Mathematics Performance Tasks
SCAVENGER HUNT: COMPARING LENGTH UNITS Overview At a Glance In this small group or partner activity, students measure classroom objects using a variety of units (inches, centimeters, and feet) and then compare the measurements to describe the relationship among units. Grade Level Grade 2 Task Format Modeled in small groups, but played in partners (2 students) Materials Needed For each student • 1 ruler (inch, centimeter, foot) • 1 Scavenger Hunt Game Sheet (either inch vs. centimeter or inch vs. foot, depending on round played) • 1 pencil • 1 clipboard or other writing surface • Extension(s) and Elaboration(s): Extension and Elaboration Task Sheets For the teacher • A variety of pre-­‐selected common classroom objects for students to measure, measuring a whole number of length units (either centimeter, inch, or foot). Note: Students may find objects around the room if this works for your classroom. • Observation Checklist (template provided) Prerequisite Concepts/Skills •
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Familiarity with direct comparison of length for two (or more) objects Experience measuring with non-­‐standard units Developing understanding of the importance of units when specifying length measure in both standard (e.g., 2 feet long) and non-­‐standard (e.g., 2 paper clips long) units Content Standards Addressed in This Task 2.MD.A.2 2.MD.A.1 Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. Extension(s) and Elaborations(s) 2.MD.A.4 Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. 1 Standards for Mathematical Practice Embedded in This Task MP5 Use appropriate tools strategically. MP6 Attend to precision. GET READY: Familiarize Yourself with the Mathematics The purpose of this task is to help students learn to measure the length of an object using various length units and understand how the measurements relate to the size of the unit chosen (2. MD.A.2). In doing so, students must select and use appropriate tools (2.MD.A.1). This task is designed to give you insight into students’ skill and understanding. Prepare to introduce the task to a small group of students, but allocate the majority of the time for students to work with their partners. Measuring length consists of two key aspects: (1) selecting a unit of measure; and (2) seeing how many of those units, laid end to end without gaps, will match the length of the object. We refer to that second process as “iterating” that unit. You can call these iterations “length units.” In the example below, the student has measured a line segment using paper clips, which is a non-­‐standard unit. The unit in this case is the paper clip. You can see the iterations, or length units, made from the paper clip. (See, e.g., Goldenberg and Clements, 2014.) STANDARD UNIT MEASUREMENTS: Counting starts at 1; measuring starts at 0. It is important that students see the “meaning of the numerals on the ruler as the number of length-­‐units” from the start of the ruler (the 0 shown only on some rulers) and, therefore, understand that there cannot be any space between successive length-­‐units (NCTM, 2011, p. 109). To many young students, this is not at all obvious. It may help to connect measuring with the ruler to measuring with the iteration of physical objects. Feel free to use your own understanding of your students’ knowledge and skills to decide if and when to have students iterate physical objects. Standards for Mathematical Practice This task exercises and builds habits of mind underlying two Standards for Mathematical Practice: MP5: Use appropriate tools strategically and MP6: Attend to precision. This task builds students’ ability to “consider the available tools when solving a mathematical problem” (MP5). These tools involve rulers, yardsticks, and meter sticks. Students must reason about which tool serves them best given the context of the problem and “make sound decisions about when each of these tools might be helpful, recognizing both the insights to be gained and their limitations” (NGA & CCSSO, 2010, p. 7). They must also reason about which length-­‐unit is most appropriate. For example, if a student selects to use a ruler, that student also must consider which length-­‐units make the most sense—small units like inches or centimeters, or large units like feet—and why. MP6 calls for students to “communicate precisely to others.” This task lets students develop precision of communication in the context of measuring the length of the same object twice using two different length-­‐units, and then comparing the two measurements by reasoning about the length-­‐units. Students get many chances to be “precise communicators”—discussing a specific length-­‐unit, the measurement, or the tool. Throughout, students describe length with the units (e.g., not saying “3,” but “3 inches”). 2 For More Information Goldenberg, E. Paul, and Clements, D. (2014) Developing Essential Understanding of Geometry and Measurement for PreK–Grade2. Reston, VA: NCTM. National Council of Teachers of Mathematics (NCTM). (2011). Focus in grade 2: Teaching with curriculum focal points. Reston, VA: Author. National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf GET SET: Prepare to Introduce the Task 1. Gather the materials listed on page 1, including a variety (approximately 4–6) of common classroom objects (e.g., crayons, markers, index cards, notebooks, paper clips, glue sticks, pencils, or other objects) ahead of time for students to measure. Students will then “hunt” for objects(either from a pre-­‐selected group or around the classroom), measure them to the nearest whole number of length-­‐
units (e.g., inches, centimeters, feet), and compare that measurement with the measure using a different length-­‐unit. Note: You might also choose to not pre-­‐select items, instead, letting the students find the objects in the room. 2. Place the classroom objects on a table for students or put them in a baggie for each pair of students. Note: if you choose to have the students hunt for objects around the room, offer guidelines about the items they choose. For example, choose items that are less than the length of the ruler. Do what makes the most sense for your management style. 3. Organize students into pairs ahead of time. Throughout this activity, specific time is given for students to “turn and talk” to their partners. “Turn-­‐and-­‐talk” is a strategy that allows students time to engage in productive discourse with a partner. Following a turn-­‐and-­‐talk, you may choose to ask the whole class to share so that partners can listen and learn from others’ ideas. Introducing the Task Introduce the task to a small group. Below is one way a teacher might explain how to play. Throughout this document, when specific language is suggested, it is shown in italics. (Note: If you have not pre-­‐
chosen objects, you might need to say this a little differently.) 1. We are going on a scavenger hunt to find objects and measure them. You’ll each need a ruler, a Scavenger Hunt Game Sheet, and a pencil. 2. For this round, you both will measure in inches. (Note: the students may also be measuring in centimeters or feet, depending on the round.) Note: Make sure students know that not all measurements will be an exact length-­‐unit. You may need to discuss this before the task and share language to support this, such as, “close to 8 inches” or “5 centimeters and a little more”. 3. (To the group) Turn and talk to your neighbor about what you know about inches. 4. You’ll take turns choosing an object. (To Player 1) You choose first. 5. Now both of you measure this object in inches using your ruler. Check to make sure that you agree 3 on the measurement. 6. Next, each of you write your measurement on your Scavenger Hunt Game Sheet. (Point to the correct spot to write the measurement-­‐ “1st Measurement”). 7. Now measure the same object using centimeters. Again, make sure that you both agree on the measurement. 8. Write your new measurement on your Scavenger Hunt Game Sheet “2nd Measurement”. (Point to the correct spot to write the measurement). 9. Now look at your results. How many inches is the object? How many centimeters is the object? Why do you think the measurements are different? 10. (To Player 2) Now you choose another object to measure and repeat what we did with the first object. Explain (Optional) 11. Once you have measured your objects, use the bottom of your summary sheet to explain why, for each object that you measured, you got a different number of centimeters than of inches? Summary Sheet When students have had ample experience with measuring in multiple length-­‐units, have each one complete the Scavenger Hunt Summary Sheet. This is an opportunity for students to capture their thinking in writing. 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. As you observe, use whichever form of the Observation Checklist that 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 choice about how to collect notes on your students and determine possible next steps for instruction. Addressing Student Misconceptions/Errors When you are observing your students, you should be aware of a few common misconceptions: •
Rulers are not all the same. There can be many differences between the rulers that students work with at school and at home. One such difference is where 0 is marked. On some rulers, the 0-­‐inch measurement begins at the left-­‐edge of the ruler (see example below). In some cases, 0 is not even notated, because it is assumed that 0 starts at the ruler’s edge. Other rulers do not have the 0 start at the beginning of the ruler. Instead, 0 may start a small distance from the ruler’s edge as seen below. 4 Students must know how to align the zero-­‐point on a ruler as the place to begin measuring (Goldenberg and Clements, 2014). The differences between rulers can easily go unnoticed if not explicitly discussed with students. You might decide that it will benefit your class to talk about similarities and differences among the rulers in your classroom prior to starting this task. •
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Some students begin to count at the numeral 1 on a ruler. These students must develop an understanding that the numeral 1 is the distance from the start of the ruler (0) to the tick mark at 1. It may benefit some students to measure by lining up physical objects—like inch cubes, centimeter cubes, or paper clips of uniform size—being sure to not leave space in between objects or to overlap objects. Some students don’t initially understand that units must be of equal size and that there must be no overlaps or gaps when measuring. Extensions and Elaborations •
One variation is to have students measure objects starting at a number other than zero on their ruler. By starting to measure from another number, students gain a deeper experience in understanding the concept of successive length-­‐units—the distance (the amount of space covered) between two endpoints. In the example below, the number of length units (4 cm) is the distance between the 2 cm and 6 cm marking. This variation can be embedded within the task itself for students who are ready for this type of elaboration. Relating distance to subtraction is a useful idea for students. It is reasonable to talk of the distance between 7 and 10 as 3, and use that to figure out that the distance from 7 to 12 is two more than that, or 5, or that the distance between 7 and 20 is ten more than the distance between 7 and 10, so it is 13. •
You may also choose to have some students try the Extension and Elaboration Task Sheet. This begins with an example of a scavenger hunt where a student has measured two different objects (a stapler and paper clip). Together, students determine how much longer one object is than another, expressing the difference in terms of a standard length-­‐unit (2.MD.A.4). The sheet then has players continue this same scavenger hunt in your classroom. They each measure using the same length-­‐
unit, finding two separate objects that are a whole number of length-­‐units. Once players find these two measurements, students work together to figure out how much longer one object is than the other. This extension helps deepen students’ understanding that they can use the arithmetic they know to figure out how much longer one object is than another.
5 GO: Carry Out the Task—Scavenger Hunt: Comparing Length-­‐Units Task Steps Keep in Mind Observations of Students 1. Model the activity with a small group of students (approximately 4–6), but have students work in pairs. 2. For each round, students will choose an object from the pre-­‐selected common classroom objects and measure that object in multiple length-­‐units. For example, if players begin to measure in inches for their “1st Measurement” then follow that by having them measure in centimeters for their “2nd Measurement.” Allow partners to discuss what they know about both length-­‐units. You choose the length-­‐units students will use. A. Student appears to demonstrate confusion This can change from one round to the next about the selected length-­‐unit and (e.g., if they measured in inches and struggles to generate a known idea about centimeters for round 1, they may measure in his length-­‐unit. inches and feet for round 2). B. Student appears to recognize the selected length-­‐unit, but may use incomplete or flawed reasoning when describing something known about the length-­‐unit (e.g., “I’ve measured with inches before. Inches are pretty big in size.”) SAY: To begin, turn and talk to your partner about what you know about the length-­‐
units that you’ll use to measure (e.g., inches and centimeters). 3. Players now begin the activity, choosing an object to measure. SAY to PLAYER 1: C. Student recognizes the selected length-­‐
unit and can share at least one known idea about the length-­‐unit that is correct. •
How does the student measure using a ruler and specific length-­‐unit? −
Choose an object to measure. 4. SAY to BOTH PLAYERS: Now both of you measure this object in inches using your ruler. Check to make sure that you agree on the measurement. 5. Have both students record the measurement on their individual −
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Does the student correctly align the 0 mark on the ruler as the beginning of length? Does the student count from 0? Does the student use physical objects to help understand how to measure with a ruler? 6 D. Student pays little to no attention to the location of the 0 mark when measuring and, therefore, is inconsistent in taking measurements (even if they are correct). E. Student correctly aligns the placement of the 0 mark on the ruler when measuring. F. Student counts on from 1 (or some other number) when measuring and is still developing the understanding that the meaning of numerals on the ruler identifies the number of length-­‐units. Task Steps Scavenger Hunt Game Sheets, pointing to the correct spot to write the measurement. Keep in Mind •
Does the student understand: −
SAY to BOTH PLAYERS: Write your measurement on your Scavenger Hunt Game Sheet. •
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G. Student correctly counts on from 0, knowing to count successive length-­‐units and understanding the meaning of numerals on the ruler as the number of length-­‐units. the concept of length-­‐unit iteration (not H. Student iterates length-­‐units when leaving any space between successive measuring, knowing that no gaps can exist length-­‐units)? between length-­‐units. − the meaning of numerals on the ruler I. Student correctly and consistently as the number of length-­‐units? measures to the nearest whole length-­‐
− that the units are (and must be) of unit. equal size? J. Student correctly and consistently Does the student correctly identify an measures to the nearest half or quarter exact number of length-­‐units? length-­‐unit. If an exact number of length-­‐units is not possible for a given object, does the student know how to read the measurement to the nearest length-­‐unit? −
6. To Both Players: Now measure the same object using centimeters (or a different length-­‐unit) using your ruler. Again, make sure that you both agree on the measurement. 7. Write your new measurement on your Scavenger Hunt Game Sheet. (Point to the correct spot to write the measurement). that length is a measurement of distance (the amount of space covered) between two endpoints? Observations of Students Note: Some students may be able to measure to the nearest half or quarter inch. Few if any will know what to call the subdivisions of the centimeter. Neither of these is a grade 2 expectation but it is certainly acceptable and worth noting if students have that skill. 8. Now take a look at your results. How many inches is the object? How many centimeters is the object? Why do you think the measurements are different? K. Student struggles to compare and describe the two measurements relative to the length-­‐units used and provides little to no explanation of his or her reasoning. L. Student attempts to compare and describe the two measurements relative 7 Task Steps Keep in Mind Observations of Students to the length-­‐units used and explains his or her reasoning; however, the reasoning is often incomplete or flawed. M. Student is able to consistently compare and describe the two measurements relative to the length-­‐units and to explain his or her reasoning. Student’s explanation is thorough and complete. 9. Students now take turns choosing other objects to measure. You might choose to declare a number of objects to measure or you might set a time limit. While this summary sheet will benefit students in providing them an opportunity to write and reflect about their experience, you choose when to have students complete this. N. Student provides little to no written explanation of his or her reasoning. Now you are ready to begin the scavenger hunt. Take turns choosing or finding other objects to measure. You will each measure both objects in inches and centimeters (or inches and feet), measuring to the nearest whole number and writing your measurements on your Scavenger Hunt Game Sheet. Note: Consider capturing students on camera or with video to document the work they are doing! Once players have had ample experience with measuring in multiple length-­‐units, they should complete a Scavenger Hunt Summary Sheet. This summary sheet acts as an “exit 8 O. Student attempts to provide a written explanation of his or her reasoning; Task Steps Keep in Mind slip” for each round—and its purpose is to provide students with the opportunity to summarize their thinking in writing. It is recommended that you keep these summary sheets in a central location in the classroom where students can pick them up as they need them. Students may sit together to complete each summary sheet; however, this is intended to be an individual endeavor. You should feel free to make the decision that best meets the needs of your students. If the summary sheet is used, you may include this as formative evidence of your students’ understanding. 9 Observations of Students however, the reasoning may be incomplete or flawed. P. Student is able to provide a written explanation of his or her reasoning that is thorough and complete. OBSERVATION CHECKLIST ASSESSING STUDENT UNDERSTANDING: SCAVENGER HUNT 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 MAKING MEANING MAKING MEANING A.
Student appears to demonstrate confusion about the selected length-­‐unit and struggles to generate a known idea about his length-­‐unit. B. Student appears to recognize the selected length-­‐unit, but may use incomplete or flawed reasoning when describing something known about the length-­‐unit (e.g., “I’ve measured with inches before. Inches are pretty big in size.”) C. Student recognizes the selected length-­‐unit and can share at least one known idea about the length-­‐unit that is correct. MEASUREMENT ACCURACY and COUNTING D. Student pays little to no attention to the location of the 0 mark when measuring and, therefore, is inconsistent in taking measurements (even if they are correct). E. Student correctly aligns the placement of the 0 mark on the ruler when measuring. F. Student counts on from 1 (or some other number) when measuring and is still developing the understanding that the meaning of numerals on the ruler identifies the number of length units. G. Student correctly counts on from 0, knowing to count successive length-­‐units and understanding the meaning of numerals on the ruler as the number of length-­‐units. H. Student iterates length-­‐units when measuring, knowing that no gaps can exist between length-­‐
units. I.
Student correctly and consistently measures to the nearest whole length-­‐unit. J.
Student correctly and consistently measures to the nearest half or quarter length-­‐unit. 10 K. Student struggles to compare and describe the two measurements relative to the length-­‐units used and provides little to no explanation of his or her reasoning. L. Student attempts to compare and describe the two measurements relative to the length-­‐units used and explains his or her reasoning; however, the reasoning is often incomplete or flawed. M. Student is able to consistently compare and describe the two measurements relative to the length-­‐units and to explain his or her reasoning. Student’s explanation is thorough and complete. EXPLAINING REASONING N. Student provides little to no written explanation of his or her reasoning. O. Student attempts to provide a written explanation of his or her reasoning; however, the reasoning may be incomplete or flawed. P. Student is able to provide a written explanation of his or her reasoning that is thorough and complete. Scavenger Hunt Game Sheet - inches/cm
Find an object that is a whole number
of inches or centimeters.
1st Measurement
Find an object that is a whole number
of inches or centimeters.
1st Measurement
Object: ____________________________
Object: ____________________________
Who measured this? _________________
Who measured this? _________________
Length: ____________ inches/cm
Length: ____________ inches/cm
2nd Measurement
2nd Measurement
Who measured this? _________________
Who measured this? _________________
Length: _____________ inches/cm
Length: _____________ inches/cm
to the nearest unit
to the nearest unit
Scavenger Hunt Game Sheet - inches/feet
Find an object that is a whole number
of inches or feet.
1st Measurement
Find an object that is a whole number
of inches or feet.
1st Measurement
Object: ____________________________
Object: ____________________________
Who measured this? _________________
Who measured this? _________________
Length: ____________ inches/feet
Length: ____________ inches/feet
2nd Measurement
2nd Measurement
Who measured this? _________________
Who measured this? _________________
Length: _____________ inches/feet
Length: _____________ inches/feet
to the nearest unit
to the nearest unit
Name: ______________________________________________________________
I measured the object to be _____________
inches
centimeters
feet
My partner measured it to be ____________
inches
centimeters
feet
Summary Sheet
What did you notice when you compared your measurements?
Which unit did you have more of?
inches
centimeters
feet Explain why.
Name: ______________________________________________________________
I measured the object to be _____________
inches
centimeters
feet
My partner measured it to be ____________
inches
centimeters
feet
Summary Sheet
What did you notice when you compared your measurements?
Which unit did you have more of?
inches
centimeters
feet Explain why.
Name ___________________________________________
Date _________________________________
Comparing Length Units: Extension and Elaboration Task
Your classroom just completed a scavenger hunt where you and your classmates measured some objects found in your
classroom. The objects were measured using the same length unit. Your job is to figure out how much longer one
object is than the other.
Round
Example
Length Unit
Object #1
Object #2
The stapler is
___ cm long.
Centimeters
The paper clip is
___ cm long.
Solve.
Now solve to find out how much longer the ________________________________ is than the
_________________________________. Show and explain your thinking in the space below.
Round
Length Unit
Object #1
Object #2
Inches
The ______________ is ______ inches long.
The ______________ is ______ inches long.
Solve.
Now solve to find out how much longer the ________________________________ is than the
_________________________________. Show and explain your thinking in the space below.
page 1
Name ___________________________________________
Date _________________________________
Comparing Length Units: Extension and Elaboration Task
Round
Length Unit
Object #1
Object #2
The ______________ is ______ feet long.
The ______________ is ______ feet long.
Feet
Solve.
Now solve to find out how much longer the ________________________________ is than the
_________________________________. Show and explain your thinking in the space below.
Round
Length Unit
Object #1
Object #2
Centimeters
The ______________ is ______ cm long.
The ______________ is ______ cm long.
Solve.
Now solve to find out how much longer the ________________________________ is than the
_________________________________. Show and explain your thinking in the space below.
page 2