Overall Frequency Distribution by Total Score Grade 4 M e an=24.62; S.D.=8.31 500
Frequency
400
300
200
100
0 0
1 2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Frequency
Fourth Grade – 2004
pg. 1
Level Frequency Distribution Chart and Frequency Distribution 2004 - Numbers of students tested in 4th grade: 10217 Grade 4 2000 - 2001 % at % at least % at % at least Level ('00) ('00) ('01) ('01) 1 30% 100% 18% 100% 2 37% 77% 23% 82% 3 26% 33% 39% 58% 4 7% 7% 19% 19% Grade 4 2002 - 2004 % at least % at % at least ('02) ('03) ('03) 100% 8% 100% 92% 14% 92% 66% 36% 78% 28% 42% 42%
% at ('02) 8% 27% 37% 28%
Level 1 2 3 4
% at ('04) 7% 18% 34% 41%
% at least ('04) 100% 93% 75% 41%
5000
4000
Frequency
3000
2000
1000
0
0-11 1 Minimal Success
12-18 2 Below Standard
19-27 3 At Standard
28-40 4 High Standard
747
1796
3456
4218
747
1796
3456
4218
Frequency
Frequency
Fourth Grade –2004
pg. 2
4th grade
Task 1
Saturday Afternoon
Student Task
Given a table of time information, solve problems involving the comparison of time, the doubling time and elapsed time.
Core Idea 5 Data Analysis
Collect, organize, represent and interpret numerical and categorical data, and clearly communicate their findings. • Interpret data to answer questions about a situation.
Core Idea 4 Geometry and Measurement
Apply appropriate techniques to determine measurement. • Choose appropriate units and use these units to measure time.
Fourth Grade – 2004
pg. 3
Fourth Grade –2004
pg. 4
Fourth Grade – 2004
pg. 5
Looking at Student Work on Saturday Afternoon The major obstacle in this task was being able to convert minutes to minutes and hours. Sometimes looking at the strategies of successful students helps to uncover good ideas for instruction. Student A breaks apart the 70 minutes into 1 hour and 10 minutes. The student uses a counting on model to add 3 hours to 1:45pm then adds the additional 20 minutes. Student A then decomposes the 65 minutes into 1 hour and 5 minutes. Student A
Fourth Grade –2004
pg. 6
Student B thinks about doubling the time by dealing with the hours and minutes separately. The student also uses a counting-on strategy to help with the addition in part 4. Student B
Fourth Grade – 2004
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Student C also thinks of the hours and minutes as separate tasks. Student C shows the subtraction for the conversion from minutes to hours and minutes. Notice how the student shows comparing the times to make the decision in part 2. Student C
Fourth Grade –2004
pg. 8
Student D uses a one-step conversion process of adding 1 hour and subtracting 60 minutes to the total times. Student D
Fourth Grade – 2004
pg. 9
Using addition for doubling can cause confusion for students. Student E adds 2 to the hours and minutes in part 3 instead of multiplying by 2. Student E
Student F had difficulty working with units of time. In part 3 the student does not make the conversion from 2 hours 70 minutes to 3hours 10 minutes. In part 4 the student attempts to use a clock model for help. The student starts with the minute hand in the wrong location. Then when attempting to count on for the hours Student F starts with the beginning hour rather than moving ahead one hour.
Fourth Grade –2004
pg. 10
Student G has some trouble thinking about elapsed time. The student tries to use subtraction instead of addition. The student struggles with how to take the large number from the small number. Notice that the 1:45 is rearranged to fit a notion about subtraction. Student G
Student H shows another common error. In trying to compare times, the student only considers the leading digit ignoring the size of the units and therefore picks skates for the shortest time. As with many students who are just starting to make sense of a concept, Student H makes a correct conversion in part 3, but fails to convert in part 4.
Fourth Grade – 2004
pg. 11
Student H
Teacher Notes:
Fourth Grade –2004
pg. 12
Frequency Distribution for each Task – Grade 4 Grade 4 – Saturday Afternoon
Saturday Afternoon M e an: 4.51, S.D.: 1.97
3000 2500
Frequency
2000 1500 1000 500 0 Frequency
0
1
2
3
4
5
6
7
188
453
1343
1375
1631
1478
1303
2446
Score
Score: %=
0 1.8% 100.0%
1 6.3% 98.2%
2 19.4% 93.7%
3 32.9% 80.6%
4 48.8% 67.1%
5 63.3% 51.2%
6 76.1% 36.7%
7 100.0% 23.9%
The maximum score available for this task is 7 points. The cut score for a level 3 response, meeting standards, is 4 points. Most students, about 91%, could compare times between two people and identify the shortest time. Many students (67%) could compare times and recognize the correct procedures for doubling time spent ice-skating and finding elapsed time. About half the students could compare times and either find elapsed time or calculate the double time and elapsed time without making conversions. About 24% of the students could meet all the demands of the task including finding elapsed times, doubling times, and making conversions between minutes and hours and minutes. Less than 2% of the students scored no points on the task. Of the students with this score 60% attempted the task.
Fourth Grade – 2004
pg. 13
Saturday Afternoon Points Understandings About 60% of the students with this 0 2
score attempted the problem. Students could compare times to find the shortest time or determine who would finish an activity first.
4
Students could compare time and either double the time for iceskating and convert units correctly or write the correct equations for doubling time and finding elapsed time.
5
Students could compare and double time, including making the conversions, and write a correct process for finding elapsed time.
Some students looked at the leading digit, ignoring the value of the units. These students picked skating for the shortest activity. The most common error for doubling time was to leave the answer as 2hr. 70 min. Of the students who made other errors, 21% just read the time off the table – 1 hour and 35 min. Another common error was to realize 70 minutes made an hour and put 3hr. 70 min. forgetting to change the minutes or 2hr. 10 minutes forgetting to change the hours. The most common errors for part 4 were to put 2:25, 4:45, or 3:45. Students either forgot to convert in part 3 or part 4.
6 7
Misunderstandings
Students could compare and double times, find elapsed time, and change units into standard notation when the minutes derived from a calculation exceeded one hour.
Teacher Notes:
Fourth Grade –2004
pg. 14
Based on teacher observations, this is what fourth grade students seemed to know and be able to do: • Interpret data from a table • Knew how to use multiplication to find twice as much • Compare times Areas of difficulty for fourth graders, fourth grade students struggled with: • Converting minutes to minutes and hours • Calculating elapsed time
Questions for Reflection on Saturday Afternoon: • • • •
• •
What opportunities have your students had to work with time this year? Could they use addition and multiplication with the different units or did they make models? Were they able to successfully convert from minutes to hours and minutes? What strategies helped them with the conversion? Could these strategies help them with conversions between other units of measurement? How is the logic of conversions the same for all units? How is the logic similar to that used in place value? What instructional activities might help students make this connection? How many of your students ignored the size of the units when trying to find the shortest time (picking 1 hour and 35 minutes instead of 55 minutes)? How does this relate to place value and conversion? What further experiences do your students need with measurement operations and understanding units?
Teacher Notes:
Instructional Implications Some students struggled with comparing times. They seemed to have the notion of comparing numbers based on the digit on the left, rather than the value of the digit. This would lead to errors like 1 hour 35 minutes is less than 55 minutes. Students need more than a procedure to think about place value, particularly in the context of different size measurement units. Understanding the “0” hours in 55 minutes is a nontrivial piece of thinking for students to discover. They need a variety of experiences and good questions to develop this idea.
Fourth Grade – 2004
pg. 15
There is a certain logic involved in how conversions and standard notation works for different measurement units. If students can learn not only the procedure for specific units, but understand the logic behind it, then they can work with other measurement units successfully without further instruction. Students at this age need lots of experiences working with units and solving problems. Elapsed time is a difficult concept and students should develop a variety of strategies for dealing with this. Successful students used timelines, clocks, or computational strategies.
Teacher Notes:
Fourth Grade –2004
pg. 16
4th grade Student Task Core Idea 5 Data Analysis Core Idea 2 Number Operations
Task 2
Chips and Soda
Read and interpret information from a bar graph. Read and complete a frequency table and bar graph representing the same data. Make and justify a prediction based on this data. Collect, organize, represent and interpret numerical and categorical data, and clearly communicate their findings. • Represent data using tables, charts, line plots, and bar graphs. • Interpret data to answer questions about a situation. •
Fourth Grade – 2004
Understand division as the inverse operation of multiplication, the operation of sharing, partitioning, repeated subtraction, and an operation to determine rates.
pg. 17
Fourth Grade –2004
pg. 18
Fourth Grade – 2004
pg. 19
Fourth Grade –2004
pg. 20
Looking at Student Work on Chips and Soda Student A does a good job of explaining her thinking. Notice how calculations are shown for part 1b and 1d. On the graph the student labels both axes and adds a title. Student A uses a chart to find not just the number of cola drinks, but the number of each kind of drink. Student A
Fourth Grade – 2004
pg. 21
Student A, continued
Fourth Grade –2004
pg. 22
Student B also verifies calculations in part 2. Student B quantifies the relationship between the 50 people surveyed and the number of people attending the party. Student B
Fourth Grade – 2004
pg. 23
Student C makes a common error of only labeling the horizontal axis on the graph. This may be a partial cause for the student thinking about the graph representing number of drinks rather than number of people. The student looks not for the number of total drinks to buy, but the number needed to add to reach a total of 25. Student C
Fourth Grade –2004
pg. 24
Student D does not see the connection between the survey information about what people like and the drinks for the party. The student only looks for some combination of drinks to make a total of 25. Student D
Fourth Grade – 2004
pg. 25
Student E also has trouble thinking about the numbers in the graph. The student uses the 16 as if it were the number of colas in a box. The student also assumes that 25 colas are needed for the party. The graph is not seen as a representation of preferences, even though the student was able to label the axis. Student E
Many students have difficulty using the graph to solve for part 3. Student F seems to associate the word “cola” as a generic word for all sodas. Secondly, the Student does not decrease the number of drinks to match the number of people attending the party, but just sums the total number of drinks in the survey. Student F
Fourth Grade –2004
pg. 26
Student G attempts to use a line graph for categorical data. There are no points between lemonade and cola. The student also uses the word “cola” as a generic word even though it is listed as one of the data groups on the survey and the graph. Student G
Student H has trouble making a bar graph. The first two bars are correct. The orange bar shows some understanding of making the odd number higher than the ten. However as the student progresses across the bars go to the top of the box with the appropriate number, instead of the bottom. Is the student thinking of the whole box representing the 14 or is it a question of following a line for a longer distance? The student also does not make allowances for the lemonade being an odd number.
Fourth Grade – 2004
pg. 27
Student H
Student I has trouble dealing with the scale for the bar graph. In section one the student interprets each box as one item on the graph, disregarding the vertical scale. When asked to make a graph of the information in part 2, Student I just recopies the frequency chart. The student also does not relate the change in numbers in the survey to the number of students attending the party in part 3.
Fourth Grade –2004
pg. 28
Student I
Fourth Grade – 2004
pg. 29
Student I, continued
Teacher’s Notes:
Fourth Grade –2004
pg. 30
Frequency Distribution for each Task – Grade 4 Grade 4 – Chips and Soda
Chips and Soda M e an: 6.06, S.D.: 1.81
3500 3000
Frequency
2500 2000 1500 1000 500 0 Frequency
0
1
2
3
4
5
6
7
8
9
10
60
108
298
519
853
1434
1997
3300
1032
410
206
Score
Score: %=
0 0.6% 100.0%
1 1.6% 99.4%
2 4.6% 98.4%
3 9.6% 95.4%
4 18.0% 90.4%
5 32.0% 82.0%
6 51.6% 68.0%
7 83.9% 48.4%
8 94.0% 16.1%
9 98.0% 6.0%
10 100.0% 2.0%
There is a maximum of 10 points available for this task. The cut score for a level 3 response, meeting standards, is 5 points. Most students (95%) could identify the largest group represented on the bar graph, use the scale to identify the number of people liking tortilla chips, and total the number of tallies on the frequency chart. Many students (82%) could read data off a bar graph, interpret tally marks on a frequency chart, and make a bar graph using data from the frequency chart. A little less than half the students (48%) could answer a variety of questions about a bar graph with a vertical scale in 2’s, read a frequency chart and use it to make a graph. Many of these students did not label the vertical axis of the graph. About 2% of the students could meet all the demands of the task including labeling both axes on the graph they made and cutting the survey information in half to determine the number of colas needed for a party. Less than 1% of the students scored no points on the task. None of the students surveyed scored zero.
Fourth Grade – 2004
pg. 31
Chips and Sodas Points 0
Understandings
3
Students with this score could identify the largest group represented on the bar graph, use the scale to identify the number of people liking tortilla chips, and total the number of tallies on the frequency chart. Students could read data off a bar graph, interpret tally marks on a frequency chart, and make a bar graph using data from the frequency chart.
5
7
8
10
Misunderstandings Less than 1% of the students scored no points on this task. Most students who could not read the number of people preferring tortilla chips picked 8 instead of 10, reading the number from the bottom of the top box instead of the number from the top of the bar.
Some students had difficulty graphing the odd numbers. Students did not label the vertical axis on the graph. Students did not read the scale when calculating the total for the first bar graph. 30% of the students who missed 1c put an answer of 25 instead of 50. Students also counted boxes to compare plain and barbeque rather than using the values for 1d. Students did not label the vertical Students could answer questions axis on their graphs. Students could about a graph with a scale other than 1, count tallies on a frequency not understand how to use the survey of 50 students to predict the number chart, and make a graph from a of drinks needed for 25 students frequency chart. attending a party. Almost half the students who missed Students could answer questions about a graph, count tallies and use part 3, read the number of cola drinks from the survey as 16, making it to make a graph with labels on no compensation for the fact that the both axes. survey was for 50 students. 8% of the students took the 16 from the graph and wanted 9 more colas to get a total of 25 drinks. Another 8% wanted 25 drinks for 25 people. 6% saw a total of 50 drinks in the survey and wanted 50 drinks for the party. Students could read and interpret a Students had difficulty relating the survey to the problem situation. graph and make their own graph Many interpreted their graphs as using a scale other than one. representing the number of drinks, Students could use proportional reasoning to adjust the information instead of number of people. They in the graph in half, as only half as did not see it as having information needed to answer the final part of the many students were attending the question. party.
Fourth Grade –2004
pg. 32
Based on teacher observations, this is what fourth grade students seemed to know and be able to do: • Read and interpret a graph • Complete a bar graph using information from a frequency table • Use a scale of 2 on a bar graph, including graphing odd numbers • Comparison subtraction • Find the total Areas of difficulty for fourth graders, fourth graders struggled with: • Labeling the axes of a graph • Using a graph to make a decision • Relating the numbers in the graph to number of drinks needed • Recognizing the relationship between 50 people and 25 people
Questions for Reflection on Chips and Soda: •
Look at student work on part one. Did your students make sense of the scale for answering questions 1c and 1d? Were students counting individual boxes? Were they reading the wrong number from the scale (e.g. 8 instead of 10)?
When making their own graphs, how many of your students: Correct graphs
• •
No labels
Horizontal labels only
Marked odd numbers incorrectly
Made bars one box too high
Other
What experiences have your students had working with graphs that have a scale other than one? How often do students make their own graphs? How could these activities promote an understanding of a need to change the scale to include more data?
Look at student work on part 3. Could your students understand the relationship between the number of students surveyed and the number of students attending the party? • • •
When doing graph activities in class, so students try to summarize what they learn by looking at the graph? Are they asked to think about how the information in survey might be used to make decisions? Do they see a purpose in have a graph?
Fourth Grade – 2004
pg. 33
Teacher’s Notes:
Implications for Instruction: Students need more work with graphing when the scale is not 1 unit. They should have practice with simple scales of 2, 5, or 10 and how to estimate amounts that do not fall on a line. This is especially difficult for them when they go to use the numbers on the graph for calculations, such as finding the total number. Students need to be exposed to a variety of questions that can be asked about the same set of data. Students also need more practice making their own graphs with appropriate titles and labels. Students need to see graphs as tools to help make decisions, such as how many sodas to buy. While mastery is not expected at this level, students should start to form ideas about scaling; like noticing that 25 is half of 50. Students should be exposed to rich problems that let them think about the idea of twice as many or half as much.
Teacher’s Notes:
Fourth Grade –2004
pg. 34
4th grade Student Task Core Idea 3 Patterns, Functions, and Algebra
Task 3
Piles of Oranges
Describe, extend and make generalizations about a growing pattern of oranges that are displayed in a grocery store. Understand patterns and use mathematical models to represent and to understand qualitative and quantitative relationships. • Represent and analyze patterns and functions using words, tables, and graphs. • Find the 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.
Fourth Grade – 2004
pg. 35
Fourth Grade –2004
pg. 36
Fourth Grade – 2004
pg. 37
Looking at Student Work on Piles of Oranges Student A is able to accurately draw the piles of oranges, while maintaining the linear rows. The student sees the pattern of growing addition underneath the drawings and uses it to solve further parts of the task. The student also uses a grouping strategy to simplify the final addition problem. Student A
Fourth Grade –2004
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Student B notices that the pattern is growing by adding a larger number each time. The student uses this generalization to make a table to solve for the number of oranges in pile #9. Student B
Many students had difficulty explaining why 44 was the wrong number of oranges. Students, like Student C, often did not give enough information. Why is 44 not enough? How did you know 44 was not enough? Student D sees the addition pattern as evidenced in part 3, but doesn’t use this to explain why the counting is incorrect or how he got the 45. Student E had some strategy for finding the total number of oranges and reasons about the missing one orange. All these students can’t connect their logic of getting a correct solution to a justification in part 4. Student C
Fourth Grade – 2004
pg. 39
Student D
Student E
Student F is not looking for mathematical sequences or functions. The student tries to find a pattern of even and odd numbers, which is not useful for working with a growth pattern. Student F
Fourth Grade –2004
pg. 40
Student G sees a pattern of adding an increasing larger number each time. However the student struggles with making a good mathematical representation and gets an incorrect value for 6. Students need lots of practice with making and seeing mathematical representations to develop the ability to see and draw in rows. Student G
Fourth Grade – 2004
pg. 41
Many students look for an easy adding on pattern in tables, rather than trying to think about how the physical model grows. Student H sees the oranges growing from 3 to 6 and decides the pattern is adding 3 every time. Almost half the students who made error on the table made this mistake. Student H
Teacher’s Notes:
Fourth Grade –2004
pg. 42
Frequency Distribution for each Task – Grade 4 Grade 4 – Piles of Oranges
Piles of Oranges M e an: 4.21, S.D.: 3.22
3500 3000
Frequency
2500 2000 1500 1000 500 0 Frequency
0
1
2
3
4
5
6
7
8
2915
324
549
393
523
603
1282
1366
2262
Score
Score: %=
0 28.5% 100.0%
1 31.7% 71.5%
2 37.1% 68.3%
3 40.9% 62.9%
4 46.0% 59.1%
5 51.9% 54.0%
6 64.5% 48.1%
7 77.9% 35.5%
8 100.0% 22.1%
There is a maximum of 8 points available for this task. The cut score for a level 3 response, meeting standard, is 4 points. Many students (about 68%) could correctly draw the next pile of oranges in the sequence. More than half the students (60%) could continue the pattern in drawings or in the table. Almost half the students (48%) could continue the pattern in drawings, in the table, and to the next number beyond the table. Almost 36% of the students could extend the pattern in pictures, tables, numbers, and use a numerical pattern to work beyond the numbers in the table from 5 to 9. 22% of the students could meet all the demands of the task including making a mathematical justification for why 44 is the incorrect number of oranges for pile 9. Almost 29% of the students scored no points on this task. All the students in the survey attempted the task.
Fourth Grade – 2004
pg. 43
Piles of Oranges Points 0
Understandings All the students with this score attempted the task.
2
Students could draw the next figure in the sequence.
4
Students could draw pile #4 and fill out the table. Students could fill in the table, find the number of oranges in pile 6 and explain how they got it, and either draw pile 4 or solve for pile 9 and explain why it is not 44 oranges.
6
7
8
Misunderstandings Students had difficulty drawing the pattern. Some students drew the total number of oranges in a straight row, made square boxes of oranges, or left out a row. These students were not able to think about the attributes of the physical pattern. When filling out the table, many students (50%) tried to add on by 3’s every time. This also led them to calculate an answer of 15 for part 3. Even though the drawing is the easiest points for most students to get, it is also one of the most difficult points for other students to get. The spatial relationships required are not trivial. Students who missed question 5 were usually thinking about groups of 3. If they added on by three’s from the table they would get an answer of 24 (7% of the students) or they multiplied 9 x 3 = 27 (12%). Students could do the entire task except explain why 44 was incorrect. Most students were too vague, “It’s not enough.” “She counted wrong.” “She forgot the one on the top.”
Students could extend a pattern using pictures, tables, and patterns. They could make a justification for why a number does not fit the sequence of piles of oranges without using a drawing.
Fourth Grade –2004
pg. 44
Based on teacher observations, this is what fourth graders knew and were able to do: • Extend pattern with drawing • Extend pattern using a table • Add on with the pattern to find number for pile 6 and for pile 9 • Show or explain thinking for how they extended the pattern Areas of difficulty for fourth graders, fourth graders struggled with: • Explaining the patterns in words • Explaining why 44 does not fit the pattern/ analyzing the thinking of other students • Generalizing using all available information • Writing equalities
Questions for Reflection on Piles of Oranges • • •
• • •
What types of growing pattern problems have your students worked with this year? When looking for patterns are students asked to describe what stays the same and what changes? How does looking at the geometrical attributes help them extend the numerical pattern and make generalizations? What activities do students do to help them build their skills at spatial visualization and mathematical representations? When looking at arrays do you think your students see rows and columns? How many are still not at that level of development? When looking at patterns, are students asked to justify their thinking? What habits of mind might have helped students realize the pattern was not just growing by 3’s? What clues were available to them? What types of opportunities do students get in your classroom to practice and develop their ability to make justifications? What strategies did students use to solve part 3 and part 4? Did they make a drawing? Extend the table? Write a number sentence? Other?
Teacher’s Notes:
Implications for Instruction: Students need to learn to describe how a pattern grows in words, by explaining what stays the same and what changes. Students need to look at all the values in a table to determine the pattern, not just the last two. Students who give good mathematical explanations often show calculations and explain how they picked those numbers. Students at fourth grade should be more specific than “I did it in my head” or “it didn’t follow the pattern.” Fourth Grade – 2004
pg. 45
4th grade Student Task Core Idea 4 Geometry and Measurement
Task 4
Symmetrical Patterns
Name the shapes found in a symmetrical pattern. Find lines of symmetry in one drawing and complete a different drawing to make it symmetrical. Use characteristics, properties, and relationships of twodimensional geometric shapes. Examine, compare, and analyze attributes of geometric figures. • Classify two-dimensional shapes according to their properties. • Understand line symmetry. • Investigate, describe, and reason about the results of combining and subdividing figures.
Fourth Grade –2004
pg. 46
Fourth Grade – 2004
pg. 47
Fourth Grade –2004
pg. 48
Looking at Student Work on Symmetrical Patterns Student A is able to meet all the demands of the task. Of the students who drew two correct lines of symmetry 11% drew the 2 diagonals and 11% drew a vertical line and a diagonal. 9% of the students drew in all lines of symmetry. Student A
Fourth Grade – 2004
pg. 49
Student B was able to show understanding of all the parts of the task, including an understanding of lines of symmetry. 23% of the students who missed the lines of symmetry only drew the vertical line of symmetry, while no child only drew one line of symmetry with any other orientation. Student B
Fourth Grade –2004
pg. 50
Student C does not appear to understand line of symmetry completely. The Student is able to use the line of symmetry to complete a pattern in part 3, but cannot mark the design with the correct lines in part 2. The student appears to just be adding two lines to complete a design. Student C
Student D shows many of the typical errors found in student work. Many students think that the pattern is a circle and the octagon is a circle. Many also omit the octagon. 52% of the students who miss the lines of symmetry do not attempt this part of the task. Student D’s pattern is common pattern for students. Student D
Fourth Grade – 2004
pg. 51
Student E makes the most common error of mistaking the octagon for a hexagon. The student draws the 2 most popular lines of symmetry used by almost 70% of the successful students. In part 3 the student does not use symmetry for making a design. Student E
Teacher Notes:
Fourth Grade –2004
pg. 52
Frequency Distribution for each Task – Grade 4 Grade 4 – Symmetrical Patterns
Symmetrical Patterns M e an: 6.43, S.D.: 1.78
3500 3000
Frequency
2500 2000 1500 1000 500 0 Frequency
0
1
2
3
4
5
6
7
8
160
44
230
469
593
647
1679
3094
3301
Score
Score: %=
0 1.6% 100.0%
1 2.0% 98.4%
2 4.2% 98.0%
3 8.8% 95.8%
4 14.6% 91.2%
5 21.0% 85.4%
6 37.4% 79.0%
7 8 67.7% 100.0% 62.6% 32.3%
The maximum score available for this task is 8 points. The cut score for a level 3 response, meeting standards, is 4 points. Most students (about 96%) could identify three shapes in the pattern. Many students (91%) could name all 4 shapes include the hexagon with no extras. A majority of students (79%) could name the three shapes in the pattern, complete a pattern when shown half, and name the shapes needed to complete the new design. 32% of the students could meet all the demands of the task including drawing in two lines of symmetry for a given pattern. Less than 2% of the students scored no points on this task. Only 1/3 of the students with this score attempted the task.
Fourth Grade – 2004
pg. 53
Symmetrical Patterns Points 0 3
4
6
7
8
Understandings
Misunderstandings
2/3 of the students with this score did not attempt the task. Students could generally name three Most students omitted octagon or thought it was a hexagon. Some shapes, usually the triangle, square, thought either the center or the and rectangle. outside of the design was a circle. For students who missed the lines Most students with this score could name all 4 shapes in the design with of symmetry, 52% did not attempt no extras or name 3 shapes and draw that part of the task and 30% drew only the vertical line of symmetry. in 2 lines of symmetry. Students with this score generally Students still had trouble with the could name 3 shapes in the drawing, octagon and drawing the lines of complete the pattern, and name the symmetry. shapes needed for the new design. 35% of the students with this score missed the lines of symmetry. 53% of the students with this score did not identify the octagon. Students could identify shapes in a pattern, including an octagon, draw in at least two lines of symmetry in a design, complete a figure given half and name the shapes in the new design. Some students drew in all 4 lines of symmetry.
Teacher Notes:
Based on teacher observations, this is what fourth graders knew and were able to do: • Identify common shapes, like triangles, squares, and rectangles • Draw a symmetrical pattern • Find a vertical line of symmetry Areas of difficulty for fourth graders, fourth grade students struggled with: • Distinguishing between octagon and hexagon • Finding more than one line of symmetry for a shape Fourth Grade –2004
pg. 54
Questions for Reflection on Symmetrical Patterns: • • • • • • • •
Some students need more work with spatial visualization. How many of your students thought there was a circle in the design in part 1? How many of your students forgot to include the triangle in part 1? What might have led to this mistake? To check for understanding requires going at an idea from several directions. Many students only drew the vertical line of symmetry. What might these students know and not know about symmetry? Most students only drew the horizontal and vertical lines of symmetry. What might these students know and not know about symmetry? What might you like to ask next to check for depth of knowledge about symmetry? How many of your students drew lines of symmetry within the individual pieces in the design? What is the big idea they are missing? How many of your students drew in lines unrelated to symmetry? What further experiences do this students need to develop the concept? In part 3 of the task, understanding of symmetry is being checked in a different way. Can students use a line of symmetry to complete a pattern? Do you think the results would have been different if the line of symmetry was horizontal or diagonal? What are some further tasks you might want to give students to check depth of knowledge on symmetry?
Teacher Notes:
Implications for Instruction: Students need exposure to the names and attributes of more complex shapes, like hexagon and octagon. Students need to be able to draw lines of symmetry on complex figures. Students need experiences with a variety shapes, making certain that not all shapes have lines of symmetry only the vertical or horizontal axes. Using mirrors to help see how the line of symmetry divides shapes into matching parts is a good tool for developing this skill. Students should also have many experiences drawing reflections along a line of symmetry.
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4th grade
Task 5
Counting Feet
Core Idea 2 Number Operations
Find the number of feet in the barn when given the number and kind of animals. Find the possible combinations of animals when given the total number of feet. Justify a conclusion based on the data. Understand the meanings of operations and how they relate to each other, make reasonable estimates, and compute fluently. • Develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems. • Develop fluency in multiplying whole numbers.
Core Idea 3 Patterns, Functions, and Algebra
Understand patterns and use mathematical models to represent and to understand qualitative and quantitative relationships. • Use inverse operations to solve multi-step problems. • Understand and use the concept of equality.
Student Task
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Looking at Student Work on Counting Feet Student A shows calculations that helped him to find the different combinations of animals in the barn. Student A also has a very complete argument in the explanation for why there can’t be 3 spiders in the barn. Some students though you couldn’t have 3 spiders because they would be too long to fit in the barn. Student A
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Student B uses one combination with less than 24 feet, which is a common misunderstanding. Student B also leaves out the combination with 6 birds, which overall was the most difficult to find. The student shows the calculations for finding the combinations. Student B
Student C uses a model to think about the number of animals in part 1. In part 2 the student makes a bar graph to show 2 dogs, 4 birds, and 1 spider (one of the possible combinations). The student is on the right track for the explanation in part 4, but doesn’t quite explain why there would be more than 24 legs.
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Student C
Student D also thinks about the table as representing only one combination of animals in the barn. The student has shown 1 dog, 2 birds, and 2 spiders (one of the possible combinations). The explanation is also close, but doesn’t quite explain why there can’t be an additional spider. Student D
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Student E makes one combination that equals 24 feet, but does not include spiders. Student E also makes one combination that is less than 24 feet. A few students think that 8 x 3 is more than 24. Student E
Student F makes the mistake of adding the feet listed in the diagram rather than thinking about the context, which includes more than one of each animal. Other students added 3 dogs + 4 birds + 2spiders to get 9 feet. These students are doing operations on numbers without considering the meaning of the numbers. In part 2 the student starts with the number of animals in part 1 and just continues adding one each time, looking for a pattern or sequence rather than thinking about number of feet. Many students gave answers about spiders biting people, being scary, the problem said there were 2 spiders, or the spiders would have babies.
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Student F
Teacher Notes:
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Frequency Distribution for each Task – Grade 4 Grade 4 – Counting Feet
Counting Feet M e an: 3.40, S.D.: 2.34
2000
Frequency
1500
1000
500
0 Frequency
0
1
2
3
4
5
6
7
1609
849
1763
1212
1038
1147
1320
1279
Score
Score: %=
0 15.7% 100.0%
1 24.1% 84.3%
2 41.3% 75.9%
3 53.2% 58.7%
4 63.3% 46.8%
5 74.6% 36.7%
6 87.5% 25.4%
7 100.0% 12.5%
The maximum score available for this task is 7 points. The cut score for a level 3 response, meeting standards, is 3 points. Many students (about 84%) could either find the total number of legs for the animals in part one or write a correct mathematical sentence for finding the number of legs. More than half the students (59%) could find the total legs in part one, show their calculations, and find one combination of animals with exactly 24 legs and including at least one of every type of animal. 25% of the students could calculate the total number of legs, find 3 combinations to make 24 legs and either find the 4th combination for 24 legs or make a mathematical justification for why there couldn’t be 3 spiders in the barn. More than 12% of the students could meet all the demands of the task. Almost 16% of the students scored no points on this task. 85% of those students attempted the task.
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Counting Feet Points 0
2 3
6
7
Understandings 85% of the students with this score attempted the problem.
Students could calculate the total feet for the animals in part 1 and show their work. Students could calculate the total number of feet in part 1 and find one combination of animals (usually 1 dog, 2 birds, and 2 spiders or 3 dogs, 2 birds, and 1 spider). Students could calculate the total number of legs, find 3 combinations to make 24 legs and either find the 4th combination for 24 legs or make a mathematical justification for why there couldn’t be 3 spiders in the barn. Students could find the total number of legs for a given combination of animals, find all the possible combinations of animals to make 24 feet while remembering the constraint that there must be one of each animal, and make a mathematical justification for why there can’t be more than 2 spiders in the barn.
Misunderstandings Students did not relate the numbers in the problem to the context. They may have added the feet in the diagram to get 14 or added the animals to get 9 feet. Many students did not show their work or had the correct number sentence but made addition errors. Some students treated the table as a recording device for only one combination. They may have used the table to make a bar graph, or put in feet for each animal separately. Others did not attempt more than one combination. Students had difficulty finding all 4 combinations. The one they were most likely to miss was either 1 dog, 6 birds, and 1 spider or 3 dogs, 2 birds, and 1 spider. Students had difficulty with mathematical justification. Many students were too vague, giving answers like it will be too big, it will equal 24, or spiders have 8 legs. Other students were concerned about spiders being scary or biting other animals. Some thought that there physically wouldn’t be enough room for more spiders in the barn.
Teacher Notes:
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Based on teacher observations this is what fourth graders knew and were able to do: • Recognize the need for multiplication to find feet for animals • Add to find the total feet • Find at least one combination of animals to equal exactly 24 feet Areas of difficulty for fourth graders, fourth grade students struggled with: • Finding solutions that held true for all the constraints of the task • Using a table set up by someone else • Justifying why there couldn’t be 3 or more spiders using the constraints of the problem
Questions for Reflection on Counting Feet: Look at student work in part 1. How many of your students answered with:
36?
14?
9?
23?
46? 34?, 33?
Other?
Why do you think students made those answers? What are the experiences needed by different students? Research shows that students have difficulty interpreting the models of others. Look at how your students answered in part 2. Did they try to make a bar graph to represent one combination? Did they put in numerical answers to represent only one combination (like making a list of feet for each animal)? Did your students give combinations that equaled less than 24 feet? What kinds of problems have students worked this year where more than one answer was possible? Have students learned strategies for solving problems like make an organized list? What other strategies might have helped students find more than one combination? What opportunities do students get to make mathematical justifications? Is this a frequent part of the mathematical discourse in your classroom? Do they evaluate the arguments of others to see if they are complete enough? What opportunities help them to improve their justifications?
Teacher Notes:
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Implications for Instruction: Students need more practice with word problems using multiplication. Students don’t have enough experience with problems where answers from one part are used to find answers in later parts. They need to do problems that are more connected. Learning to label their answers might help students to identify more clearly what they know, so they can use that information later. Understanding units or labels is a key to solving problems and making sense of operations. Students need more practice organizing their own thinking and figuring out how to display it on paper. They need to discuss different strategies for organization used by other people. This type of experience and discourse will help them develop an understanding of tables, charts, graphs. An important mathematical skill is the ability to pick out the constraints of the problem, like there must be at least one of each animal. Students should develop a habit of testing their solutions against all the constraints of the problem. Students should also have frequent opportunities to make mathematical justifications and hear and discuss the justifications of others. Engaging in rich debates helps students develop and improve their own logic skills and sharpen the level of detail to which they pay attention.
Teacher Notes:
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