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Overall Frequency Distribution by Total MARS Raw Scores, Grade 6 Mean: 20.92 StdDev: 8.91 Total MARS Raw Scores Grade 6 – 2005 pg. 1 MARS Test Performance Level Frequency Distribution Chart and Bar Graph 2005 – Numbers of Students tested in 6th grade: 7992 Frequency Distribution of MARS Test Performance Levels, Grade 6 2000 Perf. Level 1 2 3 4 % at 26% 47% 17% 10% Year of Testing 2001 % at least 100% 74% 27% 10% % at 16% 43% 28% 13% 2003 Perf. Level 1 2 3 4 % at 21% 28% 33% 18% % at least 100% 79% 50% 18% % at least 100% 84% 41% 13% 2002 % at 21% 25% 36% 19% 2004 % at 17% 27% 31% 25% % at least 100% 83% 56% 25% % at least 100% 79% 55% 19% 2005 % at 20% 22% 31% 26% % at least 100% 80% 58% 26% Bar Graph of 2005 MARS Test Performance Levels, Grade 6 Total Student Count: 7,992 Grade Six – 2005 pg. 2 6th grade Task 1 Crystal Earrings Student Task Describe and extend a pattern of crystal earrings that is increasing in size. Make generalizations around this pattern. Core Idea 3 Algebra and Functions Understand relations and functions, analyze mathematical situations, and use models to solve problems involving quantity and change. • Represent, analyze, and generalize a variety of relations and functions with tables, graphs, and words • Model and solve contextualized problems using various representations such as graphs, tables, and equations Grade 6 – 2005 pg. 3 Grade Six – 2005 pg. 4 Crystal Earrings Grade 6 Rubric The core elements of performance required by this task are: • to describe, extend, and make generalizations about a number pattern Based on these, credit for specific aspects of performance should be assigned as follows 1. Draws Pattern #4 correctly: points section points 1 1 2. Gives correct answer: 15 1 Shows work such as: 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15 3. 4. 1 2 Completes table to show: Pattern #4 has 20 crystals. Pattern #5 has 30 crystals. 1 1 Shows work such as: Pattern #7 will be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 and 28 x 2 = 56, not 50 or (accept 7 x 8) 1 1 or 2 Gives correct answer: $56 1 Total Points Grade 6 – 2005 2 3 8 pg. 5 Looking at Student Work on Crystal Earrings The purpose of this task is to give students an opportunity to show their algebraic thinking. Student A is able to meet all the demand of the task, but relies heavily on drawing and counting. For doubling the student uses repeated addition instead of multiplication. While these strategies work, the will not help the student develop generalizations and these strategies can be cumbersome and lead to errors. Student A is making use of labels to clarify the thinking and the equal size groups. Student A Grade Six – 2005 pg. 6 A different solution path is for students to notice that each row increases in size by one. This again is a pattern that will not lead to a formula and allows for more possibility of computation error. Student B has noticed the numerical structure of the pattern and uses drawing to verify his solution. Student B Grade 6 – 2005 pg. 7 Student C notices a different numerical pattern. In the table the numbers go up by an increasing even number each time. The student is able to make the argument for why fifty dollars is incorrect, by showing the number of crystals for 1 earring if $50 were correct and then showing the correct solution by continuing the table. In this solution path, the doubling has been included in the design of the table. Student C Grade Six – 2005 pg. 8 Students at this grade level should start to look for relationships between the independent variable (pattern number) and the dependent variable (number of crystals). The earring design is made of triangular numbers whose algebraic expression would be [n(n+1)]/2. Notice that Student D has found that relationship, although it is not expresses in symbolic notation to answer for part 2. Because the earrings are doubled for part 3 and 4 of the task, the expression could be simplified to n(n+1). Student D sees this relationship and uses it to find the value in part 4. Student D Grade 6 – 2005 pg. 9 Student E makes the common mistake of justifying the answer for Pattern 4 in part 2 instead of showing the way to find the crystals for Pattern 5. Student E sees that the pattern adds the pattern number for each new row to the previous total. However, the student still seems to be more comfortable with drawing and counting than using the pattern. Student E Grade Six – 2005 pg. 10 Student F is starting to look for generalizable patterns. The wording in part 2 gives a recursive pattern for finding the number of crystals. The wording in part 4 gives a generalizable pattern, which could be written in the form n(n+1). Student F Grade 6 – 2005 pg. 11 Student G sees the numerical pattern in the table [ n (n+1) ], but fails to realize that the table has already doubled the crystals to make the pair of earrings. Student G then incorrectly doubles the values a second time. Student G Grade Six – 2005 pg. 12 Student H shows the problem of relying on drawing strategies. The student omits one of the rows when attempting to draw pattern #5. For many students it is difficult to understand how tables made by other people relate to the meaning of the problem. Students often look for any common numerical pattern to fill in the table. Student H notices that 3 x 4 =12, then assumes all values will be 4 times the pattern number. The student could have found the mistake by looking at the total for 2 or by checking other values in the table, e.g. 2 x 4 ≠ 6. Student H Grade 6 – 2005 pg. 13 Student I makes the assumption that all earrings will have 3 rows. The student’s answer for part 2 would be correct if that were the pattern. However, like Student H, this student looks for a numerical pattern in the table rather than continuing her found pattern. The student sees the number of crystals going from 6 to 12, or increasing by 6. The student then continues the table by going up in groups of 6. The student did not check either the number that was found in part 2 or look at the another part of the table to test the conjecture. 2+6 ≠ 6. The student does not understand the table is for the pair of earrings, rather than the individual earring. So the student doubles the number in the table. Student I Teacher Notes: Grade Six – 2005 pg. 14 Frequency Distribution for Task 1 – Grade 6 – Crystal Earrings Crystal Earrings Mean: 4.93 StdDev: 2.77 MARS Task 1 Raw Scores Score: Student Count %<= %>= 0 1 2 3 4 5 6 7 8 845 10.6% 100.0% 499 16.8% 89.4% 441 22.3% 83.2% 815 32.5% 77.7% 658 40.8% 67.5% 701 49.5% 59.2% 1016 62.2% 50.5% 774 71.9% 37.8% 2243 100.0% 28.1% The maximum score available for this task is 8 points. The minimum score for a level 3 response, meeting standards, is 5 points. Most students, 91%, could extend the visual pattern to #4. Many students, 76%, could extend the pattern in pictures, find the correct number of crystals for pattern #5, and show how they figured it out. More than half the students, 59%, could also extend the pattern in the table for a pair of earrings, for pattern # 4 and #5. 30% of the students could meet all the demands of the task including proving why $50 would not be the cost of a pair of earrings if the pattern were extended to 7. 9% of the students scored no points on this task. All the students in the sample attempted the problem. Grade 6 – 2005 pg. 15 Crystal Earrings Points Understandings All students in the sample 0 attempted the problem. 1 Students could extend the pattern by drawing. 3 Students could extend the pattern in a drawing and could find the number of crystals for #5 and explain how they figured it out in words, pictures, or calculations. Students could extend the pattern of crystals to pattern #4 and #5 and they could extend the pattern in the table for a pair of earrings for pattern # 4 and #5. 5 6 8 Most students with this score could extend the geometric pattern with a drawing and numbers, continue the table for pairs of earrings, and find that pattern #7 would have 28 crystals. Students could extend a pattern, use a table to think about a pair of earrings, and find the cost for a pair of earrings for pattern #7. 26% of all students relied on a drawing and counting strategy. 15% of the students could continue the table to solve the problem in part 4. 9% of the students added all the rows (1+2+3+4+5+6+7=28) and doubled that answer. 6% saw a pattern of n(n+1). 3% noticed that the table was going up by the next larger even number each time. Grade Six – 2005 Misunderstandings Students did not understand the visual pattern. Half the students, who missed the drawing for pattern #4, put only 3 rows of beads for each earring. Many students thought pattern #5 should have 12 crystals (13% of the total sample). 8% of all students counted the number of crystals for pattern #4 instead of for pattern #5, putting an answer of 10 crystals. Some students did not relate the table to the context of the pattern. They made conjectures about the number patterns in the table and so filled out patterns like 12, 18, and 24 (8%); 12,24,48 (5.5%); and 12,16,20 (5%). Students had difficulty with doubling to find for a pair of earrings. 7% of the students left the answer as $28. 4% of the students doubled twice, $112, not realizing the table was for a pair of earrings. Answers varied from $1 to $300. pg. 16 Based on teacher observation, this is what sixth graders knew and were able to do: • Describing and extending a geometric pattern • Doubling • Using drawing and counting strategies to extend pattern Areas of difficulty for sixth graders: • Relating table to problem context • Writing a justification for why something is not true Strategies used by successful students: • Drawing and counting • Continuing a table • Finding a pattern for part 4: goes up by larger even numbers each time or n(n+1) • Finding a pattern for part 2: 1 + 2+3+4…n, increases by the pattern # each time, or [n(n++1)]/2 Questions for Reflection on Crystal Earrings: What kind of experiences do your students have with different types of patterns? Did students in your class have good visualization skills? Were they able to draw the pattern for pattern #4? What types of questions might help them to focus on the important attributes of the pattern? Do students get opportunities to discuss patterns and make convincing arguments for different interpretations of a pattern? What evidence might they give for why the pattern would have more than 3 rows? • What kinds of questions do you ask which might push students to look for numerical patterns instead of relying solely on drawing and counting? Why is this important for students? • When working on problem-solving activities do students in your class have the opportunity to make their own tables to record information? What evidence do you see that students did not relate the table to the context of the problem? Look at student work on the table. How many of your students put: • • 20,30 18,24 24,48 16,20 9,12 Other What patterns were students thinking about to make these errors? What evidence were they ignoring? What does this show you that they don’t understand about either tables or testing conjectures? • What are classroom norms for testing conjectures? Do students know that it takes more than one case or example to make a pattern? Look at the solutions paths that students used to solve for part 4. Did they: • Draw and count Continue the table Add (1+2+…+7) and double Add a larger even number each time (+2,+4,+6,+8…) Use a numerical pattern: n(n+1) Forget to double (answer of 28) Double twice (answer of 112) Use strategy not appropriate to mathematics of problem Grade 6 – 2005 pg. 17 • Do you see evidence of students looking for numerical patterns? Are some students ready to start thinking with algebraic symbols? Teacher Notes: Implications for Instruction: Students at younger grades can solve problems using drawing and counting strategies. Students in middle school should start to be looking for numerical patterns. This task gives them a chance to work with triangular numbers. Some students can look at the earring pattern to notice that the pattern is increasing by the row number each time, which gives them a recursive rule. Some students may notice that the table grows by an increasing even number each time, which is another recursive rule. Other students may notice that the crystals in each earring is the sum of the counting numbers from 1 to the pattern number, which is a more generalizable rule, which will work for any earring in the sequence. A few students may notice that the pattern for earrings is equal to [n(n+1)]/2. This simplifies the process for finding the number of crystals and reduces the tendency for an arithmetic slip. Some students notice that to find any value in the table, just multiply the pattern number by the pattern number plus one. This again is a generalizable rule. Students at this grade need many opportunities to work with patterns and should be striving for generalizable rules, rather than drawing and counting or using recursive rules. When thinking about generalizable rules, students need to develop the habit of mind of testing their rule against all the evidence. So just because 6 x 2= 12 could explain the jump in the table from pattern 3 to pattern 4, other evidence proves that this is an incorrect conjecture. Students should be presented with a variety of conjectures and asked which one best fits the evidence and why. Discussions will engage students in finding the most convincing argument and developing their logic skills. This will also help them start to look at the tables in relationship to the context of the problem, rather than a mindless exercise in continuing an arithmetic pattern. Students need to experience that cognitive dissonance, that there are a variety of possibilities and therefore they need to check which possibility fits the context and evidence of the problem. Teacher Notes: Grade Six – 2005 pg. 18 6th grade Student Task Core Idea 5 Statistics Grade 6 – 2005 Task 2 Money Analyze and interpret bar chart information to determine how much money was spent by four children. Write a description to fit a fifth child’s bar chart. Select and use appropriate statistical methods to display, analyze, compare and interpret different data sets. • Interpret data to answer questions about a situation • Compare data sets using measures of center and spread to understand what each indicates about the data sets • Communicate mathematical thinking clearly and coherently • Use representations to interpret physical, social, and mathematical phenomena pg. 19 Grade Six – 2005 pg. 20 Grade 6 – 2005 pg. 21 Money Grade 6 Rubric The core elements of performance required by this task are: • interpret bar charts points Based on these, credit for specific aspects of performance should be assigned as follows section points 1. Gives correct answers: Danny Chris Ben Ali All four answers correct Partial credit Three correct answers Two correct answers 3 (2) 3 (1) 2. a. Danny 1 b. Ali 1 Compares values on at least two graphs to show that the sum of Ali’s bars is larger than the other bars 2 Partial credit Makes a sensible verbal comment about the total amounts spent (1) 4 3. Gives a reasonable description such as: “I spent more and more each week” or similar 1 1 8 Total Points Grade Six – 2005 pg. 22 Looking at Student on Money: Student A assigns a scale to the graphs, finds the total money spent for each person, and then concludes that Ali spent the most. Student A also writes a good description for Ernest, showing the trend of the graph in the context of the problem. Student A Grade 6 – 2005 pg. 23 Student A, continued Student B realizes is able to think about the bars on the graph as a measurement. The scale is not important as long as it’s the same for all the graphs. The student just needs to compare the lengths of the bars for the four graphs. Student B notices the shape of the graph for Ernest, but is unable to relate it to a mathematical trend, increasing, or to the context spending more money. Grade Six – 2005 pg. 24 Student B Grade 6 – 2005 pg. 25 Student B, continued Student C makes an observation about Ernest’s graph similar to that of Student B. However, Student C is able to tie the observation back to the mathematics and context of the task. Student C Grade Six – 2005 pg. 26 Student D chooses a different scale from most students. Because the relative values of the bars remains unchanged, this gives the student a successful way of making the comparison. Notice that the student starts the scale at 10 instead of 0. Again, because this is done for all the graphs, the relative values are not effected. In part 3, Student D confuses description for total. The task was asking for the mathematical trend of the graph. Student D Grade 6 – 2005 pg. 27 Student E gives factual information about the graph for Ernest rather than summarizing the information or giving a trend of the data. Student E Student F writes a good description for the shape of Ali’s graph, but in order to justify who spent the most, the student needs to make a comparison to the values of the other graphs. In part 3, Student F describes the shape of the graph, without reference to mathematical trend or context. Student F Grade Six – 2005 pg. 28 Student G redraws the graph for Ali in an attempt to show who spent the most. The student does not give a reason why this particular shape is significant. In part 3, the student seems to confuse decreasing with small instead of the trend “getting smaller”. Student G Teacher Notes: Grade 6 – 2005 pg. 29 Frequency Distribution for Task 2 – Grade 6 - Money Money Mean: 6.41 StdDev: 1.62 MARS Task 2 Raw Scores Score: Student Count %<= %>= 0 1 2 3 4 5 6 7 8 60 0.8% 100.0% 67 1.6% 99.2% 103 2.9% 98.4% 155 4.8% 97.1% 375 9.5% 95.2% 1257 25.2% 90.5% 2097 51.5% 74.8% 915 62.9% 48.5% 2963 100.0% 37.1% The maximum score available for this task is 8 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students, about 94%, could match all the descriptions to the appropriate graphs and read and compare information from the graph to identify the student who spent the most money in the first week. Many students, 89%, could also identify who spent the most money overall. A majority of the students, 73%, could match the descriptions, decide who spent the most the first week and who spent the most overall, and write a description of the mathematical trend for Ernest’s graph. 37% of the students could meet all the demands of the task including justifying why Ali spent the most by comparing significant graphical features for all students. Less than 1% of the students scored 0 points on this task. Grade Six – 2005 pg. 30 Money Points Understandings All students in the sample 0 attempted the task. 4 Students could match the verbal descriptions to the appropriate graphs. Students could compare the graphs to identify who spent the most the first week. 5 Students could match the descriptions to the graphs, compare who spent the most the first week, identify who spent the most overall, and write a description for Ernest’s graph. Students could match descriptions to graphs, write description of the mathematical trend of a graph, and compare graphs by identifying who spent the most the first week and who spent the most overall. 6 8 Misunderstandings Students often confused the descriptions for Danny and Ali. Some students chose Ernest to fit one of the descriptions. Some students thought that Chris spent the most, because of the 3 high bars. Others though Ben spent the most, because he bought an expensive present. Less than 5% made addition errors. Students, who had difficulty with the description for Ernest, tended to give information about the shape of the data, it looks like a staircase, or give factual information like the total or the amount spent each week. Students had difficulty writing a justification for who spent the most. 12% of the students did not write a reason at all. 16% gave reasons for students other than Ali. Students who chose Ali, often did not compare her scores to the other students. (e.g. She spent a lot, she spent the same every week, or she has the most. Students could match descriptions to graphs, write description of the mathematical trend of a graph, and compare graphs by identifying who spent the most the first week and who spent the most overall. Students could also justify their comparisons by assigning a scale to the graph and totaling the values for each student. Grade 6 – 2005 pg. 31 Based on teacher observations, this is what sixth graders know and are able to do: • Read and compare graphs to match verbal descriptions of the trends to the appropriate graph • Read and compare graphs to identify who spend the most the first week • Write a verbal description to of the trend of a graph and put that trend in context with the problem • Identify who spent the most overall Areas of difficulty for sixth graders: • Assigning a scale to the graphs • Making a mathematical comparison, knowing that to justify why one graph represents the most there must be reference to why the other graphs are less Strategies of successful students: • Labeling the graph with appropriate, consistent scales • Showing the addition of the bars • Looking for trends in the graph, trying to summarize the data Questions for Reflection on Money: What opportunities have students in your class had to make comparisons of data from different graphs? What types of comparisons have they been asked to make? Why might this be important? • Were students in your class able to assign a scale to the graphs? Why is this important? Look at student justifications for part 2, who spent the most overall. How many of your students: • Assigned a scale and totaled all the graphs • • • Ali has the most or spent a lot every week Ali spent about the same every week Chris has the highest bars or Chris has the 3 highest Ben bought an expensive present Too general, Other e.g. I looked at graph or added but no values When working with data and graphs, are your students regularly asked to give a trend of the data? Are students regularly asked to give a summary of the information shown on the graph? Were your students able to write a mathematical description of the trend for the data for Ernest or did they give factual information, like the total or the size of individual bars? What other questions could you ask using these graphs to probe the understanding of your students? Grade Six – 2005 pg. 32 Implications for Instruction: Students need more experience identifying the trends in data. Is the data increasing, decreasing, or staying the same? Is there a relationship or correlation between the two variables being graphed? What is the purpose of the graph? What is the message the maker of the graph trying to convey? Students should also be in the habit of trying to quantify information from the graphs, by making statements like how much high or how much lower when comparing features of the graph or making comparisons between graphs. Students need experiences working with graphs with and without scale, so that they start to be able to assign relative values to compare relationships. This also helps them to identify ways that graphs are misleading, e.g. graphs don’t start at zero or the scale distorts minor differences in size. Comparison is a big mathematical idea for middle school. Students should think of a variety of ways to make comparisons. The first big idea is that to make a comparison, there needs to be a reference to all the items being compared – how are they different, how are they alike. At this grade level, students should be starting to make a variety of comparisons: subtraction comparison, multiplicative comparisons ( A is 5 times larger than B), and using percents to compare ratios of different size quantities ( 80% score on a test of 20 questions, 80% on a test with 5 questions instead of missing 4 questions versus 1 question). Teacher Notes: Grade 6 – 2005 pg. 33 6th grade Task 3 Winning Spinners Student Task Work with two spinners to find the probability of winning a prize. Design two new spinners that will increase the likelihood of winning. Core Idea 2 Probability Demonstrate understanding and the use of probability in problem situations. • Determine theoretical probabilities and use these to make predictions about events • Understand that the measure of the likelihood of an event can be represented by a number from 0 to 1 • Represent probabilities as ratios, proportions, decimals or percents • Represent the sample space for a given event in an organized way (e.g. table, diagram, organized list) • Use representation to model and interpret mathematical phenomena Grade Six – 2005 pg. 34 Grade 6 – 2005 pg. 35 Winning Spinners Grade 6 Rubric The core elements of performance required by this task are: • work with probability Based on these, credit for specific aspects of performance should be assigned as follows 1. Fills in the table correctly: + 8 4 2 1 7 15 11 9 8 9 17 5 3 section points 2 Spinner A Spinner B points 11 10 13 9 7 6 11 7 5 4 13 Partial credit No more than 2 mistakes or no values, only “odd” or “even” given correctly. (1) 2 2. Gives a correct answer: 4/16 = 1/4 or equivalent Accept correct decimals, percents and ’out of’. 2ft Partial credit denominator 16 (1) 2 3. Fills in spinners correctly: The even numbers split so there is one even number on one spinner and two on the other (position of the numbers does not matter) 2 Gives a correct answer: 1/2 or equivalent 2 Gives an explanation or table to show a correct method. 2 Special case 6 Correctly uses their own numbers: maximum 2 points Total Points Grade Six – 2005 10 pg. 36 Looking at Student Work on Spinners: Student A shows a strong understanding of probability. The student defines the number of possible outcomes and the number of favorable outcomes. In part 3, the student finds two ways to show the sample space. Student A Grade 6 – 2005 pg. 37 Student B is able to figure out the design for the new spinner by reasoning about number properties (an odd plus an odd makes an even). Student B Student C is able to show the sample space for part 3 by making an organized list, which includes all the possible outcomes. Student C Grade Six – 2005 pg. 38 Student D attempts to show the sample space for part 3, but her list is incomplete. Notice that the student still thinks about probability as whole numbers, how many favorable outcomes. Students at this grade should know that probability is expressed as a ratio of favorable outcomes/ all possible outcomes. Student D Grade 6 – 2005 pg. 39 Student E is able to fill out the addition chart and correctly rearrange the numbers on the spinner to get a more favorable probability. The student thinks about number combinations to reason about the arrangement of the numbers on the spinners. However, like student D the list of combinations is not exhaustive. While at younger grades students discuss probabilities in terms of likeliness, at sixth grade students need to be able to quantify the probability. Student E Grade Six – 2005 pg. 40 Student F is able to fill out the addition chart in part 1 and make a new addition chart for the spinner designed in part 3. However, the student does not realize how to use the chart to calculate probability. Student F sums each row of the chart to see if its an even number, which has nothing to do with the mathematics of this task. Student F Grade 6 – 2005 pg. 41 While Student G appears to have good understanding of the task, getting 8 out of 10 points, notice that the student is thinking about individual numbers on the spinner in part 3 rather than the outcome of combining results of two spinners used together. Student G Grade Six – 2005 pg. 42 Student H is able to fill out the addition chart, but does not make a connection between the results in the chart and finding probability. The student thinks about the spinner as an individual event, rather than the effect of combining the results of both spinners to determine an outcome. Student H Grade 6 – 2005 pg. 43 Student I thinks of probability as a “how to win”, rather than a condition of likelihood or a quantity for expressing likelihood. Student I Student J does not understand that to improve the probability for winning the numbers must be switched across spinners. Student J makes the common error of just rearranging the numbers on the same spinner. Student J Grade Six – 2005 pg. 44 Student K changes the numbers for the spinners to improve the probability for winning. While the new spinners do improve the chances, the mathematics is considerably easier than rearranging the numbers on the original spinners. Student K Student L misinterprets the task in many ways. The student does not understand probability and writes lists of numbers rather than making a numerical statement about the chances of winning. The student uses numbers from the addition chart to make the new spinner, rather than rearranging the numbers on Spinners A and B. Student L Grade 6 – 2005 pg. 45 Student M confuses probability with odds. Probably is a ratio of favorable outcomes/ total outcomes. “Odds” is a ratio of favorable outcomes/unfavorable outcomes. These are not the same thing mathematically; although they are talking about the same sample space. Student M Teacher Notes: Grade Six – 2005 pg. 46 Frequency Distribution for Task 3 – Grade 6 – Winning Spinners Winning Spinners Mean: 5.43 StdDev: 3.37 MARS Task 3 Raw Scores Score: Student Count %<= %>= 0 1 2 3 4 5 6 7 8 9 10 656 8.2% 100.0% 241 11.2% 91.8% 1346 28.1% 88.8% 269 31.4% 71.9% 1325 48.0% 68.6% 179 50.3% 52.0% 1045 63.3% 49.7% 127 64.9% 36.7% 801 74.9% 35.1% 161 77.0% 25.1% 1842 100.0% 23.0% The maximum score available for this task is 10 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students, 89%, were able to fill in the addition chart for combining the values of the two spinners. More than half the students, 69%, were able to fill in the addition chart and design a new set of spinners with a higher probability of winning. Almost half, 49%, could fill in the addition chart for the original spinners, design spinners with a higher probability of winning, and make an addition chart for the new spinners. 24% of the students could meet all the demands of the task, including giving numerical probabilities for each set of spinners. 7% of the students received no points on this task. 92% of those students attempted the task. Grade 6 – 2005 pg. 47 Winning Spinners Points Understandings 92% of the students attempted 0 2 4 6 8 10 Misunderstandings Students did not understand the purpose the task. of the addition chart. Most seemed to be looking for a pattern to the numbers in the chart, like 13, 14,15. Students did not use addition to find values. Students could use addition to Students did not know how to express a accurately fill out the chart. probability mathematically. 19% of all students gave the probability as a verbal description, like not likely, or described how to win, e.g. “get an even number”. Students could fill out addition 15% of all students rearranged the chart and design spinners with numbers on the same spinner. 15% better odds. substituted their own numbers (82% of these numbers were all even). 6% used the given numbers, but used some more than once. Many of these designs did not improve the odds. Students could fill out the 13% gave whole numbers for the addition chart, design spinners probability. 6% were thinking in with better odds, and either make eighths because there were eight a new addition chart for their numbers on the spinner. 13% gave spinner or give a probability for wrong values of sixteenths. the first set of spinners. Students could fill out the chart, Some students could make the addition design spinners, give the chart, but did not relate the chart to probability for the first spinner, finding probability. The chart was not and either make the chart for the connected to solving the task. Students new spinners or give the with the correct probability gave probability for the new spinners. incomplete explanations, usually not defining the whole sample space, or gave explanations not based on the mathematics of the task. Students could fill out an addition chart to show the sample space for spinning two spinners, using the same numbers design a spinner with a higher probability of winning, show the sample space for the new spinner, and give quantify the probabilities for each set of spinners. Of the students who gave correct probabilities: 63% used fractions,18% used percents, and 8% “out of”. Grade Six – 2005 pg. 48 Based on teacher observation, this is what sixth graders know and are able to do: • Complete an addition chart • Design a set of spinners, using the given numbers, with a higher probability of winning • Give a probability with 16ths as the denominator Areas of difficulty for sixth graders: • Quantifying probability instead of using language of likelihood • Connecting the addition chart to the probability • Understanding a compound event (Students thought of getting an even number on one of the spinners, versus the sum of numbers on both spinners.) • Identifying the constraints of the task ( using the same numbers on the new spinners, improving the odds) Strategies used by successful students: • Making an organized list or new addition chart to find the probability for the second set of spinners • Using fractions, percents, and “out of” to quantify probabilities Questions for Reflection on Winning Spinners: What experiences have your students had with probability this year? Have their experiences dealt with strictly the language of probability: likely, unlikely, equally likely, sometimes, never? Or have students had opportunities to think about quantifying probability? • Do your students commonly use ratios, percents, or “out of” to express the value of a probability? • Have students’ probability experiences been limited to thinking about a single event: spinning 1 spinner or rolling one dice? Or have students worked with compound events? • Have they used strategies like tree diagrams, organized lists, charts to help them find all possible outcomes or sample space? Look at student work on the probability for the first set of spinners. How many of your students could: • • 4/16 or 1/4 Express probability in words (likely, unlikely) Give rules of the game (get an even number) Used a Used a Other denominator denominator of 16 of 8 What does this show about their understanding of probability and compound events? Grade 6 – 2005 pg. 49 Look at the design of their spinners. How many of the students could: Design spinner with higher prob. Used their own numbers Repeated Rearranged some numbers the numbers within the same spinner Did not improve their spinner What constraints were students missing? What probability concepts were students missing? In part 3, did your students: • Make or attempt to make an organized list of possible outcomes? Did they make an addition chart for the new spinners? • Did they think about number properties (an even plus and odd equals an odd)? • Did they think just about combinations that make even numbers? • Did they ignore the combination of the two spinners and just look at values on the spinners (3/8 or 2 out of 4) • Did they appear to guess? What experiences will help students start to develop an understanding of sample space for compound events? Implications for Instruction: Students at this grade level should transition from thinking about probability in terms of likelihood to being able to quantify probabilities as a value between 0 and 1. Students might start by trying to give a variety of probability words, such as always, never, likely, equally likely, seldom and trying to assign possible values to the numbers. Students should then start to think about more exact values such as getting heads on a coin or a certain color on a single spinner. Students then need to move to compound events, like tossing two coins or spinning a spinner and flipping a coin. These combined events then need to be first expressed as possible outcomes. Students might make factor trees, organized lists, charts or tables, or use a combinatorics formula to find all the possible outcomes. Students should have discussions to find the relationships between all these representations. Having students design their own games or rules, gets them involved in understanding the context as well as an opportunity to organize the possible outcomes and find the probabilities. Giving students a game and asking them to change the rules to improve the chances of winning, also makes them think about how the events effect each other and how they change the sample space. Students should also have opportunities to investigate the difference between theoretical and experimental probabilities. Playing a game a number of times gives them a small flavor of this idea, but using a computer simulation can give them the bigger idea of how the two values become closer over a large number of repetitions. Students often have misconceptions about how spinners are made. For example they might think that a spinner 50 Grade Six – 2005 pg. with two separate 1/8 red sections will have a better chance of winning than a single 1/4 red section because it is more spread out. Computer simulations give them the opportunity to make and test these conjectures and confront these misconceptions head on. Teacher Notes: Grade 6 – 2005 pg. 51 6th grade Task 4 In The Playground Student Task Find the area of a playground sandbox and make a new design that will measure twice the area of the first sandbox. Core Idea 4 Geometry and Measurement Analyze characteristics and properties of two-dimensional geometric shapes and apply the appropriate techniques, tools, and formulas to determine measurements. • Develop, understand and use formulas to determine the area of quadrilaterals • Select and apply techniques and tools to accurately find length and area measures to appropriate levels of precision Grade Six – 2005 pg. 52 Grade 6 – 2005 pg. 53 In the Playground Grade 6 Rubric The core elements of performance required by this task are: • work with areas Based on these, credit for specific aspects of performance should be assigned as follows 1. 2. 3. Gives correct answers: 24 square feet (Accept 80 square feet) 56 square feet points 1 1 2 Draws a correct diagram: Rectangular area of sand, 48 square feet (12 squares) 2 Surrounded by a row of squares of “rubber matting” 1 Gives correct answer: their number of squares x 4 (dependent on their diagram and following the one row rule) 1ft 80 or 72 (accept 120) 4. section points 3 1 Gives a correct answer dependent on their diagram for question 2: 12 and 4 2x1ft or 8 and 6 2 or 2 and 24 Total Points Grade Six – 2005 8 pg. 54 Looking at Student Work on In the Playground Student A highlights the dimensions by labeling the sides of the sandbox and matting with two’s. The student seems to use a counting by 4’s strategy to find the area of the rubber matting in part 1. Student A is able to use the information from part 1 to design a new sandbox with twice the area of the original. The student can think about the scale for linear and area measures. Student A Grade 6 – 2005 pg. 55 Student B uses an interesting technique to find the area of the matting. The student finds the area of the total play structure and subtracts the area of the sandbox. It seems most likely that the student found the area of each by multiplying the dimensions together, because the student is able to reason correctly about the linear dimensions in part 4. However, the student struggles with the area of the new matting, multiplying the number of squares by the linear scale of 2 instead of by the area scale of 4. Student B Grade Six – 2005 pg. 56 Student C counts squares and multiplies by the scale factor of 4 to find the area of the sandbox and rubber matting in part 1. The student shows doubling the area to find the area of 48 sq. ft. for the new sandbox. However, the student cannot work backward from the area with the scale factor included to the representation of the area on the graph. Instead of dividing the 48 sq. ft. by 4 to find the number of squares, Student C draws a sandbox of with 48 squares, which would represent 192 sq. ft. The student is able to use the linear scale of 2 to find the dimensions of the sandbox in his design. Student C Grade 6 – 2005 pg. 57 Student D is also able to count squares and multiply by the area scale factor of 4 in part 1 and 3. Student D is also able to design a new sandbox with double the area of the sandbox in part 1. However the student is unable to use the linear scale to translate from the drawing of the sandbox to the actual dimensions represented by the drawing. Student D Grade Six – 2005 pg. 58 Student E is not able to find the area of the sandbox in part 1. The student multiplies the area by the linear scale factor instead of the area scale factor. The student is still able to design a new sandbox with double the area, because the number of squares doubles. In the choice of design the student has made a rectangle, which is equal to original sandboxes side by side. Student E uses the linear scale factor to find the dimensions of the new sandbox. . The student misses the constraint of the rubber matting being a strip 2 feet wide. However the student does correctly find the area for the rubber matting in his drawing. Student E Grade 6 – 2005 pg. 59 Student F makes the most common error of counting squares to find the area and ignoring the scale given in the task. The student is able to double the area, because the scale does not effect the number of squares in the representation. The student also ignores the scale for both linear and area measures in part 3 and 4 of the task. Student F Grade Six – 2005 pg. 60 Student G is able to count squares and multiply by the scale factor of 4 to find the area for the sandbox and matting in part 1. The student may have subtracted the sandbox area from the total area to find the matting. In part 2 the student designs a sandbox that is not the obvious double of anything from part 1. The student also ignores the idea that the matting is always 2 feet wide. When calculating the area of the matting, the student leaves out one set of 4 squares along the side of the sandbox, making the total off by 16 square units. The student multiplies the dimensions by the area scale factor instead of the linear scale factor. Student G Grade 6 – 2005 pg. 61 Student H miscounts the squares in the sandbox or multiplies the linear scale by the area scale to find the area of the sandbox in part 1. The student gives the width of the matting instead of the area. The student designs the sandbox by looking at the pattern of filling in the whole grid except for a surrounding frame of 1 square unit. The student gives the dimensions of his sandbox without regard to the linear scale factor. Student H Teacher Notes: Grade Six – 2005 pg. 62 Frequency Distribution for Task 4 – Grade 6 – In the Playground In the Playground Mean: 2.66 StdDev: 2.24 MARS Task 4 Raw Scores Score: Student Count %<= %>= 0 1 2 3 4 5 6 7 8 1535 19.2% 100.0% 1559 38.7% 80.8% 1246 54.3% 61.3% 1046 67.4% 45.7% 966 79.5% 32.6% 534 86.2% 20.5% 597 93.6% 13.8% 148 95.5% 6.4% 361 100.0% 4.5% The maximum score available on this task is 8 points. The minimum score for a level 3 response, meeting standards, is 4 points. Most students, 79%, can draw a new sandbox that has one row of matting surrounding it. More than half the students, 62%, can either find the area of the original sandbox and matting or draw a sandbox with one row of matting and give one of the dimensions for the new sandbox correctly using the linear scale factor. Some students, about 33%, could find the area of the original sandbox and matting and either design a new sandbox with double the area or draw a sandbox with one row of matting and give one dimension for the new sandbox. Only 4% of the students could meet all the demands of the task including correctly applying both a linear and area scale factor to a drawing and designing a new sandbox with double the area of the original while still maintaining only one row of matting. 21% of the students scored no points on this task. 86% of those students attempted the task. Grade 6 – 2005 pg. 63 In the Playground Points 0 Understandings Misunderstandings 86% of the students attempted the task. Students did not understand how to apply scale factors to a drawing. 16% of all students just counted squares in the drawing to find area for the original sandbox and matting. 8% found an area of 48 for the original sandbox, another 8% found an area of 12. About 14% drew no matting around the new sandbox. 32% drew matting with more than one row on at least one dimension. (Some of these overlap.) Students frequently confused linear and area scale factors. They might multiply the length by 4 instead of two or multiply area by 2. In some cases students multiplied the two scale factors together before applying them to drawing. 9% of all students drew a rectangle equal to the entire grid minus one row for matting, thinking of a pattern rather than an area. 8% doubled the dimensions instead of the area, drawing a rectangle that was 4 x 6. 8% doubled the area of the entire play area, 80 sq. ft., making a new sandbox 4 x 10. 1 Students could draw a new sandbox with one row of matting along the outside. 2 Students could either find the area for the original sandbox and matting or draw a sandbox with one row of matting and use the linear scale to find one of the dimensions of the new sandbox. Students could find the area of the parts of the original drawing. They could either design a sandbox with double the area or draw a sandbox with the proper matting and give one dimension of the new sandbox. Those students who could draw the sandbox with double the area were able to do so because they were thinking about number of squares. Students could find areas of the original sandbox, design a new sandbox with double the area, and find the area of the matting for the new sandbox. Students could work with linear and area scale factors to calculate areas and dimensions of a sandbox and design a scale drawing of a sandbox with double the area. 4 6 8 Grade Six – 2005 Students could not apply the linear scale factor to find the dimension for their sandbox. Most of these students counted squares (3 x4 or 2 x 6). pg. 64 Based on teacher observation this is what sixth grade students knew and were able to do: • Students could find the area of the drawing and use the appropriate scale factor to find the area of the real sandbox and matting • Many students could draw a sandbox with one row of matting, understanding the constraint that the size of the matting does not change Areas of difficulty for sixth graders: • Doubling the area of the sandbox • Using a linear scale factor to find the dimensions of the sandbox they designed • Confusing linear and area scale factors Questions for Reflection: • • • What experiences have your students had with finding area? Do they still rely on counting squares or do they use multiplication? Have students had multiple experiences with doubling or tripling shapes, so that they start to see that the dimensions don’t double or is this a new idea for them? How many of your students, do you think, could make a good justification for why the dimensions don’t double? What kind of explanation or justification would you like them to be able to make? What experiences have your students had with using scale factors? Have students worked with enlarging or shrinking shapes? Do you think they understand the difference between a linear and an area scale factor? Look at student work in part one of the task. How many of your students thought the area was: 24 sq. ft. 6 sq. ft. 12 sq. ft. 48 sq. ft. 8 sq. ft. 20 sq. ft. 40 sq. ft. Other What kind of thinking might have led to those errors? Which errors are related to understanding the sandbox? To understanding which scale factor to use? To understanding that a scale factor is needed? Look at student work on designing a sandbox with double the area. How many of your students understood the constraint of one row of matting? Did they not draw any matting? Make the matting more than one row? Now look at the area of the new sandbox. How many drew a shape with dimensions of: 3x4 or 2 x 6 4 x 12 4x6 4 x 10 Other What type of logic led to these specific errors? What were students not understanding about area, the sandbox, and using scale factors? How are these errors different? Grade 6 – 2005 pg. 65 Implications for Instruction: Students at this grade level should be comfortable with finding areas of rectangular shapes by multiplying the dimensions or counting squares. They should have a good understanding of the difference between linear measures and area measures. A big idea for middle grade students is to develop proportional reasoning, which would include understanding scale factors. Students should work with recording dimensions of objects, rooms, playgrounds, maps, etc. by making their own accurate scale drawings. They should have frequent opportunities to translate between scale drawings and giving the dimensions and areas for the real object. While students are working with growing patterns, they should be challenged to think about and justify why doubling the dimensions does not double the area. A lesson might be designed to predict the area of shape when dimensions are doubled or tripled and then make a model to test this out. Shapes should not be limited to rectangular objects, but should include triangles, trapezoids, parallelograms, and even compound shapes like an Lshaped room. Students should also work with understanding or visualizing the distortion of changing only one dimension or changing both dimensions using different scale factors. Students’ experiences should not be limited to enlarging shapes, but should also include working with shrinking shapes. These types of activities help students understand the two-dimensionality of area and build an understanding of scale. They also lay the foundation for understanding similarity at later grades. Teacher Notes: Grade Six – 2005 pg. 66 6th grade Task 5 How Much Money? Student Task Work with simple fractions to solve two money problems and use representations to organize and record the thinking. Core Idea 1 Number and Operation Understand number systems, the meanings of operations, and ways of representing numbers, relationships. And number systems. • Understand fractions as parts of unit wholes • Select appropriate methods and tools for computing with fractions from among mental computations, estimation, calculators and paper and pencil and apply selected methods • Develop and analyze algorithms for operations on fractions and develop fluency in their use • Create and use representations to organize, record, and communicate mathematical thinking Grade 6 – 2005 pg. 67 Grade Six – 2005 pg. 68 How Much Money? Grade 6 Rubric The core elements of performance required by this task are: • work with simple fractions • figure out a money problem points Based on these, credit for specific aspects of performance should be assigned as follows 1. Gives correct answer: 4 dollars and 80 cents 1 Finds 1/5 x $6.00 = $1.20 1 Shows work such as: $6.00 - $1.20 or 4/5 x $6.00 2. section points 1 2 1 dollar and 80 cents 1 or or one and a half times the incorrect answer that replaced $1.20 1 ft Shows correct work such as: Works out that $1.20 represents (1 – 1/3) or 2/3 what Rachelle started with Works out that Rachelle must have $0.60 = 1/2 of $1.20 left (adding the $1.20 + .60 = $1.80 is credited in the answer) 3 1 1 3 Total Points Grade 6 – 2005 pg. 6 69 Looking at Student Work on How Much Money? Student A is able to make sense of the part/whole relationships in this task. The student sees that 1/5 plus 4/5 equals a whole. The student also knows that the denominator divides a quantity into equal-size groups and the numerator describes the number of groups being represented. The student can think about if two-thirds is equal to $1.20, then dividing by two will find the amount in 1/3. Student A Grade Six – 2005 pg. 70 For some students this task is a straight forward exercise, for others it is a problem-solving situation. Student B shows a student doing problem-solving and really thinking about the relationships involved and making sense of the fractional parts. Student B is always verifying calculations to see if they make sense as evidenced in the work in part one, where every calculation is rechecked. In part two the student tries several paths to get the solution. The habit of mind of rechecking each calculation to see if it makes sense in the context of the problem allows the student to identify strategies that don’t make sense and finally do four calculations that use all the appropriate relationships to find the answer. Student B Grade 6 – 2005 pg. 71 Student C does a nice job of labeling all the appropriate parts. The student uses decimals to find 1/5 of $6. Notice that the student rewrites the question of what needs to be calculated before solving the task. In part 2 the student uses symbolic notation to make sense of 1/3, 2/3, and one-whole or 3/3. The student identifies what is known, 2/3, and what is missing the whole, n. Student C Grade Six – 2005 pg. 72 Student D has the habit of mind of clearly documenting work done on a calculator. In part 2 the student makes good use of labels to clarify and organize her thinking. The student uses the idea of 2/3 to help solve the problem using a bar model, where 1.20 equals two of three equal parts and showing an unknown third part. This model helps to identify the operations needed to solve for the missing part and then for finding the total. Student D Grade 6 – 2005 pg. 73 Student E is able to work with the idea of 1/5 being one of 5 equal parts to successfully solve for part 1 of the task. Student E is unable to make sense of the relationship the $1.20 in part 2 must represent 2/3 if after spending that amount only 1/3 of her money remains. This idea of 2/3 + 1/3 = the total, was very difficult for many students. Student E Grade Six – 2005 pg. 74 52% of all students scored no points on this task, so it is important to identify what about fractions and part/whole relationships they do not understand. Student F is able to meet or exceed standards on three tasks, but struggles with the idea of fractions. The student understands how to write mixed and whole numbers as fractions, but doesn’t understand operations on fractions. The 1/5 being spent is not 1/5 of 1, but 1/5 of the whole $6.00. The operation should be multiplication instead of subtraction. Just like Student E, Student F does not see the 1/3 – 2/3 relationship between what is spent and what is left. Student F Grade 6 – 2005 pg. 75 Student G is able to make sense of finding 1/5 of $6. However the student does not understand how to interpret the decimal value of 1.2 and writes $1.02. The student does not continue the thinking to find the amount left after Chuck spends the money on candy. The fractions being considered it part 2 appear to have no relationship to the 1/3 – 2/3 of the task. Student G Grade Six – 2005 pg. 76 Even with the low score on task 5, Student H is meeting standards. However the student shows limited understanding of fractional relationships. Student H is able to find 1/5 of $6 and find the amount needed to add to the 1/5 to make $6. However the student is unable to recognize which part represents the money spent and which represents the money left over. In part two the student cannot make sense of the 1/3 left after spending the money. So the student reverts to a known procedure and finds 1/3 of the original money. Student H Grade 6 – 2005 pg. 77 Student J also meets standards. The student can find 1/5 of $6, but cannot convert from fractional to decimal notation. Student J does not make sense of the relationships described in the task. While the task is not asking for students to divide with fractions, the work also allows us to see that the student does not understand how to divide fractions. The student appears to add the numerators and multiply the denominators. Again the student cannot convert between fractional and decimal notation. Student J Student K is a student working below standards. The student’s model shows that the student doesn’t have even the basic idea that fractions represent equal-sized parts. Some of the “fifths” appear to be the same size as 1/4, so the student equates 1/5 with 25¢. It is unclear what the student was considering in part 2. Student K Grade Six – 2005 pg. 78 Student L confuses 1/5 with $1.50 and 1/3 with $1.30. The student has some difficulty with subtracting decimals. Student L Student M makes a similar error. The student adds the denominators and numerators to find 4/5 = $6.00. So if 1/5 is missing, 3/5 would equal $4.50. The student does not realize that 5/5 is one whole, a very basic first step in making sense of fractional relationships. Student M Grade 6 – 2005 pg. 79 Frequency Distribution for Task 5 – Grade 6 – How Much Money? How Much Money? Mean: 1.49 StdDev: 1.97 MARS Task 5 Raw Scores Score: Student Count %<= %>= 0 1 2 3 4 5 6 4248 53.2% 100.0% 856 63.9% 46.8% 507 70.2% 36.1% 807 80.3% 29.8% 838 90.8% 19.7% 114 92.2% 9.2% 622 100.0% 7.8% The maximum score available on this task is 6 points. The minimum score for a level 3 response, meeting standards, is 3 points. Almost half the students, 48%, were able to find 1/5 of a number. Some students were able to find 1/5 of a number, recognize that the number should be subtracted from the original amount in order to correctly calculate the amount of money remaining. 8% of the students could meet all the demands of the task including understanding that if 1/3 remained, 2/3 was spent. If they knew 2/3, that amount could be divided by 2 to find the size of 1/3. This information could then be used to find the original amount of money. 52% of the students scored no points on this task. 87% of the students attempted the task. Grade Six – 2005 pg. 80 How Much Money? Points 0 Understandings 87% of the students attempted the task. 1 Students could find 1/5 of $6. 2 Students could find 1/5 of $6 and attempted to subtract this amount from $6 to find the amount of money remaining. Students could find 1/5 of a number and subtract accurately to find the amount of money remaining. 3 6 Misunderstandings Students did not understand that 1/5 means to divide a quantity into 5 equal parts. Many students, 10%, took 1/5 of $1 instead of 1/5 of $6, thinking Chuck spent $.20. Others, 6%, thought 1/5 equaled $1.50. Many students tried to use subtraction instead of multiplication or division to find 1/5. 10% of the students thought this was the money left, rather than the money spent. Students made calculation errors when subtracting decimals. Some students did not subtract accurately. Students with this score could not interpret the relationship that if 1/3 remained, then 2/3 had been spent. 13% or more just multiplied the money spent in part 1 times three.10% gave answers larger than $6. Students could find 1/5 of a number, subtract from a starting amount to find a remaining amount or change, reason that if 1/3 is left, then 2/3 is spent, and use inverse operations to go from the part to find the whole. Grade 6 – 2005 pg. 81 Based on teacher observations, this is what sixth graders know and are able to do: • Some students could find 1/5 of a number, usually by dividing by 5 or finding a number that multiplies by 5 to equal the given number • Some students understood that spending money required them to subtract the 1/5 from the starting amount Areas of difficulty for sixth graders: • Understanding that fractions are equal-size groups • Converting between fraction and decimal notation • Recognizing what quantity represented a “whole” (thinking that the whole is always 1) • Identifying the proper operations when solving problems with fractions • Seeing that if 1/3 is left, 2/3 was spent (part/part/whole) Strategies used by successful students: • Labeling quantities with the fractional amounts being represented (e.g. 1/5= , 4/5 = , 1/3 = . . .) • Labeling what is known ( e.g. spent $1.20) • Changing fractions to percents to simplify calculations • Drawing bar models to organize information and show relationships • Checking solutions to see that they match the constraints of the problem Questions for Reflection on How Much Money? When designing your unit on fractions, how much thought or time is given to assessing students’ basic understanding of fractions? How are students introduced to denominators and numerators? Are students given time to explore a variety of models for picturing and making sense of fraction; such as, circle graphs, number lines, area models, bar models, parts of sets? • When students are introduced to fractions, how much time is spent helping them to picture all the relationships involved? For example, when looking at a diagram, many texts will show a bar with 5 equal parts and 3 parts shaded in. The text will describe the bar as showing 3/5. However, it is important for students to think about the whole family of relationships. If 3/5 of the bar is shaded, then 2/5 of the bar is unshaded. 3/5 + 2/5 equals the whole bar. • Do you students do math talks, discussions about solving problems mentally? Do they have opportunities to solve problems like 1/5, 1/3, 1/4 of a number mentally and justify their answers? Do students solve more complicated fraction problems mentally? • How comfortable are your students converting between fractions, decimals and percents? How might this have helped them solve this problem? Look at student work on part one of the task. How many of your students: • $4.80 $5.99 or $5.95 Grade Six – 2005 $5.80 $5.50 $4.50 Other pg. 82 Now look at the student thinking in this part of the task. What do your students know about part/whole relationships? • What indication do you have that students understand fractions as being divided into equal parts? • What indication do you have that students don’t understand fractions as being divided into equal parts (look at things like diagrams)? • What evidence do you have that students understood the “whole” as being equal to $6? • What evidence do you have that students misunderstood the “whole” (e.g. tried to find 1/5 of $1) • What evidence do you have that students understood 1/5 of a number? How many students used an incorrect operation, like trying to subtract 1/5? • How many students had difficulty converting from fraction to decimal notation? (example, trying to make 1/5 into $1.50 or 3/15 = $.20) After reflecting on student work, what would you change about how you introduce students to the meaning of fractions or working with unit fractions? What would you do differently or give more emphasis to? Look at student work in part 2. What evidence do you have that students could make sense of the idea that if 1/3 is left, 2/3 is the amount spent? • Did students label answers to show what they knew or what fractional part was represented in a given expression? • Did students make models or diagrams to help them organize or make sense of the information? • What obvious evidence do you see that students have no idea what a fraction is ( like trying to add 1/3 + 1/5 to get 2/15, or converting 1/3 to $1.30) Implications for Instruction: Students come to middle school without a clear understanding of fractions representing equal size parts of a whole. They also don’t understand that the “whole” could be one unit, like 1 dollar, 1 cup, 1 inch, or 1 group, like 1 bag of chips, 1 group of students, or one value, the starting amount of money, like the $6 in the problem. Students need to have a clear idea of this definitional representation of a fraction before they can get into problemsolving with fractional amounts. Students should be exposed to a variety of models, such as pie graphs, bar models, sets of objects, area models, and the number line to help them reason about the meaning of fractions and their relationships. When introducing a model, the teacher and student should talk about all the relationships involved. If the shaded part represents 2/3, then the unshaded part represents 1/3. Together the 2/3 and 1/3 represent or combine to make one whole. These relationships need be explicit, rather than the text book representations where the relationships are often implicit. Having a firm grasp of models helps students to reason about the meaning of operations with fractions. If 1/5 means that something is divided into 5 equal parts, students should be able to visualize breaking an object, figure, or number into equal size parts or using division to find the size of the parts. Students should also be able to reason about spending 1/5 means 4/5 is left or inversely if 1/3 is left 2/3 was spent. Using the definition of fractions as something divided into equal size parts, students should be able to think about 83 Grade 6 – 2005 pg. the relationship between 2/3 and 1/3. Students at this grade should be able to do math talks or mental math around finding fractional parts: What is 1/5 of 30? 2/5 of 45? How do you know? Why do you think that? How could you convince your classmates? Understanding equivalencies is difficult concept for students at this grade level. Many students think of the equal sign as an indicator that the answer will follow, rather than an indicator that the two sides of the expression are the same size. While the procedures for finding equivalent fractions or converting between fractions and decimals is relatively easy, the understanding of what that means is not. Students need frequent opportunities to explore equivalency, what it means, and how it can be used to solve problems. It is in being confronted with discrepancies and trying to make sense of different ideas that students build and develop their own understanding of the concept. So having students discuss which is bigger 1/2 or 4/8, is a great activity for students. Having students discuss is 1/3 equal to $1.30? In what situation might 1/3 be equal to $1.30? In what situation would 1/3 not be equal to $1.30? It is important for students to work with fractional concepts in context. It is context that provides a reason for thinking about the whole as sometimes different from 1. It is the context that causes the need to convert between fractional notation to decimal notation or monetary notation. It is the context that helps students see the difference and similarity between the fraction of a continuous amount like 1/4 of a yard of fabric or 1/8 of a cup of sugar, versus a fraction of set 1/3 of a dozen donuts, 1/4 of the class, or part of a quantity, Sam spent 3/8 of his money on bus fare. Students need to have a variety of fraction contexts and types of fractions to get a good, conceptual understanding of this complex idea. Teacher Notes: Grade Six – 2005 pg. 84