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Learning to count by 2s is a first\n step toward learning to count objects by 2s and also lays the groundwork for understanding even and odd numbers. Counting by 10s involves relating the 1-to-10 counting\n sequence to its counterpart in the sequence of multiples of 10, 20, 30, 40, and so on. While counting by 10s by rote isn\u2019t a guarantee that students understand the role\n of 10 in our place-value system, learning this sequence is a first step.</p>",solveMentallyText:null,nextButtonText:"Next, listen to Imani count by 2s"},selectExplanation:null,listenTo:{studentId:"56",navText:"Listen to a Student Count by 2s",prompt:"LISTEN TO IMANI COUNT BY 2s",videoId:"34ni7nu8bj",nextButtonText:"Next, reflect on how Imani counted",reflect:{prompt:"Use the following prompts to think about how Imani counted.",reflection:"<ol>\n <li>What does Imani understand about counting by 2s?</li>\n <li>How would you describe Imani\u2019s confidence when counting by 2s?</li>\n <li>What might help Imani learn to count by 2s? </li>\n </ol>",nextButtonText:"Next, select a response"},select:{prompt:"Select A Response",description:"Which description matches how Imani counted?",explanationId:"3",reason:'<p>Mallika asked <b>Imani</b>, <i><b>"Can you count by 2s?"</b></i> She immediately started to count by 1s.\n When she got to 7 Mallika said she could stop and asked, <b><i>"Do you know how to count by 2s\n also?"</i></b> (By rephrasing the question Mallika wanted to make sure Imani understood what she\n was asking her to do.) Mallika began the sequence by saying <b><i>"Two, four,"</i></b> and then asked\n Imani if she could keep going with the 2s.</p>\n <p>Mallika asked Imani if she had done that before. Imani nodded her head and answered, <b><i>\u201cTwo, four, six, eight, nine.\u201d</i></b> When Mallika asked her if she could count any higher, Imani said, <b><i>"I want to stop."</i></b> Mallika smiled warmly and said,\n <b><i>"Okay.\u201d</i></b> Imani demonstrated a beginning awareness of the counting sequence with 2s, but she did not relate the counting sequence to counting objects.\n </p>',nextButtonText:"Next, listen to other students count by 2s"}},galleries:[{id:"1",displayOrder:1,navText:"Listen to Other Students Count by 2s",title:"LISTEN TO OTHER STUDENTS COUNT BY 2s",description:"After you listen to each student, you'll have the opportunity to select the description that matches how the student counted.",questionId:"1.4",studentExamples:[{studentId:"50",displayOrder:1,title:"LISTEN TO ALINA COUNT BY 2s",videoId:"yycsdwinlt",prompt:"After you listen, select the description that matches how Alina counted.",explanationId:"3",reason:"<p>\n When Marilyn asked <b>Alina</b> if she knew how to count by 2s, Alina nodded confidently and began, <b><i>'Two, four.'</i></b> She then paused to think. After a moment, she continued counting in a quiet whisper to herself. It was difficult to hear how Alina was counting or where she stopped, but it was clear that she was stumped and then shrugged her shoulders. Marilyn went on to the next question in the interview. Later in the interview when Marilyn asked her if she knew how to count by 10s, Alina wasn't able to do so. She needs more practice counting by equal groups.\n </p>",nextButtonText:"Next, listen to Alberto"},{studentId:"92",displayOrder:2,title:"LISTEN TO ALBERTO COUNT BY 2s",videoId:"mjw12u6way",prompt:"After you listen, select the description that matches how Alberto counted.",explanationId:"1",reason:"<p>\n When Lynne asked <b>Alberto</b> to count by 2s, he immediately reached for cubes that were on the table. Lynne asked him to just say the numbers first,\n but Alberto took two cubes and said, <b><i>\u201cTwo.\u201d</i></b> As he was thinking what would come next, Lynne pushed two more cubes toward him, and\n Alberto said, <b><i>\u201cFour.\u201d</i></b> From there, Alberto continued the count on his own using the cubes. He added two more and said, <b><i>\u201cSix,\u201d</i></b>\n and then two more and said, <b><i>\u201cEight.\u201d</i></b> When he added the next two cubes he said, <b><i>\u201cNine.\u201d</i></b> He quickly realized his error and said,\n <b><i>\u201cWhoops, this is 10.\u201d</i></b> He thought for a long while, and then went on to 12, 14 and 16. At that point, he ran out of cubes.\n Alberto understands the concept of skip counting, but would benefit from practice with the number sequence when counting by 2s.\n </p>",nextButtonText:"Next, listen to students count by 10s"}]}
1,{id:"2",displayOrder:2,navText:"Listen to Students Count by 10s",title:"LISTEN TO STUDENTS COUNT BY 10s",description:"After you listen to each student, you'll have the opportunity to select the description that matches how the student counted.",questionId:"1.8",studentExamples:[{studentId:"56",displayOrder:1,title:"LISTEN TO IMANI COUNT BY 10s",videoId:"vzuikxbcsp",prompt:"After you listen, select the description that matches how Imani counted.",explanationId:"1",reason:"<p>\n <b>Imani</b> correctly and deliberately counted by 10s and stopped at 100. When Mallika asked Imani if she knew what would come next if she continued to count by 10s, she shook her head no. Typically, young students first learn to count by 10s to 100 but do not have experience extending the pattern.\n </p>",nextButtonText:"Next, listen to Joaquin"},{studentId:"2",displayOrder:2,title:"LISTEN TO JOAQUIN COUNT BY 10s",videoId:"7793jvqp5q",prompt:"After you listen, select the description that matches how Joaquin counted.",explanationId:"1",reason:"<p>\n <b>Joaquin</b> confidently counted by 10s from 10 to 100. When Marilyn asked him what would come next if he continued to count by 10s, Joaquin hesitated and then said, <b><i>\u201cOne hundred and one.\u201d</i></b> He appeared to be unsure if that was correct. Joaquin has learned and practiced counting by 10s to 100. He knows what comes after 100 when counting by 1s and will benefit from additional counting experiences.\n </p>",nextButtonText:"Next, listen to Alberto"},{studentId:"92",displayOrder:3,title:"LISTEN TO ALBERTO COUNT BY 10s",videoId:"t0bzv2i3p9",prompt:"After you listen, select the description that matches how Alberto counted.",explanationId:"3",reason:"<p>\n After counting by 10s to 50, <b>Alberto</b> stopped and asked Lynne, <b><i>\u201cWas that good?\u201d</i></b>\n Lynne acknowledged what he had done so far and sat waiting. Alberto continued and said 16, but quickly\n self-corrected to 60. He continued counting by 10s correctly to 80, paused again, and had to think for a bit before saying 90.\n Alberto couldn\u2019t come up with 100, although he thought for a long time. When Lynne asked what he thought would come\n after 90, he said, <b><i>\u201cA million.\u201d</i></b> Alberto would benefit from additional practice with counting by 10s.\n </p>",nextButtonText:"Next, listen to Manoah"},{studentId:"94",displayOrder:4,title:"LISTEN TO MANOAH COUNT BY 10s",videoId:"hynyx645yn",prompt:"After you listen, select the description that matches how Manoah counted.",explanationId:"3",reason:'<p>\n When Amy asked <b>Manoah</b> if he could count by 10s, he responded that it would be easier with the ten-frames that he used in class. Amy pushed, asking him if he knew the counting numbers by 10s without the ten-frames. Manoah answered, <b><i>\u201cNot really.\u201d</i></b> Manoah would benefit from counting by 10s, with and without <span class="lab__no-wrap">ten-frames.</span>\n </p>',nextButtonText:"Next, listen to Camdynn"},{studentId:"5",displayOrder:5,title:"LISTEN TO CAMDYNN COUNT BY 10s",videoId:"p3iow3cwms",prompt:"After you listen, select the description that matches how Camdynn counted.",explanationId:"2",reason:'<p>\n <b>Camdynn</b> confidently counted by 10s and, without prompting, continued past 100. Le\u2019Vada stopped her at 160, but it seemed as if Camdynn easily would have continued further. The previous students you watched counting by 10s were in kindergarten. Camdynn was in first grade and had a year more of mathematical instruction and experiences working with 10s and 1s. It\u2019s important to provide students on-going counting activities <span class="lab__no-wrap">throughout the grades</span>.\n </p>',nextButtonText:"Next, read the suggestions for class instruction"}]}],suggestions:[{title:"<b>Class Lesson:</b> <i>Patterns on the 1-to-100 Pocket Chart</i>",displayOrder:1,text:"<p>The 1-to-100 pocket chart is a useful visual tool for helping students become comfortable with the sequence of numbers up to 100. The chart has the advantage that cards can be turned over to their blank sides to highlight number patterns. For example, turn over all the cards except for the multiples of 10 from 10 to 100. Have students count aloud as you point. Talk about the pattern that these numbers are all in the same column. Leave the chart up to help students become familiar with the sequence. Repeat at other times for the 2s and 5s.</p>"},{title:"<b>Class Lesson:</b> <i>
1Counting Cubes by 2s</i>",displayOrder:2,text:"<p>Give each student one connecting cube. Ask each to find a partner and snap their cubes together. (If there's a student without a partner, include yourself.) Collect the pairs of cubes and model counting them by 2s. It's helpful to have a student move each pair of cubes as you count while you point to the numbers on a <span class=\"lab__no-wrap\">1-to-100</span> chart.</p>"},{title:"<b>Individual or Partner Activity:</b> <i>Ten at a Time</i>",displayOrder:3,text:"<p>Ten-frames give students a visual representation when learning to count by 10s. Give partners a collection of objects, paper or plastic cups, and a ten-frame. Ask them to place objects to fill the ten-frame and then put those objects into a cup. Ask them to place objects to fill the ten-frame and then put those objects into a paper cup. They repeat until they make all the groups of 10s possible and put each into a separate cup. Then they count by 10s, count on by 1s for the extras, and record how many they have. Repeat on other days with different sets of objects. (One student observed, 'Using ten-frames makes counting easy. Just fill it in an you know you have 10, and you don't have to count.')</p>"},{title:"<b>Children's Book:</b> <i>The King's Commissioners</i> by Aileen Friedman",displayOrder:4,text:"<p>In this delightful story, 47 commissioners are counted by 2s, 5s, and 10s. The book can be used to demonstrate how things can be counted in different ways and still get to the same number. Count the number of students in your class by organizing them into 2s, 5s and 10s, then counting, and then adding on the extras by counting on by 1s.</p>"}]},{id:"9",displayOrder:2,interviewId:"3",interviewName:"Addition and Subtraction Within 20",contentFocus:a.n7.AdditionSubtraction,title:"Anna had $20. She spent $7. How much money does Anna have left?",questionIds:["3.9"],getStarted:{description:'<p>Interview 3 includes five subtraction problems, and four of the five problems are contextual. Money is the context for this question, and the problem asks the student to figure out how much money Anna has left after spending $7. The situation can be numerically represented as $20 – $7. However, it\u2019s also possible, as some students have done, to interpret the problem as a missing addend problem, $7 + <span class="lab__blank-underline small"></span> = $20.</p>',solveMentallyText:"How would you explain why $13 is the correct answer?",nextButtonText:"Next, select your explanation"},selectExplanation:{nextButtonText:"Next, listen to Katie answer the question"},listenTo:{studentId:"12",navText:"Listen to a Student Answer the Question",prompt:"LISTEN TO KATIE ANSWER THE QUESTION",videoId:"opw13dalpy",nextButtonText:"Next, reflect on Katie's explanation",reflect:{prompt:"Use the following prompts to think about Katie\u2019s explanation.",reflection:"<ol>\n <li>Katie asked Marilyn if she could use her fingers. How did Marilyn\u2019s reaction support Katie as she tried to find a solution?</li>\n <li>Compare Katie\u2019s first attempt to her final attempt. What could you infer about the value of being neutral but supportive, providing wait time, and providing the choice to return to the problem?</li>\n <li>If students were solving this problem during a class number talk, how might you numerically record Katie\u2019s second explanation?</li>\n </ol>",nextButtonText:"Next, select the explanation that is the best match"},select:{prompt:"Select an Explanation",description:"Which explanation best matches how Katie reasoned?",explanationId:"1",reason:'<p><b>Katie</b> interpreted this contextual problem as a missing addend situation and worked to solve the problem 7 + <span class="lab__blank-underline small"></span> = 20. She made several false starts, remaining stuck but demonstrating impressive perseverance. She declined Marilyn\u2019s offer to skip the problem but then agreed to return to it later. When she revisited the problem, Katie had a new insight and explained that she thought of the 20 as 10, knew that 7 plus 3 was 10, and reasoned, therefore, that 7 plus 13 was 20. It\u2019s important to note that Marilyn gave Katie all the time she needed to think, without interruption. Practicing wait time is an important aspect of interviewing.</p>',nextButtonText:"Next, listen to other students answer the question"}},galleries:[{id:"1",displayOrder:1,navText:"Listen to Other Students Answer the Question",title:"LISTEN TO OTHER STUDENTS ANSWER THE QUESTION",description:"After you listen to each student, you\u2019ll have the opportunity to select the explanation that most closely matches the student's reasoning.",questionId:"3.9",studentExamples:[{studentId:"53",displayOrder:1,title:"LISTEN TO GENESIS",videoId:"u4lu89kamw",prompt:"After you listen, select the explanation that most closely matches Genesis\u2019s reasoning.",explanationId:"2",reason:"<p>After taking time to think, <b>Genesis</b> gave the correct answer, 13. When Mallika prompted her to explain, Genesis first restated the question in her own words, <b><i>\u201cThere was 7 and then it\u2019s 20.\u201d</i></b> Then she translated the situation to a subtraction problem, <b><i>\u201cso it\u2019s 20 take away 7.\u201d</i></b> (This was different from how Katie translated the situation to a missing addend problem.) Genesis broke apart the 7 into 2 and 5 and subtracted twice, first 20 \u2212 2 and then 18 \u2212 5.<
1/p>\n <p>Here\u2019s a way to record Genesis\u2019s reasoning:</p>\n <ul>\n <li>20 \u2212 2 = 18</li>\n <li>18 \u2212 5 = 13</li>\n </ul>",nextButtonText:"Next, listen to Jeff"},{studentId:"32",displayOrder:2,title:"LISTEN TO JEFF",videoId:"iix0d3cwiw",prompt:"After you listen, select the explanation that most closely matches Jeff\u2019s reasoning.",explanationId:"2",reason:"<p>After Lynne asked the question, <b>Jeff</b> quickly and confidently responded correctly with $13. He explained, <b><i>\u201cI would go minus 5 to 15 and minus 2 to 13.\u201d</i></b> While Jeff didn\u2019t state where he got the 5 and the 2, we can assume that he split the 7 into 5 and 2, which is what Genesis did when she solved the problem. Unlike Genesis, Jeff first subtracted 5 and then 2.</p>\n <p>Here\u2019s a way to record Jeff\u2019s reasoning:</p>\n <ul>\n <li>20 \u2212 5 = 15</li>\n <li>15 \u2212 2 = 13</li>\n </ul>",nextButtonText:"Next, listen to Valeria"},{studentId:"101",displayOrder:3,title:"LISTEN TO VALERIA",videoId:"pfuvknkpq8",prompt:"After you listen, select the explanation that most closely matches Valeria's reasoning.",explanationId:"6",reason:"<p><b>Valeria</b> began to solve the problem by counting to herself, using her fingers. She came up with the incorrect answer of 17. Without giving Valeria any feedback that her solution wasn\u2019t correct, Lynne asked Valeria to count out loud so she (Lynne) could hear. This time Valeria self-corrected and got the correct answer of 13 by starting at 20 and keeping track with her fingers to count back 7. Like Katie, Valeria began by trying to use her fingers to solve the problem. However, unlike Katie, who tried counting up from 7, Valeria counted back from 20 and was able to use her fingers to solve the problem correctly.</p><p>Here\u2019s a way to record Valeria\u2019s reasoning:</p>\n <ul>\n <li>20, . . . , 19, 18, 17, 16, 15, 14, 13</li>\n </ul>",nextButtonText:"Next, listen to Holden"},{studentId:"13",displayOrder:4,title:"LISTEN TO HOLDEN",videoId:"3uvi1olp8e",prompt:"After you listen, select the explanation that most closely matches Holden's reasoning.",explanationId:"2",reason:"<p>After Marilyn read the problem, <b>Holden</b> translated the context to a numerical problem, <b><i>\u201cIt\u2019s 20 minus 7.\u201d</i></b> After having a sort of mathematical breakdown because she didn\u2019t know the answer from memory, Holden buckled down. Like Genesis and Jeff, she decomposed 7 in order to subtract it. However, while both Genesis and Jeff split 7 into 2 + 5, Holden split 7 into 2 + 2 + 2 + 1. As with Katie, Holden\u2019s persistence was impressive, and Marilyn encouraged her with her full attention, patience, and a gentle smile.</p>\n <p>Here\u2019s a way to record Holden\u2019s reasoning:</p>\n <ul>\n <li>20 \u2212 2 = 18</li>\n <li>18 \u2212 2 = 16</li>\n <li>16 \u2212 2 = 14</li>\n <li>14 \u2212 1 = 13</li>\n </ul>",nextButtonText:"Next, listen to Mark"},{studentId:"34",displayOrder:5,title:"LISTEN TO MARK",videoId:"ry88julvwz",prompt:"After you listen, select the explanation that most closely matches Mark's reasoning.",explanationId:"3",reason:"<p><b>Mark</b> reasoned the way that Katie did, but without the mental anguish that Katie demonstrated. He gave the correct answer of $13 shortly after Nishi completed reading the problem and explained that he knew that 10 \u2212 7 = 3 <b><i>\u201cand then I know if you put 10 more to the 3, it equals 13.\u201d</i></b></p>\n <p>Here\u2019s a way to numerically record Mark\u2019s thinking:</p>\n <ul>\n <li>
110 \u2212 7 = 3</li>\n <li>10 + 3 = 13</li>\n </ul>",nextButtonText:"Next, read the suggestions for class instruction"}]}],suggestions:[{title:"<b>Number Talks:</b> <i>Adding and Subtracting Within 20</i>",displayOrder:1,text:'<p>Here are some examples of problems to use in number talks that help to focus on subtraction within 20. Ask students: How would you solve these problems using subtraction? How would you solve these problems using addition?</p>\n <ul class="lab__show-bullets">\n <li>20 \u2212 11</li>\n <li>17 \u2212 9</li>\n <li>20 \u2212 14</li>\n <li>14 \u2212 9</li>\n <li>13 \u2212 8</li>\n <li>20 \u2212 7</li>\n </ul>'},{title:"<b>Individual or Partner Activity:</b> <i>Word Problems Within 20</i>",displayOrder:2,text:'<p>Ask students to write word problems that match a subtraction problem. Here are two examples for the problem 20 \u2212 11 = <span class="lab__blank-underline small"></span>.</p>\n <ul class="lab__show-bullets">\n <li>Lily borrowed 20 books from the library. She read 11 of the books. How many more books does she have left to read?</li>\n <li>Maria has 20 marbles. Eleven of them are green. The other marbles are blue. How many marbles are blue?</li>\n </ul>'},{title:"<b>Game:</b> <i>Strike It Out</i>",displayOrder:3,text:"<p>This game gives students practice with adding and subtracting within 20 and also involves them in thinking strategically. Draw a number line from 1 to 20 and then play with a student to introduce the game to the class.</p>\n <ul class=\"lab__show-bullets\">\n <li>Player 1 picks two numbers, crosses them out, and circles either their sum or difference.</li>\n <li>Player 2 crosses out the circled number and another number that\u2019s still left, and again circles the sum or the difference.</li>\n </ul>\n <p><img src='/images/tech-art-images/reasoninglab_3.1_StrikeItOut.svg' alt='Number line labeled 0-20 with tick marks for each; red diagonal slash mark on 1, 7; red circle and diagonal slash mark on 8; red circle around 20.' class=\"lab__tech-art\"/></p>\n <p>The winner is the player who makes the last possible move. When students understand how to play, have them play in pairs. This game (and others) is from\n <a href='http://nrich.maths.org' target='_blank'>The NRICH Maths Project</a>.\n </p>"}]},{id:"14",displayOrder:3,interviewId:"6",interviewName:"Foundations of Multiplication",contentFocus:a.n7.MultiDivision,title:"What is 7 x 6?",questionIds:["6.5"],getStarted:{description:"This question is one of four in the interview that spot-check fluency with multiplication\n facts. What is most important is that fluency comes after understanding. Students need to have ways\n to reason and make sense of multiplication as well as to memorize the times tables. Studying\n multiplication requires students to shift from thinking additively to thinking multiplicatively.\n Additive thinking, such as adding 7 + 6, calls for figuring out \u201chow many\u201d are in the two sets\n altogether. Multiplicative thinking, such as for multiplying 7 \xd7 6, calls for figuring how many there\n are in seven groups with six items in each group. This is a shift from having numbers represent \u201chow\n many\u201d to having numbers mean either \u201chow many groups\u201d or \u201chow many are in a group.\u201d The students\u2019\n explanations provide information about the ways in which they make sense of multiplication.",solveMentallyText:"How would you explain why 42 is the correct answer?",nextButtonText:"Next, select your explanation"},selectExplanation:{nextButtonText:"Next, listen to Michael answer the question"},listenTo:{studentId:"38",navText:"Listen to a Student Answer the Question",prompt:"LISTEN TO MICHAEL ANSWER THE QUESTION",videoId:"4g91s73381",nextButtonText:"Next, reflect on Michael's explanation",reflect:{prompt:"Use the following prompts to think about Michael's explanation.",reflection:"<ol>\n <li>What confused Michael when he first tried to solve the problem?\n </li>\n <li>What do you know about Michael\u2019s understanding of multiplication?\n </li>\n <li>How might you record Michael\u2019s reasoning?\n </li>\n </ol>",nextButtonText:"Next, select the explanation that is the best match"},select:{prompt:"Select an Explanation",description:"Which explanation best matches how Michael reasoned?",explanationId:"2",reason:"<p><b>Michael</b> made a false start and, tentatively, gave the incorrect answer of 57. Rather\n than give Michael feedback that his answer was incorrect, Marilyn asked him to\n explain how he figured. We find that students often self-correct when they have the\n opportunity to verbalize their reasoning. Michael had relied on the answer he got\n for the previous question, that 4 \xd7 6 equals 24. Then he explained that he knows\n that 3 \xd7 6 is 18, and that he needed to add 24 plus 18 to get the answer to 7 \xd7 6.\n That addition was a bit of a stretch for Michael to do mentally so he regrouped,\n both revisiting his thinking and grouping the numbers in another way. He\n skip counted to figure out 3 \xd7 7 to get 21 and then added 21 plus 21.\n </p>\n <p>Although Michael did not use precise mathematical language, he successfully\n demonstrated his ability to apply the distributive property, decomposing the 6 into\n 3 + 3, multiplying 3 \xd7 7 twice to get partial products, and then adding to get 42.\n </p>\n <p>Here\u2019s a way to record how Michael reasoned:</p>\n <ul>\n <li>7, 14, 21</li>\n <li>3 \xd7 7 = 21</li>\n <li>3 \xd7 7 = 21</li>\n <li>21 + 21 = 42</li>\n </ul>",nextButtonText:"Next, listen to other students answer the question"}},galleries:[{id:"1",displayOrder:1,navText:"Listen to Other Students Answer the Question",title:"LISTEN TO OTHER STUDENTS ANSWER THE QUESTION",description:"After you listen to each student, you'll have the opportunity to select the explanation that most closely matches the student's reasoning.",questionId:"6.5",studentExamples:[{studentId:"35",displayOrder:1,title:"LISTEN TO NATHAN",videoId:"zg6ubj7ucr",prompt:"After you listen, select the explanation that most closely matches Nathan's reasoning.",explanationId:"2",reason:"<p>\n <b>Nathan</b>\n thought for a bit and then gave the correct answer to Rusty. Nathan didn't know 7 \xd7 6 from memory, but figured correctly. While Michael figured out 7 \xd7 3 by skip counting, Nathan used addition, adding three 7s, explaining that 7 + 7 = 14 and 14 + 7 = 21. Then, instead of continuing to add 7s individually, he added another three 7s, adding 21 + 21. This shows a lot. Nathan\n understands the relationship between multiplication and addition, and that 7 times 6\n is the same as adding 7 six times. He also showed that he knows to figure out 7 \xd7 6\n by adding 7 \xd7 3 and 7 \xd7 3 again, which demonstrates an informal use of the\n distributive property.\n </p>\n <p>Here\u2019s a way to record how Nathan reasoned:</p>\n <ul>\n <li>7 + 7 = 14</li>\n <li>14 + 7 = 21</li>\n <li>7 + 7 + 7 = 21</li>\n <li>21 + 21 = 42</li>\n </ul>",nextButtonText:"Next, listen to Emilio"},{studentId:"17",displayOrder:2,title:"LISTEN TO EMILIO",videoId:"oi0cg5017a",prompt:"After you listen, select the explanation that most closely matches Emilio's reasoning.",explanationId:"2",reason:'<p>\n <b>Emilio</b>\n didn\u2019t give Rusty an answer to the question, but went right into his explanation. He\n used the known fact <sp
1an class="lab__no-wrap">7 \xd7 5</span>, and then said that you add a six or a seven, but quickly\n corrected himself and said <span class="lab__no-wrap"> <b> <i>\u201ca seven\u201d</i> </b> </span>. He then added the seven to the 35 to get the correct answer, 42.\n Students often get confused in a problem like <span class="lab__no-wrap">6 \xd7 7</span> as to whether to add a group of\n six or a group of seven when they are first learning about multiplication.\n </p>\n <p>Here\u2019s a way to record how Emilio reasoned:</p>\n <ul>\n <li>7 \xd7 5 = 35</li>\n <li>35 + 7 = 42</li>\n </ul>',nextButtonText:"Next, listen to Ella"},{studentId:"21",displayOrder:3,title:"LISTEN TO ELLA",videoId:"wu522cf9aw",prompt:"After you listen, select the explanation that most closely matches Ella's reasoning.",explanationId:"1",reason:"<p>\n <b>Ella</b>\n knew the answer immediately. When Nishi asked her to explain, she said you could\n count 7 six times or count 6 seven times.\n </p>\n <p>Here's a way to record how Ella reasoned:</p>\n <ul>\n <li>7, 14, 21, 28, 35, 42</li>\n <li>6, 12, 18, 24, 30, 36, 42</li>\n </ul>",nextButtonText:"Next, listen to Asmara"},{studentId:"95",displayOrder:5,title:"LISTEN TO ASMARA",videoId:"c33alr1ue9",prompt:"After you listen, select the explanation that most closely matches Asmara's reasoning.",explanationId:"2",reason:"<p>\n <b>Asmara</b>, like Ella, knew 7 \xd7 6 without having to figure. When Lynne asked\n Asmara how she knew, she said she had done multiplication and knew 6 \xd7 7. To learn if Asmara had access to reasoning strategies, Lynne asked her if she could think of a way to explain it to someone who hadn\u2019t learned it yet.\n Asmara decomposed 7 \xd7 6 into 6 \xd7 5 and 6 \xd7 2, demonstrating her ability to apply the\n distributive property. Asking students to further explain their thinking can\n often reveal insights into their numerical reasoning.\n </p>\n <p>Here\u2019s a way to record how Asmara reasoned:</p>\n <ul>\n <li>6 \xd7 5 = 30</li>\n <li>6 \xd7 2 = 12</li>\n <li>30 + 12 = 42</li>\n </ul>",nextButtonText:"Next, listen to Henry"},{studentId:"65",displayOrder:6,title:"LISTEN TO HENRY",videoId:"d0ovqwsmrc",prompt:"After you listen, select the explanation that most closely matches Henry's reasoning.",explanationId:"2",reason:'<p> <b>Henry</b> explained to Amy that he knew <span class="lab__no-wrap">7 \xd7 7 = 49</span> and needed to subtract seven to get 42. We have found through the years that multiplication facts with the same factors (e.g., <span class="lab__no-wrap">5 \xd7 5</span>, <span class="lab__no-wrap">6 \xd7 6</span>, etc.) are often some of the first problems students know from memory. Using a fact that you know to solve another problem is an effective strategy for solving multiplication problems. </p> <p>Here\u2019s a way to record how Henry reasoned:</p> <ul> <li>7 \xd7 7 = 49</li> <li>49 \u2212 7 = 42</li> </ul>',nextButtonText:"Next, listen to Amira"},{studentId:"22",displayOrder:7,title:"LISTEN TO AMIRA",videoId:"tnz71l5esj",prompt:"After you listen, select the explanation that most closely matches Amira's reasoning.",explanationId:"2",reason:'<p>\n <b>Amira</b>\n knew 7 \xd7 6 from memory, and her explanation to Rusty revealed that she was able to\n apply the distributive property. She used a known fact of 6 \xd7 6 and clearly explained that adding another group of 6 would equal 42, but she mistakenly said, <b><i>"and that would be six groups of seven."</i></b>\n </p>\n <p>Here\u2019s a way to record how Amira reasoned:</p>\n <ul>\n <li>6 \xd7 6 = 36</li>\n <li>36 + 6 = 42</li>\n </ul>',nextButtonText:"Next, read the suggestions for class instruction"}]}],suggestions:[{title:"<b>Number Talks:</b> <i>Multiplying with One-Digit Factors</i>",displayOrder:1,text:"<p>Use the problem of 7 \xd7 6 (or other one-digit multiplication problems) with your class with\n the goal of having the students come up with as many ways as possible to solve it. Record\n the students\u2019 methods. Then, show the video clips from this lab to the class, one at a time,\n and discuss if each video explanation matches one of their explanations.\n </p>"},{title:"<b>Game:</b> <i>Circles and Stars</i>",displayOrder:2,text:"<p>
1This two-person game is easy for students to learn and play, and is valuable for providing an\n initial experience with multiplication. The game presents a visual representation of\n multiplication as equal groups. Taking turns, each player rolls a 1\u22126 number cube to find\n out how many circles to draw, and then rolls the number cube again to find out how many\n stars to draw in each circle. The students each count their total numbers of stars and write\n matching multiplication equations. The winner is the student who draws the most stars after\n seven rounds.\n </p>"},{title:"<b>Game:</b> <i>Multiplication Bingo</i>",displayOrder:3,text:"<p>This game provides repeated practice with multiplication facts in conjunction with\n mathematical reasoning. Read about it in Marilyn\u2019s blog: <i>\n <a href='http://www.marilynburnsmathblog.com/multiplication-bingo/' target='_blank'>Multiplication Bingo</a>.</i>\n </p>"},{title:"<b>Children\u2019s Book:</b> <i>Amanda Bean\u2019s Amazing Dream</i> by Cindy Neuschwander",displayOrder:4,text:"<p>This is a fanciful story about a young girl who doesn\u2019t feel that learning multiplication is\n important until a strange dream convinces her otherwise. Here's a link for using the book\n in a class setting: <i>\n <a href='https://mathsolutions.com/ms_classroom_lessons/amanda-beans-amazing-dream-a-mathematical-story/' target='_blank'>Amanda Bean\u2019s Amazing Dream: A Mathematical Story</a>.</i>\n </p>"}]},{id:"20",displayOrder:4,interviewId:"9",interviewName:"Foundations of Fractions",contentFocus:a.n7.FracDecimal,title:'Which is greater, $$\\frac{1}{4}$$ or <span class="lab__no-wrap">$$\\frac{5}{6}$$?</span>',questionIds:["9.6"],getStarted:{description:"<p>When comparing pairs of fractions, students who have fractional number sense have the ability to\n reason in different ways. We chose these two particular fractions because we\u2019ve learned that\n students have a variety of ways to reason when comparing them. One goal of fraction instruction is\n for students to develop the ability to apply different strategies for comparing, depending on the\n fractions.\n </p>\n <p>Sometimes benchmarks are useful. For example, students might use the benchmark of $$\\frac{1}{2}$$ to\n compare $$\\frac{1}{4}$$ and $$\\frac{5}{6}$$, reasoning that since $$\\frac{1}{4}$$ is less than\n $$\\frac{1}{2}$$ and $$\\frac{5}{6}$$ is greater than $$\\frac{1}{2}$$, $$\\frac{5}{6}$$ has to be the greater\n fraction. Or students may use the benchmark of 1 and reason that $$\\frac{5}{6}$$ is closer to 1 than\n $$\\frac{1}{4}$$ is. Some apply the procedure of converting fractions to fractions with common denominators, a reliable strategy that students often apply as a procedure without understanding. Then relying on converting to common denominators could indicate that students don't have other mental reasoning strategies available.\n </p>",solveMentallyText:"How would you explain why $$\\frac{5}{6}$$ is the correct answer?",nextButtonText:"Next, select your explanation"},selectExplanation:{nextButtonText:"Next, listen to Max answer the question"},listenTo:{studentId:"15",navText:"Listen to a Student Answer the Question",prompt:"LISTEN TO MAX ANSWER THE QUESTION",videoId:"phnc0xjpra",nextButtonText:"Next, reflect on Max's explanation",reflect:{prompt:"Use the following prompts to think about Max's explanation.",reflection:"<ol>\n <li>Max first explained that $$\\frac{5}{6}$$ is greater because it\u2019s much closer to 1\n than $$\\frac{1}{4}$$ is. What information does this explanation give you about\n Max\u2019s understanding of fractions?\n </li>\n <li>After Rusty probed, Max gave a different explanation. What did you learn from\n Max\u2019s second response about how he thinks about fractions?\n </li>\n <li>If students were solving this problem during a class number talk, how might you\n record Max\u2019s explanation?\n </li>\n </ol>",nextButtonText:"Next, select the explanation that is the best match"},select:{prompt:"Select an Explanation",description:"Which explanation best matches how Max reasoned?",explanationId:"2",reason:"<p>After <b>Max</b> explained that $$\\frac{5}{6}$$ is greater because it\u2019s much closer to\n 1 than $$\\frac{1}{4}$$ is, Rusty was ready to accept his answer. However, he decided\n to probe with an additional question,\n <b>\n <i>\u201cHow do you know that?\u201d</i>\n </b>\n Max then explained using the rectangular question card as a visual model. He\n approximated dividing the card into sixths and showed Rusty about how much would be\n the leftover, $$\\frac{1}{6}$$. He then did the same with $$\\frac{1}{4}$$, showing that a\n much larger piece would be left over. Max\u2019s language was precise and clear.\n </p>\n <p>Here\u2019s a way to record Max\u2019s second explanation:</p>\n <p><img src='/images/tech-art-images/reasoninglab_9.6.svg' alt='Two rectangles; rectangle on left divided into 6 equal parts, with 5 parts shaded green and the fraction 5/6 below it; rectangle on right divided into 4 equal parts, with 1 part shaded yellow and the fraction 1/4 below it.' class=\"lab__tech-art\"/></p>",nextButtonText:"Next, listen to other students answer the question"}},galleries:[{id:"1",displayOrder:1,navText:"Listen to Other Students Answer the Question",title:"LISTEN TO OTHER STUDENTS ANSWER THE QUESTION",description:"After you listen to each student, you'll have the opportunity to select the explanation that most closely matches the student's reasoning.",questionId:"9.6",studentExamples:[{studentId:"65",displayOrder:1,title:"LISTEN TO HENRY",videoId:"enjy85t8hh",prompt:"After you listen, select the explanation that most closely matches Henry's reasoning.",explanationId:"2",reason:"<p>\n <b>Henry</b>\n confidently answered that $$\\frac{5}{6}$$ is greater. Like Max, he offered a visual\n explanation, but unlike Max, his language was imprecise. Henry explained that he\n visualized $$\\frac{1}{4}$$ as\n <b>\n <i>\u201cone square in a rectangle\u201d</i>\n </b>\n and that $$\\frac{5}{6}$$ was\n <b>\n <i>\u201calmost to a whole.\u201d</i>\n </b>\n It would have been interesting if Lynne had asked Henry how he visualized the\n $$\\frac{5}{6}$$. Using this question in a number talk would be a good opportunity to\n explore Henry\u2019s reasoning further.\n </p>",nextButtonText:"Next, listen to Stefaniya"},{studentId:"102",displayOrder:2,title:"LISTEN TO STEFANIYA",videoId:"5cqmnuxhf5",prompt:"After you listen, select the explanation that most closely matches Stefaniya's reasoning.",explanationId:"4",reason:"<p>\n <b>Stefaniya</b>\n explained to Marilyn that she converted $$\\frac{1}{4}$$ and $$\\frac{5}{6}$$ to twelfths so that the two fractions had common denominators. She then compared\n $$\\frac{3}{12}$$ with $$\\frac{10}{12}$$ and reasoned that $$\\frac{5}{6}$$ is greater than\n $$\\frac{1}{4}$$.\n </p>",nextButtonText:"Next, listen to Adrian"},{studentId:"86",displayOrder:3,title:"LISTEN TO ADRIAN",videoId:"mdu8xddet4",prompt:"After you listen, select the explanation that most closely matches Adrian's reasoning.",explanationId:"4",reason:'<p><b>Adrian</b>, like Stefaniya, also used common denominators, but he reasoned in a very different way. Adrian converted <span class="lab__no-wrap">$$\\frac{1}{6}$$ into $$\\frac{2}{12}$$</span> and <span class="lab__no-wrap">$$\\frac{1}{4}$$ into $$\\frac{3}{12}$$</span>. He then reasoned that since $$\\frac{1}{6}$$ was equal to $$\\frac{2}{12}$$, $$\\frac{2}{6}$$ would be equal to $$\\frac{4}{12}$$. He continued reasoning that $$\\frac{4}{12}$$ is more than $$\\frac{3}{12}$$ or $$\\frac{1}{4}$$, so $$\\frac{5}{6}$$ was certainly greater. This was a stunning and completely different explanation than we had ever heard and was not an easy one for Amy or anyone else to follow. When using <i>Listening to Learn</i>, this is an example of an appropriate time to ask a student to repeat the explanation and record in the Notes section. This can be helpful for reflecting later on how a student was reasoning. </p>',nextButtonText:"Next, listen to Starlie"},{studentId:"16",displayOrder:4,title:"LISTEN TO STARLIE",videoId:"83kq224k95",prompt:"After you listen, select the explanation that most closely matches Starlie's reasoning.",explanationId:"1",reason:"<p>\n <b>Starlie</b>\n began explaining to Nishi by saying that $$\\frac{5}{6}$$ is greater because it is almost a\n whole. She explained further by using the benchmark of $$\\frac{1}{2}$$, saying that\n $$\\frac{1}{4}$$ is less than $$\\frac{1}{2}$$ and $$\\frac{5}{6}$$ is more than\n $$\\frac{1}{2}$$.\n </p>",nextButtonText:"Next, listen to Silmona"},{studentId:"23",displayOrder:5,title:"LISTEN TO SILMONA",videoId:"qe138e67vp",prompt:"After you listen, select the explanation that most closely matches Silmona's reasoning.",explanationId:"2",reason:"<p>\n The explanation that <b>Silmona</b> gave was similar to Starlie\u2019s explanation, but she used the benchmark of 1 to\n compare the two fractions. She explained to Leo that $$\\frac{5}{6}$$ is greater\n than $$\\frac{1}{4}$$ because it would take three more fourths to get to 1 and $$\\frac{5}{6}$$ would take only one more sixth.\n </p>",nextButtonText:"Next, listen to Michael"},{studentId:"38",displayOrder:6,title:"LISTEN TO MICHAEL",videoId:"uuuwcdwnn1",prompt:"After you listen, select the explanation that most closely matches Michael's reasoning.",explanationId:"5",reason:'<p>\n <b>Michael</b>\n answered correctly that $$\\frac{5}{6}$$ is greater, but the reasoning he explained to\n Marilyn was faulty. He said that 5 is closer to 6, and 1 isn\u2019t that close to 4.\n Although his explanation can work for comparing some pairs of fractions, it doesn\u2019t\n always apply. For example, when comparing $$\\frac{5}{6}$$ and $$\\frac{7}{8}$$, 5 is as\n close to 6 as 7 is to 8, so that reasoning wouldn\u2019t help in deciding which fraction\n is greater. Michael\u2019s reasoning wouldn\u2019t work for comparing $$\\frac{2}{3}$$ and\n $$\\frac{98}{100}$$, where 2 is closer to 3 than 98 is to 100, but $$\\frac{98}{100}$$ is\n the greater fraction. Michael would benefit from listening to the strategies of other students during number talks that focus on comparing fractions.\n </p>\n <p>If you\u2019re interested in watching a video that includes Marilyn reflecting on how each of these students, and one other student, compared $$\\frac{1}{4}$$ and $$\\frac{5}{6}$$, watch this <wistia-lightbox videoid="2x0d16sh0j" text="video"></wistia-lightbox>.</p>',nextButtonText:"Next, read the suggestions for class instruction"}]}],suggestions:[{title:"<b>Number Talks:</b> <i>Comparing Fractions</i>",displayOrder:1,text:'<p>Compare $$\\frac{1}{4}$$ and $$\\frac{5}{6}
1$$ in a number talk with your class with the goal of\n having the students come up with as many ways as possible to explain which fraction is\n greater. Use the other four comparing fraction problems in Interview 9 for number talks:\n </p>\n <ul class="lab__show-bullets">\n <li>Which is greater, $$\\frac{1}{3}$$ or $$\\frac{1}{8}$$?</li>\n <li>Which is greater, $$\\frac{2}{5}$$ or $$\\frac{1}{2}$$?</li>\n <li>Which is greater, $$\\frac{5}{12}$$ or $$\\frac{5}{9}$$?</li>\n <li>Which is greater, $$\\frac{15}{16}$$ or $$\\frac{7}{6}$$?</li>\n </ul>'},{title:"<b>Class Lesson:</b> <i>The Fraction Kit</i>",displayOrder:2,text:"<p>The Fraction Kit is our all-time favorite for introducing, reinforcing, and extending students' understanding of fractions. We think it's extremely important for students to cut and label their own fraction kits and then use them for comparing fractions, understanding equivalence, and adding and subtracting fractions. To have students make their fraction kits, each student needs five 3-by-18-inch strips of construction paper in five different colors, a pair of scissors, and a number 10 envelope to store their cut pieces.</p>\n <ol>\n <li>Choose a color and model for the students how to fold it in half, open and label each section $$\\frac{1}{2}$$, cut on the folds so they have two pieces, and write their initials on the back of each piece.</li>\n <li>Choose a color for the second strip and model for the students how to fold it in half and then half again, open and label each section $$\\frac{1}{4}$$, cut on the folds so they have four pieces, and write their names on the back of each piece.</li>\n <li>Choose a color for them to fold into eight sections. Have them label each section $$\\frac{1}{8}$$, cut, and initial.</li>\n <li>Next, they fold a fourth strip into sixteen sections. Have them label each section $$\\frac{1}{16}$$, cut, and initial.</li>\n <li>Students leave the fifth strip whole, and label it three ways: 1 whole, 1, and $$\\frac{1}{1}$$.</li>\n </ol>\n <p>For a beginning activity, teach students how to play <i>Cover Up</i>. Here are the rules as they appear in <i>Do The Math</i>.\n </p>\n <p><img src='/images/tech-art-images/doTheMath_coverUp.png' alt='Rules for the game "Cover Up" that involves fraction pieces and a number cube' class=\"lab__tech-art\"/></p>"},{title:"<b>Class Lesson:</b> <i>Put-in-Order Problem</i>",displayOrder:3,text:'<p>This is an adaptable, repeatable activity that can be used with the whole class, in small\n groups, or individually. The activity asks students to put numbers in order from least to\n greatest and can be used with whole numbers, fractions, decimals, percents, or a combination\n of these.\n </p>\n <ul class="lab__show-bullets">\n <li>Prepare 4-by-6-inch cards with the following numbers: <span class="lab__no-wrap">$$\\frac{1}{16}$$, $$\\frac{1}{8}$$, $$\\frac{3}{16}$$, $$\\frac{1}{4}$$, $$\\frac{3}{8}$$, $$\\frac{1}{2}$$, $$\\frac{5}{8}$$, $$\\frac{3}{4}$$, $$\\frac{15}{16}$$, $$\\frac{1}{1}$$, $$\\frac{9}{8}$$, $$\\frac{3}{2}$$</span></li>\n <li>To establish a familiar benchmark, begin by showing, so all can see it, the card with $$\\frac{1}{2}$$. (The whiteboard rail works well for this.)</li>\n <li>Show the cards one at a time for students to place them and to explain their reasoning.</li>\n </ul>'},{title:"<b>Individual or Partner Activity:</b> <i>Fix-It (An Activity for Ordering Fractions)</i>",displayOrder:4,text:"<p><i>Fix-It</i> is an activity that engages students in ordering fractions. Involving students as creators of problems for classmates to solve provides an experience for them that's different from the teacher always being the source of problems. It helps students become producers as well as consumers of their math learning. In her blog post <i><a href='https://marilynburnsmath.com/fix-it-an-activity-for-ordering-fractions/' target='_blank'>Fix-It</a></i>, \n you can read about how Marilyn taught the lesson and engaged students in making their own puzzle cards. Note: <i>\"Fix It\"</i> appears in <i>Do The Math</i>, Fractions, Module B, Lesson 28.</p>"}]}]},strategy:{title:"Strategy Lab",type:a.vi.Strategy,displayOrder:2,description:"Learn about the reasoning strategies and watch how students use them to solve problems in the interviews.",colors:{lightmed:"",medium:"#592f5f",dark:"#59315e",darkest:"#452649",accent:"#004065"},labs:[{id:"4",displayOrder:1,title:"Uses the benchmark of 10",contentFocus:a.n7.AdditionSubtraction,interviewIds:["3"],getStarted:{description:'<p>The number 10 is the foundation of our place-value system. Prior to understanding the structure of place value, it\'s important for students to learn to use the number 10 as a benchmark. We sometimes talk with students about this as the "make a 10" strategy. In this interview, we provide students a range of addition and subtraction problems with numbers within 20, and we listen to learn if (and how) they use 10 as a benchmark to mentally figure out answers. For example, to add 8 + 5, a student may decompose 5 into <span class="lab__no-wrap">2 + 3,</span> add <span class="lab__no-wrap">8 + 2</span> to make 10, and then add 10 + 3 to get 13. Several ideas relate to this reasoning. One is that it requires students to be able to decompose numbers within 10 into two addends <span class="lab__no-wrap">(here 5 = 2 + 3),</span> to add on to one of the addends in the problem to make a 10 <span class="lab__no-wrap">(here 8 + 2 = 10),</span> and then to add a one-digit number to 10 <span class="lab__no-wrap">(here 10 + 3 = 13).</span> The benchmark of 10 is also a building block for adding within 100 and within 1000, where knowing <sp
1an class="lab__no-wrap">8 + 5</span>, for example, can help students reason to figure out <span class="lab__no-wrap">80 + 50 or 800 + 500.</span></p>',nextButtonText:"Next, apply the strategy"},applyTheStrategy:{questionId:"3.7",prompt:"Interview 3: What is 7 + 8?",reflection:"<ol> <li>Think about how you would explain the answer using the benchmark of 10.</li> <li>Think about how else you could reason to solve the problem.</li> <li>Which explanations demonstrate using the benchmark of 10? Select all that apply and then press Next.</li> </ol>",reason:'The first two explanations use the benchmark of 10, each in a slightly different way. In the first explanation, students make a 10 by adding on to one of the addends in the problem, here either adding 7 + 3 = 10 or 8 + 2 = 10. In the second explanation, students make a 10 by adding 5 + 5, a fact that students typically learn early on (helped by having five fingers on each hand). This calls for decomposing both of the addends in the problem, here 5 + 2 = 7 and 5 + 3 = 8. In the third explanation, students don\'t make a 10 but instead use an equally efficient strategy by first adding a "double," here either <span class="lab__no-wrap">7 + 7 = 14</span> or <span class="lab__no-wrap">8 + 8 = 16</span>, and then adjusting. (Students often know the sums of doubles from memory before they learn other facts such as 7 + 8.) The fourth explanation, counting on, is a strategy that students use when they are first learning to add and subtract, before they become comfortable with decomposing numbers within 10 and using the benchmark of 10.',explanationIds:["1","2"],nextButtonText:"Next, listen to the students answer the question"},galleries:[{id:"1",displayOrder:1,description:"After you listen to each student, you'll have the opportunity to decide if the student used the benchmark of 10.",navText:"Listen to Students Answer the Question",title:"LISTEN TO STUDENTS ANSWER THE QUESTION",studentExamples:[{studentId:"8",displayOrder:1,questionId:"3.7",showExplanations:!0,selectedExplanationId:"1",didApplyStrategy:!0,title:"LISTEN TO BLAISE ANSWER THE QUESTION",videoId:"6l3kgqf7t8",prompt:"Did Blaise use the benchmark of 10 to reason?",reason:"<p> <b>Blaise</b> demonstrated the ability to use the benchmark of 10 to add. He answered the question immediately and explained clearly to Marilyn, <b> <i>\"That one I know 'cause 8 + 2 equals 10, and 7 \u2212 2 equals 5. Then I make it a 'teen' and it's 15.\"</i> </b> We found it interesting that Blaise described adding 10 + 5 by making a <b> <i>\"teen\"</i> </b> and wondered if he had come up with this expression by himself or learned this at school or home.</p>",nextButtonText:"Next, listen to Damian"},{studentId:"47",displayOrder:2,questionId:"3.7",showExplanations:!0,selectedExplanationId:"1",didApplyStrategy:!0,title:"LISTEN TO DAMIAN ANSWER THE QUESTION",videoId:"n16jtikdfh",prompt:"Did Damian use the benchmark of 10 to reason?",reason:'<p>Like Blaise, <b>Damian</b> decomposed 7 into 5 + 2. He explained to Nishi that he took the 2 from the 7, added 8 + 2 to equal 10, and then added 10 + 5 to get 15. Unlike Blaise, Damian\'s explanation was rambling and not precise, but it was clear that he was able to "make a 10" and use the number 10 as a benchmark.</p>',nextButtonText:"Next, listen to Genesis"},{studentId:"53",displayOrder:3,questionId:"3.7",showExplanations:!0,selectedExplanationId:"2",didApplyStrategy:!0,title:"LISTEN TO GENESIS ANSWER THE QUESTION",videoId:"lheu9bs5vw",prompt:"Did Genesis use the benchmark of 10 to reason?",reason:'<p><b>Genesis</b> used 10 as a benchmark to figure out the answer, but in a different way than Blaise and Damian. She thought for a while after Mallika asked the question and then, with deliberate certainty, explained, <b><i>"I took away the 2 and the 3 from the 7 and 8, and then it\'s 5 + 5 equals 10, and 3 + 2 equals 5, and then it\'s 15."</i></b> Throughout the interview, Genesis thought carefully about every problem before giving an answer and always explained her reasoning calmly and precisely. She consistently used the strategy of using the benchmark of 10. \n You might like to watch the <wistia-lightbox videoid="81lyg2hdbm" text="full interview"></wistia-lightbox> of Genesis.</p>',nextButtonText:"Next, listen to Saniya"}
1,{studentId:"63",displayOrder:4,questionId:"3.7",showExplanations:!0,selectedExplanationId:"2",didApplyStrategy:!0,title:"LISTEN TO SANIYA ANSWER THE QUESTION",videoId:"4b39qxva1q",prompt:"Did Saniya use the benchmark of 10 to reason?",reason:'<p>Students often reason similarly but explain their thinking differently. <b>Saniya</b> solved the problem the same way that Genesis did. Also, after Mallika asked the question, Saniya thought about the problem for about as long as Genesis did before giving the answer of 15. Saniya explained, <b> <i>"I took the two 5s from both of the numbers and then equaled them, and then I added 2 plus 3 equals 5, and then I added them both together and it equals 15."</i> </b> Saniya demonstrated her ability to use 10 as a benchmark.</p>',nextButtonText:"Next, listen to Salem"},{studentId:"111",displayOrder:5,questionId:"3.7",showExplanations:!0,selectedExplanationId:"3",didApplyStrategy:!1,title:"LISTEN TO SALEM ANSWER THE QUESTION",videoId:"vkius37s2p",prompt:"Did Salem use the benchmark of 10 to reason?",reason:'<p> <b>Salem</b> didn\'t use the benchmark of 10, but instead used 8 + 8, a fact she knew from memory. She explained to Amy, <b> <i>"I know that because if 8 plus 8 equals 16, 16 minus 1 equals 15."</i> </b> Students often learn "doubles" easily and use them to solve other problems.</p>',nextButtonText:"Next, listen to Camdynn"},{studentId:"5",displayOrder:6,questionId:"3.7",showExplanations:!0,selectedExplanationId:"3",didApplyStrategy:!1,title:"LISTEN TO CAMDYNN ANSWER THE QUESTION",videoId:"43mx4jp1a3",prompt:"Did Camdynn use the benchmark of 10 to reason?",reason:'<p>Like Salem, <b>Camdynn</b> used a "double" that she knew, but instead of adding 8 + 8, she added <span class="lab__no-wrap">7 + 7.</span> Camdynn explained to Le\'Vada, in the precise and deliberate way she explained her reasoning for all problems, <b> <i>"Seven plus seven is 14, and I\'m just adding 1 more to the 14 and it\'s 15."</i> </b> </p>',nextButtonText:"Next, listen to students use the benchmark of 10 to answer other questions"}]},{id:"2",displayOrder:2,description:"After you listen to each student, read a reflection from Marilyn and Lynne.",navText:"Listen to Students Use the Benchmark of 10 to Answer Other Questions",title:"LISTEN TO STUDENTS USE THE BENCHMARK OF 10 TO ANSWER OTHER QUESTIONS",studentExamples:[{studentId:"33",displayOrder:1,questionId:null,showExplanations:!1,selectedExplanationId:null,didApplyStrategy:null,title:"LISTEN TO LILY: What is 8 + 5?",videoId:"9cuoej2pjx",prompt:"Next, read Marilyn and Lynne\u2019s reflection",reason:'<p><b>Lily</b> explained to Rusty how she knew the answer was 13, <b><i>"Because I know that 8 + 2 = 10, and then there would be 3 more, and then 10 + 3 = 13."</i></b> She didn\'t explain how she decomposed the 5 into 2 + 3, but it was clear that Lily knew how to make a 10 and use it as a benchmark number.</p>',nextButtonText:"Next, listen to Jade"},{studentId:"71",displayOrder:2,questionId:null,showExplanations:!1,selectedExplanationId:null,didApplyStrategy:null,title:"LISTEN TO JADE: What is 4 + 9?",videoId:"aj1tz072nn",prompt:"Next, read Marilyn and Lynne\u2019s reflection",reason:"<p>When <b>Jade</b> gave her answer, Lynne thought she said 30. Jade repeated her answer and explained that she didn't know how to say it in English. Lynne asked Jade to write the answer on a piece of paper, and she wrote 13. Jade began by saying 10 + 4 = 14, tilting her head with a questioning look to make sure Lynne understood. Lynne nodded her head, and Jade continued her explanation. Then Jade added 10 + 4, took 1 from the 4 and added it to the 9 to make 10, and then added 10 + 3. Jade demonstrated her ability to use the benchmark of 10 to solve this problem. It's often difficult for students to communicate their reasoning, but it's especially challenging for English learners who are developing fluency in a new language.</p>",nextButtonText:"Next, listen to Jeff"},{studentId:"32",displayOrder:3,questionId:null,showExplanations:!1,selectedExplanationId:null,didApplyStrategy:null,title:"LISTEN TO JEFF: What is 14 \u2212 8?",videoId:"y23bqqnh9r",prompt:"Next, read Marilyn and Lynne\u2019s reflection",reason:'<p>The benchmark of 10 is also useful for solving subtraction problems. <b>Jeff</b> explained to Lynne, <b> <i>"I know that because I would go minus four to 10 and minus four to six."</i> </b> Throughout the interview Jeff consistently used 10 as a benchmark when both adding and subtracting. </p>',nextButtonText:"Next, listen to Katie"}
1,{studentId:"12",displayOrder:4,questionId:null,showExplanations:!1,selectedExplanationId:null,didApplyStrategy:null,title:"LISTEN TO KATIE: I baked 20 oatmeal cookies. There were 12 cookies left. How many were eaten?",videoId:"phvets2jfm",prompt:"Next, read Marilyn and Lynne\u2019s reflection",reason:"<p>Shortly after Marilyn presented this contextual problem, <b>Katie</b> gave the correct answer of 8 without taking much time to figure. Her explanation about how she used the benchmark of 10 was different from any others we heard. Katie changed both numbers in the problem. She thought of the 20 as 10 and the 12 as 2, which changed the problem from 20 \u2212 12 to 10 \u2212 2. Katie then explained that she knew that 8 + 2 equals 10, so the answer would be 8. It seemed to us that Katie had an intuitive sense that this was an OK way to make the problem easier to solve. Katie's answers throughout the interview were interesting and often unique. For this question, Marilyn didn't probe, so we aren't sure how Katie knew this. This is an example of a situation when probing might have helped reveal how a student was thinking. </p>",nextButtonText:"Next, read the suggestions for class instruction"}]}],suggestions:[{displayOrder:1,title:"<b>Number Talks:</b> <i>Using the Benchmark of 10</i>",text:'<p>Here are some examples of problems to use in number talks that help focus on using the benchmark of 10. </p> <ul class="lab__show-bullets square"> <li>5 + 7</li> <li>13 \u2212 9</li> <li>9 + 8</li> <li>5 + __ = 16</li> <li>20 \u2212 4</li> </ul>'},{displayOrder:2,title:"<b>Number Talks:</b> <i>Combining More than Two Addends</i>",text:'<p>Ask students to think of different strategies for adding more than two addends. Here are some problems that give students the opportunity to use the benchmark number of 10. Notice that the last four examples include doubles <span class="lab__no-wrap">(e.g., 2 + 2 and 5 + 5),</span> which are often known facts for students. You might want to lead a separate number talk with repeated addends like these.</p>\n <ul class="lab__show-bullets square">\n <li>8 + 1 + 9</li>\n <li>4 + 5 + 6 + 3</li>\n <li>3 + 8 + 7</li>\n <li>1 + 6 + 3 + 9</li>\n <li>2 + 9 + 8</li>\n <li>8 + 1 + 9</li>\n <li>8 + 2 + 2 + 8</li>\n <li>1 + 1 + 9 + 5</li>\n <li>5 + 7 + 5</li>\n <li>4 + 6 + 6</li>\n </ul>'},{displayOrder:3,title:"<b>Class Lesson:</b> <i>What Is the Question?</i>",text:'<p>Instead of asking a question, give the answer and have students provide the question. To introduce the investigation, give students an example: <i>The answer is 12. What might the question be?</i> Ask students to share their ideas with a partner and then share with the class. Examples of questions we\'ve heard from students include the following:\n </p>\n <ul class="lab__show-bullets square">\n <li><i>What is 6 + 6?</i></li>\n <li><i>How many months are there in one year?</i></li>\n <li><i>How old is my brother?</i></li>\n </ul>\n <p>Then pose the following: <i>The answer is 12. Using numbers from 1 to 20 and addition, subtraction, or both, what might the question be?</i> Write the numbers 1 to 20 on the board and have students explore the possibilities with a partner and share their findings with the class. Later, this investigation can be used as a partner or individual activity with other numbers.</p>'},{displayOrder:4,title:"<b>Game:</b> <i>Addition Capture</i>",text:"<p>This partner game gives students practice with addition facts from 2 to 20. You need a deck of 40 cards (4 sets of 1–10 cards from a standard deck with the face cards removed) and an addition table for students to check their sums. Each player turns over two cards and figures out the sum. (You may want students to write the matching equations.) The player with the greater sum takes all four cards. If the sums are the same for both players, they each take their own cards. When all the cards have been played, the winner is the player who has captured more cards.<
1/p>"}]},{id:"6",displayOrder:2,title:"Applies the inverse relationship between addition and subtraction",interviewIds:["2","3","4","5"],contentFocus:a.n7.AdditionSubtraction,getStarted:{description:'<p>Applying the inverse relationship between addition and subtraction is a powerful reasoning strategy. When students first encounter subtraction, they typically learn that subtraction means "take away." While this is one interpretation of subtraction, it\'s also important for students to think of how subtraction relates to addition. For example, think about this subtraction problem: <i>If you had 5 apples and you ate 2, how many apples are left?</i> To solve it, a student might think, <b> <i>"What number is 5 minus 2?"</i> </b> This can be recorded as 5 \u2013 2 = 3. But a student might also think, <b> <i>"What number can I add on to 2 to get 5?"</i> </b> This can be recorded as 2 + <span class="lab__blank-underline small"></span> = 5.</p> <p>Both ways of reasoning can be used for any subtraction problem, and the choice of whether to subtract or add depends upon the numbers. For example, for a problem such as 11 \u2013 9, it makes sense to add up from 9 to 11; but for a problem such as 11 \u2013 2, it makes sense to subtract 2 from 11. In Interviews 2 through 5, students have multiple opportunities to demonstrate applying the inverse relationship between addition and subtraction.</p>',nextButtonText:"Next, apply the strategy"},applyTheStrategy:{questionId:"4.8",prompt:"Interview 4: What is 70 - 59?",reflection:"<ol>\n <li>Think about how you would explain the answer using the inverse relationship between addition and subtraction.</li>\n <li>Think about how else you could reason to solve the problem.</li>\n <li>Which explanations demonstrate applying the inverse relationship between addition and subtraction? Select all that apply and then press Next.</li>\n </ol>",reason:"The strategy of applying the inverse relationship between addition and subtraction is demonstrated by the two highlighted explanations. All of the first four explanations rely on reasoning numerically and using benchmarks of multiples of 10, but only the first two involve using addition to solve the problem. Some students use the standard algorithm, visualizing the problem as they might solve it with paper and pencil. We have no problem with students using the algorithms they've learned to figure out answers. We do have concerns, however, when students resort to algorithms as their only option because they don't have other ways to reason.",explanationIds:["1","2"],nextButtonText:"Next, listen to the students answer the question"},galleries:[{id:"1",displayOrder:1,description:"After you listen to each student, you'll have the opportunity to decide if the student applied the inverse relationship between addition and subtraction to solve a subtraction problem.",navText:"Listen to Students Answer the Question",title:"LISTEN TO STUDENTS ANSWER THE QUESTION",studentExamples:[{studentId:"18",displayOrder:1,questionId:"4.8",showExplanations:!0,selectedExplanationId:"1",didApplyStrategy:!0,title:"LISTEN TO ROCCO: What is 70 \u2212 59?",videoId:"bwg4nzdelh",prompt:"Did Rocco apply the inverse relationship between addition and subtraction to reason?",reason:"<p><b>Rocco</b> gave the answer of 11 and explained to Lynne, <b><i>\"I know when there's a 9 . . . and then there's something that's rounded to like a 10, like 70, I know to add 1, I usually round to a friendly number with the 9 and get, like, 60, and then I added 10 and then plus the other 1 because I added 1 from 59. So, I had 11.\"</i></b> Although Rocco's language was not precise, his reasoning was solid and he demonstrated his ability to use addition to solve a subtraction problem. From this one problem, it's not clear whether it was the 9 in the problem that triggered this strategy for Rocco or whether it's a strategy he would use in other situations.</p>",nextButtonText:"Next, listen to Sloan"},{studentId:"4",displayOrder:2,questionId:"4.8",showExplanations:!0,selectedExplanationId:"4",didApplyStrategy:!1,title:"LISTEN TO SLOAN: What is 70 \u2212 59?",videoId:"v1736bu7vd",prompt:"Did Sloan use the inverse relationship between addition and subtraction to reason?",reason:"<p> <b>Sloan</b> gave Marilyn a clear explanation of how he solved the problem, first subtracting 50 from 70 to get 20, and then subtracting 9 from 20 to get 11. This is an efficie
1nt reasoning strategy for solving this subtraction problem, but Sloan didn't apply the inverse relationship between addition and subtraction to figure out the answer. His explanation shows his ability to decompose a number into its place-value parts in order to compute. In this interview, we included eight subtraction problems, so we provide ample opportunities for students to demonstrate applying the inverse relationship. Sloan applied the inverse relationship to solve five of the eight problems, which demonstrated that he has access to this important strategy but chose not to use it for this particular problem. (Rocco applied the inverse relationship to solve four of the eight problems.)</p>",nextButtonText:"Next, listen to Lamia"},{studentId:"19",displayOrder:3,questionId:"4.8",showExplanations:!0,selectedExplanationId:"1",didApplyStrategy:!0,title:"LISTEN TO LAMIA: What is 70 \u2212 59?",videoId:"ikxyr1iv2l",prompt:"Did Lamia use the inverse relationship between addition and subtraction to reason?",reason:'<p><b>Lamia</b> used the same strategy as Rocco, but her language was clear and she gave a concise explanation of how she applied the inverse relationship between addition and subtraction to figure out the answer. She explained to Nishi, <b><i>"If you add 1 to 59 you get 60, and 60 plus 10 is 70, so 10 plus 1 is 11."</i></b> In the interview, as Sloan did, Lamia demonstrated applying the inverse relationship between addition and subtraction to solve five of the eight subtraction problems, an indication that she is comfortable applying this strategy.</p>',nextButtonText:"Next, listen to students apply the inverse relationship between addition and subtraction to answer other questions"}]},{id:"2",displayOrder:2,description:"Before you listen to each student, think about how you could apply the inverse relationship between addition and subtraction to answer the question.",navText:"Listen to Students Apply the Inverse Relationship Between Addition and Subtraction to Answer Other Questions",title:"LISTEN TO STUDENTS APPLY THE INVERSE RELATIONSHIP BETWEEN ADDITION AND SUBTRACTION TO ANSWER OTHER QUESTIONS",studentExamples:[{studentId:"75",displayOrder:1,questionId:null,showExplanations:!1,selectedExplanationId:null,didApplyStrategy:null,title:"LISTEN TO MATIAS: What is 9 \u2212 7?",videoId:"ib0ufku6va",prompt:"Next, read Marilyn and Lynne\u2019s reflection",reason:'<p>For seven of the questions in Interview 2 (Addition and Subtraction Within 10), students can figure out the answer by applying the inverse relationship between addition and subtraction. <b>Matias</b> is an emerging English speaker and needed time to collect his thoughts in order to explain his reasoning. When Lynne asked Matias how he knew the answer was 2, he said, <b> <i>"Because 7 more 2 equals 9."</i> </b> (In Spanish, the word "mas" can mean either "more" or "add" and Matias\'s explanation shows that he\'s in the process of learning English.) </p>',nextButtonText:"Next, listen to Katie"},{studentId:"12",displayOrder:2,questionId:null,showExplanations:!1,selectedExplanationId:null,didApplyStrategy:null,title:"LISTEN TO KATIE: What is 14 \u2212 8?",videoId:"jao5rex7c2",prompt:"Next, read Marilyn and Lynne\u2019s reflection",reason:'<p> <b>Katie</b> applied the inverse relationship between addition and subtraction, using addition to solve this subtraction problem from Interview 3 (Addition and Subtraction Within 20). Her explanation to Marilyn demonstrates her understanding that addition and subtraction are related operations. She also used the strategy of adding up to the benchmark of 10. She said, <b> <i>"I know <span class="lab__no-wrap">8 + 2 = 10,</span> and there\'s about four more, and <span class="lab__no-wrap">4 + 2 = 6</span>."</i> </b> </p>',nextButtonText:"Next, listen to Meeyah"},{studentId:"82",displayOrder:3,questionId:null,showExplanations:!1,selectedExplanationId:null,didApplyStrategy:null,title:"LISTEN TO MEEYAH: What is 100 \u2212 98?",videoId:"n577bh4333",prompt:"Next, read Marilyn and Lynne\u2019s reflection",reason:"<p>With this problem from Interview 4 (Addition and Subtraction Within 100), <b>Meeyah</b> explained to Amy that you just have to add 2 to get to 100. She clearly saw that 98 and 100 are close to each other, and she chose to add up rather than to think about subtracting 98 from 100. The inverse relationship is especially useful when the numbers in the problem are close, such as for this problem and also for problems such as 9 \u2013 7, 11 \u2013 9, and 1000 \u2013 998, which are problems from other interviews. Subtracting or counting back seems more efficient when the numbers in the problem are farther apart, such as 9 \u2013 2, 11 \u2013 1, and 1000 \u2013 3. </p>",nextButtonText:"Next, listen to Santos"}
1,{studentId:"88",displayOrder:4,questionId:null,showExplanations:!1,selectedExplanationId:null,didApplyStrategy:null,title:"LISTEN TO SANTOS: My school has 600 students. My friend\u2019s school has 499 students. How many more students are in my school?",videoId:"zh7nyiz2vr",prompt:"Next, read Marilyn and Lynne\u2019s reflection",reason:'<p>This comparing problem from Interview 5 (Addition and Subtraction Within 1000), can be solved by using either addition (499 + <span class="lab__blank-underline small"> </span> = 600) or subtraction (600 \u2013 499). <b>Santos</b> thought for a few moments after Nishi read the word problem and then explained how he used addition. He said, <b> <i>"So . . . 499 plus 1 will equal 500, then plus 100 would equal 600, so the answer is 101."</i> </b> We worry about students being taught \u201ckey words\u201d for deciding how to solve word problems, for example, teaching that "how many more" and "how many less" signal subtraction. We prefer to keep the instructional emphasis on having students interpret contextual situations and then decide how they\'d like to reason in order to figure out the answer. </p>',nextButtonText:"Next, read the suggestions for class instruction"}]}],suggestions:[{displayOrder:1,title:"<b>Number Talks:</b> <i>Applying the Inverse Relationship Between Addition and Subtraction</i>",text:'<p>Here are some examples of problems to use in number talks to help focus on applying the inverse relationship between addition and subtraction. </p> <p>Within 10</p> <ul class="lab__show-bullets square"> <li>6 \u2013 4</li> <li>8 \u2013 5</li> <li>10 \u2013 7</li> </ul> <p>Within 20</p> <ul class="lab__show-bullets square"> <li>11 \u2013 8</li> <li>13 \u2013 9</li> <li>14 \u2013 7</li> </ul> <p>Within 100</p> <ul class="lab__show-bullets square"> <li>50 \u2013 39</li> <li>100 \u2013 76</li> <li>70 \u2013 55</li> </ul> <p>Within 1000</p> <ul class="lab__show-bullets square"> <li>1000 \u2013 775</li> <li>800 \u2013 450</li> <li>625 \u2013 47</li> </ul>'},{displayOrder:2,title:"<b>Individual or Partner Activity:</b> <i>Connecting Addition and Subtraction</i>",text:'<p>Choose a set of numbers that\u2019s appropriate for your class (see below) and write the numbers on the board. Ask the students to find as many different pairs of addition and subtraction equations as they can. For example, for the first set, one pair of equations might be <span class="lab__no-wrap">7 + 2 = 9</span> and 9 \u2013 2 = 7. </p> <ul class="lab__show-bullets square"> <li>Numbers for sums and differences within 10 or within 20:</li> </ul> <ul> <li> <img src=\'/images/tech-art-images/strategylab_inversesums_differences1.svg\' alt="first row of numbers: 3, 2, 7, 5, 10, 8; second row of numbers: 6, 0, 4, 1, 9" class="lab__tech-art"/> </li> </ul> <ul class="lab__show-bullets square"> <li>Numbers for sums and differences within 100:</li> </ul> <ul> <li> <img src=\'/images/tech-art-images/strategylab_inversesums_differences2.svg\' alt="first row of numbers: 40, 70, 10, 90, 20, 30; second row of number: 100, 80, 50, 0, 60" class="lab__tech-art"/> </li> </ul> <ul class="lab__show-bullets square"> <li>Numbers for sums and differences within 1000:</li> </ul> <ul> <li> <img src=\'/images/tech-art-images/strategylab_inversesums_differences3.svg\' alt="first row of numbers: 900, 100, 700, 200, 800, 400; second row of numbers: 600, 300, 500, 1000, 0" class="lab__tech-art"/> </li> </ul>'},{displayOrder:3,title:"<b>Game:</b> <i>Making Connections</i>",text:'<p>This is a version of the Individual or Partner Activity, <i>Connecting Addition and Subtraction,</i> that presents a game for students to play in pairs. Choose a set of numbers (see above) that\'s appropriate for students. Ask the students to look for three numbers in the set that they can use both for an addition equation and for a related subtraction equation <span class="lab__no-wrap">(e.g., 3 + 7 = 10 and 10 \u2013 3 = 7).</span> It\'s OK to use the same number twice <span class="lab__no-wrap">(e.g., 4 + 4 = 8 and 8 \u2013 4 = 4)</span>.</p><p>Then have them play the game in pairs. <span class="lab__no-wrap"><b>Player A</b> selects three numbers,</span> writes down the addition and subtraction equations, and crosses out the three numbers used. <b>Player B</b> does the same, using numbers that haven\'t been crossed out. Play continues until it\'s no longer possible to write an equation. Students play for four rounds, alternating who goes first. The player with the most equations is the winner.</p>'}]}]},interview:{title:"Interview Lab",type:a.vi.Interview,displayOrder:3,description:"Get specific suggestions about how to conduct interviews.",colors:{lightmed:"#347975",medium:"#006b68",dark:"#377975",darkest:"#005b56",accent:"#004065"},labs:[{id:"1",displayOrder:1,title:"What's a good way to put a student at ease when you begin an interview?",videoId:"3d392r9kqn",getStarted:{description:"<p> There\u2019s no one right way to begin an interview, but our first goal is to help students feel comfortable. We always communicate that we\u2019re interested in learning from them about how they think about math, so that we can be better teachers. Our introductions vary depending on the age of students and how well we know them, and we take special care when it\u2019s a student\u2019s first experience with a one-on-one math interview. Also, it\u2019s helpful to be sure that students understand that we\u2019ll always ask them to explain how they figured out answers. </p> <p> The video clips included mostly show teachers introducing the interview to students they haven\u2019t met before (a result of our gathering video footage from a large sample of students). In the last video, you'll listen to Marilyn begin an interview with Jacob, a student she knew from teaching math lessons in Jacob\u2019s classroom. This clip shows some \u201cbehind the scenes\u201d footage about how we made these videos and also shows Jacob answering the first question in the interview. </p>",nextButtonText:"Next, listen to examples of beginning an interview"},galleries:[{id:"1",displayOrder:1,description:"As you listen to each introduction, reflect on what the interviewer said to explain the interview process and to help put the student at ease.",navText:"Listen to Examples of How to Begin an Interview",title:"LISTEN TO EXAMPLES OF HOW TO BEGIN AN INTERVIEW",studentExamples:[{studentId:"69",displayOrder:1,title:"LISTEN TO AMY BEGIN AN INTERVIEW WITH ADRIANA",interviewerId:"1",videoId:"djvqqss4m0",reason:null,reasonButtonText:null,nextButtonText:"Next, listen to Nishi begin an interview with Carlos"},{studentId:"55",displayOrder:2,title:"LISTEN TO NISHI BEGIN AN INTERVIEW WITH CARLOS",interviewerId:"10",videoId:"lica6yo2xt",reason:null,reasonButtonText:null,nextButtonText:"Next, listen to Lynne begin an interview with Jeff"}
1,{studentId:"32",displayOrder:3,title:"LISTEN TO LYNNE BEGIN AN INTERVIEW WITH JEFF",interviewerId:"7",videoId:"pfoqftxkud",reason:null,reasonButtonText:null,nextButtonText:"Next, listen to Rusty begin an interview with Nathan"},{studentId:"35",displayOrder:4,title:"LISTEN TO RUSTY BEGIN AN INTERVIEW WITH NATHAN",interviewerId:"12",videoId:"rpkjg7w3le",reason:null,reasonButtonText:null,nextButtonText:"Next, listen to Marilyn begin an interview with Stefaniya"},{studentId:"102",displayOrder:5,title:"LISTEN TO MARILYN BEGIN AN INTERVIEW WITH STEFANIYA",interviewerId:"9",videoId:"whqpqz8cs7",reason:null,reasonButtonText:null,nextButtonText:"Next, listen to Marilyn begin an interview with Jacob"},{studentId:"39",displayOrder:6,title:"LISTEN TO MARILYN BEGIN AN INTERVIEW WITH JACOB",interviewerId:"9",videoId:"zfxqa84oo6",reason:null,reasonButtonText:null,nextButtonText:"Next, listen to a final reflection from Marilyn"}]}]},{id:"2",displayOrder:2,title:"When is it helpful to ask additional probing questions during an interview?",videoId:"w9bapx4rfj",getStarted:{description:"<p>When you listen to students explain, there are times when you\u2019d like to hear more. Sometimes you\u2019re not sure that you understand students\u2019 explanations and you aren\u2019t clear about how they reasoned. Sometimes students give an answer and explain that they <b><i>\u201cjust know\u201d</i></b> and didn\u2019t have to figure out the answer. Sometimes students offer something that makes you mathematically curious about their thinking. These are all examples of times when we should ask a probing question to encourage students to give us a little more information in order to learn as much as possible about how they reason.</p>\n <p>How we frame a probing question will differ according to the student and the situation. To learn more, we might ask, <b><i>\u201cCan you tell more about that?\u201d</i></b> Or <b><i>\u201cWould you please explain that again so I\u2019m sure I understand your thinking?\u201d</i></b> When students respond that they <b><i>\u201cjust know,\u201d</i></b> you might ask, <b><i>\u201cHow would you explain that to someone who didn\u2019t know how to figure it out?\u201d</i></b> When you\u2019re particularly curious and want to capture a student\u2019s thinking in the Notes section to reflect on later, you might ask, <b><i>\u201cExplain once more so I can write down how you figured and then think more about it later.\u201d</i></b></p>\n <p>It\u2019s important to remember that we avoid probing in order to guide a student\u2019s thinking or to lead a student to a correct answer, both of which we often do when we teach. When we interview, we avoid teaching to further students\u2019 learning and, instead, focus on probing to learn as much as possible about how students reason, how they bring meaning to the mathematics, and how they make sense of questions.</p>",nextButtonText:"Next, listen to examples of probing"},galleries:[{id:"1",displayOrder:1,description:"As you watch these video clips, pay attention to how the interviewers probe and how their probing questions give insights into students\u2019 reasoning.",navText:"Listen To Examples of Probing",title:"LISTEN TO EXAMPLES OF PROBING",studentExamples:[{studentId:"7",displayOrder:1,title:"LISTEN TO LEO PROBE TO LEARN MORE FROM GEORGIA LEE: Marco wants to save $100. He saved $47 so far. How much more money does Marco need to save?",interviewerId:null,videoId:"nyjzbm1dza",reason:"<p>Leo asked <b>Georgia Lee</b> to solve this word problem from Interview 4. She gave the answer of $53 quickly; but when Leo asked her how she had figured it out, her lengthy explanation was difficult to follow and she talked herself into a different answer, $57. Leo repeated the question, and this time Georgia Lee again gave the correct answer of $53. Leo then probed, <b> <i>\u201cCan you explain one more time how you got this?\u201d </i> </b>Georgia Lee clearly stated, <b> <i>\u201cI got it because I know 100 minus 40 equals 60 and 60 minus 7 equals 53.\u201d</i> </b> Asking students to re-explain their thinking can be a useful way to help students clarify their thinking. </p>",reasonButtonText:"Next, read Marilyn and Lynne's reflection",nextButtonText:"Next, listen to Rusty probe to learn more from Max"},{studentId:"15",title:"LISTEN TO RUSTY PROBE TO LEARN MORE FROM MAX: Which is greater, $$\\frac{1}{4}$$ or $$\\frac{5}{6}$$?",interviewerId:null,displayOrder:2,videoId:"phnc0xjpra",reason:"<p>When answering this question from Interview 9, <b>Max</b> said that $$\\frac{5}{6}$$ is greater than $$\\frac{1}{4}$$ and he explained that $$\\frac{5}{6}$$ is <b> <i>\u201cmuch closer to 1.\u201d</i></b> Max was correct, but Rusty probed a little further and asked, <b> <i>\u201cHow do you know that?\u201d</i> </b> Max then gave a detailed explanation that revealed how he looked at fractions geometrically. He showed with his hands how he\u2019d divide a rectangle into sixths, showed Rusty $$\\frac{5}{6}$$, and approximated the distance from $$\\frac{5}{6}$$ to 1. Max then divided the same rectangle into fourths, approximated the distance from $$\\frac{1}{4}$$ to 1, and reasoned that $$\\frac{5}{6}$$ was greater than $$\\frac{1}{4}$$.</p>",reasonButtonText:"Next, read Marilyn and Lynne's reflection",nextButtonText:"Next, listen to Marilyn probe to learn more from Juliette"},{studentId:"43",title:"LISTEN TO MARILYN PROBE TO LEARN MORE FROM JULIETTE: What is 99 + 99 + 5?",interviewerId:null,displayOrder:3,videoId:"3m4zxewk73",reason:'<p>When Marilyn asked <b>Juliette</b> to answer this question from Interview 5, Juliette gave the correct answer of 203 almost immediately. She explained that 99 + 99 = 198 and 198 + 5 = 203. Maril
1yn was interested in how Juliette had added 99 + 99 so quickly and said, <b> <i>\u201cI\u2019m curious how you added 99 plus 99.\u201d</i> </b> Juliette responded, <b> <i>\u201cIt was kind of easy. You take 1 from the other 99 and put it to the 99 and that\u2019s 100.\u201d</i> </b> This allowed Juliette to add <span class="lab__no-wrap">100 + 98.</span> By probing into how Juliette reasoned, we learned about a strategy we had never heard before from students solving this particular problem. </p>',reasonButtonText:"Next, read Marilyn and Lynne's reflection",nextButtonText:"Next, listen to Lynne probe to learn more from Asmara"},{studentId:"95",title:"LISTEN TO LYNNE PROBE TO LEARN MORE FROM ASMARA: What is 7 \xd7 6?",interviewerId:null,displayOrder:4,videoId:"c33alr1ue9",reason:"<p>When Lynne presented this problem from Interview 6, <b>Asmara</b> knew the answer without having to figure, an indication that she knew the answer from memory. Lynne then asked Asmara to explain how she knew. Asmara said she had done multiplication and knew 7 \xd7 6. Lynne probed more to find about if Asmara had access to a reasoning strategy for solving this problem. She asked Asamara, <b> <i>\u201cHow would you explain it to someone who didn\u2019t know that? Can you think of a way that would make it easier for them to learn it?\u201d</i> </b> Asmara responded, <b> <i>\u201cYou can do 6 times 5, which is 30, and then 6 times 2, which is 12, and then add them together.\u201d</i> </b> Probing further helped Lynne learn that Asmara was able to apply the distributive property, an important reasoning strategy. </p>",reasonButtonText:"Next, read Marilyn and Lynne\u2019s reflection",nextButtonText:"Next, listen to a final reflection from Marilyn"}]}]}]}},o=[{id:"1",text:"Yes",displayOrder:1},{id:"2",text:"No",displayOrder:2}]},7675:function(e,t,n){n.d(t,{O:function(){return a}});var 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Line numbers count LF bytes from the start of the resource, as the search results do. Vendor segments are library code the classifier recognised; they are stored but not indexed. Bytes are shown as Latin1 characters, one per byte.