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Grade 1 Mathematics Performance Task
UNKNOWN NUMBERS
Overview
At a Glance This instructional task helps students develop their ability to determine an unknown number in an addition or subtraction equation relating three whole numbers and gives teachers information to help guide instruction.
Grade Level Grade 1
Task Format Partner (2 students); modeled whole class (optional) Played over a series of 3-5 days
Materials Needed For each student 1 colored pencil or crayon (different colors) 1 pencil (or dry erase marker; see game board below) For each pair of students 1 game board (either A or B) (You may choose to laminate each game board so it becomes a dry-erase surface. In substitute of a pencil, students must use a dry-erase marker) numeral cards (7– 20 ; template provided; include 21–30 for Extension) 1 numeral cube (2 cubes for Extension) Student Sentence Frames (template provided), cut to one half-sheet to be available for each pair of students (optional) 20 counters, 1 number line (optional)*, available to pairs of students upon request or teacher decision For the teacher Observation Checklist
*Although the formal use of number lines do not appear in the CCSS until grade 2, number lines may be used in this task as a tool for grade-1 students.
Prerequisite Concepts/Skills
Recognizing and describing parts of quantities to 10 in a variety of configurations Familiarity with the concept of addition as “adding to” Familiarity with the concept of subtraction as “taking apart” or “taking from” Representing equations with symbolic notation: addition and subtraction as (+, – ) and equality as (=)
Content Standards Addressed in This Task
1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 +? = 11, 5 =? – 3, 6 + 6 = ?.
Extensions and Elaborations
2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawing and equations with a symbol for the unknown number to represent the problem.
Standards for Mathematical Practice Embedded in This Task
MP 3 Construct viable arguments and critique the reasoning of others.
MP 6 Attend to precision.
MP 7 Look for and make use of structure_._
GET READY: Familiarize Yourself with the Mathematics
Solving for an Unknown Number and the Meaning of Equality
The purpose of this task is to give students practice working with addition and subtraction equations and to give you insight into students’ understanding of the relationship among three numbers in these equations (1.OA.D.8), and their understanding of equality and the meaning of the equal sign (1.OA.D.7).
An “=” sign placed between two expressions indicates that they are intended to have the same numerical value. It is therefore just as correct to write ____ = _____ + _____ as to write ___ + ___ = ____. It is important that students understand this and not treat the “=” sign as an indicator that “an answer” must come next. Among other things, that means that when they see 3 + 4 = ____ + 5, they do not think “three plus four equals” and then immediately write “7.” The equals sign means that the blank must be filled in so that 3 + 4 is equal to ____ + 5. Understanding the meaning of the “=” symbol allows even _____ = _____ + _____ – _____ to make sense. What is critical is the notion of equality. The image of balance is sometimes useful. For example, 3 + 2 = 5 and 5 = 3 + 2 might be thought of as just switching which side of the scale these two equal quantities are on.
In the equation x + 5 = 8, the x represents an unknown number. However, the overall concept remains the same. If the scale is balanced, the quantities on either side must be equal. We need three more blocks to make the two sides equal, so the unknown number is 3. To accurately solve for unknown numbers, students must fully understand the notion of equality and also see the relationships among numbers.
One misuse of the equal sign, common even among well-educated adults, is illustrated below. To calculate 4 + 7 + 3 - 2, the writer wrote:
4 + 7 = 11 + 3 = 14 – 2 = 12
In this case, the writer used the equal sign to show the value of the expression on the left, but then continued the calculation without careful attention to the meaning of that equal sign. This is exactly the
unknown number. As a result, they must look for the structure needed to achieve “balance” or “equality” in each equation.
GET SET: Prepare for Introducing the Task
- Gather the materials listed on page 1.
- There are two game boards (A and B) for students to use. It is recommended that your students all begin with Game Board A. Game Board A includes four equation forms, all of which will always generate a solution that will result in a positive number.
- Cut one set of number cards (7–20) for each pair of students. Copy one game board per pair of students. Be sure to have 20 counters on hand per pair of students in case they are needed.
- Have students sit next to each other at a table. Sit close to the students so you can carefully observe their game. While the students are engaged in this activity, you should have the opportunity to observe several pairs of students.
- To begin, place the pile of number cards face-down on a table or hard surface. Each student should have a colored pencil or crayon, a standard pencil, and access to counters, if needed.
Introducing the Task
The task is game-like to maintain students’ curiosity and attention_._ If you choose, you may introduce the game to the class by having two students up front modeling. You may also model the game directly to a small group. Throughout this document, when specific language is suggested, it is shown in italics.
- Today we’ll play a game called “Unknown Numbers.” You will share a game board and use one set of cards and one numeral cube. Each of you will need a different colored pencil or crayon and a regular pencil. Let’s play!
- To Player 1: Choose a blank equation from your game board. Now turn over a number card and write that number in the big box for your equation with your crayon/colored pencil.
Player 1 writes the number from the card selected in the card-space box provided.
- To Player 2: Roll the number cube and write the number in one of the blank spaces in the same equation. Use your crayon/colored pencil.
- To Player 1: Now, figure out the number that goes in the other blank space that makes the equation true. Write it in pencil.
- To Player 2: Check Player 1’s work and explain why the equation is or is not true. “This equation is true because __________ .” “This equation is not true (false) because ___________.”
- To Both Players: Now, you can switch roles and continue to play the game until all spaces on the game board have been completed. (You may also decide to have students play until time is up and you feel you have collected sufficient data).
Note: This task presumes that students know what an equation is and looks like. If it becomes clear in the introduction of this task that this is not the case for your students, you might take a few minutes to introduce this to them or delay the use of this task until they have a better understanding.
Preparing to Gather Observation Data and Determine Next Steps in Instruction
As students engage in the task, the notes in the next section will help you identify students’ current strengths and possible next steps for instruction. Use whichever form of the Observation Checklist best helps you record your observations of students and other relevant evidence as you see it: Individual, Partner, or Class. These varied forms, available at the end of this document and in a separate MS Excel file, are intended to give you a choice about how to collect notes on your students and determine possible next steps for instruction.
Addressing Student Misconceptions/Errors
A common student misconception is that an equation is constructed only with the equals sign at the end of the equation (e.g., ___ + ___ = ____). Many students who think this, or who see equations written only in this form, come to think that the equal sign (=) means “now write the answer.” Therefore, students do not understand the meaning of equality—an equation must “balance.” It is important for students to be flexible with their understanding of equality and how it can be represented. For example, 6 + 4 = 10 is the same as 10 = 6 + 4.
Extensions and Elaborations
This task can be extended in a variety of ways.
One variation is to have students play with two number dice or draw two number cards. Students may then choose to use either of the numbers as is or they may use the sum or difference of the numbers as one of the addends or subtrahends. For example, if 6 and 4 are rolled or drawn, the
Player 2 writes the number rolled in either of the blank spaces.
GO: Carry Out the Task
Task Steps Keep in Mind Observations of Students
- Have students choose who will be Player 1 and Player 2.
- Ask Player 1 to select any blank equation on the game board to build and solve for Round 1.
- Ask Player 1 to select a number card from the deck. Player 1 should look at the card and read the number aloud. SAY to PLAYER 1:
What number did you choose?
- Prompt Player 1 to write the number from the number card in the box labeled “card- space” of the equation on the game board using a crayon/colored pencil.
At the start of grade 1, it is very unusual for students not to recognize numerals 0 – 10 , but some students may continue to struggle with recognizing and reading numerals 11 – 20. Teen numbers, especially numbers 11 and 12, can be challenging for some students. While there is a linguistic pattern to the count sequence beginning at 13 (thirteen, fourteen, fifteen,…), 11 and 12 do not follow the same pattern, and none of these numbers (11–19) follow the pattern that starts, and remains, after 20. Make a note if any of your students struggles to read any of these numbers.
A. Student reads all numerals 0 – 10 fluently and independently. B. Student reads all numerals 0 – 20 independently.
- Next, have Player 2 roll the number cube and write the number rolled in one of the remaining blank spaces in the equation with a different crayon/colored pencil. SAY to PLAYER 2: What number did you roll? Write that number in one of the blank spaces in your equation using your color.
Please read your equation aloud. What do you notice?
Notice how students count after rolling the die. Do students… count the dots? recognize the individual numbers without obvious counting? Are students able to correctly write the corresponding numeral from the number rolled? Writing a number backwards is still common and is not a mathematical error.
C. Student counts the dots with one-to-one correspondence to determine the total. D. Student “knows” the total instantly. E. Student correctly writes the corresponding numeral to match the number rolled.
Task Steps Keep in Mind Observations of Students
- At this point in the game, the players have created an incomplete equation with only 2 whole numbers. One space remains blank. Remind Player 1 that the goal is to now solve for the unknown number. Suggested prompts to use include:
What strategies will you use to solve for the unknown number? How will you make this equation true? Player 1 should write the unknown number in pencil. This is so the option remains available to self-correct his or her thinking while working. If you choose, you may suggest that players trace over their solution with their crayon or marker, so that you can more easily determine which student has solved each equation.
How do students solve for the unknown number? Do students appear to use specific tools, including a number line, counters, or fingers? If so, do students use the selected tool appropriately? What strategy or strategies do students use to determine the unknown number? Related combinations of numbers, including pairs whose sum is 10? Forming 10? (e.g., 9+3 = 9+(1+2) = (9+1)+2 = 10+2 = 12) Using helpful known sums? (e.g., using doubles to add doubles +1 or doubles – 1; 6 + 7 = 6 + 6 + 1) Decomposing a number leading to a 10? (14 – 5 = 14 – 4 – 1 = 10 – 1 = 9) Using the relationship between addition and subtraction? (e.g., knowing 7 + 4 = 11, one also knows 11 – 4 = 7)
- Do students instantly recognize the unknown number? Are they able to compute it mentally?
F. Student makes “guesses” as to the unknown number, but no efficient strategy is observed. G. Student attempts to use strategies, but may misuse them or overly rely on one. H. Student solves by decomposing and finding 10s. (e.g., 9+3 = 9+(1+2) = (9+1)+2 = 10+2 =
- OR (14 – 5 = 14 – 4 – 1 = 10 – 1 = 9) I. Student uses helpful known sums (doubles, near doubles, etc.). J. Student uses the relationship between addition and subtraction. (e.g., knowing 7 + 4 = 11, one also knows 11 – 4 = 7)
- Once completed, prompt Player 1 to justify his or her thinking. If this has already been done, move to Step 8.
See Step 6. K. Student provides little to no explanation for the reasoning used to solve for the unknown number and is unsure whether
Task Steps Keep in Mind Observations of Students
subsequent rounds.
The “game” ends when both players have completed or attempted to complete all of the equations on their game board or when you deem that time is up.
OBSERVATION CHECKLIST
ASSESSING STUDENT UNDERSTANDING: UNKNOWN NUMBERS
Use this page to record individual student observations. Use the letters to note each event as you see it unfold. This record is intended to help you plan next steps in your instruction for your students.
Student Name Observations of Student Possible Individual Student Observations FLUENCY A. Student reads all numerals 0– 10 fluently and independently. B. Student reads all numerals 0– 20 independently. COUNTING and FLUENCY C. Student counts the dots with one-to- one correspondence to determine the total. D. Student “knows” the total instantly. E. Student correctly writes the corresponding numeral to match the number rolled. STRATEGIES F. Student makes “guesses” as to the unknown number, but no efficient strategy is observed. G. Student attempts to use strategies, but may misuse them or overly rely on one. H. Student solves by decomposing and finding 10s. {E.g. 9+3 = 9+(1+2) = (9+1)+2 = 10+2 = 12} OR (14 – 5 = 14 – 4 – 1 = 10 – 1 = 9) I. Student uses helpful known sums (doubles, near doubles, etc.). J. Student uses the relationship between addition and subtraction. (Knowing 7 + 4 = 11, one also knows 11 – 4 = 7)
EXPLAINING REASONING K. Student provides little to no explanation for the reasoning used to solve for the unknown number and is unsure whether the equation is true or false. L. Student justification is incomplete, flawed, or uses vocabulary inaccurately. Student may be able to correctly identify whether the equation is true or false, but does not provide a complete explanation as to why. M. Student is able to consistently explain her reasoning and provide a clear justification. Student is able to correctly determine whether the equation is true or false. Student’s explanation is thorough and complete. CRITIQUING REASONING N. Student requires support in critiquing partner’s work. O. Student correctly agrees or disagrees with partner, but does not provide justification. P. Student correctly agrees or disagrees with partner and is able to justify the reason for this critique.
Unknown Numbers — Game Board B
Names: Date:
Directions: Choose an equation. Player 1 begins by choosing a number card from the deck and writing that number in the card space box. Player 2 rolls a number cube and writes that number in either of the blank spaces. Player 1 then solves for the unknown number and writes it in the remaining blank and Player 2 checks Player 1’s work. Players switch roles and the game continues. For equations marked with a star ( ), you must explain your thinking. Pay attention to whether or not an equation results in a number below zero and, if so, how you could change the equation to solve it.
= +
card space
6 1
3
= -
card space
6 1
3
= +
card space
6 1
3
= -
card space
6 1
3
= +
card space
6 1
3
= -
card space
6 1
3
= +
card space
(^6 )
= -
card space
(^6 )
= +
card space
6 1
3
= -
card space
6 1
3
I used the strategy of _____ to solve for the unknown number because ________________.
The equation is true (false) because _____________________________________________.
I agree that the equation is true (false) because ____________________________________.
I disagree that the equation is true (false) because _________________________________.
Unknown Numbers - Student Sentence Frames
Unknown Numbers - Student Sentence Frames
Teacher directions: Cut and provide 1 copy to each pair of students to scaffold their responses.
Cut along the line.
I used the strategy of _____ to solve for the unknown number because ________________.
The equation is true (false) because _____________________________________________.
I agree that the equation is true (false) because ____________________________________.
I disagree that the equation is true (false) because __________________________________.
