GRADE 3 MATH: COOKIE DOUGH - Web.stanford.edu

Third Grade: Cookie DoughAnnotated Student WorkThis student’swork is not onlycorrect, but theexplanations arevery clear andcomplete. Thisstudentreceived a scoreof 7 andperformed atLevel 4.In part 1 and 2, thestudent showscorrect equationsthat models thesituations, andlabels the units thatare part ofreasoningquantitatively.3.OAMP2, MP4In part 3, thestudent links theoperations ofmultiplication anddivision.3.OA.5&6MP2The student writesa completeexplanation forpart 4, explainshis/her thinkingprocess, andassociates the tubwith the dollaramount at eachinterval. Inaddition, thestudent explainshis/herjustification towhy 21 is toomuch. At thirdgrade level, this isan example of aconvincingargument. MP311

Third Grade: Cookie DoughAnnotated Student WorkLevel 3: Performance at Standard (Score Range 4 – 5)For most of the task, the student’s response shows the main elements of performance that the tasksdemand, and is organized as a coherent attack on the core of the problem. There are errors oromissions, some of which may be important, but of a kind that the student could well fix with more timefor checking and revision, and some limited help. The student explains the problem and identifiesconstraints. The student makes sense of the quantities and their relationship in the problem situations.S/he often uses abstractions to represent a problem symbolically or with other mathematicalrepresentations. The student response may use assumptions, definitions, and previously establishedresults in constructing arguments. S/he may make conjectures and build a logical progression ofstatements to explore the truth of their conjectures. The student might discern patterns or structuresand make connections between representations.This studentmet standard(Level 3) andscored 4. Thisstudent showedunderstanding,but made errorsin bothcalculating andinterpretingtheir own work.In part two, thestudent interpretedthe situation,reasoned through themulti-steps, andmodeled the numbersentences correctly,but when multiplying4 times 5 made acalculation error. Therest of the reasoningis sound.3.OA.1 to 3, MP2,MP4The answer topart 3 is correct,yet the erasedwork does notindicate how thestudent arrived atthe answer.In part 4, the studentused a chartassociating 3 (dollars)to circles (tubs). Thestudent correctlydetermined when tostop the pattern andstated that 21 wentpassed 20, but thenthe student did notdistinguish correctlybetween the amountof tubs and dollars.The reasoning andproblem solving wassound. MP1, MP212

Third Grade: Cookie DoughAnnotated Student WorkThis studentperformed atLevel 3(MeetingStandards) witha rubric score of5. The workshown islimited, butappears to showmathematicalunderstanding.The student wasable to get thefirst partcorrect, and thisindicates thestudent canreasonquantitatively.3.OA, MP2The studentprovided incorrectanswers for part 2and showed nowork, so it is notclear how thestudent arrived at 34.In part 3, thestudent wrote avery faint numbersentence that wascorrect 32 4.Although the workdoes not show theactual calculation,the student didarrive at thecorrect answer.3.OA.5&6In part 4, thestudent found thecorrect answer byrepeatedly countinggroups of three.This involvesreasoningquantitatively andunderstanding aproblem situation.MP2, MP113

Third Grade: Cookie DoughAnnotated Student WorkLevel 3 Implications for InstructionStudents who met standard on the task can still improve their performance by being attentive toprocess and by making complete explanations and justifications. Students must learn to providecomplete arguments when justifying why a statement is correct. It is equally important that studentsalso explain why a value is not correct. When there is only one answer, the students should justify why.This requires the student not only to tell how an answer was arrived at, but also why another valuecannot be a solution. In part four, students were asked to determine the greatest number of tubs ofOatmeal Dough that could be purchased with 20. A complete explanation isn’t limited to the answer ofnumber of tubs. It should also address why that answer is the maximum amount possible. Studentsneed experiences in writing complete explanations and justification. They can benefit from readingother students’ explanations and critique explanations to improve them. Students should follow thereasoning of sound justifications and write conclusions from deductive arguments. Routinely askingstudents to justify and explain their approaches, solutions, justifications and generalizations in class willfoster the development of this skill. Students should be held accountable for complete explanations andmust learn that mere answers are unsatisfactory. Explaining “why”, “how you know” and “why ananswer cannot be something else” should be part of every student’s argumentation.14

Used in the moduleThird Grade: Cookie DoughAnnotated Student WorkLevel 2: Performance below Standard (Score Range 2 - 3)The student’s response shows some of the elements of performance that the tasks demand and somesigns of a coherent attack on the core of some of the problems. However, the shortcomings aresubstantial, and the evidence suggests that the student would not be able to produce high-qualitysolutions without significant further instruction. The student might ignore or fail to address some of theconstraints. The student may occasionally make sense of quantities in relationships in the problem, buttheir use of quantity is limited or not fully developed. The student response may not state assumptions,definitions, and previously established results. While the student makes an attack on the problem it isincomplete. The student may recognize some patterns or structures, but has trouble generalizing orusing them to solve their problem.The student wasable to get thecorrect answers tothe first two partsof the task byshowing a numbersentence in thesecond part. 3.OA& MP4The student eithermisread ormisinterpreted part 4.At times, when studentsare unsure, they look tofriendly numbers toperform calculations.Here, the studentshowed nounderstanding for asense of division, andremainders wereinvolved. 3.OA.8&9The student couldnot workbackwards and didnot use the inverseoperation ofmultiplication(division). Insteadthe studentincorrectlymultiplied by 3.3.OA.5 & 3.OA.615

Third Grade: Cookie DoughAnnotated Student WorkLevel 2 Implications for InstructionStudents need help understanding the relationship between multiplicative operations (scaling up,repeated addition, skip counting up, multiplying and reasoning about the number of equal groups, etc.)and its inverse operation. The inverse process can be understood in several ways including, repeatedsubtraction, partitioning a set into equal groups, counting backwards, counting up knowing thetermination number, and keeping track of groups, etc. Students come to understand that division is theinverse of multiplication, or from a student’s perspective, division undoes what multiplication did.Students at this level must understand the inverse relationship to be successful in parts 3 and 4.Students can create numeric tables, use number lines and skip count (one direction or the other),and/or use sets of counters to understand division. Students who were successful approached theproblems using different strategies, yet regardless of the strategy, they all reasoned about therelationship of the numbers. Students who were unsuccessful merely chose operations to use.Therefore students need to learn the relationships of operations, what they mean, and how twoquantities are related. Understanding fact families of multiplication and their related divisionrelationship is an important tool students should know. Students should learn to reflect on theiranswers, check units of measure and their calculations to verify their solutions. This requires studentsto privately ask, “does this answer make sense in the context of the problem?”16

Third Grade: Cookie DoughAnnotated Student WorkLevel 1: Demonstrates Minimal Success (Score Range 0 – 1)The student’s response shows few of the elements of performance the tasks demand. The work shows aminimal attempt on the problem, and a struggle to make a coherent attack on the problem.Communication is limited and work shows minimal reasoning. The student’s response rarely usesdefinitions in their explanation. The student struggles to recognize patterns or the structure of theproblem situation.This studentperformed at Level1, minimal successwith a score of 2.The studentanswered twoquestionscorrectly, but thework showedmisunderstandingsand flaws inreasoning.In part two, thestudent does userepeatedaddition to find4x4 but thenmodels anincorrect numbersentence for theChocolate ChipCookie Dough.The student doesseem to knowthe two amountsthat need to beadded.MP2.In part 4, thestudent appears tomisinterpret theproblem. Insteadof finding howmany OatmealCookie tubs youcan buy for 20,the student tries tofind numbers of 4in 20. Thestudent’s recordingis not clear foreven the student,and he/she arrivesat 7 in 20. Thestudent has troublereasoningquantitatively.17

Third Grade: Cookie DoughAnnotated Student WorkLevel 1 Implications for InstructionStudents need support with reasoning quantitatively. They may need to start with repeated addition tounderstand what the total of several equal groups mean. Having student use counters to understandskip counting may be a foundational experience for the students.Part two requires a more complex chain of reasoning. Students need to sort and reason about twodifferent types of cookie dough, creating number sentences to find 4 x 4 and 4 x 5. After arriving atthose products (or sums using repeated addition), the student must perform a third step adding thosetwo amounts to get a total. Students learn to solve multi-step problems only by engaging in multi-stepproblems. All students need to regularly experience solving problems with more than one or two stepsand need to experience problem solving that requires longer chains of reasoning.Students need experience in creating mathematical models for contextual situations. The modelingexperience in third grade may involve writing number sentences, making numerical tables, creatingcharts or diagrams and/or drawing pictures to characterize a situation. Students will benefit fromconnecting and linking the representations to make sense of how the representations model thesituation. Students learn to model situations by making sense of the problem and then tinkering withrepresentations through trial and error.Students need to talk through why a mathematical model makes sense in representing a problem. Thiscan be done in pairs, small groups or by the whole class. More instructional emphasis should be placedon finding and understanding why a representation makes sense, than merely finding the answer toproblems in context.18

GRADE 3 MATH: COOKIE DOUGHINSTRUCTIONAL SUPPORTSThe instructional supports on the following pages include a unit outline with formativeassessments and suggested learning activities. Teachers may use this unit outline as it isdescribed, integrate parts of it into a currently existing curriculum unit, or use it as a model orchecklist for a currently existing unit on a different topic.UNIT OUTLINE .20INITIAL ASSESSMENT: SPONSORED WALK 27FORMATIVE ASSESSMENT:INTERPRETING MULTIPLICATION & DIVISION . 30FORMATIVE ASSESSMENT: SQUIRRELING IT AWAY . 5719

Unit Outline TemplateINTRODUCTION: This unit outline provides an example of how teachers may integrateperformance tasks into a unit. Teachers may (a) use this unit outline as it is described below;(b) integrate parts of it into a currently existing curriculum unit; or (c) use it as a model orchecklist for a currently existing unit on a different topic.Grade 3: Interpreting Multiplication and DivisionUNIT TOPIC AND LENGTH:This unit should run between 20 and 25 standard periods of instruction. One of the periodswill involve the pre-assessment (0.5 period), introducing and supporting problem solvingon the long lesson (2 periods), teaching the formative assessment lesson (2.5 periods) andthe final assessment (0.5 period). This unit should be taught after or during the timestudents have learned about multiplication and division.COMMON CORE LEARNING STANDARDS:3.OA.1 Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number ofobjects in 5 groups of 7 objects each.3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 8 as thenumber of objects in each share when 56 objects and partitioned equally into 8 shares, or asa number of shares when 56 objects are partitioned into equal shares of 8 objects each.3.OA.3 Use multiplication and division within 100 to solve word problems in situationsinvolving equal groups, arrays, and measurement quantities, e.g., by using drawings andequations with a symbol for the unknown number to represent the problem.3.OA.4 Determine the unknown whole number in a multiplication or division equationrelating three whole numbers.3.OA.5 Apply properties or operations as strategies to multiply and divide.3.OA.6 Understand division as an unknown-factor problem.3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationshipbetween multiplication and division (e.g., knowing that 8 x 5 40, one knows 40 5 8) orproperties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers.3.OA.8 Solve two-step word problems using the four operations. Represent these problemsusing equations with a letter standing for the unknown quantity. Assess the reasonablenessof answers using mental computation and estimation strategies including rounding.3.OA.9 Identify arithmetic patterns (including patterns in the addition table ormultiplication table), and explain them using properties of operations.MP.1 Make sense of problems and persevere in solving them.MP.2 Reason abstractly and quantitatively.MP.3 Construct viable arguments and critique the reasoning of others.MP.4 Model using mathematics.20

BIGIDEAS/ENDURING UNDERSTANDINGS:ESSENTIAL QUESTIONS:Student will understand:How to match equations to languagedescriptions.What are the various representations ofmultiplication and division (finding anunknown product, and finding an unknownfactor)?How to describe multiplicative anddivision relationships with area modelsand discrete models.How to connect naked numberequations to contextual word problems.How to convince others that differentmultiplication and divisionrepresentations can be equivalent.SKILLS:CONTENT:The targeted proficiencies; technicalactions and strategies. Starting with anaction verb.The big idea of the unit is to understandthe meaning of multiplication and itsinverse relationship to division. Theunderstanding should include the abilityto translate between differentrepresentations of multiplication anddivision including the language ofmathematics: understanding equationswritten in terms of equal-sized groups,area models, discrete models, and usingcontextual word situations to modelmultiplication and division.Students match multiplication anddivision equations with the language ofmathematics, area models, discretemodels, and contextual word problems.Students apply what they know aboutthe meaning of multiplication anddivision to generate visual, verbal andnumerical representations.Students use understanding ofequivalency to match multiplication anddivision equations to various visualrepresentations and verbal descriptions.Students look at problem situations andtranslate them into numerical and visualrepresentations.Students demonstrate knowledgethrough the expert investigation, theperformance assessment task in theformative assessment lesson and thefinal assessment.Students apply the new knowledge toconstructing arguments, makingconnections between representationsand carefully evaluate the argumentsand justifications of others.21

ASSESSMENT EVIDENCE AND ACTIVITIES:INITIAL ASSESSMENT :The unit begins with the performance task Sponsored Walk. The task is designed to measure whatstudents bring to the unit with regard to their knowledge and skill at working with multiplicationand division. Please reference Sponsored Walk for full details.FORMATIVE ASSESSMENT:About 3/4 of the way through the unit, teachers would use the formative assessment lesson(FAL).The FAL is entitled Interpreting Multiplication and Division. A different pre-assessment taskshould be administered in class at least two days prior to the two-day lesson. Students shouldspend no more than 20 minutes on the task. Teachers should review the student work prior toteaching the lesson. The FAL comes with complete teacher notes and student pages. Pleasereference Interpreting Multiplication and Division for full details.FINAL PERFORMANCE TASK:The final performance assessment is entitled Cookie Dough. It should be administered during aclass period. Most students will complete the task in about 10 – 20 minutes, although time shouldnot be a factor. The teacher should provide a reasonable amount of time for all students to finish.The students should be allowed to use any tools or materials they normally use in their classroom.The task can be read to the students and all accommodations delineated in an IEP should befollowed. Please reference Cookie Dough for full details.LEARNING PLAN & ACTIVITIES:The unit is designed with a pre-assessment task, an expert task/investigation, a formativeassessment lesson and a final assessment. The mat

GRADE 3 MATH: COOKIE DOUGH RUBRIC The rubric section contains a scoring guide and performance level descriptions for the Cookie Dough task. Scoring Guide: The scoring guide is designed specifically to each small performance task.The points highlight each specific piece of student thinking and explanation required of the task and