which shape? - noycefdn.org · 3. i have 4 sides. my opposite sides ... the level of detail needed...

Copyright © 2007 by Mathematics Assessment 74 Which Shape? Test 3Resource Service. All rights reserved.

Which Shape?

This problem gives you the chance to:• identify and describe shapes• use clues to solve riddles

Use shapes A, B, or C to solve the riddles.

1. I have 4 sides.

My opposite sides are equal.

I have 4 right angles.

Which shape am I? _____________________________

2. I have 4 sides.

I have only 1 pair of parallel sides.

Which shape am I? _____________________________

3. I have 4 sides.

My opposite sides are parallel.

I do not have any right angles.

a. Which shape am I? _____________________________

b. Draw lines that divide my shape into a rectangle and two right triangles.

A B C

Copyright © 2007 by Mathematics Assessment 75 Which Shape? Test 3Resource Service. All rights reserved.

4. Write three different clues for shape D.

Clue 1:

_________________________________________________________________

Clue 2:

_________________________________________________________________

Clue 3:

_________________________________________________________________

5. Look closely at shapes E and F.

Write a statement that tells how they are alike.

_________________________________________________________________

Write a statement that tells how they are different.

_________________________________________________________________

D

E F

9

3rd

grade - 2007Copyright © 2007 by Noyce Foundation 76Resource Service. All rights reserved.

Task 4: Which Shape? Rubric

The core elements of performance required by this task are:• identify and describe shapes• use clues to solve riddles

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

pointssectionpoints

1. Gives correct answer: C or rectangle 11

2. Gives correct answer: B or trapezoid 11

3. Gives correct answer: A or parallelogram

Draws lines that are correct.

1

1ft 2

4. Gives three correct statements such as:

I have 4 sides.I have 4 right angles.All my sides are equal.

3x1

3

5. Gives two correct statements such as:

They both have 4 sides.

Shape F has a line of symmetry, and E does not.

Shape F has four equal sides, and shape E does not.

2x1

2

Total Points 9

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Which Shape?Work the task. What are some of the attributes that you might expect students to use todescribe the shapes in parts 4 and 5? What are some of the attributes or vocabulary thatyou think will give them difficulty?

Mathematical literacy requires attention to details and constraints that is different fromthe level of detail needed to make sense of a story. Look at the first part of the task,identifying shapes from clues.In part one, approximately 11% of the students saw the 4 sides and picked shape A. Howmany of your students made this choice?In part two, students saw the parallel sides, but ignored the one pair. 23% picked shape Aand 11% picked shape C. How did your students do?30% of the students saw the no right angles, the last clue read, and picked shape B forpart 3.30% of the students did not attempt to draw lines for part 3b. What types of errors do yousee in the student drawings? Which attributes confused students? (It might be helpful todraw a few examples, so that you can categorize mistakes or think how to set up adiscussion about the errors with the class.)

Look at the clues students used for part 4. List them into categories of correct orincorrect, incomplete. Students often think of mathematical ideas not anticipated by therubric, so add those to the list as you find them.Correct Attributes Incorrect Attributes4 sidesEqual sides4 anglesRight anglesParallel sidesLine of symmetry

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How does this help you think about further instruction? What are students confusedabout? What are their limitations with vocabulary? How can you design activities wherestudents have a reason to use and to practice vocabulary in a meaningful way rather thana memorization of definitions?

Developing the ability to see relevant attributes is related to opportunity and practice. Inreading development, students are often given opportunities to compare and contrast. Dostudents get these same opportunities to be critical thinkers in mathematics?

What are the geometrical attributes that we want students to notice about the shape? Canthey see number of sides, size of sides, number and size of angles? Are they thinkingabout properties like parallel lines and symmetry? How do students think about theconcave or reflex angle in shape E? What terms do they use to describe these ideas? Arethey talking about points or corners instead of angles? Are some of your studentsthinking about non-geometrical properties, such as it points to the right or it looks like anarrow?List some of the incorrect choices that students used to describe the differences.

What are the implications for instruction? Do students have enough opportunities to sortor categorize shapes for themselves? Do they get to make their own labels when sorting?What other activities help students to develop this attention to attributes?

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Looking at Student Work on Which Shape?Student A is able to use the clues to identify the shapes being described and names theshapes. The student draws lines on shape A to meet the constraints or demands of thetask. The student uses geometrical vocabulary to describe the attributes in parts 4 and 5.Student A

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Student A, part 2

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Recognizing the Seeds of Mathematical IdeasStudents sometimes come up with mathematical ideas not anticipated by the rubric orusing imprecise language. How do we, as teachers, develop the flexibility to seemathematical thinking in student work? Look at the work of Student B. Notice that onescorer has given the student credit and one has not. Has the student noticed importantcharacteristics about the shapes?Student B

Look at the work of Student C. When taken alone, 2 right angles is not enough todescribe shape D; neither is 4 sides or 4 equal sides. Clues need to be considered as a set.Can you make a shape with 4 sides and 2 right angles that is not a square? Can youmake a shape with 4 equal sides and 2 right angles that is not a square?Student C

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Student D has the seeds for the mathematical idea of rotational symmetry, but doesn’thave the mathematical language to name the property. How do we encourage students tothink about important ideas, which might not be introduced formally until much latergrades? How do we keep them looking for and discovering relationships?Student D

What’s wrong with this picture?Students had difficulty drawing in the lines to make a rectangle and two right triangles.For each piece of student work, try to identify the attributes that they overlooked orincomplete definitions about a shape.

Student E makes two triangles and a four-sided shape. What is the student notunderstanding or focusing on? Now look at part 4. The student comes up with formathematical ideas about the square, but then uses none of those ideas in thinking aboutshapes E and F. What might be a next instructional move for this child?Student E

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Shape E, part 3

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Student F has trouble with the drawing in part 3. What is the student forgetting?Examine the student’s description of differences. The student is probably noticing thereflex angle in E, but does not know how to talk about it and leaves the descriptionvague. We sometimes want students to have precise language. When encountering a newidea, students need to rely on everyday language and some mathematical language usedin new ways. What are some ways that we might hope third graders would describe thisangle? What are the attributes from part 4 that might help the student in thinking aboutthe differences between the two shapes? What could the student build on?Student F

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What are each of these students understanding and not understanding?Student G

Student H

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Do you think Student I is only asked to draw lines when looking for symmetry?Student I

What are the important attributes of shape that we want students to payattention to? How do we help them develop their sense of what to payattention to?Look at the work of Student J. Does the student only think about angles as acute shapes?Notice that the student in part 5 is thinking about the shape as a totality, rather thancomparing parts. How is this student thinking about the reflex angle?Student J

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Thinking about the angle in E is difficult for students. While we don’t want students tomemorize formal language about the angle, students should have opportunities toencounter and think about these types of big mathematical ideas and try to describe them.What questions might you want to ask Student K to probe his thinking?

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Student K

Look at the descriptions by Student L. Would you have given them credit? Why are whynot? Are the attributes accurate? Relevant?Student L

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Some students struggled with just counting the sides and angles for the two shapes. Seesome typical responses by Students M and N.Student M

Student N

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Finally some students, like Student O, are still operating at the lowest van Hiele level.They are thinking of what the shape looks like and how it can be subdivided.Student O

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3rd Grade Task 4 Which Shape?

Student Task Identify and describe shapes, using key attributes. Use clues to solveriddles. Identify how shapes are alike and different using geometricattributes.

Core Idea 4GeometryandMeasurement

Recognize and use characteristics, properties, andrelationships of two-dimensional geometric shapes.

• Identify and compare attributes of two-dimensional shapes anddevelop vocabulary to describe the attributes.

• Use visualization, spatial reasoning, and geometric modeling tosolve problems.

• Recognize geometric ideas and relationships and apply them toproblems.

Mathematics in this task:• Understanding geometric attributes, such as angle size, number of angles, number

of sides, parallel sides.• Sorting and describing shapes using attributes.• Comparing and contrasting shapes on geometric attributes.• Composing and decomposing shapes.

Based on teacher observations, this is what third graders know and are able to do:• Identify shapes by their attributes• Describe attributes of a square, particularly 4 sides and 4 angles

Areas of difficulty for third graders:• Drawing in lines to decompose a shape into a rectangle and two triangles• Describing similarities and differences for E and F• Correct vocabulary

o Students used point or corner for angleo Closed shape, open shapeo How to classify angleso They confused side and face and side and edgeo Parallel sides

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The maximum score available on this task is 9 points.The minimum score needed for a level 3 response, meeting standards, is 4 points.

Most students, 88%, could give 3 clues for describing a rectangle. 77% could alsoidentify the shape with 4 right angles and opposite sides equal. Almost half the students,47%, could identify all 3 shapes and give 3 clues for describing a rectangle. 30% couldalso draw in lines for subdividing a shape into a rectangle and two right triangles. 5%could meet all the demands of the task including comparing and contrasting shapes E andF, using geometric attributes. About 2% of the students scored no points on the task. Halfof these students attempted the task.

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Which Shape?

Points Understandings Misunderstandings0 Half the students with this score

attempted the task.They had difficulty describing attributes fora square. They gave information, such as:“no angles or no right angle, “looks like abox or piece of paper or a rectangle”, or“its pointy”.

3 Students could give 3 cluesabout a square.

They had difficulty matching a set of cluesto a shape. For part 1 (4 sides, 4 rightangles) 11% of the students picked theparallelogram.

4 Students could identify therectangle from clues and give 3clues for a square.

For part 2 (4 sides, only 1 pair of parallelsides) 23% picked the parallelogram and11% picked the rectangle. For part 3 (4sides, no rightAngles, opposite sides parallel) 30%picked the trapezoid.

6 Students could identifyrectangle, parallelogram, andtrapezoid from their attributesand give attributes for a square.

30% of all students did not draw lines tosubdivide a shape into a rectangle and 2right angles. Students did not usually thinkcarefully about angle size.

7 Students could identify shapesfrom attributes and describeattributes of shapes. Studentscould decompose a shape intorectangle and two righttriangles.

Students had difficulty describing thesimilarities and differences between twoshapes. 31% described the shapes as sharpor pointy.16% just noted that the shapeswere different. 16% thought that E hadonly 3 angles or 3 sides. 7% tried todescribe the concave angle with words likeindent or goes in and goes out.

9 Students could identify shapesfrom attributes and describeattributes of a square. Theycould decompose shapes intocomponent parts by attributes.They could also compare andcontrast shapes based ongeometric attributes.

Implications for InstructionStudents should be able to think about the geometric attributes of shapes: number ofsides, number of angles, types of angles, length of sides, and parallel sides. Studentsshould be able to both identify a shape from its attributes and describe the attributes of ashape.

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Students need experiences with a rich variety of 2- and 3- dimensional shapes. Studentsshould be pressed to describe how shapes are the same or different. This helps thembecome of aware of a variety of attributes and lays the foundation for understandingformal definitions at a later time.Ideas for Action Research – The Logic of Classifying and SortingLook at the lesson below (from Teaching Student-Centered Mathematics, Grades K-3 byJohn Van de Walle). How might this lesson be modified to help students think about 3-dimensional shapes? Can you design your own set of shapes? Make sure you thinkcarefully about the question posed by the Stop Sign.

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A Question of Numbers

This problem gives you the chance to:• show you can compare and order numbers

Holly’s class is learning about big numbers.

1. Here is part of the class number line.

650 660

Put an X on the number line, on the place that is halfway between 650 and 660.

What is the number that should be there? _______________

2. Holly knows a pony weighs between 365 pounds and 425 pounds.

Write a possible number for the weight of a pony. ______________ pounds

3. The activity center swimming pool holds between 1,875 gallons and 1,940 gallons

of water.

Write a possible number for the amount of water

the swimming pool holds. ______________ gallons

4. A school computer could cost between $2,950 and $3,055.

Give three possible prices for the computer.

$ _______________ $ _______________ $ _______________

Copyright © 2007 by Mathematics Assessment 96 A Question of Numbers Test 3Resource Service. All rights reserved.

5. The American Revolution started in 1775 and finished in 1783.

In which year was it halfway through? _________________

Show how you know using this number line.

6. Holly’s teacher says the school library has 1000 books.Holly thinks that the library may have 40 more books.

What is the greatest number of books that the library could have? __________

Show how you figured this out.

Tom thinks that the library may have 40 fewer books than 1000.What is the smallest number of books that the library could have? ____________

Show how you figured this out.


1770 1780 1790

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Task 5: A Question of Numbers Rubric•

• The core elements of performance required by this task are:• • show you can compare and order numbers•Based on these, credit for specific aspects of performance should be assigned as follows

pointssectionpoints

1. Gives correct answer: 655

Draws an X in the correct place on the number line.

1

1 2

2. Gives correct answer such as: 380 pounds. 11

3 Gives correct answer: such as 1,920 gallons. 1

1

4. Gives correct answers such as: $2955 $3000 $3053 11

5 Gives correct answer: 1779 and

Marks the correct year on the number line. 11

6. Gives correct answer: 1040 and

Shows 1000 + 40

Gives correct answer: 960 and

Shows 1000 - 40

1

1 2Total Points 8

A Question of NumbersWork the task and look at the rubric. What do are the big mathematical ideas being assessed by thistask?

In part 1, students were asked to use a number line to find a number between 650 and 660. Howmany of your students put:655 markon # line

655 no markon # line

655.5 656 651 1310 Other

Think about each of these mistakes. How are they showing different misunderstandings aboutnumber and place value?To work parts 2,3,and 4 students had to think about several things. What does between mean?Should I do a computation? How do I make a number larger or smaller? In looking at place value,Liping Ma and Kathy Richards talk about numbers going around ten, numbers going aroundhundreds, and numbers going around a thousand. Look carefully at how your students thought aboutthese ideas. How many of your students:Correct Answers Misunderstood

“between”Put in the end

values

Added orsubtracted the two

numbers

Values for part 4were too high or

too low

Other

Now look at work for part 5. Have your students worked with number lines? If they haven’tworked with number lines, what other strategies could they use to make sense of the problem?

Look at student work.1179 1779, but no

use of numberline

1778 1780 1775/1790 Other

Did students have trouble drawing the scale?Did students have trouble counting marks instead of spaces?What other misconceptions did you see?

What are some ways that you could incorporate use of the number line in warm ups? Problem-solving? Number talks? How does the number line help to develop a deeper understanding of placevalue? Of number sense? Of how the base-ten system is built?

Finally look at work for part 6. Did students know to add and subtract? Did they pick 1000 formost and 40 for least? When subtracting what types of place-value understanding errors did yousee?

3rd

grade - 2007

Copyright © 2007 by Noyce Foundation 99Resource Service. All rights reserved.

Looking at Student Work on A Question of NumbersStudent A is able to use the number line to find numbers within a given range or to divide a rangehalfway. The student understands the mathematical use of the word between and uses landmark orfriendly numbers to name values between. Notice how Student A puts all the values on the numberline, identifies the section being considered, then labels the midpoint.Student A

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Student A, part 2

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Student B uses the number line as a tool. See the marking off of equal distances to locate themidpoint in part 1. The student sees the number line as a series of numbers between two end points,but has not quite mastered how to scale in all the values or measure in equal units. (See part 5. Thestudent is trying to count backwards to the middle, but makes a stumble. The student has a veryclear, clean explanation for finding the most and least books.Student B

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Student B, part 2

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This idea of measuring off equal units or scaling the number line was difficult for students. NoticeStudent C attempted the idea and then simplified the number line to just the relevant numbers.Student D was able to think about the locations for starting and ending numbers and then put in theappropriate midpoint without counting every number. Student E combines measuring equal units,with marking off the starting, ending and midpoints.Student C

Student D

Student E

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Student F may or may not have had experience with a number line, but is able to use other strategiesto find the midpoint. Then Student F is able to estimate the location of that value on the numberline. Notice the full explanation for finding the values in part 6.Student F

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Student G tries to use subtraction to find the midpoint. But does not use that information to find themidpoint. How could the “8” be used to find the midpoint? What would be the next step(s)? Noticehow the student uses place value units to find the total number of books in the library.Student G

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Student H is able to measure the scale and locate the endpoints for the war, but doesn’t count off thespaces to locate an exact midpoint. Notice that student H seems to understand the concept of ± 40,but has further work of finding values within the range of possibility.Student H

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Some students are having difficulty applying ideas about place value to computation. Student I has4 points on page 1 of the task, but lines up thousands and tens when adding and subtracting. StudentJ puts the 4 tens in the hundreds column. Student K and L have trouble with place value which leadsto procedural errors in using the standard algorithm. Notice Student K also has trouble thinkingabout what comes after 1778.Student I

Student J

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Student K

Student L

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Understanding mathematical terminology or academic language is often different from commonword usage. Student M relies too heavily on finding and circling key words, rather than teasing outwhat is being asked in the question. In this particular problem, “and” does not mean the combiningof two values, but is identifying two points that are similar because that represent endpoints.Student M

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Student N has trouble distinguishing between the number with the greatest absolute value andadding more books to obtain the greatest possible value. This example shows the subtleties requiredfor reading academic material.Student N

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3rd Grade Task 5 A Question of Numbers

Student Task Compare and order numbers using values and number lines. Make areasonable argument to find the midpoint of a range of numbers.

Core Idea 1NumberProperties

Understand numbers, ways of representing numbers, relationshipsamong numbers and number systems.

• Develop understanding of the relative magnitude of wholenumbers and the concepts of sequence, quantity, and the relativepositions of numbers.

Mathematics in this task:• Ability to measure in equal size units on a number line• Ability to count around tens, hundreds, and thousands• Ability to find a numerical midpoint and to find and compare numbers within a range• Understand most and least in the mathematical sense of ± 40.

Based on teacher observations, this is what third graders knew and were able to do:• Find and name the middle number on a number line (part 1) when the scale marks are drawn

in• Find the most books that could be in a library• Find points between two numbers

Areas of difficulty for students:• Subtracting from 1000• Finding a midpoint between two numbers on a number line, when the intervals are not

marked and the endpoints for the task are different from the endpoints on a number line• Marking off equal intervals, dividing something into ten parts• Understanding that between does not include the endpoints

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The maximum score available for this task is 8 points.The minimum score needed for a level 3 response, meeting standard, is 4 points.

Most students, about 87%, could find and name the midpoint on a scaled number line. Manystudents, about 71%, could locate the midpoint on a number line, and find values between twoendpoints (parts 2 and 3). More than half, 58%, could also find three values between two endpoints.26% could also find solutions for thinking about the most and least books in the library. 8% couldmeet all the demands of the task including scaling their own number line to find a midpoint wherethe endpoints were not the ends of the number line. 7% of the students scored no points on this task.Only 38% of those students attempted the task.

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A Question of Numbers

Points Understandings Misunderstandings0 38% of the students attempted

the task.Time may have been an issue for many students.Some students struggled in trying to find the midpointbetween 650 and 660 on the number line. 7.5% of thestudents marked 656. Some students thought thenumber was 655.5. Students may have been countinglines instead of spaces to count of the distance.

2 Students could mark and namethe midpoint between 650 and660.

Students had difficulty interpreting between in parts 2and 3. 7.5% gave endpoints instead. 3% added theendpoints.

4 Students find a midpoint on anumber line and give valuesbetween two endpoints. (parts 2and 3)

Students had difficulty naming three values betweentwo endpoints. Almost 18% gave at least one valuethat was above the highest endpoint. 15% gave valuesthat were too low. 5% included one or both endpoints.

6 Students could find midpoints,locate values in between twoendpoints (parts 2,3, and 5), andfind the most books that could belocated in the library.

About 7% gave the correct answer for most, butshowed no calculations or explanation. 7% thoughtthe answer for most was 1000 because it was thegreatest number in the problem. Some students haddifficulty with place value, getting answers like 5000or 1400. Students had even more difficulty with thesubtraction (1000-40). 4% had the correct answer butno work. 6% thought the answer was 40. 6% hadplace value errors getting either 60 or 600. Others hadplace value errors like 3000, 996, or 9504.

7 Students could find midpoints,locate values in between twoendpoints, and find the most andthe least books that could belocated in the library.

Students had difficulty using a number line to find amidpoint when there was no scale. They could figureout how to divide the sections into 10 equal parts andmay have only put 2 or intervals. 21% picked 1780,the middle of the given number line but not the middleof the war(1775 to 1783). 7.5% knew the midpointwas 1779, but used the number line incorrectly or didnot use the number line. 5% thought the midpoint was1778. 5% thought the midpoint was 1775 (midpointfor left half of the number line). 4% picked one of theendpoints on the given number line.

8 Students could find midpoints,locate values in between twoendpoints, and find the most andthe least books that could belocated in the library. Studentscould also mark in intervals on anumber line and use it to find themidpoint between two numbers,which were not the endpoints onthe number line.

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Implications for InstructionStudents need to understand place value through the thousands and be able to use place value to findnumbers falling between two values. Students should be able to use mathematical models, like anumber line, to help them think about the size and order of numbers and to help them solve simpleproblems. A good article that talks about developing number line through measurement is in theNCTM Yearbook, Developing Mathematical Reasoning in Grades K-12: “Mathematical Reasoningwithin the Context of Measurement”

Students often have trouble counting beyond a certain number:28,29,30, 40,50, 60 . . .98,99,100, 105,110,120 . . . .

Students need activities like what number is one more than 399 or what number is one less than 590.Students should also be thinking about what is 10 more or 10 less. This helps them to developlarger ideas about place value and how the number system works.Students need to understand the logic behind procedures. Put up some examples (without studentnames) of the place value errors. (See Student I and J) Ask students if these methods are correct.Try to have them develop a convincing argument for their opinions.Ideas For Action Research:Try to develop students use of number line to think about quantity of numbers by doing number linetalks like the ones on the next page. Also think about using number line to record student strategiesfor adding and subtracting numbers during number talks. See the examples below for:Maria has 87 cookies in a box. How many will be left if she eats 18 of them?

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Activities from website for San Diego School District: Mathematics Dept. Third grade

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Reflecting on the Results for Third Grade as a Whole

Think about student - Work through the collection of tasks and the implications for instruction.What are some of the big misconceptions or difficulties that really hit home for you?

If you were to describe one or two big ideas to take away and use for the planning for next year whatwould they be?

What were some of the qualities that you saw in good work or strategies used by good students thatyou would like to help other students develop?

Three areas that stood out for the Collaborative as a whole for third grade mathematics were:1. Understanding Scale – Students had difficulty working with a “unit” or equal size group. In

Square Patterns we wanted students to notice that there were four arms so white squaresgrew in groups of 4. In Parking Cars, students could often read a value off a graph with ascale or intervals of 10, but couldn’t use scale to do comparison subtraction or add data to thegraph. They also had difficulty with thinking about what the value of a bar between the gridlines represented. The same is true in A Question of Numbers. Students had difficulty withunderstanding between. It is not clear if this is a counting issue, place value issue, orvocabulary issue. Teachers should be encouraged to investigate these ideas in theirclassrooms. Students had difficulty subdividing the number line into equal size groups orunits.

2. Place Value – Place value issues arose in student work for Adding Numbers. Students didnot know where to put numbers when writing their own addition and subtraction problems.The process of writing your own problem to solve is different than working a problem that isalready set up on a worksheet. Similar problems arose in a Question of Numbers whenstudents had to write their own problems for adding and subtracting 40 from 1000. Studentsalso had difficulty finding values that are between. Some students only considered theleading digit when comparing values.

3. Attributes - In Square Patterns students were to look at attributes of a growing pattern andtry to describe them. Students who could do this well were able to make generalizations thathelped them solve other parts of the task. In What Shape? students struggled with findingattributes of shape: They may have not used all the attributes to identify a shape. They mayhave trouble giving enough attributes to define a square. Students had difficulty describingsimilarities and differences in the final two shapes.

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which shape? - noycefdn.org · 3. i have 4 sides. my opposite sides ... the level of detail needed...

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