eight grade mars 2006 task descriptions overview of … · measurement aaron’s designs ... the...

Eighth Grade – 2006(c) Noyce Foundation 2006. To reproduce this document, permission must be granted by the Noyce Foundation:[email protected].

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Eight Grade Mars 2006 Task DescriptionsOverview of Exam

Core Idea Task Score

Geometry and

Measurement

Aaron’s Designs

This task asks students to draw reflections and rotations of a given figure on a grid.

Students describe transformations needed to make a given pattern. Successful students

could draw and describe reflections, flips, and slides. Students working at a high

level could draw rotations and quantify the transformations.

Algebra and Functions Squares and CirclesThis task asks students to work with perimeter and circumference of squares and

circles. Students use and interpret line graphs and their equations. Successful students

could reason about perimeter of squares and plot points on a graph. They could

identify the equation for perimeters and explain why the graph was a straight line.

Students working at a high level could compare and contrast the graphs of two

functions.

Data Analysis TemperaturesThis task asks students to understand and interpret statistical graphs and diagrams

showing real data. Students compare and contrast data sets. Successful students

compare and contrast line graphs and match the line graphs to box and whisker

diagrams by noting key features of the data. Students working at a high level

understood the upper and lower quartiles on a box and whisker diagram.

Number and Operation 25% SaleThis task asks students to work with percentage increase and decrease in the context

of a sale and develop mathematical arguments on the effects of decreasing a price by

25% four times. Successful students could calculate the cost of a 25% reduction and

reason why reducing something by 25% four times does not result in zero cost.

Students working at a high level could calculate the percent reduction given the

amount of reduction and original cost.

Mathematical Reasoning Going to TownThis task asks students to interpret and complete a distance/time graph for a described

situation and work with rates in the context of slope. Successful students could

calculate rates from a graph, relate the meaning of slope to the problem context, and

use information about the situation to continue the graph.


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Aaron’s Designs Rubric

The core elements of performance required by this task are:• draw reflections and rotations of a given figure on a grid• describe transformations needed to make a given pattern

Based on these, credit for specific aspects of performance should be assigned as follows pointssectionpoints

1. Draws all 3 shapes correctly.

Partial credit

Draws shape 2 and one other correctly.Draws shape 2 or shape 4 correctly.

3

(2)

(1)

3

2. Draws all 3 rotations correctly.

Partial credit

Draws shape 2 correctly.

2

(1)

2

3. Gives a correct description such as:Reflects the shape over the vertical line,then translates/slides the 2 shapes down4 squares.

1

1

13

Total Points 8

Shape 2

Shape 2

Shape 4


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8th Grade – Task 1: Aaron’s DesignsWork the task and examine the rubric.What do you think are the key mathematics the task is trying to assess?

What opportunities do students in your class have to work with transformations:reflections, rotations, flips, turns, slides?How are these skills relevant to today’s job market?What kind of work have your students done with symmetry? Do they get practicemaking their own shapes or drawing in the second part of a shape? How is drawing ina shape different from drawing in a line of symmetry? What mathematics come up inmaking your own drawing?Look at student work on part one, using reflections across two lines of symmetry.How many of your students:Drew all 4

shapescorrectly

Slid the topquadrants to

the lowerquadrants

Slid theshape to

right

Put nothingin thelower

quadrants

Made adifferent

shape

Didn’tdraw theshape to

scale

Other

Rotating the shape was difficult for students. What opportunities have they had towork with rotations? What strategies might students use to help them solve this task?Try working this part with colleagues. Did you all use the same way? See if you canshare some of this variety with students.

Look at part two of the task, rotating a shape. How many of your students:Made all the

rotationscorrectly

No response Reflected theshape 4 times

Made a differentshape

Other


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Many students are using common language rather than mathematical language todescribe transformations such as copy and move. What type of work have studentsdone with the transformation?Look at student work on part three, describing a transformation. How many of yourstudents:Describe

therotation

Describethe slide

down

Use quantityfor the slide

Usenonmathematical

language

Talkedabout

copying

Inaccuratemeasurement

or scale

Other

As you looked at student work, did you see evidence of students who were operatingat a low van Hiele level?

What effort do you make to design and put activities into your curriculum to helpstudents to progress through the van Hiele levels?

What are some implications for instruction that you want to remember after looking atyour student work?


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Looking at Student Work on Aaron’s DesignsStudent A is able to do both the reflection and rotations, keeping the shapes in scale as they aremoved. Notice in part three the student numbers the quadrants to make the meaning clearer to thereader. The student is able to quantify the distance and direction to move the shapes.Student A


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Student B is able to draw the reflection for part 1 of the task and attempts to do a reflection in parttwo instead of a rotation. The student distorts the shape, not quite able to track all the measurementsof the original design.Student B

Student C tries to slide the shape in part two, but again can’t accurately locate one of the vertices.Notice that not even a distorted version of the shape is maintained when the shape is moved to the4th quadrant.Student C


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Student D is able to do the reflection across the vertical axis, but changes the shape and size of thedesign when trying to reflect across the horizontal axis. The student does not understand rotationand the shape also is not scale in part 2.Student D

Student E seems do a horizontal slide in part 1, but is unable to correctly locate two of the vertices.In part two, the student seems to do a diagonal slide, but again distorts the shape by mislocating avertex.Student E


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Student F does one correct rotation and then tried to slide the shape down to the lower quadrants,maybe using the drawing in part three as a guide. Notice that the lower right shape is not to scale.In part two the student draws shapes different from the original figure. Do you think this is causedbecause the student knows there is some movement but doesn’t understand the language enough tofinish the end part of each design?Student F

Student G seems to have some vague idea of reflection over the vertical axis, but cannot maintainthe original shape with the reflection over the vertical axis.Student G


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Students had trouble with the formal language for transformation. Student H is able to use thelanguage correctly in the first two sentences but then uses reflection incorrectly in the last sentence.The student gives the direction of the slide but not the distance in the second sentence.Student H

Student B uses the vocabulary of transformation, but uses it incorrectly. There is no mention ofdirection or distance in the student’s directions.Student B

Here are some more typical responses to part 3. What is missing or incorrect in each one?


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Eighth Grade

8th Grade Task 1 Aaron’s Designs

Student Task Draw reflections and rotations of a given figure on a grid. Describetransformations needed to make a given pattern.

Core Idea 4Geometry &Measurement

Apply transformations and use symmetry to analyze mathematicalsituations.

• Describe sizes, positions, and orientations of shapes underinformal transformations such as flips, turns, slides, and scaling.

Based on teacher observation, this is what eighth graders knew and were able to do:• Reflect shapes over the vertical axis and slightly less often over the horizontal axis• Describe a reflection

Areas of difficulty for eighth graders:• Rotations• Describing direction and distance for slides or translations• Maintaining shape and size or scale when making their drawings


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The maximum score available for this task is 8 points.The minimum score for a level 3 response, meeting standards, is 5 points.

Many students, about 78%, could reflect a design across a vertical axis while maintaining the scaleand shape of the original design. More than half the students, 54%, could accurately reflect a designacross both a vertical and a horizontal axis. A few students, about 20%, could do reflections andalso describe how a design had been reflected as well as give a direction for a slide or translation.Less than 2% of the students could meet all the demands of the task, including reflecting androtating shapes, describing reflections and translations with direction and quantity for the moves.Almost 22% of the students scored no points on this task. 80% of the students with this scoreattempted the task.


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Aaron’s Designs

Points Understandings Misunderstandings0 80% of the students with this

score attempted the task.In attempting the reflections in part one,12% of the students could not maintainthe original shape when making theirdrawing or did not draw it to scale. 5% ofthe students slid the shape to the rightinstead of making a reflection.

1 Students could make areflection across a vertical axis.

19% of the students slid the design fromthe top quadrants to the lower quadrantsinstead of making a reflection. Another5% put nothing in the lower quadrants.

3 Students could make areflection across a vertical and ahorizontal axis, whilemaintaining shape and size.

10% of the students did not attempt todescribe the translation in part 3 of thetask. 19% of the students usednonmathematical language in theirdescriptions in part 3.

5 Student could make reflectionsand describe reflections in atransformation.

They had difficulty giving direction andquantity to slides. 18% of the studentsdid not give a reference as to what wasbeing moved or transformed or an axisfor making a reflection. 28% of thestudents gave instructions about copyingthe design. Students could not dorotations. Almost 50% of them drewreflections for part two instead ofrotations. 19% changed the shapecompletely in their attempt to do areflection. Almost 10% were unwillingto even attempt a rotation.

6 Students did not give a direction for theslide or “move”. They did not quantifythe size of the slide or “move”.

7 Students did not quantify the size of themove.

8 Students could make reflectionsand rotations across vertical andhorizontal axes, whilemaintaining shape and size.Students could describereflections with reference toaxis of reflection and slideswith reference to direction andsize of the slide.


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Implications for InstructionStudents at this grade level need frequent exposure to activities that develop their spatial reasoningand ability to distinguish geometric shapes, properties of shapes, and to develop their reasoning andgeneralizing skills between properties of shapes. Research suggests that everyone develops throughlevels of understanding (van Hiele levels) based not on maturity, but on experiences. Researchfurther suggests that the ability to make formal and informal deductions, such as that required in ahigh school geometry class, without first moving through these lower levels. More than half thestudents entering a geometry class may still be operating at a level 0 (visualization) or 1(analysis).

Students need opportunities to sort and categorize shapes by their properties. For middle schoolworking with a software program, like Geometer’s Sketchpad, can be useful for exploring examplesof classes of shapes and can further help students start to build and test conjectures.

Students need more work with drawing rotations, slides, and reflections. Students need to be ableto describe the line of reflection or the distance of a slide or other transformation. By making theirown drawings and transformations, students learn about the importance of scale and start to seemore of the detail in the shapes or designs. Students need more opportunity to work withtransformations on a coordinate grid. In a world dominated by special effects in movies, video cellphones, graphic design, missile technology, hdtv, sending images on computers, defense systems, aswell as traditional work of engineers, architects, carpenters, it is more important than ever forstudents to have the visualization skills to function in today’s world.

Working with geometry can be very enjoyable for students and give some students a chance toshine, who may have not be so successful in other areas of mathematics. For most students thesetypes of activities and learning experiences are very motivating. Their success can transfer to apositive attitude when they then attempt other activities in the classroom. There are a number ofgreat resources for working with transformations: John Van de Walle – Teaching Student-CenteredMathematics, Connected Mathematics, and Mathematics in Context. Developing spatial skills isdirectly related to opportunity to learn.

Ideas for Action ResearchClue CardsStudents need to understand the importance of detail, reference to line of reflection, direction, anddistance in giving directions. Try making clue cards from some of the student responses to partthree in Aarons design and give students grids with only the top left quadrant filled in. Ask studentsto use the clue cards to try and complete Aaron’s third design. Don’t give them a lot of definitionsor explanations, but tell them that they can look up any words that are unfamiliar to them or use anyresources available in the room. Let them take some initiative in their own learning and what theyneed to know.

When students have completed their designs, have them compare what they made with each other.Did they all draw the same thing for each clue card? Why or why not? Have students discuss whichclues were most helpful or most confusing. What would make the clues more helpful?

Now have students make their own simple design that can be made in the 4 quadrants using flips,slides and rotations. Then ask students to write their own clue card for their design. Have themtrade with each other or pick three or four to give to the class. See how their thinking has improved.


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How did the students do with the idea of being responsible to look up things that they didn’tunderstand? What evidence did you see that they used this opportunity? How is this type oflearning different from you “explaining” something to them?

How did the discussion about the clues focus students on the geometric ideas of distance anddirection? What other important ideas came up during the discussion?


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Squares and Circles Rubric

The core elements of performance required by this task are:• work with perimeter and circumference of squares and circles• use and interpret line graphs and their equations

Based on these, credit for specific aspects of performance should be assigned as follows pointssectionpoints

1. Gives correct answer: 20 (inches) and correct point marked on graph. 11

2. Gives correct answer: 3 (inches) and correct point marked on graph. 11

3. Correct line drawn.

(a) Gives correct explanation such as:The perimeter is zero if the side length is zero.

(b) Gives correct explanation such as:The perimeter is always four times side length.orThe perimeter is proportional to side length.

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1

1

3

4. Gives correct answer: y = 4x 11

5. Correct points marked and line drawn. 11

6. Writes correct statements such as:

Both lines go through (0,0)

The line for the squares is steeper than the line for the circles.

1

12

Total Points 9

Eighth Grade – 2006(c) Noyce Foundation 2006. To reproduce this document, permission must be granted by the NoyceFoundation: [email protected].

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Squares and CirclesWork the task. Look at the rubric. What do you think are the majordifficulties students might have with this task?

In part one and two, a few students had trouble with graphing. They mayhave made zigzagging lines because one of their points was off or connectthe points to one of the axes. Did any of your students have difficulty withthe graphing issue?Almost 35% of the students didn’t graph the line fordiameter/circumference. If you get a chance, interview a few of yourstudents. Did they give up before they got that far in the task? Did theyoverlook the instructions in part five? Do you think trying to graph numberswith decimals confused them?For part 3a, many students thought that lines or all lines go through zero orthat it goes through zero because it slopes. They did not relate the origin tothe context. What were the common misconceptions that you found in yourstudents’ work?For part 3b, many students thought that the points were linear because theylined up that way or because there was a pattern? What were the commonmisconceptions that you found in your students’ work?What kinds of graphing activities have your students done this year? Whatopportunities have they had to graph from a context and discuss how thecontext effects the shape of the graph?Look at student work in part six. How many of your students wrote answersthat discussed:

• Slope (going upward, quantifying amount, steepness of the line)• Origin• Whole numbers/decimals• Nonmathematical ideas: easy to read, ones easy ones hard, the dots

are large

Did your students know what where critical or important things to considerwhen comparing and contrasting graphs?What are some of the implications for instruction?


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Looking at Student Work on Squares and CirclesStudent A shows how he calculated perimeter and side length. The student is able torelate the origin to the context of the problem in 3a. Notice the reference to origin andslope in part 6.

Student A


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Student A, continued

Student B has an almost perfect paper, but then when finding differences looks at thenumbers plotted rather than properties of the line or graph.Student B


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Student C also has a score of 8 points. The student is not thinking about the issue ofrounding in finding circumference, but instead notices that for the values in the table thedifference is slightly different between points. What kind of activity would help studentsunderstand level of accuracy as it relates to this type of function?Student C

Student D has a score of 9. Notice the drawing to explain why the line is straight.Student D

Student E is another example of an “8”. The student understands the properties thatshould be compared and is able to give the equations. However the student does not seethat the rate of increase is different for circles than squares.Student E


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As scores drop from 8 or 9, there is a change in the types of arguments presented in partthree and six. Student F is able to calculate the perimeter and side length in part one andtwo and draw the graphs for circles and squares. While the student can graph, theunderstanding of meaning and purpose seem to get lost. The student’s line for circlemisses the origin by a hair. How would you compare the reasoning in this paper to that ofthe previous students?Student F


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While Student G is able to calculate the perimeter and side length in part one and two, thestudent does not plot the points for either correctly. Notice that there is an idea thatstraight lines start at (0,0), which is prevalent in many papers. Why do you think studentshave this idea? What kinds of experiences would help them to understand why not alllines go through the origin? What do students need to understand to know why a functionwill be linear or nonlinear?Student G


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Student H is able to find perimeter and side length, but does not notice that the finalpointfor circles is graphed incorrectly. The student, like many others with scores of 4 orless, is no longer focused on the mathematics. Student H is concerned primarily withaccuracy. The student understands that the square pattern is increasing by 4 every timeand gives the recursive equation rather than the seeing the multiplicative relationship.Student H


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Student H, continued

As students get to scores of 3 or less, many students do not attempt page two of the task.They seem unwilling to even attempt making explanations. For those students whoattempted this part, work might look like Students I and J. How would you compare theirresponses to 8’s and 9’s? to the 4,5,6’s? What is the difference in their mathematicalthinking?


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Student I


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Student J


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Eighth Grade

8th Grade Task 2 Squares and Circles

Student Task Work with perimeter and circumference of squares and circles. Use andinterpret line graphs and their equations.

Core Idea 3Algebra andFunctions

Understand relations and functions, analyze mathematicalsituations, and use models to solve problems involving quantity andchange.

• Identify functions as linear or nonlinear, and contrast theirproperties from tables, graphs, or equations.

• Explore relationships between symbolic expressions and graphsof lines, paying particular attention to the meaning of interceptand slope.

Based on teacher observation, this is what eighth graders knew and were able to do:• Find perimeter of a square given the side length• Graph the points• Find the side length of a square given the perimeter

Areas of difficulty for eighth graders:• Understanding slope or rate of change• Connecting the origin to the context of the perimeter• Using mathematical vocabulary to describe patterns or trends• Understanding the “continuousness” of a line• Explaining why a line is straight• Comparing and contrasting linear functions, picking mathematically significant

details of the graphs


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The maximum score available on this task is 9 points.The minimum score for a level 3 response, meeting standards, is 4 points.Many students, 76%, were able to calculate the perimeter for a square with a side of 5”,plot the point, and draw a line through all the points on the graph. More than half thestudents, 59%, were able to find the perimeter, graph point one, connect the line, choosethe equation for the square, and either find the side length or state what was the samebetween the square and circle graphs. About 1/4 of the students could find the perimeterand side length, draw the lines for squares and circles, explain why the line for squareswas the same as circles, choose an equation to represent the squares, and state somethingthat was the same about the square and circle graphs. About 5% of the students couldmeet all the demands of the task, including explaining why the graphs for squares wentthrough the origin and finding a significant feature that was different between the squareand circle graph. 15% of the students scored no points on this task. 90% of themattempted the task.


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Squares and Circles

Points Understandings Misunderstandings0 90% of the students with this

score attempted the task.Almost 6% of the students calculated theperimeter, but failed to put in on thegraph or didn’t locate the point correctly.5% of the students thought the perimeterwas 5. 6% of the students did not attemptto connect the points on the graph. 5%made zigzagging lines instead of straightlines.

2 Students were able to calculatethe perimeter for a square with aside of 5”, plot the point, anddraw a line through all the pointson the graph.

13% of the students calculated perimeterof a square with a side of 12 instead offinding the side length of a square with aperimeter of 12. 6% found the sidelength but did not or could not plot thepoint.

4 Students were able to find theperimeter, graph point one,connect the line, choose theequation for the square, and eitherfind the side length or state whatwas the same between the squareand circle graphs.

10% of the students explained that theline was straight because the points linedup. 7% thought the line was straightbecause it was a pattern. 17% of thestudents thought the equation for sidelength and perimeter of a square wasy = x + 4, 9% picked each of the otherincorrect choices.

7 Students could find the perimeterand side length, draw the lines forsquares and circles, explain whythe line for squares was the sameas the line for circles, choose anequation to represent the squares,and state something that was thesame about the square and circlegraphs.

Students could not explain why the linefor perimeter goes through the origin.About 30% think all lines go through theorigin. Students had trouble explainingwhat was different about the two graphs.10% thought the difference was wholenumbers and decimals.

9 Students understood perimeterand side length and could explainwhy it would pass through theorigin as well as why the graphwas linear. Students could plotdecimal values and compare andcontrast two graphs usingsignificant features, like originand slope.


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Implications for InstructionStudents need frequent opportunities to relate equations to a familiar context, so that theycan make conclusions about how the graph, equation, and context fit together. They needto think about what in the context causes the graph to go through the origin. Why doesthat make sense? When do graphs not go through (0,0)? What about the situation causessome other y-intercept to make sense?

Students need to have opportunities to compare and contrast graphs of differentfunctions. Through rich classroom discussion they develop insights into the importantfeatures of graphs and why they are relevant. They should be comfortable with importantproperties of graphs: positive or negative slope, degree of steepness or rate of change, andy-intercept.

Some students had difficulty identifying the graph showing the relationship between theside length and perimeter of a square. They could usually state that the value wasincreasing by 4’s. However many students saw this is additive, y= x + 4, rather thanmultiplicative, y = 4x. They do not see the relationship between “equal groups” andmultiplication.

Some students had difficulty graphing decimal values. Others had difficulty thinkingabout how rounding might have effected the values in the table for circumference.Students need to work with a variety of functions, which should include those wherenumbers are not always whole numbers and where rounding might be helpful to developa more robust understanding of rounding and growth rates.

Ideas for Action ResearchLooking at Classes of Graphs: Starting point for generalizationsDesign an exploration about comparing and contrasting graphs. Give students a series ofgraphs, such as: y = x + 4, y = 4x, y = 4x + 4, y = x – 4, y = x/4, y = 4 – x

Ask students in pairs to think of situations that could fit each equation. Have groupsshare out their situations and be prepared to prove why that situation matches theequations. Try to get them to discuss why in each situation the “x” represents a variable.What does that mean in the specific context?

Now ask students to make a table of values and to plot the graphs. Have them write asummary statement about patterns in the tables. Then ask them to compare the graphs forthe various equations: How they are alike and how are they different? The groups shouldmake posters to show their findings. Students should start to think about slope andy-intercept as they compare graphs. During discussion, try to push them to explain whytables that go up by 4 are not the same as the equation y = x + 4. Can they give of reasonfor deciding which graph might be the steepest? Really try to let them come up with theideas. Try to capture some of these ideas on a poster for later reference.


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Now ask them to predict what a graph for a different equation might look like and givereasons for their prediction. New equations might be: y = 3x, y = x + 5, y = x – 7, y = 3x+9. Then have the students graph the equations and compare the graphs to theirpredications.

Why is having an investigation around graphing different that giving a lesson on theequation for a line? What things do you think students were picking up from this activitythat are different from the traditional lesson? What things do you feel are lost?

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