Form: "PACT - Mathematics - 4. Assessment Commentary Form v. 2009"

Created with: Taskstream

Author: Braden PauleDate submitted: 03/15/2014 10:07 pm (PDT)

Write a commentary that addresses the following prompts.

(REQUIRED) 1.

Identify the specific standards/objectives measured by the assessment chosen for analysis. You may just cite the appropriate lesson(s) if you are assessing all of the

standards/objectives listed.

Lesson 1

1. SWBAT interpret and draw diagrams showing mixed numbers and improper fractions.

(Q1-4) 2. SWBAT convert improper fractions to mixed numbers and vice versa. (Q6-7)

3. SWBAT accurately add mixed numbers with like denominators (Q4, 10) and put their answers in the simplest form. (Q8)

Lesson 2

4. SWBAT accurately add mixed numbers with unlike denominators (Q11) and put their

answer in the simplest form. (Q8) 5. SWBAT describe how to add mixed numbers with unlike denominators using

appropriate mathematical vocabulary. (describe: Q12-13, vocabulary: Q5a-5e)

Lesson 3&4

6. SWBAT accurately add mixed numbers with like and unlike denominators and put their answer in the simplest form. (Q8, 10, 11) Note: objective 6 (from lessons

3 and 4) is a combination of objectives 3 and 4 (from lessons 1 and 2). I evaluate objectives 3 and 4 below, but not objective 6 because there are

substantial differences in students’ ability to deal with the two different types of mixed numbers and the results are more informative when

categorized separately.

To get a better sense of how students are dealing with the fractional portion of adding mixed numbers, the assessment also contains two questions (Q3 and Q9) designed to

measure how students add fractions with like denominators when the answer will be less than and greater than one whole. These two questions are not directly linked to the

objectives above, but seen as a precursor to objective 3.

Standards

CA- California K-12 Academic Content Standards

Subject: Mathematics

Grade: Grade FiveBy the end of grade five, students increase their facility with the four basic arithmetic operations applied to fractions, decimals, and positive and negative

numbers. They know and use common measuring units to determine length and area and know and use formulas to determine the volume of simple geometric figures. Students know

the concept of angle measurement and use a protractor and compass to solve problems. They use grids, tables, graphs, and charts to record and analyze data.

Area: Number Sense

Sub-Strand 2.0: Students perform calculations and solve problems involving addition,

subtraction, and simple multiplication and division of fractions and decimals:

Standard 2.3 (Key Standard): Solve simple problems, including ones arising in concrete situations, involving the addition and subtraction of fractions and mixed numbers (like and

unlike denominators of 20 or less), and express answers in the simplest form.

(REQUIRED) 2.

Create a summary of student learning across the whole class relative to your evaluative criteria (or rubric). Summarize the results in narrative and/or graphic form (e.g., table or

chart). Attach your rubric or evaluative criteria, and note any changes from what was planned as described in Planning commentary, prompt 6. (You may use the optional chart

provided following the Assessment Commentary prompts to provide the evaluative criteria, including descriptions of student performance at different levels.) (TPEs 3, 5)

See table attached as pdf.

(REQUIRED) 3.

Discuss what most students appear to understand well, and, if relevant, any misunderstandings, confusions, or needs (including a need for greater challenge) that were

apparent for some or most students. Cite evidence to support your analysis from the three student work samples you selected. (TPE 3)

Group 1: There are nine students in the class who appear to have command over procedures and relatively clear conceptions around mixed numbers and fractions. This group

includes five GATE students, one EL, and no students with IEPs. Five students in this group are fourth graders, and four are fifth graders.

These nine students answered most, if not all, of the questions on the assessment correctly.

Among the group of nine there were one or two mistakes in each section of the test. Most of these mistakes represented simple procedural errors (such as addition that was off by one)

rather than a misunderstanding of the procedures or concepts.

This group is confident and finished the test quickly. Judging from the scattered simple errors, they probably should have taken more time to check their work. All students in this

group showed clear understanding of the procedures related to adding mixed numbers with like and unlike denominators, putting their answers in simplest form, renaming fractions

and mixed numbers to various equivalents and forms, and a firm grasp of the vocabulary words. All students described the process of renaming the fractional portion of mixed

numbers to common denominators as their favored approach to adding mixed numbers with unlike denominators, though many students skipped over key steps of the process when

describing it in words, perhaps also indicating a tendency to rush through their work.

While these students have all mastered the procedure, they could use more practice explaining their mathematical thinking, and for some students some of the concepts may be

incomplete or there may be lurking misconceptions. For example one student in this group still is still working to overcome a misconception that equivalent fractions can be found by

subtracting the same number from numerator and denominator. In the unit test referenced in the Context for Learning, he used this strategy to find equivalents throughout and scored

quite low. On the more recent assessment he used the accurate strategy of multiplying the numerator and denominator by the same number instead and his other work shows clear

concepts about fractions. However he still has not entirely disproven his initial misconception to himself. He listed the “subtract from numerator and denominator” method

as a second method that he could use to add mixed numbers with unlike denominators, and then asked why that strategy does not work on an exit slip from the same day, revealing

that he is still struggling to understand wether this method works or not.

The sample attached (labeled S5) typifies this group of nine students with clear, accurate answers with a not-entirely-clear description of her process.

Note there was one student absent the day of the assessment who would very likely fit in

this group based on observations and other student work samples.

Group 2: There are nine students in the class who have are fairly confident and competent adding mixed numbers and fractions with like denominators, but use an incomplete

procedure to add mixed numbers with unlike denominators. This group includes five ELs, two students with IEPs, and one GATE student. There are five fourth graders and four fifth

graders in this group.

Students in this group demonstrated their ability to add mixed numbers and fractions with like denominators. Mistakes on these types of problems appeared to be mistakes made

adding (i.e. answers that were off by one) rather than mistakes appearing to be based on a misconception about the nature of mixed numbers or fractions. Only one student in this

group was able to convert a mixed number to its simplest form, and many students in this group had a hard time representing improper fractions (as in sample S6, on questions 2b

and 7). On average students in this group made 1 mistake in representing key vocabulary words, though some students (as in sample S6) got all of these problems right, and the two

students with IEPs both got four out of five wrong.

Most students in this group attempted to add mixed numbers by finding common denominators, but failed to adjust the numerators when renaming their fractions indicating

both a procedural and conceptual misunderstanding. Some students tried other inaccurate strategies, such as the sample student who added the numerators and kept the largest

denominator. Most of the students in this group used the word “common denominator” or described part of the process to find a common denominator when describing their favorite

way to add mixed numbers with unlike denominators. However none of the descriptions were complete and their work showed that they did not understand why it is important to

adjust the numerator when adjusting the denominator to make sure the mixed number is still equivalent.

Note there was one student who was absent for the assessment and for most of the

learning segment who would likely fall into this group based on in-class observations.

Group 3: There are seven students in the class whose work shows serious misconceptions regarding the meaning and form of fractions and mixed numbers. This group includes five

ELs, two students with IEPs, and no GATE students. There are five fourth graders and two fifth graders in this group.

All of the students in this group made mistakes either interpreting or representing mixed

numbers and improper fractions with diagrams. Their mistakes indicate a lack of understanding about what the numerator and denominator of a fraction represent. For

example the answers to question 1 in the attached sample (from student S4) shows a pattern where the student counted the number of pieces in the whole(s) and wrote this

number as the numerator and consistently wrote 2 as the denominator regardless of the diagram. Other students followed a similar pattern counting the number of pieces in the

whole(s) as the numerator, but writing the number of “left-over” pieces as the denominator.

All but one student in this group correctly added fractions with like denominators with a given diagram, and most students were also able to add fractions whose sum was greater

than one whole. Two students (including the attached sample) accurately added mixed numbers with like denominators.

None of the students in this group were able to convert a mixed number to simplest form or

convert mixed numbers to improper fractions or vice versa. Students showed a variety of invented strategies in this section including adding the whole

number+numerator+denominator (as in the attached sample, questions 6-8).

Three students in this group added mixed numbers with unlike denominators by adding the wholes, the numerators and the denominators together (as in the sample, questions 11 and

12). Others used other inaccurate student-invented algorithms or appear to have copied answers from a neighbor. This last observation is not made lightly, and is based on correct

answers to question 11 that shows no work, does not match student description of how to do that type of problem, and the existence of other answers on the test, both correct and

incorrect, that match a more proficient neighbors’ answers.

On average students in this group were able to give examples of three of the five vocabulary words. Five out of seven students in this group gave clear and mostly complete

descriptions of their strategies for adding mixed numbers with unlike denominators using both words and numerical examples (as in the sample). Only one of these strategies was

likely to result in an accurate answer, though. The student with the strategy likely to result in an accurate answer was rather surprising because her explanation on how to add mixed

numbers with unlike denominators was clear, accurate and concise, but she was not able to answer any of the mixed number addition problems correctly. Her work elsewhere on the

test points to an incomplete understanding of the term “common denominator” though she knows a part of the process for finding one.

(REQUIRED) 4.

From the three students whose work samples were selected, choose two students, at least one of which is an English Learner. For these two students, describe their prior knowledge of

the content and their individual learning strengths and challenges (e.g., academic development, language proficiency, special needs). What did you conclude about their

learning during the learning segment? Cite specific evidence from the work samples and from other classroom assessments relevant to the same evaluative criteria (or rubric). (TPE

3)

S6 is a fourth grader. Her first language is Spanish, but she is also fluent in English and has

never been enrolled in ELD classes. Her CST test results from third grade place her at the Basic level for ELA and the Advanced level for Math. While she did quite well on the CST,

she is not very confident in math class and almost never raises her hand. On the unit test referenced in the Context for Learning, she rated her own ability as low in all categories and

in fact scored fairly low in each section with a 20% overall score. S6 is talkative enjoying conversation with all the students at her table.

S6 appears to have learned more about the concept and form of fractions and mixed

numbers as well as the procedure for adding fractions and mixed numbers with like denominators. She has learned to interpret diagrams showing mixed numbers and write

them in numerical form. She gave examples indicating and understanding of key vocabulary surrounding mixed numbers and fractions and used clear language, though an incomplete

and inaccurate description, to describe her favorite way to add mixed numbers with unlike denominators. She is still working on the concept of equivalence. A better grasp of

equivalence should help her to learn the procedure for adding mixed numbers with unlike denominators since renaming the fractional part plays such an important role.

A comparison of the test results for S6 is summarized in the table below.

BEFORE

(unit test mentioned in Context for

Learning)

AFTER

(current assessment)

S6 was unable to interpret any diagrams

showing mixed numbers.

S6 is able to accurately interpret diagrams showing mixed numbers and draw or shade her

own diagrams to illustrate a fraction or mixed number, but not an improper fraction. (Q. 1-4)

S6 got every fraction addition problem wrong because she added the

denominators as well as the numerators.

S6 generally added fractions and mixed numbers

with like denominators accurately (Q. 3, 9, and 10), though she added one inaccurately for a

reason I cannot determine (Q.4).

S6 was unable to find equivalent fractions

because she changed the numerators without changing the denominators.

S6 shows some recognition of fractions equivalent to ½ (Q. 1 and Q5c), but continues to change the

numerator without changing the denominator at other times (Q. 8).

S6 answered about half of the problems

requiring her to convert improper fractions to mixed numbers and vice

S6 converted improper fractions to mixed

numbers accurately (Q. 6 and 9), but did not convert from mixed number to improper fraction

versa (given multiple choices). (Q. 7).

S4 is a fifth grader. His primary language is Spanish and he is in the Intermediate CELDT level. He has a speech and language impairment that makes it difficult for him to

comprehend spoken information. This effects his retention, recall and processing of curricular language and verbal directions. He tries hard in class and is cooperative and

good-natured. Because comprehension is challenging for him he needs frequent checks for understanding, lots of repetition and review of curricular material, and repetition and

rephrasing of directions. He wears glasses to correct his 20/40 vision and receives preferential seating close to the board from an angle with limited glare. His test scores show

slow and steady progress over the last few years, but he is still in the Below Basic level for the CST in ELA and Mathematics. S4's IEP for the current school year mentions that he

needs to work on his multiplication facts. On the unit test referenced in the Context for Learning, S4 scored 20%. He demonstrated ability in counting and adding, but his invented

strategies for solving other problems did not reflect an understanding of fractions as parts of a whole.

S4 has expanded his prior procedural ability to add fractions with like denominators and can

now add mixed numbers with like denominators as well. His strategies for interpreting diagrams and adding mixed numbers with unlike denominators have changed, but have not

become more accurate. He does not understand many of the terms and concepts related to fractions and mixed numbers.

BEFORE

(unit test mentioned in

Context for Learning)

AFTER

(current assessment)

S4 was unable to interpret any

diagrams showing mixed numbers. He counted all of the

pieces the same whether they were wholes or parts.

S4 was unable to interpret any diagrams showing mixed numbers. He counted the numbers of pieces in the

whole(s) and wrote it as the numerator. His denominator was always 2. In one problem he counted the number of

wholes and showed it as the whole part of a mixed number (Q. 1, part 3).

S4 was unable to find equivalent

fractions. For every problem he added one to the denominator,

but did not change the numerator.

S4 did not answer any of the equivalent fraction questions

in fraction form (Q. 5c, 8).

S4 accurately converted between

fractions and mixed numbers about half of the time (given

multiple choices).

S4 did not convert between improper fraction and mixed number form (Q. 6-7).

S4 accurately added fractions S4 accurately added fractions and mixed numbers with

with like denominators. like denominators (Q. 3, 4, 9, 10).

S4 did not accurately add

fractions with unlike denominators. When the

numerators were the same, he kept the same numerator and

added the denominators.

S4 still does not accurately add fractions with unlike denominators, but his strategy has changed. He now adds

numerators together and denominators together regardless of whether the numerator is the same or not

(Q. 11-12). He also wonders if he can get the answer by multiplying the denominators together and the numerators

together (Q. 13).

(REQUIRED) 5.

What oral and/or written feedback was provided to individual students and/or the group as a whole (refer the reviewer to any feedback written directly on submitted student work

samples)? How and why do your approaches to feedback support students’ further learning? In what ways does your feedback address individual students’ needs and learning goals?

Cite specific examples of oral or written feedback, and reference the three student work samples as evidence to support your explanation.

Oral feedback for groups

The week after teaching my learning segment, I pulled groups of students to go over the results of the quiz while my master teacher taught the rest of the class. I broke up each of the groups above into

two groups called them to the kidney table. They got a chance to look over their tests and ask me questions about what I had written. Then I asked them follow up with notes I had left such as trying

their strategies with their fraction kits. Finally we discussed possibilities for next steps toward our goal of having everyone reach a solid understanding of fractions as we continue into subtracting mixed

numbers, and multiplying and dividing fractions.

Individual feedback written on work

I marked answers as correct or incorrect using the symbols used in this class (star/circle) and wrote the correct answers where students made a mistake. I chose certain questions to model if mistaken. I

modeled the diagram questions (Q. 1-4) when mistaken because I think that incorporating this concrete, visual representation is critical for many of my students to develop a better understanding of

fractions and mixed numbers. I gave examples for the vocabulary words as well (Q5), using at least one of the numbers the student used whenever possible. I did not model (Q.6-8). Based on the results

and on class feedback, I believe I did not teach these concepts and procedures clearly enough and plan to reteach with the whole class as we move forward. I did model Q. 11 for every student who

missed it, because we have spent a lot of time talking about it already and done a lot of examples in class. I thought that one more example of the procedure to solving this type of problem that the

students could keep might be useful. Finally I gave comments and asked questions about the students’ descriptions in Q. 12-13 to prompt them to write more detail about their answer or try out

their invented strategies with a fraction kit and an equation or problem that would be easy to model and would challenge inaccurate theories.

S5

Because of her outstanding performance on the test, I found very little to comment on with S5's

test. Her description in Q12 was not entirely clear so I showed that I think that she meant “common

denominator” not “a number they can both divide or multiply by”. I also left one comment encouraging her use of simplest form (something much of the class is struggling to understand) and to

consider division to rename fractions as well as multiplication. Many students in the class only use multiplication and quite a few only use quick common denominators even when they could find a lower

common denominator, so I hope that S5 will share her division strategy with her classmates.

S6

For S6 I modeled questions 2b, 4, and 11 for the reasons described above. In Q12 I asked her to

test her invented strategy with an equation that followed the rules of her algorithm, but would clearly

be incorrect with the fraction kit. I also left her a happy face to show that I *love* the fraction kit (a fact I have told the students many times), and think it can really help her even though I can see from

the faces she drew on her test that she much prefers her invented algorithm.

S4

For S4 I used arrows and words to show how the diagrams can be interpreted as mixed numbers. I chose one diagram

each from Q. 1 and 2 because I thought that writing more than that would be overwhelming for him. I think that understanding

where each of these parts of the mixed number is critical to S4s understanding of mixed numbers and progress. I also drew

arrows to show how to shade the fraction bar diagram in Q4 because I think it may be a useful tool for him to figure out these

types of problems. I used S4s numbers to give examples for Q. 5 and underlined the word “fractions” in each answer that he had

answered with a whole number. I modeled Q. 11 as I had for the rest of the class. I wrote an expression to test his algorithms

described in questions 12 and 13 using his fraction kit.

(REQUIRED) 6.

Based on the student performance on this assessment, describe the next steps for instruction for your students. If different, describe any individualized next steps for the two

students whose individual learning you analyzed. These next steps may include a specific instructional activity or other forms of re-teaching to support or extend continued learning

of objectives, standards, central focus, and/or relevant academic language for the learning segment. In your description, be sure to explain how these next steps follow from your

analysis of the student performances. (TPEs 2, 3, 4, 13)

Whole class

In addition to reteaching simplest form, I want to continue working with the whole

class on the practice of asking questions and explaining their thinking. In the short term I see highly structured turn and talk conversations as a valuable tool. The adapted turn-and-

talk with a specific script that I used in Lesson 2 seemed much more successful than the more vague one in Lesson 1. I would like to do more highly-scripted turn-and-talks and

then have students share what they said with the whole class. Instead of always using the same turn-and-talk structure, I can keep it interesting by using a variety of conversation-

based activities like Numbered Heads Together, Fishbowl, Inside/Outside, Talking Chips, Snowball, etc.

There are already many questions in the students' textbook requiring them to

explain their thinking, but the students often gloss over these questions and do not see

them as "real" questions because they do not have a "right" answer. I think that taking more time to review and discuss their answers to these questions will be valuable.

In the long run I hope to slowly remove the turn-and-talk scaffold and have students

share ideas and question with more flexibility and less direction. I believe this will require a shift

in classroom culture where students are comfortable sharing ideas that may not be fully formed

and may be changed or disproven later. This will require an explicit set of values based on

respect for everyone's ideas and unique ways of understanding mathematics.

Group 1

Group 1 is ready to go deeper with the ideas we are currently working on. While the students appear to still have some minor misconceptions and make errors from time to

time, the activities and material designed for the whole class will not provide them enough challenge. As we move into subtraction with mixed numbers and then multiplication and

division of fractions, I would like to have the students in this group work with a partner or two to create informational posters around topics related to fractions and mixed numbers

such as the concepts of equivalence, fraction-whole relationship, parts of a mixed number, and procedures for the various types of problems we work on in class. Each poster would

need to contain illustrations, examples, and student explanations. I think that this work will be both creative and challenging for the students in Group 1. I have often thought that I

understand a topic well until I try to explain it to someone else. I believe that trying to get their ideas clearly on paper will reveal any misconceptions and lead to deeper

understandings. Of course the posters will also be helpful to display around the classroom for the benefit of all students.

Group 2

Group 2 is trying to understand the procedures for adding mixed numbers (and

fractions) with unlike denominators, however they are remembering only part of the procedure. I believe that without a better understanding of the underlying concept of

fractional equivalence, these students will have a hard time learning and remembering the procedures. I would like to have this group spend some time using physical models (such as

their fraction kits, grid or dot paper, paper folding, and sets of counters) to find as many different names for a fraction as they can and then record their results in collections of

equivalent fractions.

After they get some practice and facility with these physical models, I want them to start developing and testing algorithms for finding equivalent fractions. I think this is

particularly important for this group that has memorized part of the standard algorithm (multiplying or dividing the numerator and denominator), but do not remember to apply

every step and end up with fractional parts that are not equivalent. One way to support students in developing algorithms is to have them look through their equivalent fraction

collections to find patterns. I believe this strategy will be helpful. My students appear to have a lot of experience and skill in finding and defining patterns. Another way I can

support them in defining an algorithm for renaming fractions to equivalent forms is to show students enough of a physical or visual model so that they can determine the number of

pieces, but not enough of it to count pieces. For example, I could have students watch as I shade 2/3 of a piece of paper with the three sections delineated by vertical lines. Then, out

of students' sight I could draw a number of horizontal lines and then show students two edges of the paper so that they can count the number of rows and columns it is divided into

and see which ones are shaded, but cannot count each piece. Students would then need to multiply (or serially add) the number of pieces they see to figure out what fraction of the

paper is shaded. By following a similar procedure students in Group 2 could quiz each other to try out their algorithms and strategies and then uncover the paper to check their

answers.

Group 3

Students in Group 3 appear to have misconceptions regarding the meaning of the parts of a fraction and mixed number. In particular the meaning of the denominator appears to be the

most challenging. I think these students will benefit from fair share activities where they decide how to break up a whole (or multiple wholes) into a given number of “shares”. They

could model these shares using their fraction kits, chip sets, folded paper, etc. Once they have their physical answers, I would help students write these answers in fraction or mixed

number form while discussing what each part of the numerical symbolic form represents. Once students have shown some facility and understanding of what each part of the fraction

means, I would challenge them and have them challenge each other to write number forms based on diagrams or models and create diagrams, models and story contexts based on

numerical forms.

The students in Group 3 also need more practice with multiplication and division. Without a mastery over these operations, they are easily overwhelmed by the multi-step problems

involved in working with fractions. I can give these students some time to practice multiplication and division during the Math Minute and in small groups on centers days

without interrupting their continued access to the whole class curriculum.

Vocab group

There are six students (including four from group 3 and two from group 2, and also including all four students with IEPs) in the class who got almost all of the vocabulary

questions wrong on the assessment. Not understanding these key terms will make it difficult for these students to learn from class discussion and lectures. For this reason, I plan to

work with these six students to create personal word banks using index cards and a binder ring to keep them all together. These students will write down new mathematical vocabulary

words along with a definition and examples. At first these cards may be based entirely on the words we track on anchor charts, but over time I will prompt them to start adding new

words or words they don't understand and then checking with me or with their regular classroom teacher to figure out the meaning and some examples.

% of students meeting all criteria Included skills/concepts Performance Level 1 Performance Level 2 Performance Level 3

Interpret diagrams

76% of students were able to interpret and write as a mixed number all three diagrams in question 1.

8% of students were able to interpret and write as a mixed number one or two of the three circle/slice diagrams in question 1. Mistakes were due to mistaking the denominator of the fractional part in one or two diagrams.

16% of students were able to interpret and write as a mixed number none of the diagrams in question 1. All students in this category were able to count the wholes and/or fractional pieces, but placement of these numbers in mixed number or fractional form show a lack of understanding of the meaning of the parts of a mixed number (whole, numerator, denominator) and how these parts relate to the diagram.

Draw diagrams

40% of students were able to accurately draw diagrams showing a mixed number and an improper fraction, and shade both a circle diagram and fraction bar to help solve or represent their answer to fraction and mixed number addition.

24% of students drew pieces that were too small for the proportion of the whole that they were designed to represent (i.e. drawing a 1/4 piece that looked closer to 1/8 of the area of the whole) but otherwise drew diagrams accurately showing a mixed number and an improper fraction, and shaded both a circle diagram and fraction bar to help solve or represent their answer to fraction and mixed number addition.

46% of students made more fundamental errors in drawing diagrams showing a mixed number and an improper fraction, or shading a circle diagram and fraction bar to help solve or represent their answer to fraction and mixed number addition, such as failing to represent the entire number or confusing the role of the numerator and denominator.

2 SWBAT convert improper fractions to mixed numbers and vice versa. (Q6-7)

40% of students answered all questions related to this objective correctly.

Convert improper fraction to mixed number and vice versa

40% of students were able to both convert an improper fraction to a mixed number AND convert a mixed number to an improper fraction

20% of students were able to accurately convert an improper fraction to a mixed number, but were not able to convert a mixed number to an improper fraction.

40% of students were not able to accurately convert an improper fraction to a mixed number OR convert a mixed number to an improper fraction. Errors varied significantly with several student-invented algorithms and several students answering in the wrong form.

Add mixed numbers with like denominators

60% of students were able to accurately add a mixed number with a fraction and two mixed numbers with like denominators.

24% of students were able to accurately add a mixed number with a fraction OR two mixed numbers with like denominators. Most mistaken answers were very close, seemingly due to simple errors in addition. When adding mixed numbers, two students answered with the largest whole number, numerator, and denominator from the two mixed numbers being added.

16% of students were not able to accurately add either a mixed number with a fraction or two mixed numbers with like denominators. Several students added the fractional parts, but left off the whole parts. Others left the answer fields blank, made errors in addition, or answered with numbers not seemingly related to the problem.

Put answer in the simplest form

36% of students were able to rename a mixed number to its simplest form.

8% of students renamed a mixed number to an equivalent stated in greater terms rather than simpler terms.

56% of students answered with a number that was not equivalent to the mixed number given. Several students modified the whole number portion of the mixed number and several used student-invented algorithms adding various parts of the mixed number together to find an answer in either whole number or fraction form.

Add mixed numbers with unlike denominators

40% of students were able to add two mixed numbers with unlike denominators. All of these students renamed the fractional part of the mixed numbers to make a common denominator. One student made an error in arithmetic, but showed understanding of the process of renaming fractional parts and adding.

20% of students multiplied the denominators to a common number, but did not adjust the numerator before adding.

40% of students added wholes, numerators and denominators together, OR added numerators and kept the largest denominator, OR left the field blank, OR strong evidence suggests the student copied the correct answer from a neighbor.

Put answer in the simplest form (Note: this skill is the same as the one listed in objective 3b)

36% of students were able to rename a mixed number to its simplest form.

8% of students renamed a mixed number to an equivalent stated in greater terms rather than simpler terms.

56% of students answered with a number that was not equivalent to the mixed number given. Several students modified the whole number portion of the mixed number and several used student-invented algorithms adding various parts of the mixed number together to find an answer in either whole number or fraction form.

4SWBAT accurately add mixed

numbers with unlike denominators (Q11) and put their answer in the

simplest form. (Q8)

28% of students answered all questions related to this standard correctly.

3SWBAT accurately add mixed

numbers with like denominators (Q4, 10) and put their answers in the

simplest form. (Q8)

28% of students answered all questions related to this standard correctly.

Objective

1

SWBAT interpret and draw diagrams showing mixed numbers and improper

fractions. (Q1-4)

36% of students answered all questions related to this objective correctly.

Student describes a method for adding mixed numbers with unlike denominators that will result in an accurate answer.

24% of students described a strategy for adding mixed numbers with unlike denominators that was complete, detailed, and likely to result in a correct answer. Every student in this category described the method of renaming the fractional parts with like denominators as their favorite method. Many students in every category correctly used the terms "denominator" and/or "common denominator" in their description.

32% of students described a strategy for adding mixed numbers with unlike denominators that was likely to result in a correct answer, but left out key steps or failed to describe how to perform steps critical to the process. Every student in this category described the method of renaming the fractional parts with like denominators as their favorite method. Many students in every category correctly used the terms "denominator" and/or "common denominator" in their description.

44% of students described strategies for adding mixed numbers with like denominators, OR student-invented algorithms not likely to result in a correct answer, OR descriptions were not clear enough to be categorized. The most common student-invented algorithm was to add the two wholes, the two numerators, and the two denominators together. Many students in every category correctly used the terms "denominator" and/or "common denominator" in their description.

Student gives examples indicating an understanding the following terms: improper fraction, mixed number, equivalent fractions, unlike denominators, like/common denominators

48% of students gave examples that fit each of each of the italicized vocabulary words.

28% of students gave examples for 1 or 2 of the italicized vocabulary words that were incorrect.

24% of students gave examples for 3 or more of the italicized vocabulary words that were incorrect. There were no clear patterns within the words themselves as all were missed with similar frequency.

6SWBAT accurately add mixed numbers with like and unlike denominators and put their answer in the simplest form. (Q8, 10, 11)

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Note: this objective (from Lessons 3 & 4) is a combination of objectives 3 and 4 (from Lessons 1 and 2). Combining results from those two objectives reveals no patterns more revealing than those already detailed above under objectives 3 and 4. <see above> <see above> <see above>

5

SWBAT describe how to add mixed numbers with unlike denominators

using appropriate mathematical vocabulary. (describe: Q12-13,

vocabulary: Q5a-5e)

20% of students answered every question related to this objective completely and accurately.

4th Graders 4th graders were significantly less likely than 5th graders to accurately add mixed numbers with unlike denominators (Obj. 4). Otherwise results were similar.

GATE students scored significantly higher than the class average in every area except in describing how to add a mixed number with unlike denominators (Obj. 5) where every student had described a strategy likely to result in a correct answer, but many were missing key steps in their description.GATE

Identified groups

Students with IEPs Students with IEPs performed significantly more poorly than other students in three areas: adding mixed numbers with unlike denominators (Obj. 4), giving examples of vocabulary words (Obj. 5), and describing how to add mixed numbers with unlike denominators (Obj. 5). No clear and significant differences in performance are seen in other areas.

EL students EL students' performance was significantly lower than the class as a whole in the following areas: drawing diagrams of improper fractions (Obj. 1), Converting mixed numbers to improper fractions (Obj. 2), Adding mixed numbers with unlike denominators (Obj. 4), and Describing how to add mixed numbers with unlike denominators (Obj. 5). Many of the descriptions in this last area described strategies likely to work, but were incomplete, missing one or more steps.

Characteristics of identified group performance