In this lesson, students will work with partners to find ...
left564515In this lesson, students will work with partners to find different arrangements that add to 100. They will count to 100 by counting and combining groups to get to 100, using what they know about money (nickels, dimes, quarters, half-dollar), skip counting, and combining groups using multiplication and addition. 00In this lesson, students will work with partners to find different arrangements that add to 100. They will count to 100 by counting and combining groups to get to 100, using what they know about money (nickels, dimes, quarters, half-dollar), skip counting, and combining groups using multiplication and addition. Seeing Arrays as Equal GroupsNC Mathematics Standard(s):Represent and solve problems involving multiplication and division.NC.3.OA.1 For products of whole numbers with two factors up to and including 10:Interpret the factors as representing the number of equal groups and the number of objects in each group.Illustrate and explain strategies including arrays, repeated addition, decomposing a factor, and applying the commutative and associative properties.NC.3.OA.2 For whole-number quotients of whole numbers with a one-digit divisor and a one-digit quotient: Interpret the divisor and quotient in a division equation as representing the number of equal groups and the number of objects in each group.Illustrate and explain strategies including arrays, repeated addition or subtraction, and decomposing a factor. Standards for Mathematical PracticeMake sense of problems and persevere in solving themReason abstractly and quantitativelyConstruct viable arguments and critique the reasoning of othersModel with mathematicsUse appropriate tools strategicallyAttend to precisionLook for and make use of structureStudent Outcomes:I can work with partners to find different arrangements that add to 100.I can count to 100 by counting and combining groups to get to 100.I can use what I know about money (nickels, dimes, quarters, half-dollar).I can skip count to get to 100.I can combine groups using multiplication and addition.I can multiply by 2s, 5s, 10s.Materials:One Hundred Hungry Ants, by Elinor J. Pinczes (Houghton Mifflin, 1993100 objects per groups (color tiles, 2 color counters, paper clips, pennies)dot paper (each dot represents an ant)tape, scissorsAdvance Preparation:100 items for each groupIf possible find a copy of the book, One Hundred Hungry Ants or accessYouTube ( or other sites.If you do not have a copy of One Hundred Ants and you cannot access You Tube,skip to step B in this lesson.Directions:Activity A: (Begin here if you have the book or access to YouTube.)If you have the book, One Hundred Hungry Ants, read the book aloud to the class.Each time the ants rearrange themselves, ask students to predict what the next arrangement might be. If you do NOT have the book or access to YouTube, you might read about the book online.Lesson:Show an array to the whole group. Ask students to describe the arrangement of counters or tiles. Ask students to build the array on blank paper. Ask students to describe the arrangement. Be sure students are able to describe rows and columns. Ask students to rotate the array by turning the paper. Ask: What do you notice? (It is important that students focus on equal groups. Students should be able to skip count by the number in each group or use a combination of doubles and other counting strategies. Some students may count by ones. This is an opportunity for formative assessment.Tell students, we are going to imagine how 100 ants might march in different rows.Ask students, “How many different arrangements do you think we could make with 100 counters or 100 tiles?”Ask students to work in groups to use cubes or counters to build an arrangement to show 5 equal groups with 20 cubes or counters in each group. Groups should be in straight rows and columns.Different groups share solution strategies.Students should show how they know they have exactly 100 counters.Students work in groups of four. Each group should represent each array on dot paper by thinking about each dot as an ant. Draw a frame around an array to illustrate the arrangement. Label each array. (See the end of the lesson for blackline master of dot paper.) Students who represent 1 x 100 and 2 x 50 will have to cut and paste the dot paper to show the array.Each group of students should be able to explain how they know they have found all possible arrangements.Questions to Pose:Before:How would you describe your array? How many objects did you use in your arrangement?If you could rotate your arrangement, you would see 20 groups of 5 objects each. Explain why?During:If an array of four rows of twenty-five is possible, then is an array of twenty- five rows of four also possible? Explain your reasoning?After:How many different arrangements did you find for the 100 Hungry Ants?Explain how arrays with the same numbers but in different order—for example, twenty lines of five (20 x 5) and five lines of twenty (5 x 20)—would be different formations in the storyHow do some arrays relate to other arrays with the same number in reverse?Possible Misconceptions/Suggestions:Possible MisconceptionsSuggestionsStudents may not know that 5 x 20 is the same amount as 20 x 5When counting in equal groups, students lose track.Students may have difficulty seeing that each arrangement can be rotated to show the commutative property.Students need many opportunities to rotate an array to see the commutative property.Students might separate the rows and columns to give them more space to count objects or to more easily see one equal group at a time.Students need many opportunities to find the total number of objects when building arrays. Students can learn by listening to other students as they share strategies.Special Notes:It is important for third grade students to count large amounts of objects. Students are more likely to realize that grouping objects is a more efficient way to find a total.Teachers should notice and name the strategies students are using.Teachers ask questions that provoke children to think, articulate their thinking, and sometimes try a new strategy; and we extend their thinking.Solutions:1 x 100; 2 x 50; 4 x 25; 5 x 20; 10 x 10 and the reverse.100 x1; 50 x 2; 25 x 4; 20 x 5 (commutative property)Adapted from Math Solutions professional development.638083205969106532220596914985412059691925781205969235303120596927802582059693213477205969364070320596940759062059694503145205969493635720596953635972059695790848205969621808720596966512872059697078527205969638083178620106532217862014985411786201925781178620235303117862027802581786203213477178620364070317862040759061786204503145178620493635717862053635971786205790848178620621808717862066512871786207078527178620638083187553106532218755314985411875531925781187553235303118755327802581875533213477187553364070318755340759061875534503145187553493635718755353635971875535790848187553621808718755366512871875537078527187553638083148843106532214884314985411488431925781148843235303114884327802581488433213477148843364070314884340759061488434503145148843493635714884353635971488435790848148843621808714884366512871488437078527148843638083187553106532218755314985411875531925781187553235303118755327802581875533213477187553364070318755340759061875534503145187553493635718755353635971875535790848187553621808718755366512871875537078527187553638083178620106532217862014985411786201925781178620235303117862027802581786203213477178620364070317862040759061786204503145178620493635717862053635971786205790848178620621808717862066512871786207078527178620638083187554106532218755414985411875541925781187554235303118755427802581875543213477187554364070318755440759061875544503145187554493635718755453635971875545790848187554621808718755466512871875547078527187554638083172665106532217266514985411726651925781172665235303117266527802581726653213477172665364070317266540759061726654503145172665493635717266553635971726655790848172665621808717266566512871726657078527172665638083187554106532218755414985411875541925781187554235303118755427802581875543213477187554364070318755440759061875544503145187554493635718755453635971875545790848187554621808718755466512871875547078527187554638083178620106532217862014985411786201925781178620235303117862027802581786203213477178620364070317862040759061786204503145178620493635717862053635971786205790848178620621808717862066512871786207078527178620638083187546106532218754614985411875461925781187546235303118754627802581875463213477187546364070318754640759061875464503145187546493635718754653635971875465790848187546621808718754666512871875467078527187546638083160753106532216075314985411607531925781160753235303116075327802581607533213477160753364070316075340759061607534503145160753493635716075353635971607535790848160753621808716075366512871607537078527160753638083175643106532217564314985411756431925781175643235303117564327802581756433213477175643364070317564340759061756434503145175643493635717564353635971756435790848175643621808717564366512871756437078527175643638083178620106532217862014985411786201925781178620235303117862027802581786203213477178620364070317862040759061786204503145178620493635717862053635971786205790848178620621808717862066512871786207078527178620638083187554106532218755414985411875541925781187554235303118755427802581875543213477187554364070318755440759061875544503145187554493635718755453635971875545790848187554621808718755466512871875547078527187554638083171826106532217182614985411718261925781171826235303117182627802581718263213477171826364070317182640759061718264503145171826493635717182653635971718265790848171826621808717182666512871718267078527171826638083184576106532218457614985411845761925781184576235303118457627802581845763213477184576364070318457640759061845764503145184576493635718457653635971845765790848184576621808718457666512871845767078527184576638083175643106532217564314985411756431925781175643235303117564327802581756433213477175643364070317564340759061756434503145175643493635717564353635971756435790848175643621808717564366512871756437078527175643638083178620106532217862014985411786201925781178620235303117862027802581786203213477178620364070317862040759061786204503145178620493635717862053635971786205790848178620621808717862066512871786207078527178620638083187553106532218755314985411875531925781187553235303118755327802581875533213477187553364070318755340759061875534503145187553493635718755353635971875535790848187553621808718755366512871875537078527187553 ................
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