Dinosaur Data - noycefdn.org · Dinosaur Data Rubric The core elements of performance required by...

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

49

Dinosaur Data

This problem gives you the chance to:• relate a table of data and a bar graph• derive information

Sangita and Zach are doing a project about dinosaurs.

They have discovered the facts shown in the table below.

Name of dinosaur Food dinosaurs eat Estimated length in meters

tyrannosaurus meat 12

seismosaurus plants 40

diplodocus plants 27

allosaurus meat 10

brachiosaurus plants 25

They use the numbers in the table to draw this bar graph.

Copyright © 2006 by Mathematics Assessment Page 4 Dinosaur Data Test 4Resource Service. All rights reserved.

tyrannosaurus seismosaurus diplodocus allosaurus brachiosaurus

40

35

30

25

20

15

10

5

0

Estimated

length

in

meters


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1. The students have made a mistake.

One of the bars on the graph is wrong.

Which one is it? ___________________________________

Explain your answer.

________________________________________________________________

________________________________________________________________

2. Using the table on the opposite page, the students have written some

information about these dinosaurs.

One of the sentences is not correct. Underline the incorrect sentence.

• A brachiosaurus can be about twice as long as a tyrannosaurus.

• Meat eating dinosaurs are longer than plant eaters.

• An allosaurus is shorter in length than a seismosaurus.

Explain what is wrong with the information in the sentence you have underlined.

________________________________________________________________

________________________________________________________________

________________________________________________________________

3. Using the facts on the opposite page, write two sentences of your own

comparing the length of these dinosaurs.

________________________________________________________________

________________________________________________________________

________________________________________________________________

________________________________________________________________Copyright © 2006 by Mathematics Assessment Page 5 Dinosaur Data Test 4Resource Service. All rights reserved.

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Dinosaur Data Rubric

The core elements of performance required by this task are:• relate a table of data and a bar graph• derive information

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

1. Gives correct answer: diplodocus

Gives correct explanation such as:

On the bar chart this is recorded at about 37 (accept 36 to 38 inclusive)

rather than 27 meters.

Partial credit

For a partially correct/inaccurate answer.

1

2

(1) 3

2. Underlines “Meat eating dinosaurs are longer than plant eaters”.

Gives correct explanation such as:

From the table it would appear that the plant eating dinosaurs are longer

than the meat eaters.

1

1

2

3. Writes correct statements such as:

The seismosaurus is (about 15 meters) longer than the brachiosaurus.

The shortest of these dinosaurs is the allosaurus.

1

12

Total Points 7


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Dinosaur DataWork the task and examine the rubric.Did your students use quantity when making mathematical statements?Did your students have the habit of mind of writing numbers or values on thegraph? Why might this have been helpful?What evidence do you have that students understood or were confused by the scaleof the graph?How many of your students used a quantity to explain the error in part one?How often do students in your class get the opportunity to combine or summarizedata on a graph? What are some examples of questions that push forgeneralizations about graphs?Look at their reasoning for part 2 correcting the sentence. What stood out for youabout student reasoning or logic?Did your students get confused by the use of estimation in context? (Look at thework of students who picked sentence one, “A brachiosaurus can be about twice aslong as a tyrannosaurus”. In what contexts do your students use estimation? Doyou have discussions about when rounding is okay or useful and when they needexact answers?What are the types of comparisons that you would want students to make? Whyare they significant?

What is the difference between making a comparison and stating a fact?Could your students make a complete comparison, giving what made two or moreitems different?Did they use quantity, like how much more or less, using comparison subtraction?Did they use multiplicative relationships, like about three times more or abouttwice as much?


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Look at student work for part 3. How many of your students:Could make a relative comparison using the graph aboutrelative size? (larger, largest, smaller, smallest)Could make a quantitative comparison using subtraction?Could make a quantitative comparison using multiplication?Made a statement that was not a comparison or stated a fact?Make a comparison with no reference to something else?Made a nonmathematical statement (“All dinosaurs eat plantsand meat, not just plants.” or “Tyrannosaurus is thestrongest.”)?Made an incorrect statement?Made statements that were not about the length?Repeated statements from part two or made the reverse of theirfirst comparison (tyrannosaurus is larger than allosaurus,allosaurus are shorter than tyrannosaurus)

What are some simple ways to give students more practice making comparisons?


54

Looking at Student Work on Dinosaur DataStudent A quantifies the mistake in part one. The student can use the logic of part two to correctthe generalization about the size of plant-eaters. In the part three, the student makes acomparison about relative size, smallest. The student also is able to use multiplicativerelationships to make a comparison.Student A


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Student B writes the lengths on the graph to help analyze the mistake and make comparisons.The student confuses the idea of estimation in part two, mistaking “about” for an exact amount.The student makes only comparisons about relative sizes.Student B


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Student C describes a process for finding the error, but does not quantify the error in part one. Inpart three, the comparisons, while interesting, are not related to the mathematics of the table orgraph.Student C

Students made several interesting comments, that didn’t necessarily fit into the format of acomparison. Consider the student who is trying to reason about the plant eaters are longer: “Planteaters are longer probably because when they stand it would help them reach higher branches toeat.” Or “ It is interesting that plant eaters weigh more (notice that it should be are longer) thanmeat eaters.” “I would think that meat eaters would weigh more than plant eaters.” Others try toreason about previous knowledge or ideas about dinosaurs. : “The tyrannosaurus should be thebiggest.” Or “Dinosaurs can come in all different shapes and sizes.” “Many dinosaurs could belong and skinny but some are strong, but short. It doesn’t matter but how they are.” Somestudents stated facts instead of making a comparison. “Tyrannosaurus is 40 meters. Siesmosaurus


57

is 11 meters.” Or “Diplodocus eats plants.” Some students confused making a comparison withwriting a question or giving clues. “Which two make 22? Which two make 65?”

In 2006, fourth graders were given a task with data about dinosaurs. To get them to comparegroups of the data students were asked to pick an incorrect statement about the graph and thenrewrite it to make a correct statement. 87% could identify the incorrect statement. 70% of thestudents were able to generalize that plant eaters were larger than meat eaters. Incorrectstatements contained ideas like, “plant eaters might eat meat”, “It doesn’t make sense for meateaters to be smaller”, “It depends on how old they are”, or “I checked and compared informationin the graph”.

The task then asks students to write 2 comparisons about the length of dinosaurs in the graph. 5%did not attempt to write a comparison. 16% only wrote one comparison.

Below is a breakdown of comparison statements by thinking styles.

Type of Comparison %Repeat of previous comparison (meat eaters are smaller or planteaters are larger)

9%

Largest/Smallest 20%Relative Size (Tyrannosaurus larger than Allosaurus, Seismosauruslarger than Diplododaucus

11%

Relative Size (about the same) 5%Comparison Subtraction 7.6%Multiplicative Comparison (“seismosaurus is 4 times allosaurus”,“brachisaurus is 1/2 of seismosaurus”)

11%

Attempt at range (“plant eaters larger than 20”, “meat eaters are lessthan 15”)

1.5%

Categories of Incorrect StatementsAttempts at facts (“Allosaurus is 12 meters”, “some eat meat”, “planteaters are longer”, “plant eaters are longer to get the high plants,”“dinosaurs come in all shapes and sizes”)

11%

Incorrect statements 6%Insignificant comparisons (“all are dinosaurs”, “all eat”) 1%Not a comparison 4%


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Fourth Grade

4th Grade Task 3 Dinosaur Data

Student Task Relate a table of data and a bar graph. Analyze data and makemeaningful comparisons about the data.

Core Idea 5Data Analysis

Collect, organize, represent and interpret numerical and categoricaldata, and clearly communicate their findings.

• Represent data using tables, charts, line plots, and bar graphs.• Interpret data to answer questions about a situation• Describe the shape and important features of a set of data

The Mathematics in the Task:• Ability to relate a table of data and a bar graph• Ability to interpret a table and read a graph with a vertical scale of 5• Ability to make and quantify comparison statements using relative size, comparison

subtraction, multiplicative comparison

Based on teacher observation, this is what fourth graders knew and were able to do:• Read scale on a bar graph• Find the discrepancy between facts in a table and graph• Summarize information in a table and write a correct generalization about the data

Areas of difficulty for fourth graders:• Making a comparison of data, rather than stating opinions or facts• Quantifying their ideas: what is the difference between the table and graph• Using subtraction and multiplication to make a comparison• Understanding estimation in context

Strategies used by successful students:• Writing values by each bar in the table• Grouping data into like groups, meat-eaters and plant eaters


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MARS Test Task 3 Frequency Distribution and Bar Graph, Grade 4

Task 3 – Dinosaur Data

Mean: 5.04 StdDev: 1.94

MARS Task 3 Raw Scores

The maximum score available for this task is 7 points.The minimum score for a level 3 response, meeting standards, is 4points.

Most students, about 93%, could find the incorrect data on the graph, and write one comparison.Many students, 87%, could also correct the generalization about plant-eaters and meat-eaters.Almost 80% of the students could identify the incorrect data with a partial quantification of theerror (giving either the value in the table or the graph), identify the incorrect generalization intwo, and give one comparison in three. 28% of the students could meet all the demands of thetask including identifying and quantifying the incorrect information in the bar graph, correct ageneralization about the graph, and make two comparisons about the length of the dinosaurs. 4%of the students scored no points on this task. All the students in the sample with this scoreattempted the task.


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Dinosaur Data

Points Understandings Misunderstandings0 All students in the sample with

this score attempted the task.6% of the students thought that thetyrannosaurus was incorrect. Also studentshad trouble quantifying the error, so 8%made statements that had nothing to do withthe size of the error.

2 Students could find theincorrect data on the graph, andwrite one comparison.

5% just mentioned the correct size, 27meters, of the value in the table. Another5% read the scale incorrectly and talkedabout 32 or 35 instead or 37.

3 Students could find theincorrect data on a graph,identify the incorrect thegeneralization about plant-eaters and meat-eaters, andmake one comparison.

Some students had difficulty with the ideaof estimation in context. They didn’tunderstand the word “about” or in thecontext of multiplication. They wanted thenumbers to be exactly 2 times larger for thebrachiosaurus.

4 Students could identify theincorrect data with a partialquantification of the error(giving either the value in thetable or the graph), identify theincorrect generalization in two,and give one comparison inthree.

Students had difficulty correcting thegeneralization in part 2. They thought thatlike humans, dinosaurs could eat both plantsand meat.

5 Students with this score usually missed bothcomparisons. (See table from previoussection.)

7 Students could identify and

quantify the incorrect

information in the bar graph,

correct a generalization about

the graph, and make two

comparisons about the length of

the dinosaurs.


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Implications for InstructionStudents need to look at a variety of graphs for understanding and describing trends. Theyshould think about how to combine different pieces of data into groups or categories. In thiscase, they were asked to think about meat-eaters and plant-eaters, but in a graph about howstudents come to school they might sort them by those requiring fuel or high-cost fuel (riding acar or bus) and those that don’t (walking, skateboarding, bicycling). Looking at favorite tvshows, it might be helpful to sort the data by type of show (drama, news, cartoons), length ofshows, time of shows depending on the types of questions the data is supposed to be answering.

Students at this grade level should start to think about important features of the data, largest,smallest, and range. They should also think about how to compare the data. Why are thesecomparisons useful? It might be interesting to know the something is 15 feet longer (comparisonsubtraction), but they should also start to think relationally. Does that 15 feet represent a smallchange, like 100 to 115, or is it a significant change like twice as large or three times as large.Warm up graphs, entry graphs, graphs from other subjects, newspapers allow students to quicklypractice and keep up on the comparison skills. At later grades this relational thinking will moveto making fractional comparisons or comparisons using percents.

Some students do not understand the logic of comparisons. Many just stated opinions orinformation not related to the data. Others gave facts in isolation, not giving a referent: Thetyrannosaurus is 12 meters, instead of the tyrannosaurus is 2 meters larger than the allosaurus orabout 1/3 of the size of the diplodocus. Again, using graphs as warm ups can quickly build upthese skills.

Ideas for Action Research:Using Attendance Graphs to Improve ComparisonsHave students do entry graphs such as:

• Did you come to school in a car, by bus, on a bike, walk?• How many nights a week does your family sit down at a table together for dinner?• What time do you get up in the morning? (6:00, 7:00, 8:00, . . .)• How many brothers and sisters do you have?

Use these graphs to get students to learn about comparisons. Can students talk about relativesize: most, least, and same. Can students make comparisons using subtraction (how much more,how much less)? Can students start to see multiplicative relationships (about twice as much,about 1/4 of, about three times as much? How do these skills improve over time?

When doing work in other areas, such as science or social studies, ask students to writesummaries of the graphs. Do these statements include comparisons?


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Stars

This problem gives you the chance to:• work with symmetry and area of shapes in a tessellation

1. How many sides does this shaded star

shape have?

____________________

2. Explain why the area of the star is 3 squares.

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

This pattern is made from lots of stars. Some of them are flipped.

Hexagons are made between the stars.

Copyright © 2006 by Mathematics Assessment Page 6 Stars Test 4Resource Service. All rights reserved.


63

3. Shade three of the hexagons in the diagram.

4. How many stars are there in the diagram? ________________________

5. Is the area of the hexagon bigger or smaller than the area of the star?

________________________

Explain how you figured it out.

________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________


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Stars Rubric

The core elements of performance required by this task are:• work with symmetry and area of shapes in a tessellation


1. Gives correct answer: 8 1 1

2. Gives a correct explanation such as:

There is one whole square and four half squares. 2 2

3. Shades three hexagons. 1 1

4. Gives correct answer: 16 1 1

5. Gives correct answer: bigger

and

Explains that there are 4 whole squares and 4 half squares, or 6 whole

squares

2

2

Total Points 7

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

65

StarsWork the task and examine the rubric.What are the core mathematics of this task? How does spatial visualization play inthe students’ ability to solve this task?

Some students had difficulty counting the sides of the star. Why do you think astudent might put 4 sides?

Explaining the area of the triangle posed problems for students. How many of yourstudents gave answers that:Explained

all theparts

Becausethat’s howbig it is!

Incomplete:1 whole &4 halves

Incomplete:triangles

are halves

Countedsides

Other

Did your students show reasoning about 1/2’s and wholes? What was yourevidence?For part four, how many stars are in the picture, the most common incorrect answerwas 12. What might the student be thinking?

Comparing sizes can be done by giving the total areas and showing which is biggeror by showing enough of one shape to prove it is larger. Look at your students’explanations in part 5. How many of your students gave answers such as:

The area ofthe hexagon

is 6

I justlooked. (noquantity)

I counted.(no

quantity)

Thehexagon is

smaller.

Hexagonsare justbigger!

Other

What was missing in students’ explanations that you wanted to see? What types ofdiscussions or experiences help students develop and improve their ability to makeexplanations? How do you help foster this development?


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Looking at Student Work on StarsStudent A does a good job of explaining the area for the star, taking into account all of thetriangles. Then the student shows how to find the area of the hexagon in order to compare theareas of the two shapes.Student A


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Student B seems to understand the big idea of area, but has not quite developed the formallanguage for describing it. In part 2 the student uses the word “sides” to mean the triangles on thesides. In part 5 the student actually does a quite sophisticated piece of reasoning. Part ofmeasurement is defining or identifying the unit. While most students use a unit of one square,this student uses a unit of one star or 3 square units. With this unit established the hexagon equals2 stars, so the student should probably have gotten full credit for this piece of the task. The rest ofthe answer should be considered further work, as the student after answering the first questionnow correctly compares the area of all the hexagons to all the squares.Student B


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Student C has the idea of finding area, but doesn’t recognize the correct shape for the hexagon.Student C


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Not all students are considering the unit of the shape as a square or group of squares. Student Dseems to be thinking about how many parts can I divide the shape into? In part 5, the studentseems to lose the logic of the problem. Instead of finding which is larger or smaller, the studentmight be saying they are different sizes. What do you think?Student D

As you read part 2 for Student E, you might think there is some understanding of putting trianglestogether. But as you read further in part 5, the student seems to be confusing area and perimeter.Student E


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Student F sees all four triangles being put together to make a large square. The student does notrealize that the square is a different size from the square on the grid paper. This is an importantpiece of measurement that the units need to be the same size. It is not enough to just be the sameshape.Student F

Student G does not understand the logic of making a justification. In part 5 the student just statesthe facts, not how the sizes were obtained or why the parts are relevant to the decision. Whatkinds of questions might push this student to give more detail or make a more convincingargument?


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Student G

Student H also has difficulty with making an argument. Look carefully at part 2. What do youthink the student is thinking? In part 5, the student actually finds the correct area for the hexagon,but still can’t explain why it is significant. What does this student need to understand in order tobuild a better argument or justification? What is the missing logic?

Student H


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Student I seems to be explaining the “3-ness” as helping to make the design instead of a functionof area. In part 5 the student again seems to ignore the relevance of the number part of area andgives a definition. Student J seems to waiver between the concept of area and perimeter. Again,not seeing the importance of quantifying the amounts.Student I

Student J


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In the small star diagram, it would appear that Student K understands the idea of area. The pointsare correctly marked as 1/2 and center as one. However, if you read the whole work carefully thestudent is really thinking of each side of the triangle as 1/2, giving you a total of 4 units as you goaround the outside. Notice that the student does not see the stars in the middle of the tessellation.Student K


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Fourth Grade

4th Grade Task 4 Stars

Student Task Work with geometric properties including spatial visualization,symmetry, and area of shapes in a tessellation.

Core Idea 4GeometryandMeasurement

Use characteristics, properties, and relationships of two-dimensionalgeometric shapes and apply appropriate techniques to determinemeasurements. Examine, compare, and analyze attributes ofgeometric figures.

• Understand line symmetry and predict the results of sliding,flipping, or turning two-dimensional figures.

• Investigate, describe, and reason about the results of combiningand subdividing figures

• Understand and use appropriate techniques to determinemeasurements.

Mathematics in the Task:• Ability to work with area of shapes on a grid• Ability to pick out individual shapes in a tessellation• Ability to quantify area and explain how each part of a shape contributes to area• Ability to build a convincing argument comparing the areas of two shapes

Based on teacher observation, this is what fourth graders knew and were able to do:• Identify a hexagon or a star within a tessellation, by counting the total number of the

shape or shading in individual shapes• Count the sides of the stars

Areas of difficulty for fourth graders:• Confusing finding the area of all the hexagons compared to all the stars, instead of

comparing the area of one hexagon and one star• Confusing area and perimeter• Quantifying their ideas• Understanding the logic of making a convincing argument, often showing some

understanding of the concept of area but not how to explain finding it in part two ormaking a complete justification of the difference in part five

Strategies used by successful students:• Numbering the parts of each star to find the area• Being able to compose and decompose shapes and then matching individual pieces


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Task 4 – Stars



The maximum score available for this task is 7 points.The minimum score for a level 3 response, meeting standards, is 4 points.

Most students, 89%, could shade in the 3 hexagons and count the total number of stars in thetessellation. Many students, 73%, could also count the number of sides on a star. More than halfthe students, 53%, could also explain why the area of the star was a 3. 25% of the students couldmeet all the demands of task including counting the sides of a star, decomposing the star toexplain how to find the area, identifying hexagons and stars within a tessellation, and comparingthe areas of two shapes. 3% of the students scored no points on this task. All of the students inthe sample with this score attempted the task.


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Stars

Points Understandings Misunderstandings0 All the students in the sample

with this score attempted thetask.

Some students confused pentagon andhexagon and shaded in the incorrect shape.Many students, 7%, only counted the 12stars around the outside of the tessellation.

2 Students could shade in 3hexagons and count the numberof stars in the tessellation.

Students had difficulty counting the sidesof the star. 12% of all students thoughtthere were 4 sides. A few thought therewere 8 sides or 3 sides.

3 Students could count sides andnumbers of stars and identifythe hexagons.

Students did not know how to explain whythe area of a star was 3 squares. Theytalked about the properties of the shape: itfits the pattern, its pointy, otherwise itwouldn’t be a star, that’s how big it is(8%). Some students disagreed with theanswer and said it was equal to 4 squares.

4 Students could begin todescribe why the area was 3squares.

They may have been able to describe that 2triangles make a square, but didn’tcomplete the rest of the argument (9%) ortalked about combining some triangles butnot quantifying how much that made orhow it contributed to the total (3%).

5 Students could count the sidesand number of stars, shade inthe hexagons, and either explainwhy the hexagon was larger orexplain why the area of star was3 squares.

13% of the students stated that the hexagonwas smaller. 6% said something like,“they’re just bigger!” Many students talkedabout a procedure: 8% - “I just counted”.6% - “I could tell by just looking”.

7 Students could count the sidesand number of stars, shade inthe hexagons, explain why thehexagon was larger, decomposethe star to explain why the areawas 3 squares.


78

Implications for InstructionSome students need more practice with spatial visualization and learning attributes of shapes.They should have opportunities to pick out shapes that have been flipped or rotated from within amore complex design. They may need to work with puzzle pieces, where they can actually feelthe sides of shape and the size of angles. These types of experiences help to build theirunderstanding of geometric properties and progress through the van Hiele levels.

The work on this task suggests that many students need more opportunities to explain whysomething is true or develop a convincing argument. While most students had the idea of area“taking up space”, they did not know how to decompose the shape to find area or build aconvincing argument that accounted for all the parts. Classroom discussions about how youfigured it out, should include looking at how much detail is needed to complete the justification orconvince others that an answer is correct. Questions like, “Is it enough to convince someone tosay that two triangles make a square? Why or why not? What could make this argument moreconvincing? Why do you think that?” , help students develop the logic of making an explanationor justification.

The same holds true for making a comparison. Students need to understand that it is not enoughto talk about one choice alone. Reference must be made to how or why it is different fromsomething else or from a group of objects. How is it different? Is there a way to quantify thatdifference? Can the comparison be made by matching parts and then by seeing what is left over?A lesson could be designed around giving three or four different justifications (See the work ofStudents A,G, H, and I for part 5.) and having students discuss which is more convincing andwhy. What are the qualities they prefer in a justification? This type of discussion really focusesthem on not only the mathematics of a comparison but on the qualities of a good justification.

Ideas for Action Research:Is Counting Enough?Visual CluesGive students some common optical illusions and ask them which is longer or shorter, smaller orlarger. Ask them to explain how they knew. After all students have made a commitment to one ofthe choices, have students measure to find the correct answer. Then discuss why it is important tomeasure.


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Now give students some shapes on a grid. Ask them to make a guess based on how the shapeslook about which one is larger. Then have them count the area of the shapes.

Do students still think you can tell from looking? What have they learned about the importance ofquantifying answers? Were students able to think about the top of the long shape as 1/2 of asquare? How did they build their argument?


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Building the Idea of Comparisons, Helping Students Build Justifications:Think of some situations that require comparison, like getting the better deal at the store, findingthe largest area of a shape, who had the most hits on the team. Then design some tasks to helpstudents build the logic of making a complete comparison. How do they learn to show which isbetter, bigger, or more by quantifying all the objects being compared?

Consider this task.

18 students are going on a field trip. Three students can fit in each car and the cost of gas for thetrip is $12. Six students can fit in a minivan and the cost of gas for each minivan is $25. Thestudents can get a round trip city bus ticket for $4. Which is the best deal? How do you know?

How do students justify their choice?

Consider this task.

Ralph, Fred, and Bob were on the same little league team. Ralph hit one out of every two timeshe was at bat. He was at bat 24 times during the season. Fred hit three out of every four at-bats.He was at bat 20 times. Bob had a total of 13 hits. Who had the fewest hits? How did you know?

Were the students able to quantify their solutions? Did they give numbers for all the students?

For both problems, pick 3 or 4 different types of justifications and have students discuss whichone is the correct solution and decide what convinced them.


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Bikes and Trikes

This problem gives you the chance to:• solve number problems in a real context

The cycle shop on Main Street sells bikes (two wheels) and trikes (three wheels).

1. Yesterday, Sarah counted all of the cycles in the shop.

There were seven bikes and four trikes in the shop.

How many wheels were there on these eleven cycles? ______________

Show your calculation.

2. Today, Sarah counted all of the wheels of all of the cycles in the shop.

She found that there were 30 wheels in all.

There were the same number of bikes as there were trikes.

How many bikes were there? ______________

How many trikes were there? ______________

Show how you figured it out.

Copyright © 2006 by Mathematics Assessment Page 8 Bikes and Trikes Test 4Resource Service. All rights reserved.

8


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Bikes and Trikes Rubric

The core elements of performance required by this task are:• solve number problems in a real context


1. Gives correct answer: 26 wheels

Shows work such as:7 x 2 and 4 x 314 + 12 =

Accept repeated addition or diagrams

1

2

3

2. Gives correct answers: 6 bikes and 6 trikes

Gives correct explanation such as:6 bikes = 12 wheels6 trikes = 18 wheelsin all 30 wheels

May list or draw diagrams1 bike and 1 trike = 2 + 3 = 5 wheels2 bikes and 2 trikes = 4 + 6 or 2 x 5 = 10 wheels3 bikes and 3 trikes = 6 + 9 or 3 x 5 = 15 wheels4 bikes and 4 trikes = 8 + 12 or 4 x 5 = 20 wheels5 bikes and 5 trikes = 10 + 15 or 5 x 5 = 25 wheels6 bikes and 6 trikes = 12 + 18 or 6 x 5 = 30 wheels

2

3

5Total Points 8


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Bikes and TrikesWork the task and examine the rubric. How does this task get students to focus on multiplegroups and using multiplication in context? What strategies do you think students would use tosolve this task?

How often do students in your class get problems with multiple constraints? What strategies doyou have them use to keep track of what they know and what they are trying to find out?

Look at student work in part 1. What might be the thinking of students who gave an answer of22? Of 28? Of 14? What strategies did your students use to think about and solve this part of thetask? Did your students use:

Calculationsor number

sentences only

Repeatedaddition

Diagrams orpictures

Combinationcalculations& diagrams

Labeling ofanswers

Other

In part two, the students needed to keep track of several constraints: the number of wheels on abike and on a trike, the number of cycles had to be equal, and altogether there had to be 30wheels. How many of your students put:

6,6, 4,7 Combination that yieldscorrect # wheels, such as 9, 4

or 12, 2

15,10 Other

What might be the errors in logic for some of these errors?

Now look at strategies. Did your students use:Calculations like:6x2=12, 6x3=18,

12+18=30

Counting by 5’s ordividing by 5’s

Making a table Drawing orDiagrams

Other

What might have confused students about the mathematics of the task?What types of experiences do students need?


84

Looking at Student Work on Bikes and TrikesMany students seem to understand the multiple groups and combining groups. For them, the taskis just a series of multiplication and addition steps. See the work of Student A.

Student A


85

Student B uses pictures to think about labeling the computations in part one and then transitionsinto using a diagram in part two. The student knows that for the bikes and trikes to be equal therewill be sets consisting of a bike and a trike. The diagram shows that for each set there are 5wheels. This allows the student to find how many sets of 5 fit into 30 wheels.

Student B


86

Student C shows similar thinking about the 5 wheels for each set, but solves the task by making atable. The numbers on the left also make equivalent fractions, 2/3=4/6=6/9. At later grades thiswould be a useful connection to help students make.Student C


87

Student D also uses a table to think about all the combinations of equal bikes and trikes. Then thestudent can check for combinations that yield 30 wheels.

Student D

Student E and F both use diagrams to find the number of bikes and trikes that yield 30 wheels.For part two the diagrams look similar. The difference is in how Student E makes sense of theconstraint, equal number of each, and matches the 2’s and 3’s.


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Student E

Student F


89

Scores for this task, clustered around 8, 3, and 0. Students with a score of 3,like Student F,struggled with part three of the task. Student F was able to find a solution that yielded the rightnumber of wheels, but missed the constraint “same number of each”. There were other types ofconceptual errors around part two of the task. Student G did not make sense of the 30 wheels andjust used listed the information given about bikes and trikes in previous part of the task.Student G

Student H did not understand that the two types of bikes combined needed to equal 30 wheels andsolved the simpler problem of how many bikes equal 30 wheels and how many trikes equal 30wheels.


90

Student H

Many students, like Student I, found equal amounts of bikes and trikes to yield 30 “cycles”instead of 30 “wheels”.Student I


91

Why are some students scoring no points on the task? Are there any understandings to build on?What misconceptions do they need to overcome? Look at Student J. The student adds the bikesand trikes together and multiplies by 2 wheels per cycle. The student doesn’t pick up on thedifference in wheels between the two types of cycles. In the part two, the student seems to havelearned an underline strategy to help identify what is being asked, but then mistakes 30 for bikesinstead of wheels.

Student J

Student K shows similar thinking. The student draws out the 11 bicycles (The top dot mayrepresent the seat or handle bars on the bike) and counts the wheels. In part 2, Student K splits 30cycles into two equal parts.


92

Student K

Student L seems to have the strategy of underlining important information. The student sees twowheels and three wheels and adds them together to get 5 wheels, ignoring the other information inthe story. In part two the student says, “Read the directions, it says there are 30.” The studentdoesn’t perceive that a question is being asked that requires calculations.


93

Student L

Student M uses numbers and number sentences that appear unrelated to the context of theproblem. Where do you start to help this student understand basic ideas about operation andquantity?Student M


94

Student N makes no sense of the first part of the task or the meaning of multiplication. Thestudent multiplies 4 trikes by 7 bikes to get 28_______(?). However, the student does have someunderstanding of the context, because in part two the solution would yield 30 wheels. It isimportant not to generalize about all the students with the same low score, because what they aremaking sense of varies widely and requires different intervention strategies.Student N


95

Fourth Grade

4th Grade Task 5 Bikes and Trikes

Student Task Use multiplication and division to solve problems about wheels per bikeand total wheels in a bike shop.

Core Idea 3Patterns,Functions,and Algebra

Understand patterns and use mathematical models to represent andto understand qualitative relationships.

• Find results of a rule for a specific value.• Use inverse operations to solve multi-step problems• Use concrete, pictorial, and verbal representations to solve

problems involving unknowns.• Understand and use the concept of equality.

Mathematics of the task:• Ability to add and multiply• Ability to work with equal-sized groups of objects• Ability to use multiple constraints• Ability to begin reasoning proportionally

Based on teacher observations, this is what fourth graders knew and were able to do:• Knew multiplication facts• Could draw diagrams or make number sentences to help them solve the task• Knew the difference between bikes and trikes

Areas of difficulty for fourth graders:• Tracking all the constraints in part two• Confusing wheels and cycles• Making sense of the entire set of information before beginning computation

Strategies used by successful students:• Making diagrams• Labeling answers to keep track of what each calculation represented• Counting or dividing by 5’s (seeing the incremental number of wheels)• Making tables


96


Task 5 – Bikes and Trikes



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

Most students, 70%, understood the process for finding the total number of wheels in part one, butsome may have made computational errors. More than half the students, 65%, could solve partone with no computational errors. 30% of the students could make sense of the entire taskincluding finding equal numbers of bikes and trikes to make 30 wheels. 27% of the studentsscored no points on this task. All of the students in the sample with this score attempted the task.


97

Bikes and Trikes

Points Understandings Misunderstandings0 All the students in the sample

with this score attempted thetask.

Students were confused by the constraints.They may have added bikes and trikesbefore multiplying, added just the wheels,or multiplied bikes times trikes.

2 Students understood the processfor finding the total number ofwheels in part one, but madecomputational errors.

Basic addition and multiplication errors.

3 Students could find the numberof wheels for 7 bikes and 3 trikesusing drawings/diagrams (7%),or multiplication and addition(32%).

Students did not understand some of theconstraints in part 3. They confused 30wheels for 30 cycles. They worried aboutgetting 30 wheels and forgot about gettingthe “same number” of each kind. Somesolved the simpler problem. How manybikes make 30 wheels? How many trikesmake 30 wheels?

5 Students could find the equalnumber of bikes and trikes tomake 30 wheels, but could notsolve anything in part 1 of thetask.

8 Students could deal withmultiple constraints and thinkabout equal, repeated groups:groups of bikes with two-wheelsor groups of trikes with three-wheels. They could find thetotal number of wheels for agiven number of bikes and trikesor work backwards from thenumber of wheels to the numberof cycles.


98

Implications for InstructionStudents need to look to recognize contexts, which involve multiple groups, as multiplicationsituations. Students need practice picking out key pieces of information and organizing theirwork to solve problems. The grappling with setting up the problem and then discussing waysother students set up the problem helps students build an understanding of the meaning of theoperation of multiplication. Learning to make diagrams (students should be transitioning frompictures to diagrams), using a bar model, or working with a number line help students to “see” theaction of the story problem.

Another important tool for making sense of calculations is the use of labels. It is not that labelsare a rule to please the teacher. When working several steps at once, labels can help the studentknow what has been found and think about what still needs to be done. In looking at studentwork, many of them quit before the final step. Could putting a post-it with a quick note aboutlabels help them see that they aren’t done yet? How would using labels help students think morecarefully in part 2?

Ideas for Action Research

Learning from Good Mistakes:Sometimes looking at a mistake can help uncover some deeper mathematics and confrontstudents’ misconceptions and can promote good, productive discussions. The process of trying toreconcile what is in the mistake and what would work to solve the problems, helps students tofirm up their ideas and cement their learning. It also allows students to see their own logic errorsand revise their thinking.

Consider posing the following problem to your class after everyone has had a chance to try andmake sense of this problem by themselves:Cynthia thinks that it is important to add the number of wheels together. Three wheels plus 2wheels equal 5. Could this help her solve part one? Why or why not? Could it help her solvepart two? Why or why not? How are these two situations different? Why can we use it in one partbut not both?Then have students think about:Conner thinks there is an add and multiply part to the problem. He does 7+4 = 11 and then 11x2 = 22. What would be the labels for the 7, 4, and 11? What does it mean when we multiply the11 by 2? What is being found? Is this what we want to know? Why or why not?

Make an overhead transparency of the work of Student E and F. Both students have solutions thatyield an answer of 30 wheels. Can they both be right?

After students have discussed the question, state that both drawings look almost alike. Why didone give the correct answer for this task and one give only a partially correct solution? What isdifferent about the two? This important idea of what is the same and what is different helpsstudents think about diagram literacy and ways to use diagrams productively in the future.

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