2Nd Grade Tasks 12/2/2012

www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012

2.OA A Pencil and a Sticker Alignment 1: 2.OA.A.1 Domain OA: Operations and Algebraic Thinking Cluster Represent and solve problems involving addition and subtraction. Standard 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 drawings and equations with a symbol for the unknown number to represent the problem.See Glossary, Table 1.

A pencil costs 59 cents, and a sticker costs 20 cents less. How much do a pencil and a sticker cost together?

Solution:Bar diagram The pencil costs 59 cents, and the sticker costs 20 cents less than that:

So the sticker costs 59-20 = 39 cents. The cost of the two together:

is 59 + 39 = 98 cents.

1 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 2.OA Red and Blue Tiles Alignment 1: 2.OA.C.3 Domain OA: Operations and Algebraic Thinking Cluster Work with equal groups of objects to gain foundations for multiplication. Standard Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Lin wants to put some red and blue tiles on a wall for decoration. She is thinking about several different patterns of tiles she could create. She wants to choose a pattern that would let her use exactly as many red tiles as blue tiles. a. Is it possible to create the pattern below using the same number of red tiles as blue tiles? Explain.

b. Is it possible to create the pattern below using the same number of red tiles as blue tiles? Explain.

c. Can you figure out how many tiles are in the pattern below without counting them one by one? Is it possible to create this pattern using the same number of red tiles as blue tiles? Explain.

d. Of the patterns above, which ones have an even number of tiles? Which ones have an odd number of tiles? If Lin wants to use an equal number of red tiles and blue tiles, should she use a pattern with an even number of tiles, or one with an odd number of tiles? Explain. Commentary:

2 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 An even number is a whole number that is evenly divisible by 2. There are two interpretations of division: "unknown number of groups " and the "the unknown group size." Standard 2.OA.3 calls for students to determine whether a set of objects contains an even or odd number of members using strategies such as pairing objects or attempting to divide the objects into two equal subgroups; these two strategies correspond to the two interpretations of division that students will study in more depth in third grade. This task provides opportunities for students to use different strategies to determine whether a set has an even or odd number of objects. In particular, part (b) invites students to pair tiles until it is clear that one unpaired tile is left over; in part (c) the tiling arrangement could easily be divided into two equal parts (e.g., the top half and the bottom half). It would be a good idea for the teacher to draw attention to these two different strategies when they are used by the students. In addition, part (c) asks students to determine the number of tiles in an array without counting each tile individually. This can be accomplished using repeated addition, as suggested by 2.OA.4. This foreshadows the introduction of multiplication in third grade. The reading level for this task may be above that of many second graders, so it is likely that the teacher will need to verbally introduce the task to the students. The figures are intentionally drawn so that students can color the tiles, so it would be good to have red and blue crayons or pencils on hand.

Solution:Solution a. The first arrangement can be divided into two equal parts; for example, we can divide the pattern into the top half and the bottom half. This means that the number of tiles in this arrangement is even. b. If we try pairing tiles in the second seating arrangement, we end up with one tile left over. Therefore, the number of tiles in this arrangement is odd. c. The tiles in this arrangement are in a 4-by-4 array, so the total number of tiles is 4 + 4 + 4 + 4 = 16. Again, we can divide this arrangement into a top half and a bottom half, so the number of tiles is even. d. When the number of tiles is even, Leslie can divide the tiles into two equal-sized groups, one to be made of red tiles and one to be made of blue tiles. She can then create the pattern using an equal number of tiles of each color. When the number of tiles is odd, there will have to be a color (red or blue) that has at least one more tile than the other color. Therefore, Leslie should use a pattern with an even number of tiles if she wants to use the same number of red and blue tiles.

2.OA Counting Dots in Arrays Alignment 1: 2.OA.C.4 Domain OA: Operations and Algebraic Thinking Cluster Work with equal groups of objects to gain foundations for multiplication. Standard Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Which of the following are equal to the number of dots in the picture below? (Choose all that apply.)

3 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012

a. 3 + 3 + 3

b. 3 + 4

c. 4 + 4 + 4

d. 4 + 4 + 4 + 4

e. 3 + 3 + 3 + 3

Commentary: Students who work on this task will benefit in seeing that given a quantity, there is often more than one way to represent it, which is a precursor to understanding the concept of equivalent expressions. This particular question also lays a foundation for students to understand the commutative property of multiplication in third grade. This task would be much more valuable if included in an appropriate place in an instructional sequence than as an isolated task.

Solution:Solution We can see 3 rows with 4 dots in each row, so (c) 4+4+4 can represent the number of dots in the array. We can also see 4 columns with 3 dots in each column, so (e) 3+3+3+3 can represent the number of dots in the array.

2.NBT Boxes and Cartons of Pencils Alignment 1: 2.NBT.A.1 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Pencils are packed 10 in a box. A classroom carton has 10 boxes. a. Jem has 1 carton and 4 boxes. How many pencils does Jem have all together?

b. Lee needs to pack 370 pencils. i. How many boxes does Lee need?

ii. If Lee puts the boxes in cartons, how many cartons can he completely fill?

c. Ms. Kato needs 10 pencils for each of her 26 students. i. If she can only buy boxes, how many boxes does she need?

ii. She finds out that it is cheaper to buy pencils in cartons. How many cartons should she buy? How many additional boxes will she need?

Commentary: This task is a kernel for an instructional task that could be elaborated with commentary about teaching strategies and examples of student work. When this website is fully functional, teachers will be able to submit tasks and related materials for review. Illustrative Mathematics invites teachers to help us with this.

Solution:Answers a. Jem has a total of 140 pencils.

b. To pack 370 pencils, i. Lee needs 37 boxes.

ii. If he puts the boxes in cartons, he can completely fill 3 cartons and will have 7 boxes left over.

c. Ms. Kato needs 10 pencils for each of her 26 students. i. If she can only buy boxes, she needs 26 boxes.

ii. She can buy 2 cartons and will need 6 additional boxes. Alternatively, she can buy 3 cartons, but she will have some boxes left over.

5 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 2.NBT Bundling and Unbundling

Alignment 1: 2.NBT.A.1 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: Make true equations. Write one number in every space. Draw a picture if it helps. a. 1 hundred + 4 tens = ______4 tens + 1 hundred = ______b. 14 tens = 10 tens + _____ tens 14 tens = _____ hundred + 4 tens 14 tens = _____ ones c. 7 ones + 5 hundreds = ______

d. 8 hundreds = ______

e. 106 = 1 hundred + _____tens + _____ones 106 = _____tens + _____ones 106 = _____ones f. 90 + 300 + 4 = ______

Commentary: Students determine the number of hundreds, tens and ones that are necessary to write equations when some digits are provided. Student must, in some cases, decompose hundreds to tens and tens to ones. The order of the summands does not always correspond to the place value, making these problems less routine than they might seem at first glance. See the solution for detailed information about the parts of this task.

Solution:Annotated solutions a. 140, 140. The first problem asks for the same number (140) in different ways. This emphasizes that order doesn’t matter in addition – yet order is everything when using place-value notation. b. 14 tens = 10 tens + 4 tens 14 tens = 1 hundred+4 tens 14 tens = 140.

In this problem, the base‐ten units in 140 are bundled in different ways. In the first line, “tens” are

thought of as units: 14 things = 10 things + 4 things. c. 507. By scrambling the usual order, the third problem requires students to link the values of the parts with the order of the digits in the positional system. Also, to encode the quantity, the student will have to think: “no tens,” emphasizing the role of 0. 7 ones + 5 hundreds = 507 d. 800. In the fourth problem, the zeros come with a silent “no tens and no ones”: 8 hundreds = 800 e. 106 = 1 hundred + 0 tens + 6 ones 106 = 10 tens + 6 ones 106 = 106 ones In this problem, the base-ten units in 106 are bundled in different ways. This is helpful when learning how to subtract in a problem like 106 – 34 by thinking about 106 as 100 tens and 6 ones. f. 394. The sixth problem is meant to illustrate the notion that If the order is always given “correctly,” then all we do is teach students rote strategies without thinking about the size of the units or how to encode them in positional notation. 90 + 300 + 4 = 394

2.NBT Counting Stamps Alignment 1: 2.NBT.A.1 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

The post office packages stamps like this:  10 stamps in each strip.

 10 strips of 10 in each sheet.

a. Yesterday Mike saw 4 full sheets, 7 strips, and 2 extra stamps in the drawer. He counted all the stamps and found out that there were 472 stamps in all. He said, The number 472 matches the 4 sheets, 7 strips, and 2 stamps. Cool! Why did Mike’s number match up with the numbers of sheets, strips, and extra stamps? Draw a picture to help explain your answer. b. Today Mike found 3 extra stamps, 1 sheet, and 5 strips. He said, Because of how things matched up yesterday, I guess there are 315 stamps total. i. Find the total number of stamps.

ii. Explain why Mike's guess is incorrect. What could he have done to guess correctly?

Commentary: This is an instructional task related to deepening place-value concepts. The important piece of knowledge upon which students need to draw is that 10 tens is 1 hundred. So each sheet contains 100 stamps. If students do not recall this fact readily, one way to review it is to have them draw a strip of ten stamps on graph paper (so they don't have to draw all the individual stamps) and then draw ten strips that are side-by-side to represent a sheet and ask how many stamps there are in one sheet. Given how closely pictures in this problem correspond to base-ten blocks, having experience with those and/or having them on hand would be helpful. The second part of the problem highlights the fact that while reordering is allowed in addition (because of commutativity), we cannot reorder the digits of a number, since these digits are attached to different place values. Its solution engages Standard for Mathematical Practice 6, Attend to precision.

Solution:Solution a. Since one full sheet is 100 stamps, 4 sheets will be 400 stamps. Since a strip is 10 stamps, seven will be 70 stamps. So 4 sheets, 7 strips, and 2 extra stamps is a total of 400 + 70 + 2 = 472. One way to picture this is as follows:

The number of sheets, strips, and extra stamps are matching up with the digits in the final number because listing by number of sheets, strips and extras is also listing by hundreds, tens and ones, respectively, which is how base-ten numbers are defined. b.

i. One full sheet is 100 stamps, and 5 strips is 50 stamps, so when Mike finds 3 extra stamps, 1 sheet and 5 strips, that makes for a total of 3+ 100 + 50 = 100 + 50 + 3 = 153 . ii.

The number of sheets, strips, and extra stamps did not match up with the digits of the final number because listing by extras, sheets and strips is listing by ones, hundreds and tens, respectively, which is not the order used in the base-ten number system. The difference from yesterday is the order Mike used. If he had found (or chosen to list) the 1 sheet first, then the 5 strips and then the 3 extra stamps, the numbers would match up again.

2.NBT Largest Number Game Alignment 1: 2.NBT.A.1 - Domain NBT: Number and Operations in Base Ten - Cluster Understand place value. Standard Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Dona had cards with the numbers 0 to 9 written on them. She flipped over three of them. Her teacher said: If those three numbers are the digits in another number, what is the largest three-digit number you can make? a. First Dona put the 8 in the hundreds place. Is this the right choice for the hundreds place? Explain why or why not.

b. Next, Dona said, “It doesn’t matter what number I choose for the other places, because I put the biggest number in the hundreds place, and hundreds are bigger than tens and ones.” Is she correct? Explain.

Commentary: It is important that students be asked to explain well beyond saying something like “She should choose the 8 because it is the biggest.” They should be asked to think through the other possibilities and then draw on their ability to compare three digit numbers (as developed in 2.NBT.4) to complete the task. In the second part, students are presented with an incorrect statement supported by a correct one. It is worth pausing to ask students to carefully sort this through, since attending to reasoning that is partially true and partially false lends itself to the SMP.3: Constructing viable arguments and critiquing the reasoning of others. One can ask students if they know how to build the biggest three-digit number given any three numbers between 0 and 9 to use as digits. If students can’t explain the best strategy at the greatest level of generality, one could have them play the game and explain how their method works in examples.

Solution:1 a. Dona is correct in putting the 8 in the hundreds place. If the 8 is in the hundreds place, the number will be bigger than 800. If she puts the 5 in the hundreds place, the number must be smaller than 600. If she puts the 1 in the hundreds place, the number must be smaller than 200.

b. Dona is not correct; all the digits matter. Tens are greater than ones, so she needs to choose the next largest number for the tens place. If she chooses 1 for the tens place and then 5 for the ones place the result is 815. The only other possibility is if she chooses 5 for the tens place and then 1 for the ones place, yielding 851, which is greater than 815. So the choice matters (and 851 is the “winning” total).

2.NBT Making 124 Alignment 1: 2.NBT.A.1 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Lamar and Siri had some base-ten blocks. Lamar said, "I can make 124 using 1 hundred, 2 tens, and 4 ones." Siri said, "I can make 124 using 124 ones." a. Can you find a way to make 124 using only tens and ones? Can you find a different way?

b. Find as many ways as you can to make 124 using hundreds, tens, and ones. If you think you have found all the ways, explain how you know your list is complete.

Commentary: Not all students have seen base-ten blocks. This task should only be used with students who know what they are or have some on-hand to use themselves. Because this task asks students to explain how they know the list is complete, it aligns with Standard for Mathematical Practice 3 Construct viable arguments and critique the reasoning of others. A systematic approach to listing the solutions is not required to meet the standard, but it's a nice way for students to explain how they found all the possible ways to make 124 using base-ten blocks

Solution:systematic exchanges The list of all ways using 1 hundred is:  1 hundred, 2 tens, 4 ones.

 1 hundred, 1 ten, 14 ones

 1 hundred, 0 tens, 24 ones.

The list of all ways not using any hundreds is:  12 tens, 4 ones.

 11 tens, 14 ones

 9 tens, 34 ones

 8 tens, 44 ones

 7 tens, 54 ones

11 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012  6 tens, 64 ones

 5 tens, 74 ones

 4 tens, 84 ones

 3 tens, 94 ones

 1 ten, 114 ones

 0 tens, 124 ones.

To know the list is complete as we make it, we can start with the standard way, namely 1 hundred, 2 tens, and 4 ones, and exchange tens for ones, one at a time, to get the first list. Then we exchange the hundred for 10 tens, to get a total of 12 tens along with 4 ones. Once again, we can exchange tens for 10 ones step by step in order to get the second list.

2.NBT One, Ten, and One Hundred More and Less Alignment 1: 2.NBT.A.1 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. What number is 1 more than 99?

b. What number is 1 less than 600?

c. What number is 10 more than 90?

d. What number is 10 less than 300?

e. What number is 100 more than 570?

f. What number is 100 less than 149?

Commentary: This task acts as a bridge between understanding place value and using strategies based on place value for addition and subtraction. Within the classroom context, this activity can be differentiated using numbers that are either simpler or more difficult to manipulate across tens and hundreds.

Solution:answers a. 100

b. 599

12 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 c. 100

d. 290

e. 670

2.NBT Regrouping Alignment 1: 2.NBT.A.1 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. What number represents the same amount as 2 tens + 7 ones?

b. What number represents the same amount as 4 tens + 0 ones?

c. What number represents the same amount as 5 tens + 12 ones?

d. What number represents the same amount as 3 hundreds + 18 tens + 5 ones?

e. What number represents the same amount as 7 hundreds + 19 tens?

Commentary: This task serves as a bridge between understanding place-value and using strategies based on place-value structure for addition. Place-value notation leaves a lot of information implicit. The way that the numbers are represented in this task makes this information explicit, which can help students transition to adding standard base-ten numerals.

Solution:Answers a. 27

d. 485

e. 890

2.NBT Ten $10s make $100 Alignment 1: 2.NBT.A.1 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. 13 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 Standard Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. How many ten-dollar bills equal a hundred-dollar bill?

b. Jem had 20 ten-dollar bills. How many hundred-dollar bills can she trade them for?

c. Dan had 6 hundred-dollar bills. How many ten-dollar bills can he trade them for?

Commentary: People often use pennies, dimes, and dollars as a context for thinking about bundling by ten. However, using one, ten, and one-hundred dollar bills is easier for students to understand initially because the names of the bundles match the names of the numbers. While dimes are bundles of ten pennies and dollars are bundles of ten dimes, the translation between base-ten numbers and pennies, dimes, and dollars is more complex because of the name differences. Furthermore, there is the potential for confusion since a penny can be thought of as both 1/100 of a dollar and as 1 cent. As a result students may require more support from the teacher to understand contexts involving e.g. dimes and dollars than they would for this task context.

Solution:answers a. 10 ten-dollar bills equal 1 hundred-dollar bill.

b. Jem can trade 20 ten-dollar bills for 2 hundred-dollar bills.

c. Dan can trade 6 hundred-dollar bills for 60 ten-dollar bills.

2.NBT Three composing/decomposing problems Alignment 1: 2.NBT.A.1 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

Some students are working with base-ten blocks. a. Nina has 3 hundreds, 8 tens, and 23 ones. How many ones would this be?

b. Lamar wants to make the number 261 . He has plenty of hundreds blocks and ones blocks to work with, but only 4 tens blocks. His friend Jose said,

You can still make 261 with the blocks you have. Explain how he can.

c. Find at least three different ways to can make 124 using hundreds, tens and ones. 14 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 Commentary: The purpose of this task is to help students understand composing and decomposing ones, tens, and hundreds. This task is meant to be used in an instructional setting and would only be appropriate to use if students actually have base-ten blocks on hand. The last two tasks full engage the notion of composing and decomposing as needed for algorithms for addition and subtraction. Both parts require persistence, as in the Standard for Mathematical Practice 1. After seeing the first two tasks, students have the ideas needed to start listing possibilities in the third task. The idea of exchanging a ten for ten ones and a hundred for ten tens is needed in order to complete the task.

Solution:Solution a. While some students might try to simply add, others will recognize that 23 ones is 2 tens and 3 ones. When we combine the 2 tens with the 8 tens we already have we get 10 tens, which is one hundred. So we have 3 hundreds and another hundred and three ones, which is 403.

b. Lamar could use ten ones for each ten-block which he was missing. So instead of 2 hundreds, 6 tens and 1 one as he wanted, he can start with the 2 hundreds and 4 tens which he has and then use two sets of ten ones instead of the two more needed tens. Those make 20 ones, which we add to the 1 one needed to get 21 ones. Collecting all of these we get 2 hundreds, 4 tens and 21 ones. There are many possible solutions – for example using 2 hundreds, 3 tens and 31 ones – but the one given is the most likely.

c. The list of all ways using 1 hundred is:

o 1 hundred, 2 tens, 4 ones.

o 1 hundred, 1 ten, 14 ones

o 1 hundred, 0 tens, 24 ones. The list of all ways not using any hundreds is: o 12 tens, 4 ones.

o 11 tens, 14 ones

o 10 tens, 24 ones

o 9 tens, 34 ones

o 8 tens, 44 ones

o 7 tens, 54 ones

o 6 tens, 64 ones 15 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 o 5 tens, 74 ones

o 4 tens, 84 ones

o 3 tens, 94 ones

o 2 tens, 104 ones

o 1 tens, 114 ones

o 124 ones.

To know the list is complete as we make it, we can start with the standard way, namely 1 hundred, 2 tens, and 4 ones, and exchange tens for ones one at a time to get the first list. Then we exchange the hundred for 10 tens, to get a total of 12 tens along with 4 ones. Once again, we can exchange tens for 10 ones step by step in order to get the second list. Because we cannot use two or more hundreds, these two lists contain all possibilities.

2.NBT Party Favors Alignment 1: 2.NBT.A.1.a Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: 100 can be thought of as a bundle of ten tens–-called a “hundred.”

Pia was having a party. She put 10 stickers in each party bag. a. On the first day she made 10 bags. How many stickers were in her 10 bags all together?

b. On the second day she made 3 more bags with ten stickers in each one. How many stickers total were in her 10 bags plus 3 more bags?

c. On the third day she made 7 more bags with ten stickers in each one. How many stickers total are in her 20 bags of ten?

d. On the fourth day, she made another 10 bags with ten stickers in each one. How many stickers are in her 30 bags of ten?

e. After one week, she had made a total of 50 bags with ten stickers in each one. How many stickers total are in her 50 bags of ten?

Commentary: The point of this task is to emphasize the grouping structure of the base-ten number system, and in particular the crucial fact that 10 tens make 1 hundred. Second graders should have been given opportunities to work with 16 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 objects and pictures that represent the grouping structure of the base-ten number system, which would help prepare them for doing this task. This task is meant to be encountered early in the progression for 2.NBT.1, building on some facility with 2.NBT.2. The task has many parts, especially for this grade level, and while they all establish or use the fact that ten tens is one-hundred, they differ slightly. So one could work on this task in multiple sessions, but one should avoid cutting it short altogether, to more likely engage MP.8 look for regularity in repeated reasoning. As appropriate, students should be asked to explain their process. Because the task requires a lot of reading, it might be best used as a story that unfolds where the teacher tells the class what happens next and the students figure out what she did each day.

Solution:1 a. With the help of a picture such as the following:

one can count by tens to see that ten groups of ten prizes is one-hundred prizes. b. There are two main possible approaches. One can either add three bags to the picture above and continue to count. Or one can notice that the three new bags constitute 30 prizes, and since there were already 100 that would make 130 prizes.

c. Since we counted that ten boxes of ten is one hundred, twice that would be two hundreds, or 200.

d. Ten boxes of ten is 100 more. Since we had 200 – or two hundreds – before, adding one more hundred will give three hundreds, or 300.

e. Fifty boxes of ten can be broken up into five collections of ten boxes of ten. Since ten boxes of ten is 100, five of those is five hundreds, or 500.

2.NBT Comparisons 1 Alignment 1: 2.NBT.A.4 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard - Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using > , =, and < symbols to record the results of comparisons.

17 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 Are these comparisons true or false? a. 2 hundreds + 3 ones >5 tens + 9 ones

b. 9 tens + 2 hundreds + 4 ones <924

c. 456 <5 hundreds

d. 4 hundreds + 9 ones + 3 ones <491

e. 3 hundreds + 4 tens <7 tens + 9 ones + 2 hundred

f. 7 ones + 3 hundreds >370

g. 2 hundreds + 7 tens = 3 hundreds - 2 tens

Commentary: This task requires students to compare numbers that are identified by word names and not just digits. The order of the numbers described in words are intentionally placed in a different order than their base-ten counterparts so that students need to think carefully about the value of the numbers. Some students might need to write the equivalent numeral as an intermediate step to solving the problem.

Solution:Solution and explanations a. True. It reads, “Two hundreds and three ones is greater than five tens and nine ones.” 203>59 203 is, in fact, greater than 59 because 203 has two 100 s (a two in the hundred’s place), while 59 has no 100 s (a zero in the hundred’s place). b. True. It reads, “Nine tens, two hundreds, and four ones is less than 924.” 294<924 294 is, in fact less than 924 because 294 has two 100 s (a two in the hundred’s place), while 924 has nine 100 s (a nine in the hundred’s place). c. True. It reads, “456 is less than five hundreds.” 456<500 456 is, in fact, less than 500 because 456 has four 100 s (a four in the hundred’s place) and some tens and ones that total less than one hundred, while 500 has five 100 s (a five in the hundred’s place). d. True. It reads, “Four hundreds and 9+3=12 ones is less than 491 .” 12 ones is the same as one ten and three ones, so let’s rewrite the previous sentence. “Four hundreds, one ten, and two ones is less than 491 .” 18 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 412<491 412 is, in fact, less than 491 . Although both numbers have four 100 s (fours in the hundred’s place), 412 only has one ten (a one in the ten’s place), while 491 has nine tens (a nine in the ten’s place). e. False. It reads, “Three hundreds and four tens is less then seven tens, nine ones, and two hundreds.” 340<279 340 is, in fact, greater than 279 because 340 has three 100 s (a three in the hundred’s place), while 279 has two 100 s (a two in the hundred’s place). f. False. It reads, “Seven ones and three hundreds is greater than 370.” 307>370 307 is, in fact, less than 370 . Although both numbers have three 100 s (threes in the hundreds place), 307 has no 10 s (a zero in the ten’s place), while 370 has seven 10 s (a seven in the ten’s place). g. False. It reads 2 hundreds + 7 tens = 3 hundreds - 2 tens 270=280 While both numbers have 2 hundreds (a two in the hundreds place), 270 has 7 tens (a seven in the tens place) while 280 has an 8 in the tens place. Therefore 270<280

2.NBT Comparisons 2 Alignment 1: 2.NBT.A.4 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using > , =, and < symbols to record the results of comparisons.

Use < , =, or > to complete the following number sentences. a. 657 ______457+100+100

b. 926 ______726+100+10

c. 511+10+10+10 ______531−10−10

d. 923+10 ______953−10−10

e. 100+100 300+10 ______510

19 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 f. 347+30 ______397−10−10

g. 126 - 10−10−10−10 ______96−10 Solution:Answers and explanations

a. 657=457+100+100 To find the answer to part (a), we must add the numbers on the right side. 457 is the same as 4 hundreds plus 5 tens plus 7 ones. The right is then 4 hundreds plus 5 tens plus 7 ones plus 1 hundred plus 1 more hundred. We combine all of the hundreds to have a total of 6 hundreds plus 5 tens plus 7 ones, or 657 . Thus the left side is the same as the right side. 657=657 b. 926>726+100+10 To find the answer to part (b), we add the numbers on the right side. 726 is the same as 7 hundreds plus 2 tens plus 6 ones. The right side can now be described as 7 hundreds plus 2 tens plus 6 ones plus 1 more hundred and 1 more ten. We combine all of the hundreds to have a total of 8 hundreds plus 3 tens plus 6 ones, or 836 . Thus the left side is greater than the right side. 926>836 c. 923<953−10−10 To find the answer to part (c), we can subtract the numbers on the right side. 953 is the same as 9 hundreds plus 5 tens plus 3 ones. We also need to subtract 2 tens. If we have 5 tens and subtract 2 tens, we are left with 3 tens. Finally, we have 9 hundreds plus tens plus 3 ones, or 933 . 923 is, in fact, less than 933 . Although both numbers have 9 hundreds (9 in the hundred’s place) 923 has only 2 tens (2 in the ten’s place), while 933 has 3 hundreds (3 in the hundred’s place). 923<933 d. 100+100+300+10=510 To find the answer to part (d), we can add the numbers on the left side. The left side is 1 hundred plus 1 hundred plus 3 hundreds plus 1 ten. We combine all of the hundreds to have a total of 5 hundreds and 1 ten, or 510 . Thus the left side is the same as the right side. 510=510 e. 347+30=397−10−10 To find the answer to part (e), we can add the numbers on the left and subtract the numbers on the right. Let’s begin with the left side. 347 is the same as 3 hundreds plus 4 tens plus 7 ones. 30 is the same 3 tens. The left side can now be described as 3 hundreds plus 4 tens plus 7 ones plus 3 more tens. We combine all of the tens to have a total of 3 hundreds plus 7 tens plus 7 ones, or 377 . Now we can work on the right side. 397 is the same as 3 hundreds plus 9 tens plus 7 ones. −10 means take 1 ten away. If we have 9 tens and subtract 2 tens, we are left with 7 tens. So we have 3 hundreds plus 7 tens plus 7 ones, or 377. Thus the left side is the same as the right side. 377=377 f. 126−10−10−10−10=86

20 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 To find the answer to part (f), we must subtract the numbers on the left side. 126 is the same as 1 hundred plus 2 tens plus 6 ones. We also need to subtract 4 tens. If we have 1 hundred plus 2 tens plus 6 ones and subtract 2 tens, we have 1 hundred plus no tens plus 6ones. We still have to subtract 2 more tens, but have no tens left, so we must break down (unbundle) the 1 hundred. One hundred is the same as 10 tens. We now have 10 tens and 6 ones and if we subtract 2 tens, we have 86 for the left side. On the right side of the equation we have 96 (with its 9 in the tens place) - 10 (with its 1 in the tens place), leaving 86 . With 86 on both sides of the equation, the equation is balanced as indicated by the = sign. 86=86 Solution:Answers and explanations

a. 657=457+100+100 To find the answer to part (a), we must simplify the right side. 457 is the same as four hundreds, five tens, and seven ones. 100 is the same as one hundred. The right side can now be described as four hundreds, five tens, seven ones, one hundred and one more hundred. We combine all of the hundreds to have a total of six hundreds, five tens, and seven ones, or 657 . Thus the left side is the same as the right side. 657=657 b. 923=723+100+100 To find the answer to part (b), we must simplify the right side. 723 is the same as seven hundreds, two tens, and three ones. 100 is the same as one hundred. The right side can now be described as seven hundreds, two tens, three ones, one hundred and one more hundred. We combine all of the hundreds to have a total of nine hundreds, two tens, and three ones, or 923 . Thus the left side is the same as the right side. 923=923 c. 923<953−10−10 To find the answer to part (c), we must simplify the right side. 953 is the same as nine hundreds, five tens, and three ones. −10 means take one ten away. The right side now reads, “Nine hundreds, five tens, three ones. Take away one ten. Take away one more ten.” If we have five tens and take one ten away, we are left with four tens. We then take one more ten away and have three tens. Finally, we have nine hundreds, three tens, and three ones, or 933 . 923 is, in fact, less than 933 . Although both numbers have nine 100 s (nines in the hundred’s place) 923 has only two 10 s (a two in the ten’s place), while 933 has three 100 s (a three in the hundred’s place). $923 d. 100+100+300+10=510 To find the answer to part (d) , we must simplify the left side. 100 is the same as one hundred. 300 is the same as three hundreds. 10 is the same as one ten. The left side can now be described as one hundred, one hundred, three hundreds, and one ten. We combine all of the hundreds to have a total of five hundreds and one ten, or 510 . Thus the left side is the same as the right side. 510=510 e. 347+30=397−10−10

21 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 To find the answer to part (e), we must simplify both sides. Let’s begin with the left side. 347 is the same as three hundreds, four tens, and seven ones. 30 is the same three tens. The left side can now be described as three hundreds, four tens, seven ones, and three tens. We combine all of the tens to have a total of three hundreds, seven tens, and seven ones, or 377 . Now we can simplify the right side. 397 is the same as three hundreds, nine tens, and seven ones. −10 means take one ten away. The right side now reads, “Three hundreds, nine tens, seven ones. Take away one ten. Take away one more ten.” If we have nine tens and take one ten away, we are left with eight tens. We then take one more ten away and have seven tens. Finally, we have three hundreds, seven tens, and seven ones, or 377. Thus the left side is the same as the right side. 377=377 f. 126−10−10−10−10=86 To find the answer to part (f), we must simplify the left side. 126 is the same as one hundred, two tens, and six ones. −10 means take one ten away. The left side now reads, “One hundred, two tens, six ones. Take away one ten. Take away another ten. Take away one more ten.” If we have One hundred, two tens and six ones and take one ten away, we have one hundred, one ten, and six ones left. Now we take another ten away and are left with one hundred, no tens, and six ones. We still have to take one more ten away, but have no tens left, so we must break down (unbundle) the one hundred. One hundred is the same as ten tens. We now have ten tens and six ones and take one ten away, leaving nine tens and six ones, or 96 . But there is one more ten remaining, so we take one ten from the nine tens, leaving 86 for the left side. On the right side of the equation we have 96 (with its nine in the tens place) - 10 (with its one in the tens place), leaving 86 . With 86 on both sides of the equation, the equation is balanced as indicated by the = sign. 86=86

2.NBT Digits 2-5-7 Alignment 1: 2.NBT.A.4 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using > , =, and < symbols to record the results of comparisons.

a. Use all the digits 5, 7, and 2 to create different 3-digit numbers. b. What is the greatest number you can make using all of the digits? ______c. What is the smallest number you can make using all of the digits? ______22 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 Commentary:

Students who are struggling to build an understanding of the relationship between digit placement and the value of the number may still need concrete manipulatives such as grid paper and Base Ten Blocks. As a classroom extension, after students have worked independently or in small groups to solve the problem, the teacher can ask students to share their numbers, until all six possibilities are listed. Then, independently or as a whole group, students can order the six numbers from smallest to largest (or largest to smallest).

Solution:Solution a. The correct answers are 752, 725, 572, 527, 275, and 257. b. 752 is the greatest number with seven hundreds, five tens, and two ones. c. 257 is the smallest number with two hundreds, five tens, and seven ones.

2.NBT Ordering 3-digit numbers Alignment 1: 2.NBT.A.4 Domain NBT: Number and Operations in Base Ten Cluster Understand place value. Standard Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using > , =, and < symbols to record the results of comparisons.

1. Arrange the following numbers from least to greatest: 476 647 74 674 467 ______2. Arrange the following numbers from greatest to least: 326 362 63 623 632 ______Commentary: Each number has at most 3 digits so that students have the opportunity to think about how digit placement affects the size of the number. Each group also contains a two-digit number so that students have to do more than just compare the first digit, the second digit, etc.

Solution:Consider digit placement 1. Least to greatest: Arranging the numbers from least to greatest means writing the numbers in an ordered list according to their values. The smallest number should be written on the left, with the next smallest number written to its immediate right. The process continues for all the given numbers, the largest of which should be on the right.

23 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 First we look for the smallest number given. 74 is the smallest number given because it is the only number with no 100s. (There is an implied zero in the hundred’s place.) We now look for the second smallest number. There are two numbers, 476 and 467 , with four 100 s (fours in the hundred’s place). We must now consider the ten’s place. 467 is the next smallest number because it only has six 10s (a six in the ten’s place), while 476 has seven 10 s (a seven in the ten’s place). The next smallest number is 476 because it only has four 100 s (a four in the hundred’s place), while all the other remaining numbers have six 100 s (sixes in the hundred’s place). We now have two numbers remaining, 647 and 674 . Both numbers have six 100 s (sixes in the hundred’s place), so we must compare their 10 s. 647 is smaller than 674 because in has just four 10 s (a four in the ten’s place), rather than seven 100 s (a seven in the hundred’s place). This leaves 674 as the largest number. 74 467 479 647 674 2. Greatest to least Arranging the numbers from greatest to least means writing the numbers in an ordered list according to their values. The largest number should be written on the left, with the next largest number written to its immediate right. The process continues for all the given numbers, the smallest of which should be on the right. First we look for the largest number given. There are three numbers that begin with six: 63 , 623 and 632 . However, the 6 in 63 is in the tens place, not in the hundreds place. There are zero hundreds, making 63 the smallest number, the only number that is less than 100 . 623 and 632 both have six 100 s (sixes in the hundred’s place), so it is necessary to compare the digits in the 10 ’s place. 623 has two 10 s (a two in the ten’s place), while 632 has three 10s (a tens in the ten’s place. Since the value of the three in the 10 's place is larger than two in the 10 's place, 632 is the largest number and 623 is the second largest number. The two remaining numbers both start with three. Comparing 326 and 362 , which both have 3 hundreds (a three in the hundred’s place), means comparing the numbers in the ten’s place. Six tens (a six in the ten’s place) is larger than three tens (a three in the ten’s place), so 326 is smaller than 362 . 632 623 362 326 63

2.MD 2.NBT Jamir's Penny Jar Alignment 1: 2.MD.C.8, 2.NBT.B.5 Domain MD: Measurement and Data Cluster Work with time and money. Standard Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have? Domain NBT: Number and Operations in Base Ten 24 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 Cluster Use place value understanding and properties of operations to add and subtract. Standard Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Jamir has collected some pennies in a jar. Recently, he added coins other than pennies to his jar. Jamir reached his hand into the jar and pulled out this combination:

a. Jamir wants to count the total value of these coins. What coin do you suggest he start with? Why would Jamir want to start counting with this coin?

b. What is the total value of these coins? Write a number sentence that represents the total value of the coins.

c. Jamir reached into the jar again and was surprised to pull out a different combination of coins with the same total value as before. Draw a collection of coins that Jamir could have pulled from the jar. Write a number sentence that represents the total value of the coins.

Commentary: The purpose of this task is to help students articulate their addition strategies as in 2.NBT.5 and would be most appropriately used once students have a solid understanding of coin values. It also provides a context where it makes sense to “skip count by 5s and 10s” (as described in 2.NBT.2) for the combinations that involve more than one nickel or dime. The context provides a link to 2.MD.8. This task would be best used in an instructional setting especially since the language is somewhat complex and the teacher might need to help students decode the task statement. Students could work in pairs, taking turns explaining their approach to determining the value of the coins and comparing their solutions to the last question. The task is open-ended and offers multiple solutions allowing for student opportunities to critique each other's reasoning as in standard for mathematical practice 3.

Solution:Solution a. Jamir should start with a quarter because that has the largest value of all the coins. It is easiest to count up starting with the largest value and adding lesser values. b. The total value of the coins is 33¢. One way to write the number sentence that would reflect the strategy of starting with the largest value and adding on the next largest values and so on would be 25+5+1+1+1=33

25 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 although students can write the addends in any order. a. Note that only one picture is shown but students should draw a picture for the combination they chose. Students should be encouraged to write the cent symbol on their pictures. Possible answers:

 1 quarter and 8 pennies

25+1+1+1+1+1+1+1+1=33

 3 dimes and 3 pennies

10+10++10+1+1+1=33

 2 dimes, 2 nickels, and 3 pennies

10+10+5+5+1+1+1=33

 2 dimes, 1 nickel, and 8 pennies

10+10+5+1+1+1+1+1+1+1+1=33

 1 dime, 4 nickels, and 3 pennies

10+5+5+5+5+1+1+1=33

10+5+5+5+1+1+1+1+1+1+1+1=33

10+5+5+1+1+1+1+1+1+1+1+1+1+1+1+1=33

 1 dime, 1 nickel, and 18 pennies

10+5+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=33

 1 dime and 23 pennies

10 +1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 +1+1+1+1+1=33  6 nickels and 3 pennies

26 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 5+5+5+5+5+5+1+1+1=33

 5 nickels and 8 pennies

5+5+5+5+5+5+1+1+1+1+1+1+1+1=33

5+5+5+5+1+1+1+1+1+1+1+1+1+1+1+1+1=33

5+5+5+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=33

5+5 +1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 +1+1+1+1+1=33  1 nickel and 28 pennies

5 +1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 +1+1+1+1+1+1+1+1+1+1=33  33 pennies

1 +1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 +1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=33

2.NBT.6 Toll Bridge Puzzle Alignment 1: 2.NBT.B.6 Domain NBT: Number and Operations in Base Ten Cluster Use place value understanding and properties of operations to add and subtract. Standard Add up to four two-digit numbers using strategies based on place value and properties of operations.

The picture shows islands connected by bridges. To cross a bridge, you must pay a toll in coins. If you start on the island marked in blue with 100 coins, how can you make it to the island marked in red?

Commentary: This task is intended to assess adding of four numbers as given in the standard while still being placed in a problem-solving context. As written this task is instructional; due to the random aspect regarding when the correct route is found, it is not appropriate for assessment. This puzzle works well as a physical re-enactment, with paper plates marking the islands and strings with papers attached for the tolls. Students will often be tempted by the single digit numbers to assume the route has to pass that way. They may also miss the "crossover" using the central island, finding the 23 + 25 + 29 + 24 route and the 32 + 15 + 40 + 38 route but not the 32 + 15 + 40 + 38 route or the 32 + 15 + 29 + 24 one. Technically, paths that run from right to left along some bridges could be considered as well (for example, 32 + 5 + 41 + 40 + 29 + 24). However, for this particular example, such paths can be ignored since students will find a cheap enough path by only investigating paths that run from left to right. One might be tempted to modify the problem by having the student start with 99 coins, and ask if it is possible to reach the red island. However, such a modification would be inappropriate since a mathematically valid solution would require that the student consider all paths, including those that cross some bridges from right to left.

Solution:Solution

28 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 32 + 15 + 29 + 24 = 100 Other possible routes: 23 + 51 + 3 + 24 = 101 23 + 25 + 29 + 24 = 101 23 + 25 + 40 + 38 = 126 32 + 15 + 40 + 38 = 116 32 + 5 + 41 + 38 = 116

2.NBT How Many Days Until Summer Vacation? Alignment 1: 2.NBT.B.7 Domain NBT: Number and Operations in Base Ten Cluster Use place value understanding and properties of operations to add and subtract. Standard Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

MATERIALS  Paper

 Pencil

 Hundreds board

 Base-ten blocks

ACTIONS Pose this problem to the children: We are in school 180 days. Today is the 124th day of school. How many more days until we are out of school for summer vacation? Explain how you know. Commentary:

The purpose of the task is to allow children an opportunity to subtract a three-digit number including a zero that requires regrouping. The solutions show how students can solve this problem before they have learned the traditional algorithm. Children need to be familiar with the 100s board, base ten blocks, counting on, and counting backwards. The solutions given make sense to children and are often easier for them to explain and justify than using the traditional algorithm. The complexity of this task can be modified depending on the day such a problem is given. For example, on the 110th day of school, the subtraction is very simple: once students can count by tens, they can easily solve it. This task could also be used to give students extra practice outside of math instructional time; for example, if

29 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 teachers go over the calendar at some point during the day, such a problem could be given then as a way to get them doing math, even though they're not in math time.

Solution:1 Count by ones to the next decade: 125, 126, 127, 128, 129, 130. So we have added 6. Then count by tens: 140, 150, 160, 170, 180, which makes 5 tens or 50. Since 50 + 6 = 56, there are 56 days left before we get out of school for summer vacation. Students can use the 100s board or draw an "empty number line" to help them count.

Solution:2

Count by tens from 124: 134, 144, 154, 164, 174, 184. That is 6 tens or 60. But 184 is too big so count back to 180, which is 4. Subtract 4 from 60 to get 56. We have 56 days of school left.

Solution:3

Using base-ten blocks: Start with 1 hundred, 8 tens, 0 ones. We can’t take away 4 ones from 0 ones so we have to break a ten into 10 ones. I now have 7 tens 10 ones. Now we can subtract using the take-away model. 10 - 4 = 6. 70 - 20 = 50. 100 - 100 = 0. So there are 56 days of school before we get out for summer vacation. Here is a picturing showing the 100s, 10s and 1s.

2.MD Growing Bean Plants Alignment 1: 2.MD.D.9 Domain MD: Measurement and Data Cluster Represent and interpret data. Standard Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

a. Students in pairs grow bean plants from seed. In a small glass jar, put a roll of paper towel inside, cut to the height of the jar. Place a bean seed between the paper towel and the glass. Pour around 1 centimeter of water into the bottom of the jar. Place the jar on a window ledge where it will receive light, but preferably not direct sunlight. b. When the bean seeds start to grow, have the students measure their bean plant every couple of days or so. Students record each measurement with the date in their science books. c. Record the student pairs’ data on a line plot every few days, depending on how fast the plants grow. Prepare a blank line plot for each ‘measurement day’, and invite each student pair to record the height of their plant on the line plot. d. Each time the line plot is finished, discuss the patterns evident in the line plot, and compare with earlier line plots. Commentary:

Growing bean plants is a common science activity at this age. This task adds some rigor to the activity, by collecting actual growth data, providing practice for students in measuring and recording length measurements. Centimeters are an appropriate unit for these measurements, as they provide a good level of precision for these measurements, while being easy enough for students to handle. Students will need instruction on how to measure accurately with a ruler. Provide students with rulers marked in centimeters, and point out that a measurement has to begin at the ‘0’ line. Teach students to measure from the bottom of the seed to the top of the longest shoot to the nearest centimeter. The line plot provides some useful conceptual scaffolding for this task, helping students to understand how the ‘X’s marked on the graph each represent a specific bean plant. Having student pairs record their own ‘X’ will help them to identify with ‘their’ mark on the graph.

31 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 The discussion about the line plots helps to prepare students for more in-depth investigations in later grades, looking at data sets and calculating means, modes standard deviations and so on. At the grade two level, students can be expected to understand that the ‘shape’ of the graph shows features of the plants as a group (in scientific language, a ‘population’). In the discussion, highlight features such as the outliers (plants that are much taller or shorter than the majority) and the most common sizes. As the plants grow, successive line plots should show all the plants’ data points on the line plots appearing further and further to the right. Other suggested questions or discussion topics:  Why is a specific bean plant taller than the rest, or shorter than the rest? Could it be due to different amounts of water, different light levels, or something else?  What else do you know that grows to different sizes? Talk about pets, farmers’ crops, and people - each individual is a different height, and variation in heights is normal in any population.  What will the graph look like the next time we measure the plants?  How tall will the plants become? How could we find out the expected height (check the seed packet, look online, etc.)? Submitted by Peter Price to the Illustrative Mathematics Task Writing Contest Jan 17 – Jan 30, 2012

Solution:Solution (example) Week 1 Line Plot:

Week 2 Line Plot:

2.MD Hand Span Measures Alignment 1: 2.MD.D.9

Domain MD: Measurement and Data

Cluster Represent and interpret data.

Standard Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

Hand span is a measure of distance from the tip of the thumb to the tip of the little finger with the hand fully extended.  Each student places his or her dominant hand on the edge of a piece of paper with the hand fully extended.

 The student should make a mark at the tip of the thumb and the tip of the little finger. The distance between marks is the length of the hand span.

 The student should measure his or her hand span with a centimeter ruler and round the measurement to the nearest whole centimeter.

 Each student should record his or her measurement on a piece of paper.

The teacher can ask each student for his or her measurement and record the data using a line plot with a horizontal scale marked off in whole centimeters. Alternatively, the teacher can set up the line plot and ask each student to come record his or her own hand span, showing the students how by recording the teacher's hand span. Students should comment about patterns they observe the line plot and write or discuss the answers to these two questions: 33 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 a. What are the largest and smallest spans? What is the difference between the largest and smallest spans? b. Use words to describe the shape of the data set. Does it appear taller in the center like a mountain? Are there peaks in more than one place? Is the shape of the data flat like a table top? Are there gaps? Are some hand spans much bigger or smaller than the others? Commentary:

The size of the hand makes a difference in some sports that involve throwing or catching and some activities such as playing the piano. Hand span is a measure that has been used for many years. By placing the hand on the edge of a piece of paper and marking the tips of the thumb and little finger, the student can measure a straight line. This is a better method than placing the hand directly on the ruler. Discuss rounding conventions. This could be used as a class activity, or students could gather and plot data on separate line plots from different age groups. Submitted by Miriam L. Clifford for Illustrative Mathematics Task Writing Contest Jan 17 - Jan 30, 2012

Solution:Solution Students from ages 6 – 12 usually have a span in the range of 15 to 20 cm, with an average of about 17 cm. The teacher or students should record class data by placing an X above the appropriate number on the horizontal scale for each hand span that is measured. Or, provide a square post-it note to each student. Each student records his or her hand span on the note and posts it directly above the appropriate number on the horizontal line plot. The scale on the graph should match the size of the post-it notes. Post-it notes should be placed edge- to-edge and should not overlap. Here is data from Jackie Giacalone's 2nd grade classes at Waukesha STEM Academy – Randall Campus in Waukesha, WI:

a. The difference between the largest and smallest measures is the range. This number is helpful in describing the “spread” of the data. b. Encourage students to think of the data as a collection of points that form a shape or picture. They should use their own words to describe the shape and any interesting features of the data set.

2.MD The Longest Walk Alignment 1: 2.MD.D.9 Domain MD: Measurement and Data Cluster Represent and interpret data. Standard Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

a. Pick two points on the outside borders of the United States map (excluding Hawaii and Alaska) so that the line between them stays within the borders. Draw the line. How far apart are the points? Measure the length of the line to find out. Do this 10 times and make a line-plot of your data. b. Starting anywhere on the map of the United States and drawing in a straight line until hitting a border, what is the longest line you can draw? It might help to ask your classmates what lengths they found as well.

Commentary:

The language in this task is above the reading level of many second grade students, so it is best for the teacher to explain the task to the students verbally. After students have drawn and measured their ten line segments, it might be more useful for the class to discuss part (b) as a whole group. It is a good idea to have the students use color to help them keep track of the connection between a line that they have drawn and the corresponding data point on the graph. If students draw 20 data points rather than 10, the lines will start to be hard to see so spare maps are recommended for students who need them. As this is a second grade standard, students are expected to round rather than use fractions, but some classroom discussion can include how some distances might be more than others even though they round to the same thing. The idea of the scale map is beyond this grade level but can be discussed in a holistic sense: how the longest line on the map will represent the longest line in the real United States. If a more concrete version of the activity is desired, it can be performed in an outdoor setting -- a playground, for instance -- on any irregular shape. Public domain US map, with different sizes: https://commons.wikimedia.org/wiki/File:Blank_US_map_borders.svg Doing the activity on a large wipeable mat will allow students to experiment. The size of the map and the type of units (centimeters or inches) should be chosen at the instructor's discretion to allow a reasonable difficulty level. (The example given in the answers uses the 1000px size from the link above.) Cliff Pickover has written about the problem: http://sprott.physics.wisc.edu/pickover/pc/american-line.html Submitted by Jason Dyer for Illustrative Mathematics Task Writing Contest Jan 17 - Jan 30, 2012

Solution:Solution a. Sample of data collection: 37 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012

b. The longest walk is from Washington to Florida. Note that getting the correct answer is not a trivial task, even for adults, so any sufficiently long line should be acceptable and will allow for a good classroom discussion as students will likely have alternate answers.

2.G Which Pictures Represent One Half? Alignment 1: 2.G.A.3 Domain G: Geometry Cluster Reason with shapes and their attributes. Standard Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

a. Which pictures show one half of the shape shaded? Explain.

b. Is more or less than one half of the shape shaded in (ii)? Explain.

c. Is more or less than one half of the shape shaded in (iv)? Explain.

Commentary: 39 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 The purpose of this task is for students to see different ways of partitioning a figure into two or more equal shares, by which we mean decomposing the figure into "pieces" with equal area. In third grade, students should be able to reason in general about the relative size of unit fractions based on the idea that they are created by partitioning a whole into a certain number of equal shares; in second grade they begin to build this understanding by working with geometric figures that represent the unit fractions 1/2 ,1/3 , and 1/4 . For picture (i), students will intuitively see that the circle is composed of two pieces with equal area and that one half of the circle is shaded. From here, it is not a very big stretch for students to see that the circle in (ii) is composed of three pieces with equal area and that one of these is shaded and that this is less than one half. It is less intuitive for students to think of the shaded region in picture (iii) as representing one half. This is in part due to the fact that the pieces are less symmetrical, and the transformation needed to show they are the same size and same shape is harder to visualize for some students. Students struggling with this would benefit from having some tracing paper that would allow them to physically demonstrate that one can be rotated to exactly match up with the other. For picture (iv), students need to compare areas of different shapes and decide whether the shaded area is more or less than one half. If some students suggest that one half of this picture is shaded, the teacher may need to remind student that the shaded and unshaded regions must have equal area.

Solution:1

i. One half of the circle is shaded because the shaded region and the unshaded region have the same shape and size, making each one half of the full circle. This is perhaps the picture most frequently associated with the words "half of a circle" or "semi-circle."

ii. Less than one half of this circle is shaded. Visually, the shaded area is far less then the unshaded area while the two would be equal if the shaded area represented one half of the circle. Mathematically, the shaded area represents one third of the circle since one of three equal pieces is shaded. One way to see this is to note that if a cookie is divided into 3 equal shares, each share is less than if the same cookie is divided into two equal shares.

iii. One half of this rectangle is shaded. This can be seen geometrically as the shape and size of the black triangle is identical to the shape and size of the white triangle, making each one half of the whole rectangle. Concretely, if the rectangle is cut in half and then the white triangle is rotated by 180 degrees about the center of the rectangle, then it will lie precisely on top of the black triangle.

iv. More than one half of this rectangle is shaded. Visually the black shaded piece is far larger than the white unshaded piece and so this picture does not show one half.

b. In picture (ii), the shaded area is equal to each of the two unshaded pieces and so the unshaded area is larger than the shaded area (twice as large). So the shaded area represents less than one half. c. In picture (iv), the shaded part of the rectangle is larger than the unshaded part and so the shaded area is more than one half of the area of the rectangle. Unlike picture (ii) which represents one third, however, the lower picture does not provide enough information to conclude precisely what fraction of the rectangle is shaded.

40 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 2.G Representing Half of a Rectangle Alignment 1: 2.G.A.3 Domain G: Geometry Cluster Reason with shapes and their attributes. Standard Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

Ms. Nim gave her students a picture of a rectangle. Then she asked them to shade in one half of the rectangle. Here are three pictures:

Which ones show 12/ ? Explain. Commentary:

This task is for assessment purposes, providing a context for indentifying different ways of representing half of an object, a rectangle in this case. The task may also be used for instructional purposes but if so the teacher may wish to introduce some other ways of showing one half of the rectangle, such as dividing along a diagonal (and shading in one piece) or dividing it into four equal pieces, shading in two pieces that only touch at a corner. Teachers are referred to ``Which pictures show half of a circle?'' for more variants on this theme.

Solution:1

41 [email protected] www.illustrativemathematics.org 2nd Grade Tasks 12/2/2012 a. In the picture on the top left, the rectangle has been divided in half vertically while in the picture on the top right it has been divided in half horizontally. In each case, the big rectangle has been divided into two equal pieces, one shaded and one unshaded. So the shaded area represents one half of the big rectangle in both cases. In the third picture, each side of the rectangle has been divided in half but the small rectangle does not represent one half of the area of the large rectangle. As can be seen in the picture below, the large rectangle can be divided into four rectangles, each equal to the small one so the shaded area only represents one fourth of the area of the large rectangle:

The student who drew the lower incorrect picture does show some understanding of the fraction one half. Each side of the rectangle has been cut in half, demonstrating an understanding of the fraction one half in the context of measuring a length. The mistake is that when both linear measurements are cut in half, the area is only one fourth the area of the original shape.

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