. a) 3. b)

Student NameGender(M/F)

IEP/ELL 2 2 4 1 1 1 1

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CLASS TRACKING SHEET – PART B

Board: _____________________________________________ Teacher Name: _____________________________________

7m32 (Measurement Relationships)Determine the relationships among units and measurable attributes, including the area of a trapezoid

and the volume of a right prism.

7m36 7m37 7m38 7m39 7m40 7m41 7m42

4. a) 4. b) 4. c) 5. a) 5. b) 5. c) 6. a) 6. b) 7. a) 7. b) 8. a) 8. b) 9. a) 9. b) 10. a) 10. b) 10. c)

1 1 1 1 1 2 1 1 2 2 2 1 2 1 1 2 1 27

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Overall Expectation 7m31 (Attributes, Units, and Measurement Sense):

Report on research into real-life applications of area measurements.

ONTARIO CURRICULUM CORRELATION TO ONAP MEASUREMENT 8 — PART BNOTE: This correlation is to the Grade 7 Ontario Curriculum Expectations.

QuestionNumber Specific Expectation

1. a)–b) 7m33: research and report on real-life applications of area measurements (e.g., building a skateboard;painting a room)

Overall Expectation 7m32 (Measurement Relationships):

Determine the relationships among units and measurable attributes, including the area of a trapezoid and thevolume of a right prism.

QuestionNumber Specific Expectation

2. a)–b) 7m34: sketch different polygonal prisms that share the same volume

3. a)–b) 7m35: solve problems that require conversion between metric units of measure (e.g., millimetres andcentimetres, grams and kilograms, millilitres and litres)

4. a)–c) 7m36: solve problems that require conversion between metric units of area (i.e., square centimetres,square metres)

5. a)–c) 7m37: determine, through investigation using a variety of tools (e.g., concrete materials, dynamicgeometry software) and strategies, the relationship for calculating the area of a trapezoid, andgeneralize to develop the formula [i.e., Area = (sum of lengths of parallel sides � height) ÷ 2]

6. a)–b) 7m38: solve problems involving the estimation and calculation of the area of a trapezoid

7. a)–b) 7m39: estimate and calculate the area of composite two-dimensional shapes by decomposing intoshapes with known area relationships (e.g., rectangle, parallelogram, triangle)

8. a)–b) 7m40: determine, through investigation using a variety of tools and strategies (e.g., decomposing right prisms; stacking congruent layers of concrete materials to form a right prism), the relationshipbetween the height, the area of the base, and the volume of right prisms with simple polygonal bases (e.g., parallelograms, trapezoids), and generalize to develop the formula (i.e., Volume = area of base � height)

9. a)–b) 7m41: determine, through investigation using a variety of tools (e.g., nets, concrete materials, dynamicgeometry software, Polydrons), the surface area of right prisms

10. a)–c) 7m42: solve problems that involve the surface area and volume of right prisms and that requireconversion between metric measures of capacity and volume (i.e., millilitres and cubic centimetres)

Part C: Performance-Based Assessment

Administration

The two performance tasks in Part C are designed to provide insight into how wellstudents are able to perform in terms of the categories of the Ontario AchievementChart: Knowledge and Understanding, Thinking, Communication, and Application.

Since all of the specific and overall expectations for this strand have been assessedthrough the Concepts and Skills Assessment in Part B, it is recommended that youselect one performance task for the Measurement strand.

Read all parts of the problem orally to students. Tell students that they should providedetailed answers to the problem, including showing how they solved the problem.Remind students that they may use pictures, numbers, words, diagrams, and/or chartsto explain effectively how they solved the problem.

Timing

Each task is designed to be completed in a 45- to 60-minute period. If necessary,provide additional time as long as students complete the task in one sitting.


If individual students have difficulties explaining their thinking in writing, considerproviding scribes to record for the students or encourage students to show and explaintheir thinking using concrete materials.

Scoring the Assessment

A generic rubric based on the Ontario Achievement Chart for Mathematics is providedon page 135 to assist with scoring student responses to the tasks. Spend some timereviewing the anchors and rationales provided for each level of achievement on pages136 to 151. The four categories should be considered as interrelated, reflecting thewholeness and interconnectedness of learning. Each student’s performance shouldtherefore be determined holistically by selecting the level that best describes thestudent’s overall achievement.

Sometimes a student will not achieve at the same level for each criterion within acategory or across categories. For example, a student may perform at Level 3 onKnowledge and Understanding, Thinking, and Application but at Level 2 onCommunication. While you may determine that, overall, the student performed mostconsistently at Level 3, you may want to make a note that this student would benefitfrom additional instruction in the area of Communication.

Note: When scoring student work on the performance tasks, it is appropriate to note whatyou observed and heard while the student worked on the task.

Once you have completed scoring the students’ assessments, you may record theresults electronically using the ONAP 8 CD-ROM, or using the Performance ClassTracking Sheet provided on page 134.

Next Steps

Strategies for improving performance in the four areas of the Achievement Chart areprovided in the ONAP introduction, pages 18 and 19.



Mea

sure

men

t

Performance Task 1: Activity Fair—What’s My Sign?

Materials

Introducing the Task

For this task, students apply their measurement skills to design a promotional sign.This sign must be between 1.5 m2 and 2.5 m2 and must include at least two of thefollowing polygons: parallelograms, triangles, or trapezoids.

Tell students that they will

• design a sign based on both geometrical design and measurement guidelines

• explain their design strategy

Display the design from BLM C1: Performance Task 1: Activity Fair—Sample Sign Designon the overhead. Point out the scale you are using and record it:1 grid length represents 10 cmThen work through the area calculation with the class.

Area of 2 triangles: � 9800 cm2

Area of trapezoid: (140 cm � 280 cm) � 70 cm � 2 � 14 700 cm2

Total area: 9800 cm2 � 14 700 cm2 � 24 500 cm2, or 2.45 m2

Have students use BLM C2: Performance Task 1: Activity Fair—What’s My Sign? tocomplete this activity. Explain that their signs should be very different from thesample.

2 � (140 cm � 70 cm)��

2


• BLM C1: PerformanceTask 1: Activity Fair—Sample Sign Design:p. 129

• Performance Task ClassTracking Sheet: p. 134

• Performance TaskRubric: p. 135

• Anchors and rationales:pp. 136–151


• BLM C1: Performance—Task 1: Activity Fair—What's My Sign? pp. 130–131

• pencil

• eraser

• ruler

• pencil crayons


127Ontario Numeracy Assessment

Performance Task 2: Design Glass Boxes

Materials

Introducing the Task

In this activity, students will be designing three boxes, each being a rectangular prismwith a volume of 1000 cm3. The boxes must all have different dimensions. Studentswill identify the design with the least surface area. Then they will determine theheight of a box with a specific hexagonal base and a specific volume. Finally, they willdesign a pentagonal prism box with a volume within a specific range.

Tell students that they will

• sketch different rectangular prisms that have the same volume

• determine the surface area of each rectangular prism, and identify the box withthe least surface area

• determine the height for a hexagonal prism with a given base and volume

• design a triangular prism with a volume within a given range and explain theirdesign strategy

Have students use BLM C2: Performance Task 2: Design Glass Boxes to complete thisactivity.

Answers

1. a) Possible solutions:

b) The 10 cm � 10 cm � 10 cm cube should be made of red glass.

NEL


• Performance Task ClassTracking Sheet: p. 134

• Performance TaskRubric: p. 135

• Anchors and rationales:pp. 136–151


• BLM C2: PerformanceTask 2: Design GlassBoxes pp. 132–133

• pencil

• eraser

• protractor


Box Length Width Height Volume (cm3) Surface area (cm2)

1 20 5 10 1000 700

2 25 4 10 1000 780

3 10 10 10 1000 600


Mea

sure

men

t

2. Area of the base:Area � (10 cm � 6 cm) � 4 cm

� 64 cm2

Volume: 770 cm3

Height: 770 cm3 � 64 cm2 � 12.03125 or about 12 cmNOTE: Accept an estimate of 11 or 12 cm.

3. Strategy: The hexagonal prism inquestion 2, with a height of 12 cm andthe given base, has a volume close to770 cm3, which is between 600 cm3

and 800 cm3.

So, I can use similar dimensions for a triangular base, sketch it, andcalculate its area.Area of base: Triangle: (16 cm � 9 cm) � 2 � 72 cm2

With this base area, a height of 10 cm would give a volume of 720 cm3,which is between 600 cm3 and 800 cm3.

16 cm

9 cm


Performance Task 1: Activity Fair—SampleSign Design

Provided for teachers to create overhead transparency.

ONAP MEA C-1


ONAP MEA C-2.1

Performance Task 1: Activity Fair—What’s MySign? (page 1)

Glenview School is hosting an Activity Fair for all the clubs in the schoolto recruit new members. Each club will make an interestingly shaped signfor its display area. The committee has provided these guidelines:

• The signs must be about 2 m2 (no smaller than 1.5 m2 and no largerthan 2.5 m2).

• The signs should have an interesting shape that can be decomposedinto other polygons such as parallelograms, triangles, and trapezoids.

• The sign may not be a single rectangle, triangle, or trapezoid.

Design your own sign. Calculate the area of your sign and remember toinclude the scale for your drawing below.

Scale: __________________________________________

Name: __________________________________________ Date: ____________________________


Name: __________________________________________ Date: ____________________________

NEL

Performance Task 1: Activity Fair—What’s MySign? (page 2)

ONAP MEA C-2.2

Explain your design strategy.


Performance Task 2: Design Glass Boxes (page 1)

Rebecca is taking stained-glass lessons. She wants to make threedecorative glass boxes with lids. Each box will be a rectangular prism and have a volume of 1000 cm3, but will have different dimensions. For one of the boxes, she plans to use red glass, which is the mostexpensive. She wants this box to have the least surface area possible.

1. a) Design and draw three different boxes with lids and label whichbox should be made of red glass. Fill in the chart with thedimensions of each of your boxes.

b) Which box should be made of red glass? _______________________

ONAP MEA C-3.1

Name: __________________________________________ Date: ____________________________

Box Length Width Height Volume (cm3) Surface area (cm2)

1

2

3


ONAP MEA C-3.2

Performance Task 2: Design Glass Boxes (page 2)

2. Rebecca has enough blue glass left to make a box that is a hexagonalprism. The dimensions of the base are shown on the diagram below.Rebecca wants the box to have a volume of close to 770 cm3. How tall should she make it?

3. Design a box that is a triangular prism with a volume of between 600 cm3 and 800 cm3. Explain your design strategy.

Name: __________________________________________ Date: ____________________________

Show your work.

6 cm

10 cm

4 cm


ONAP PERFORMANCE TASK CLASS TRACKING SHEETGRADE 8: MEASUREMENT

PART CDate: _____________________________________________________________ Grade: __________________________________

School: _________________________________________________ Board: ___________________________________________

Teacher Name: _____________________________________________________________________________________________

Performance Task Title: ___________________________________________________________

Student Name Level 1–4 Comments


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and

does

not

sat

isfy

th

ede

sign

cri

teri

a)

App

licat

ion

•di

spla

ys s

ome

abil

ity

to a

pp

ly m

ath

emat

ical

kn

owle

dge

rela

ted

to t

he

area

of

com

pos

ite

shap

es b

ut

is n

ot a

ble

to a

pp

ly c

orre

ct c

onve

rsio

ns

to t

he

area

mea

sure

men

ts


Mea

sure

men

t

Perf

orm

ance

Tas

k 1:

Act

ivit

y Fa

ir—

Wha

t’s M

y Si

gn?

LEVE

L3

(Anc

hor

1)K

now

ledg

e an

d U

nder

stan

ding

•re

veal

s co

nsi

dera

ble

abil

ity

to e

stim

ate

and

calc

ula

te t

he

area

of

com

pos

ite

two-

dim

ensi

onal

sh

apes

by

deco

mp

osin

g in

to s

hap

es w

ith

know

n a

rea

rela

tion

ship

s•

dem

onst

rate

s co

nsi

dera

ble

un

ders

tan

din

g of

con

vert

ing

betw

een

met

ric

un

its

of m

easu

re f

or a

rea

Com

mun

icat

ion

•co

mm

un

icat

es w

ith

som

e ef

fect

iven

ess

usi

ng

a di

agra

m l

inke

d by

let

ters

to c

orre

spon

din

g ca

lcu

lati

ons;

min

or t

ran

scri

bin

g er

ror

in t

he

scal

e•

pro

vide

s a

clea

r an

d lo

gica

l ex

pla

nat

ion

for

th

e de

sign

str

ateg

y bu

tin

accu

rate

ly n

otat

es s

cale

, an

d ca

lcu

late

s on

e ar

ea a

s 20

0 cm

2in

stea

dof

400

cm

2

Thin

king

•

exh

ibit

s co

nsi

dera

ble

un

ders

tan

din

g of

th

e p

robl

em b

y dr

awin

g a

sign

com

pos

ed o

f m

ore

than

on

e sh

ape

and

then

dev

elop

ing

a st

rate

gy t

ose

lect

a s

cale

th

at w

ill

mee

t th

e ar

ea r

equ

irem

ents

•tr

ansc

ribe

s th

e ar

eas

of t

he

deco

mp

osed

sh

apes

in

accu

rate

ly, w

hic

hle

ads

to a

n i

nco

rrec

t co

mbi

ned

are

a; t

he

tota

l ar

ea i

s ac

tual

ly 2

.68

m2 ,

wh

ich

is

beyo

nd

the

desi

gn p

aram

eter

s•

does

not

tak

e ad

van

tage

of

the

sym

met

ry o

f th

e bo

ttom

par

t of

th

esh

ape

to s

imp

lify

cal

cula

tion

s

App

licat

ion

•ex

pre

sses

con

side

rabl

e ab

ilit

y to

ap

ply

mat

hem

atic

al k

now

ledg

e of

both

are

a u

nit

con

vers

ion

s an

d ar

ea r

elat

ion

ship

s


Perf

orm

ance

Tas

k 1:

Act

ivit

y Fa

ir—

Wha

t’s M

y Si

gn?

LEVE

L3

(Anc

hor

2)K

now

ledg

e an

d U

nder

stan

ding

•de

mon

stra

tes

con

side

rabl

e ab

ilit

y to

est

imat

e an

d ca

lcu

late

th

e ar

ea o

fco

mp

osit

e tw

o-di

men

sion

al s

hap

es b

y de

com

pos

ing

into

sh

apes

wit

hkn

own

are

a re

lati

onsh

ips

•di

spla

ys c

onsi

dera

ble

un

ders

tan

din

g of

con

vert

ing

betw

een

met

ric

un

its

of m

easu

re f

or a

rea

Com

mun

icat

ion

•co

mm

un

icat

es w

ith

con

side

rabl

e ef

fect

iven

ess

usi

ng

a cl

ear

diag

ram

and

wel

l-re

fere

nce

d ca

lcu

lati

ons;

th

ere

is a

sli

ght

erro

r in

th

eco

mm

un

icat

ion

of

the

com

bin

atio

n o

f p

arti

al a

reas

(th

e an

swer

is

corr

ect

but

not

all

par

tial

are

as a

re l

iste

d)•

pro

vide

s n

o ex

pla

nat

ion

for

des

ign

str

ateg

y

Thin

king

•

exh

ibit

s co

nsi

dera

ble

un

ders

tan

din

g of

th

e p

robl

em•

show

s ev

iden

ce o

f an

ap

pro

pri

ate

pla

n a

lth

ough

it

is n

otco

mm

un

icat

ed t

hor

ough

ly o

r ac

cura

tely

(st

ates

div

isio

n b

y 10

0, b

ut

actu

ally

div

ides

by

10 0

00, w

hic

h i

s co

rrec

t)

App

licat

ion

•de

mon

stra

tes

a co

nsi

dera

ble

abil

ity

to a

pp

ly m

ath

emat

ical

kn

owle

dge

of b

oth

are

a u

nit

con

vers

ion

s an

d ar

ea r

elat

ion

ship

s


Mea

sure

men

t

Perf

orm

ance

Tas

k 1:

Act

ivit

y Fa

ir—

Wha

t’s M

y Si

gn?

LEVE

L4

(Anc

hor

1)K

now

ledg

e an

d U

nder

stan

ding

•de

mon

stra

tes

a th

orou

gh u

nde

rsta

ndi

ng

of a

pro

blem

th

at r

equ

ires

con

vers

ion

bet

wee

n m

etri

c u

nit

s of

mea

sure

for

are

a•

exh

ibit

s a

hig

h l

evel

of

abil

ity

to e

stim

ate

and

calc

ula

te t

he

area

of

com

pos

ite

two-

dim

ensi

onal

sh

apes

by

deco

mp

osin

g in

to s

hap

es w

ith

know

n a

rea

rela

tion

ship

s

Com

mun

icat

ion

•co

mm

un

icat

es w

ith

som

e ef

fect

iven

ess;

in

clu

des

an a

pp

rop

riat

edi

agra

m b

ut

does

not

sh

ow c

alcu

lati

ons

•p

rovi

des

a cl

ear

and

insi

ghtf

ul

exp

lan

atio

n o

f th

e de

sign

str

ateg

y,p

arti

cula

rly

wit

h r

esp

ect

to t

he

dete

rmin

atio

n o

f sc

ale;

not

e th

at s

cale

shou

ld r

ead

eith

er 1

cm

:10

cm o

r 1

:10

Thin

king

•

show

s ev

iden

ce o

f a

thor

ough

pla

n•

disp

lays

fle

xibi

lity

an

d in

sigh

t w

hen

car

ryin

g ou

t th

e p

lan

, par

ticu

larl

yw

hen

det

erm

inin

g sc

ale

App

licat

ion

•de

mon

stra

tes

a so

ph

isti

cate

d ab

ilit

y to

ap

ply

mat

hem

atic

al k

now

ledg

eof

bot

h a

rea

un

it c

onve

rsio

ns

and

area

rel

atio

nsh

ips


Perf

orm

ance

Tas

k 1:

Act

ivit

y Fa

ir—

Wha

t’s M

y Si

gn?

LEVE

L4

(Anc

hor

2)K

now

ledg

e an

d U

nder

stan

ding

•de

mon

stra

tes

an u

nde

rsta

ndi

ng

of a

pro

blem

th

at r

equ

ires

an

un

ders

tan

din

g of

rel

atio

nsh

ips

amon

g li

nea

r an

d ar

ea u

nit

s•

dem

onst

rate

s ab

ilit

y to

cal

cula

te t

he

area

of

com

pos

ite

two-

dim

ensi

onal

sh

apes

by

deco

mp

osin

g in

to s

hap

es w

ith

kn

own

ar

ea r

elat

ion

ship

s

Com

mun

icat

ion

•p

rovi

des

a cl

ear

exp

lan

atio

n i

ncl

udi

ng

app

rop

riat

e di

agra

ms

and

aju

stif

icat

ion

of

the

solu

tion

’s s

imp

lici

ty; n

ote

that

un

its

wer

e u

sed

onon

ly o

ne

term

; not

e th

at s

cale

sh

ould

be

eith

er 1

:100

or

1 cm

:100

cm

, n

ot 1

:100

cm

Thin

king

•

exh

ibit

s a

thor

ough

un

ders

tan

din

g of

th

e p

robl

em a

nd,

in

fac

t,su

gges

ts a

sol

uti

on t

hat

is

mat

hem

atic

ally

sim

ple

wh

ile

stil

l sa

tisf

yin

gal

l of

th

e cr

iter

ia

App

licat

ion

•de

mon

stra

tes

a so

ph

isti

cate

d ab

ilit

y to

ap

ply

mat

hem

atic

al k

now

ledg

eof

bot

h a

rea

con

vers

ion

s an

d ar

ea r

elat

ion

ship

s


Mea

sure

men

t

Perf

orm

ance

Tas

k 2:

Des

ign

Gla

ss B

oxes

LEV

EL1

(Anc

hor

1)K

now

ledg

e an

d U

nder

stan

ding

•de

mon

stra

tes

som

e u

nde

rsta

ndi

ng

of v

olu

me

but

no

un

ders

tan

din

g of

surf

ace

area

of

righ

t p

rism

s •

exh

ibit

s li

mit

ed u

nde

rsta

ndi

ng

of c

alcu

lati

ng

the

volu

me

of r

igh

tp

rism

s w

ith

sim

ple

pol

ygon

al b

ases

Com

mun

icat

ion

•p

rovi

des

lim

ited

exp

lan

atio

ns

•p

rovi

des

som

e ap

pro

pri

ate

diag

ram

s an

d ot

her

s th

at a

re i

nac

cura

te (

inq

ues

tion

3, c

orre

ct h

eigh

t is

not

in

dica

ted

and

diag

ram

is

not

dra

wn

reas

onab

ly t

o sc

ale)

Thin

king

•

dem

onst

rate

s li

mit

ed u

nde

rsta

ndi

ng

of t

he

pro

blem

(in

qu

esti

on 2

,do

es n

ot u

se t

he

req

uir

ed i

nfo

rmat

ion

pro

vide

d)

App

licat

ion

•di

spla

ys s

ome

abil

ity

to s

olve

pro

blem

s re

late

d to

th

e vo

lum

e, b

ut

not

the

surf

ace

area

, of

rect

angu

lar

pri

sms;

dis

pla

ys l

imit

ed a

bili

ty w

ith

form

ula

s fo

r h

exag

onal

pri

sms

and

tria

ngu

lar

pri

sms


Perf

orm

ance

Tas

k 2:

Des

ign

Gla

ss B

oxes

LEV

EL1

(Anc

hor

2)K

now

ledg

e an

d U

nder

stan

ding

•de

mon

stra

tes

som

e u

nde

rsta

ndi

ng

of v

olu

me

and

surf

ace

area

of

righ

t p

rism

s•

show

s li

mit

ed u

nde

rsta

ndi

ng

of c

alcu

lati

ng

the

volu

me

of r

igh

t p

rism

sw

ith

sim

ple

pol

ygon

al b

ases

th

at a

re n

ot r

ecta

ngl

es•

pro

vide

s ca

lcu

lati

ons

that

are

fre

qu

entl

y in

accu

rate

Com

mun

icat

ion

•of

fers

lim

ited

exp

lan

atio

ns

•p

rovi

des

som

e ap

pro

pri

ate

diag

ram

s an

d ot

her

s th

at a

re i

nac

cura

te

Thin

king

•

dem

onst

rate

s li

mit

ed u

nde

rsta

ndi

ng

of t

he

pro

blem

(fa

ils

to a

nsw

erq

ues

tion

1b)

an

d se

lect

s in

app

rop

riat

e st

rate

gies

in

qu

esti

ons

2 an

d 3)

App

licat

ion

•di

spla

ys s

ome

abil

ity

to s

olve

pro

blem

s re

late

d to

th

e vo

lum

e an

dsu

rfac

e ar

ea o

f re

ctan

gula

r p

rism

s bu

t di

spla

ys l

imit

ed a

bili

ty w

ith

tria

ngu

lar

pri

sms

and

hex

agon

al p

rism

s


Mea

sure

men

t

Perf

orm

ance

Tas

k 2:

Des

ign

Gla

ss B

oxes

LEV

EL2

(Anc

hor

1)K

now

ledg

e an

d U

nder

stan

ding

•de

mon

stra

tes

som

e u

nde

rsta

ndi

ng

of t

he

volu

me

of r

igh

t p

rism

s w

ith

sim

ple

pol

ygon

al b

ases

as

wel

l as

su

rfac

e ar

ea o

f ri

ght

pri

sms

Com

mun

icat

ion

•co

mm

un

icat

es w

ith

som

e ef

fect

iven

ess

usi

ng

both

dia

gram

s an

dfo

rmu

las

(doe

s n

ot e

xpla

in s

trat

egy

in q

ues

tion

3)

Thin

king

•

dem

onst

rate

s so

me

un

ders

tan

din

g of

th

e p

robl

em, b

ut

use

sin

app

rop

riat

e st

rate

gies

in

qu

esti

ons

2 an

d 3

(div

ides

an

swer

by

2 at

the

end,

in

stea

d of

mu

ltip

lyin

g 10

�9)

App

licat

ion

•ex

hib

its

som

e ab

ilit

y to

ap

ply

kn

owle

dge

of c

alcu

lati

ng

the

volu

me

of r

igh

t p

rism

s to

sol

ve p

robl

ems,

par

ticu

larl

y w

hen

th

e p

rism

is

rec

tan

gula

r


Perf

orm

ance

Tas

k 2:

Des

ign

Gla

ss B

oxes

LEV

EL2

(Anc

hor

2)K

now

ledg

e an

d U

nder

stan

ding

•de

mon

stra

tes

som

e u

nde

rsta

ndi

ng

of v

olu

me

and

surf

ace

area

of

righ

t p

rism

s•

exp

ress

es l

imit

ed u

nde

rsta

ndi

ng

of c

alcu

lati

ng

the

volu

me

of r

igh

tp

rism

s w

ith

sim

ple

pol

ygon

al b

ases

•u

ses

orde

r of

op

erat

ion

s in

corr

ectl

y in

qu

esti

on 2

Com

mun

icat

ion

•p

rovi

des

som

e ex

pla

nat

ion

th

rou

gh c

alcu

lati

ons

and

diag

ram

s; d

oes

not

pro

vide

str

ateg

y in

qu

esti

on 3

•in

clu

des

extr

aneo

us

deta

ils

for

som

e an

swer

s

Thin

king

•

reve

als

a go

od u

nde

rsta

ndi

ng

of v

olu

me

and

surf

ace

area

of

rect

angu

lar

pri

sms;

in

clu

des

one

calc

ula

tion

err

or•

dem

onst

rate

s li

mit

ed a

nd

inco

mp

lete

str

ateg

ies

for

dete

rmin

ing

volu

me

of n

on-r

ecta

ngu

lar

righ

t p

rism

s

App

licat

ion

•di

spla

ys s

ome

abil

ity

to a

pp

ly m

ath

emat

ical

kn

owle

dge

of v

olu

me

of p

rism

s•

show

s li

mit

ed a

bili

ty t

o ap

ply

for

mu

las

for

calc

ula

tin

g th

e ar

ea o

f a

trap

ezoi

d or

a t

rian

gle;

qu

esti

on 2

is

not

fu

lly

answ

ered


Mea

sure

men

t

Perf

orm

ance

Tas

k 2:

Des

ign

Gla

ss B

oxes

LEV

EL3

(Anc

hor

1)K

now

ledg

e an

d U

nder

stan

ding

•de

mon

stra

tes

a co

nsi

dera

ble

un

ders

tan

din

g of

th

e vo

lum

e of

rig

ht

pri

sms

wit

h s

imp

le p

olyg

onal

bas

es a

s w

ell

as t

he

abil

ity

to c

alcu

late

surf

ace

area

of

righ

t p

rism

s

Com

mun

icat

ion

•co

mm

un

icat

es w

ith

som

e ef

fect

iven

ess

•u

ses

diag

ram

s ef

fect

ivel

y an

d cl

earl

y id

enti

fies

mos

t ca

lcu

lati

ons

(wri

tten

exp

lan

atio

n i

n q

ues

tion

3 l

acks

cla

rity

)

Thin

king

•

exh

ibit

s co

nsi

dera

ble

un

ders

tan

din

g of

th

e p

robl

em a

nd

show

sev

iden

ce o

f ap

pro

pri

ate

pla

ns

•ca

rrie

s ou

t so

luti

on p

lan

s u

sin

g ef

fect

ive

stra

tegi

es; t

he

exce

pti

on i

sq

ues

tion

3, w

her

e th

e ex

pla

nat

ion

see

ms

to i

ndi

cate

th

at t

he

corr

ect

answ

er m

ay h

ave

been

arr

ived

at

for

the

wro

ng

reas

ons

(no

nee

d to

divi

de 7

0 by

2, s

ince

70

was

cal

cula

ted

by d

ivid

ing

140

by 2

)

App

licat

ion

•di

spla

ys c

onsi

dera

ble

abil

ity

to a

pp

ly k

now

ledg

e of

vo

lum

e an

dsu

rfac

e ar

ea t

o so

lve

pro

blem

s


Perf

orm

ance

Tas

k 2:

Des

ign

Gla

ss B

oxes

LEV

EL3

(Anc

hor

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Next Steps for Measurement Instructional Next Steps for Overall Expectations After summarizing individual and class performance on each the of overallexpectations, you may find that there are areas that could be retaught to somestudents. The following suggestions have been provided to assist you in preparingtasks for individuals or small groups of students.

Background

This overall expectation is about applying an understanding of area to real life. By theend of Grade 7, students will have had many experiences calculating the area of two-dimensional shapes using a variety of strategies. For this expectation, students buildon their foundational learning from earlier grades. By the end of Grade 7, studentsshould• understand that the area of an object is a two-dimensional attribute• be familiar with standard units of area (square centimetre, square metre, and

square kilometre)• develop referents for each standard unit of area and be able to use units

appropriately• be able to calculate the area of composite shapes by applying area formulas for

familiar shapes drawn inside or outside the shape in question• recognize that the surface area of 3-D shapes may be calculated• recognize that there are many real-life situations that require knowledge of area

Strategies

Revisiting Formula DevelopmentA formula describes a shape in terms of relationships among its componentmeasurement attributes. By Grade 8, students will have developed and be able to useformulas for determining the areas of rectangles, parallelograms, triangles, andtrapezoids. Students who apply a formula to the wrong shape or who have difficultyremembering a formula may need to review the development of the formula. Forexample, provide a rectangle on a centimetre grid and have students relate thenumber of rows and columns to the formula for area. Then have them draw adiagonal of the rectangle and determine the area of one of the resulting triangles.Have them relate this area to the area of the rectangle and develop the formula for thearea of a triangle. Continue with the development of the formulas for the area of aparallelogram and a trapezoid as necessary.


Mea

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Overall Expectation 7m31 (Attributes, Units, and Measurement Sense)Report on research into real-life applications of area measurements.

ComposingGive students opportunities to decompose and/or redraw shapes onto shapes thathave areas that can be calculated using a formula. For example,

A � area of square � area of 3 triangles� 16 � (3 � 1 � 0.5) � 11.5 square units

A � area of triangle � area of rectangle � area of trapezoid� 10 � 5 � 8� 23 square units

Students need many experiences calculating the areas for composite shapes bydecomposing them into shapes for which there are area formulas. In the first exampleabove, the area of the rectangle that will enclose the shape is determined and then the“extra” areas are subtracted from the whole. This is not an easy relationship for somestudents to identify. In the second example, a shape is decomposed into three shapesfor which there are known area formulas.

Have students practise determining area through decomposition and then sharestrategies so that everyone can see that there is more than one way to decompose ashape. Make mathematical connections between the strategies so that students willbegin to understand area relationships and expand on the ways in which they cansolve area problems independently.

ConstructingHelp students make the connection between real life and area measurement by usingcontexts and references beyond formal mathematics instruction. For example, havestudents do one of the following:• build a ramp for model cars/trains/hamsters, or design skateboard or snowboard

ramps


3 1

0.5

• using centimetre grid paper, plan a number of interesting garden shapes for a newpark or horticultural centre. Students can determine the area of each garden aswell as the area of the paths between the gardens. Each garden could have adecorative stone border, and students could calculate the area of the stone border.Students should include a scale on their plans.

Background

This overall expectation deals with relationships among units and measurableattributes. In Grade 7, students solve problems involving estimating and calculatingthe areas of familiar geometric shapes. They investigate and apply their understandingof relationships between metric units, areas of familiar shapes, base area and volumeof right prisms, and surface area of right prisms. By the end of Grade 7, studentsshould be able to• solve problems that require conversion between metric units of measure • develop the formula for the area of a trapezoid• estimate and calculate the area of composite two-dimensional shapes by

decomposing into shapes with known area relationships• understand the relationship between the area of the base and the volume of

right prisms• sketch polygonal prisms• determine surface area of right prismsHaving students use a hands-on approach means that the formulas are less likely to beforgotten, will be used in a flexible way, and are more likely to be expanded andadapted appropriately.

Strategies

Unit RelationshipsUse base ten blocks to help students visualize area units. Have them examine ahundreds block and relate this to a 10 cm by 10 cm area, or 100 cm2. Have themoutline a square metre using metre sticks and/or masking tape on the floor. Then havethem place as many hundreds blocks as are available in that square, and imagine howmany hundreds blocks it will take to completely cover the square. Relate this to a 100 cm by 100 cm area, which is the same as a 1 m by 1 m area, or 10 000 cm2 or 1 m2.

Have students design two possible mural sizes for the school hall. Explain that themurals will be rectangles and must have a total area of between 1.5 m2 and 2.5 m2.Explain that the murals will be created using square centimetre tiles. Have studentsthen calculate the number of tiles that will be required for each of the murals.


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Overall Expectation 7m32 (Measurement Relationships)Determine the relationships among units and measurable attributes, including thearea of a trapezoid and the volume of a right prism.

Developing FormulasOne development of the formula for a trapezoid depends on an understanding of theformula for a parallelogram.

A triangle can be one half the area of a parallelogram with the same base and height.

A trapezoid can be one half of the area of a parallelogram.

Students need time to manipulate these relationships concretely and makeconnections to the formula before they will understand both the similarities anddifferences between the two formulas. Students will then be able to reconstruct theirlearning if memorization fails.

Have students investigate formula relationships using Power Polygons. Ask them toselect pairs of congruent triangles and trapezoids. Have students combine each pair toform a parallelogram and then trace the pairs on paper. Then ask students to measurethe dimensions of the triangles and trapezoids as well as the parallelograms that theyhave created. Finally, have students explain how the measurements relate to theformulas for triangles, trapezoids, and parallelograms. If Power Polygons are notavailable, students may use polygons cut from cardstock.


h

b

h

b

A � bh

h

a b

ab

A � h ��a �

2b

��

A � �12

�bh

Not Always BiggestHave students draw nets for 3-D objects for which they are determining the surfacearea. Knowing that the surface area of a prism is simply the sum of the area of all ofthe faces is generally more helpful than memorizing a formula for a specific shape.

Provide students with three empty boxes (e.g., cereal boxes, cracker boxes, pencilboxes, etc.). Have them estimate the order for both volume and surface area. Thenhave students determine volume and surface area. Allow students to cut the boxes tocreate nets as they determine the surface area. Discuss any observations. Use theexamples to link to the formula for surface area as simply a way of systematicallyorganizing the areas of all the faces.

Building PrismsStudents who have difficulty determining the volume of right prisms that do not havea rectangular base may use cubes to help with their understanding. They can use thecubes to construct rectangular prisms, calculate the volume, and then use thatinformation to generalize to other right prisms. Have students build a 5 cm � 4 cmbase with centimetre cubes. They can place this on grid paper to confirm the area ofthe base. This prism has a volume of 5 � 4 � 1. As layers are stacked, the totalnumber of cubes changes predictably; 5 � 4 � 2, 5 � 4 � 3, 5 � 4 � 4, and so on.Understanding this relationship, where cubes can be counted to verify calculations,will help students to understand that the volume of any prism is always the area ofthe base times the height. Students could use pattern blocks to create bases that aretriangles, hexagons, trapezoids etc., and then stack layers to confirm that the formulafor the volume of prisms (base area � height) applies to any base.


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onap - nelson · · 2009-07-20the measurement strand of the ontario curriculum for grade 7...

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