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PARCC Draft Grade 1 Formative Task Prototype Comparing Numbers

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Overview

Purpose Formative assessment of students’ ability to compare two collections of objects or two numbers and determine whether they are equal and, if they are not, which one is less.

Grade Level(s) Kindergarten to 1st Grade

Task Format • Small group (preferably three students)

• Task will be repeated over three to five consecutive days (for 10–15 minutes per day) until all students have had the opportunity to participate and be assessed.

Materials Needed Kindergarten Counters Quantity cards 0–10 Numeral cards 0–10

1st Grade Base-10 blocks Base-10 number cards Numeral cards 2 pencils to create =, > and < symbols Personal erasable white boards and markers

Prerequisite Concepts and Skills

Counting with one-to-one correspondence.

Distinguishing when a concrete representation of two sets of quantities are the same or different.

Subitizing (instantly recognizing) numbers to 5.

For example, when shown five objects briefly, the student says, “Five.”

For numbers greater than 5, objects are usually in standard arrangements (e.g., the six dots on a die). Kindergarten and 1st grade students learn to subitize numbers greater than 5 by making groups. For example, a student might say, “In my mind, I made two groups of four and one more, which makes nine.”

See kindergarten “Counting and Cardinality” in the Common Core State Standards for Mathematics (K.CC) for these and related skills and concepts.


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Standards Assessed

K.CC.C.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.1

K.CC.C.7 Compare two numbers between 1 and 10 presented as written numerals.

1.NBT.B.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

EXTENSION

2.NBT.A.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits using >, =, and < symbols to record the results of comparisons.

This task belongs to the major work of the K–2 grade band. For more information, see the PARCC Model Content Frameworks for Mathematics.

Standards for Mathematical Practice Embedded in This Task

MP2 Reason abstractly and quantitatively.

Students reason abstractly and quantitatively when they describe, read or write a number sentence that compares two numbers represented in various forms. They also engage in this reasoning when they relate the symbolic numerals to a particular quantity.

MP5 Use appropriate tools strategically.

Students work with tools, including counters and Base-10 blocks, to help them make sense of the quantities and the relationships. They might also make use of the layout of the pictorial representation to help subitize the quantity and to make the comparison between the two quantities.

Preparing for the Task

This task assesses where students are in their ability to compare two collections of objects or two numbers following three modes of representation: concrete, pictorial and abstract. The picture below illustrates this triad relationship among these modes — which are interdependent upon one another. Students do not necessarily pass from one mode to another; instead, it is the relationships between these modes that matters most in developing students’ conceptual understanding of numbers.

The tasks are therefore set up in an order, which builds this conceptual understanding (concrete, pictorial) and attaches it to the abstract number or symbol.

1 Include groups with up to 10 objects.


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Concrete: Students are taught and assessed using hands-on manipulatives to make the abstract idea tangible. For example, a student shows four fingers to represent the number 4.

Pictorial: Students are able to visualize and represent quantities pictorially. For example, a student shows four dots in an array to represent the number 4.

Abstract: Students use written numbers and symbols because they understand their meaning. For example, a student sees (or writes) the numeral 4 and already possesses a mental image of four objects. There is no need for a physical manipulative or picture representation because this student is able to internally visualize four.

For larger numbers like 47, essentially nobody has a mental image or a collection of an exact number of objects, but numerous studies show that a sense for the magnitude of numbers (even for young students) typically involves some number-line-like visualization of rough distance (ahead, behind, left, right) or other placement relative to other numbers.

Grade Materials

Kindergarten Round 1: Have 10 counters available. Use counters that are all the same color.

Round 2: Copy and cut one set of quantity cards on page 20.

Round 3: Copy and cut one set of numeral cards on page 21.

1st Grade Round 1: Use Base-10 blocks. Have at least six 10s and fifteen 1s available.

Round 2: Copy and cut one set of Base-10 number cards found on pages 22–23.

Round 3: Copy and cut one set of numeral cards found on page 24.

Concrete

Pictorial

Abstract


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Implementing the Task

Explain to students that the goal of this task is to compare quantities using counters (or other objects), pictures and/or numbers (depending on grade level and the student’s prior knowledge and understanding).

Grade Level Numbers Used in the Task by Grade Level

Mode of Representation Possible Materials Needed

Kindergarten • Counters 0–10

• Written numerals 0–10

Concrete • Counters • Quantity cards Pictorial

Abstract Numeral cards

1st Grade Two-digit numbers Concrete • Base-10 blocks (used to model two-digit numbers)

• Base-10 block cards Pictorial

Abstract Numeral cards

Invite three students to sit next to one another in a row either on the floor or at a table. They will need a surface for placing the cards and other tools. Students A and B are on the outside. Student C sits in the middle. You sit facing the students. (See illustration below.)

The task consists of three separate rounds. Throughout the three rounds, each student should have a chance to play the role of Student A, Student B and Student C.

Round 1: Concrete

Round 2: Pictorial

Round 3: Abstract

Throughout this document, when specific language is suggested, it is shown in blue text.

Student A Student B Student C

Teacher


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Role of Students A and B: Building Numbers

1. Give the two “outside” students (A and B) numbers to make using counters (kindergarten) or Base-10 blocks (1st grade).

2. Say, “Please show me ____ using counters.” (kindergarten)

Say, “Please build ____ using Base-10 blocks.” (grade 1)

See the section Illuminating the Task for numbers specific to each grade level.

Note: Let students A and B model the number in whatever way spontaneously suits them. Questions to consider:

Does the student start from 1 and count out objects? Count them out by 2s or use some other strategy?

How does the student decompose the number given? For example, when you say, “Can you make 8 using counters,” the student may create two groups of 4. Or the student may make one group of 5 and another of 3.

In building a two-digit number, does the student begin counting 10s followed by 1s? For example, when you say, “Please show me 21 using Base-10 blocks,” the student counts 10 (rod), 20 (rod), 21 (unit).

Role of Student C: Comparing Numbers — Are They Equal?

3. Explain to student C that he or she will decide whether the two quantities — whether shown as collections of counters, quantity cards, Base-10 blocks or numbers — are equal and will show that by making an equal sign with the pencils.

Say, “Are these numbers ____ and ____ equal? Tell my why you think so.” Point to the numbers represented by students A and B. The student should identify the quantities as being equal (or not equal) and provide justification for his or her reasoning.

Student C: If Not, Which Number Is Less?

If the collections (or numbers) are not equal, student C might require further prompting.

4. Say, “How would you compare these numbers?” Give the student wait time to think about this word “compare” and see whether he or she can generate a comparative statement.

5. If the student requires further prompting, you may say, “Which number is less: ____ or ____? Tell me why you think this is true.” Here you can use the words “greater” or “less” interchangeably.

As you casually use this language of greater or less than, you build students’ vocabulary. Throughout this task, encourage students to use comparative language when making mathematical comparisons.

1st Grade Student C: Representing =, > and < with Pencils

Note: In kindergarten, the teacher can write equations and inequalities for the students. The standards do not require kindergarten students to write equations or inequalities. (Of course, kindergarten students may do this, it is just not expected in the kindergarten standards.)


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6. Prompt student C to arrange two pencils to form the proper inequality sign (pointing to the smaller of the two quantities).

7. Say, “We can use these pencils to make a mathematical symbol to make this number sentence true.”

8. Guide this student to place the symbol appropriately between the two collections of fingers or numbers. Encourage all students watching to notice that the wide part of the symbol is next to the larger number and the small part, the point, is next to the smaller number.

1st Grade Student C: Reading (and Writing) the Number Sentence

9. Encourage student C to read aloud the number sentence using the level of language formality with which he or she is most comfortable — informal terms (more/bigger, less/smaller, the same) or technical vocabulary (greater than, less than, equal). For example, the number sentence 7 > 6 can be read correctly as “seven is greater than six” or “six is less than seven.” Words like “bigger” and “smaller” are perfectly adequate in place of “greater” or “less.” The use of these words tells us where on the spectrum the student is in understanding the formal vocabulary.

10. Offer a small whiteboard or piece of paper and ask student C to write a number sentence that matches what has just been acted out. If students have not already learned how, show them. Say, “Please write a number sentence to match what you just said aloud.”

11. If student C is able to write an equation/inequality to match correctly, say to student C, “Is there another way we can read this number sentence?”

Any number sentence of equality or inequality can be read left to right or right to left as long as the correct comparison is made. For example, 5 < 3 + 3 can be read “five is less than three + three” or “three + three is greater than five.”

Next Steps

12. Have students take turns filling each role of student A, B and C.

13. Do the same kind of activity with the quantity cards, handing the outside students each a card and having the middle student C show equality or inequality with the pencils. Again, let students take turns being in the middle.


14. Do the same activity with the written numbers, using numbers appropriate for your grade level.


Note: To shorten the amount of time it takes to administer the task, start at the level (concrete, pictorial or abstract) where students may be able to have success and administer both that level and the next level. If the students do not have success on the first level chosen, administer the previous level for the next round.

1st Grade Students: Ordering Numbers

15. For 1st grade students only, finish this task by having students order three two-digit numbers from least to greatest. All students (A, B and C) will order numbers individually.


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Choose three numeral cards. Shuffle them and place them face up in a mixed order in front of each student. Say, “Please put these numbers in order from least to greatest.”

16. Repeat this three times so that each student takes a turn ordering numbers.

Ordering three numbers is not required in the 1st grade standards per se (see 1.NBT.B.3), but this part of the task helps to embed mathematical practice standard MP.1 in the task (“Make sense of problems and persevere in solving them”). It does so in a way that is well connected to the major work of the grade.

Final Thoughts

In all cases, guide your assessment, just as you would guide your teaching, to suit the students’ needs. If students in either kindergarten or 1st grade find smaller numbers “too easy,” ask if they want to try bigger numbers and offer what they feel brave enough to try. If 1st grade students are not succeeding with small numbers of concrete objects, try to determine whether it is the language or understanding the task that is getting in the way or whether the failure is actually a mathematical misconception.


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Illuminating the Task

Kindergarten

1. To begin Round 1, ask student A to count out five counters. Say, “Please show me the number 5 using counters.”

2. Ask student B to count out three counters. Say, “Please show me the number 3 using counters.”

Next, prompt student C to make sense as to whether the collections are equal. Say, “Are these numbers — 5 and 3 — equal? Tell me why you think so.” Be sure to point to the numbers represented by students A and B.

3. If the collections are not equal, use further prompting. Say, “How can you compare these numbers?” Give the student a moment to think about this word “compare” and see whether he or she can generate a comparative statement.

If the student requires further prompting, you may say, “Which number is greater/less: ____ or ____? Tell me why you think so.”

4. The teacher may summarize by writing the sentence “5 is greater than 3” and/or the sentence “3 is less than 5” (or both).

Note: The symbols for greater than and less than are not required by the kindergarten standards.

a. In cases of equality, the teacher may write equations such as “5 = 5.” It can be helpful for students to see the equals sign used in a context other than computation results such as 3 + 2 = 5.)

5. Repeat the task with other students rotating to fill the roles of students A, B and C to complete Round 1.

6. Follow with Round 2 using quantity cards. Each student will have the opportunity to rotate as student A, B and C.

7. End with Round 3 using numeral cards.


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Kindergarten

Student A Student B Student C or Teacher

Round 1: Concrete Use counters

5 3 “5 is more than 3” “3 is less than 5”


4 4 “4 is equal to (the same as) 4” and/or “4 = 4”


Round 2: Pictorial Use quantity cards


9 9 “9 is equal to (the same as) 9” and/or “9 = 9”



Round 3: Abstract Use numeral cards

10 10 “10 is equal to (same as) 10” and/or “10 = 10”




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Illuminating the Task

1st Grade

1. To begin Round 1 (concrete), ask student A to represent 27 using Base-10 blocks. Say, “Please build 27 using Base-10 blocks.”

2. Ask student B to represent 37 using Base-10 block. Say, “Please build 37 using Base-10 blocks.”

Next, prompt student C to make sense as to whether the collections are equal. Say, “Are these numbers — 27 and 37 — equal? Tell me why you think so.” Be sure to point to the numbers represented by students A and B.

3. If the collections are not equal, use further prompting. Say, “How can you compare these numbers?” Give the student a moment to think about this word “compare” and see whether he or she can generate a comparative statement.

If the student requires further prompting, you may say, “Which number is greater/less: ____ or ____? Tell me why you think this is true.”

4. Have student C arrange two pencils to form the greater than or less than symbol and position it appropriately between the numbers. Encourage all students to notice that the wide part of the symbol is next to the larger number and the closed part is next to the lesser number.

5. Say to student C, “Can you read this number sentence aloud?”

Here, student C may use the formal mathematical language (greater than, less than, equal) when reading the expression, or the student may use more informal terms (more/bigger, less/smaller, the same). These data will be important to note as they provides information about the student’s understanding of the language.

One sample response could be, “37 is more than 27.”

6. Next, prompt student C to write a number sentence to match. Say, “Please write a number sentence to match what you just said aloud.” Ask student C to record his or her number sentence in written form (37 > 27).

7. Ask student C, “Is there another way that you could read this number sentence?” Here, you are looking to see if the student understands that the converse is also true: 27 < 37.

8. Repeat the task with other students rotating to fill the roles of students A, B and C to complete Round 1.

9. Follow with Round 2 using Base-10 number cards. Each student will have the opportunity to rotate as student A, B and C.

10. End with Round 3 using numeral cards.

11. Finish this task by having students order three two-digit numbers from least to greatest. All students (A, B and C) will order numbers individually.

Choose three numeral cards. Shuffle them and place them face up in a mixed order, in front of each student. Ask each student to put his or her numbers in order from least to greatest. Say, “Please put these numbers in order from least to greatest.”


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12. Repeat this three times so that all students take a turn ordering numbers.


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1st Grade


Round 1: Concrete Use Base-10 blocks

27 37 “37 is greater than 27” “27 is less than 37”

20 20 “20 is equal to 20” and/or “20 = 20”



Round 2: Pictorial Use Base-10 cards





Round 3: Abstract Use numeral cards





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Scoring Rubric

Comparing Numbers: Kindergarten

KINDERGARTEN BEGINNING DEVELOPING SECURE EXTENDING

Uses counting strategies

Student counts starting at 1 to create the number provided. Number(s) may also be shown with counters incorrectly.

Student demonstrates partial understanding of using one particular counting strategy:

matching counting on

or may use this strategy inconsistently. Number(s) might also be shown with counters incorrectly.

Check the appropriate box to indicate the specific counting strategy used.

Student demonstrates concrete understanding of using one particular counting strategy:


and uses the preferred strategy consistently to accurately arrive at the given number(s).

Check the appropriate box to indicate the specific counting strategy used.

Student demonstrates concrete understanding of using multiple counting strategies, including


and uses the preferred strategy consistently to accurately create the given number(s).

Compares numbers using words and symbols

Student requires teacher support to make sense of the relationship.

Student is able to make sense of the relationship inconsistently.

For example, student means to say, “5 is greater than 3,” but actually says “3 is greater than 5.”

Student is able to make sense of the relationship using words consistently.

When prompted to compare the numbers in another way, student is able to provide the inverse statement.

For example, understanding the statement “5 is greater than 3,” student can conclude quickly (without


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KINDERGARTEN BEGINNING DEVELOPING SECURE EXTENDING

a new investigation) that “3 is less than 5.”

Uses appropriate vocabulary

Student shows little evidence of the concept of comparison. Student does not use words such as “more,” “less,” “greater,” “fewer,” “same” or “equal.”

Student uses informal language to describe comparison, including “more,” “less” or “same.”

Or student attempts to use more formal language, but misuses it in context.

For example, student says “5 is less than 3.” Here the student knows to compare and tries to use the formal language, but the number sentence does not make sense.

Student correctly uses terms “greater than,” “less than” or “equal to” to describe the number sentence.

For example, student says, “5 is greater than 3.”

When prompted to compare the numbers in another way, student is able to provide the inverse statement using the appropriate language.

For example, student understands that the statement “5 is greater than 3” also means “3 is less than 5.”

Records number relationships using symbols

Fill out for “high-ceiling” kindergarten students when appropriate.

Student understands that symbols are used to represent quantities and relationships but does not demonstrate a sense of how to use them.

Student understands that symbols are used to represent quantities and relationships and attempts to use them. Student may use symbols (=, > or <) incorrectly.

Student can correctly record the relationship using symbols.

Student can correctly record the relationship using symbols in more than one way.

For example, student is able to write 5 > 3 and mean 3 < 5.


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Scoring Rubric

Comparing Numbers: 1st Grade

1ST GRADE BEGINNING DEVELOPING SECURE EXTENDING

Compares two-digit numbers using words and symbols

Student requires teacher support to make sense of the relationship. Requires teacher support to use the symbols (=, > or <).

Student is able to make sense of the relationship using words only; student misuses symbols.

For example, student says, “63 is greater than 57,” but then misuses the symbols.

Student is able to make sense of the relationship using words and symbols (=, > or <).

For example, student says, “83 is greater than 57,” and uses the > symbol appropriately.

When prompted to compare the numbers in another way, the student is able to provide the inverse statement.

For example, student understands that the statement 43 > 27 also means 27 < 43.


Student uses informal language to describe comparison, including “more,” “less” or “same.”

Or student attempts to use more formal language but misuses it in context.

For example, student says, “5 is less than 3.” Here the student knows to compare and tries to use the formal language, but the number sentence is not true.

Student inconsistently uses formal terms “greater than,” “less than” or “equal” to describe the relationship.

Student consistently and correctly uses formal terms “greater than,” “less than” or “equal” to describe the relationship.

For example, student may say, “52 is greater than 37.”

Student consistently and correctly uses formal terms “greater than,” “less than” or “equal” to describe the relationship.


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1ST GRADE BEGINNING DEVELOPING SECURE EXTENDING


Student understands that symbols are used to represent quantities and relationships but does not demonstrate an ability to use these symbols.

Student understands that symbols are used to represent quantities and relationships and attempts to use them. Student may use symbols (=, > or <) incorrectly or inconsistently.

Student can correctly record the relationship using symbols.

Student can correctly record the relation using symbols in more than one way.

For example, student is able to write 35 > 31 and mean 31 < 35.

Orders numbers Student orders three numbers incorrectly from least to greatest.

Student orders at least two numbers correctly from least to greatest, but the other number is placed in the incorrect order.

Student orders all three numbers correctly from least to greatest.

Student orders three numbers correctly from least to greatest. Student might show evidence of only using the 10s digit (when possible) in this process.


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Recording Sheet

Comparing Numbers Rubric: Kindergarten

STUDENT NAME:

Beginning Developing Secure Extending




Records number relationships using symbols (high ceiling only)

Notes:

STUDENT NAME:






Notes:


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STUDENT NAME:






Notes:

STUDENT NAME:






Notes:


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Recording Sheet

Comparing Numbers Rubric: 1st Grade

STUDENT NAME:





Orders numbers

Notes:

STUDENT NAME:





Orders numbers

Notes:


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STUDENT NAME:





Orders numbers

Notes:

STUDENT NAME:





Orders numbers

Notes:


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Kindergarten: Round 3 — Numeral Cards

4 7

8 9

10 10


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1st Grade: Round 2 — Base-10 Number Cards


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27

1st Grade: Round 2 — Base-10 Number Cards


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1st Grade: Round 3 — Numeral Cards

63 36

59 60

18 81

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