Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 Critical Area(s): Multiplicatin and Divisin FOCUS fr Grade 3 Majr Wrk 70% f time Supprting Wrk 20% f time Additinal Wrk 10% f time 3.OA.A.1-2-3-4 3.MD.B.3-4 3.NBT.A.1-2-3 3.OA.B.5-6 3.G.A.1-2 3.MD.D.8 3.OA.C.7 3.OA.D.8-9 3.NF.A.1-2-3 3.MD.A.1-2 3.MD.C.5-6-7 Fluency standards: 3.OA.C.7 and 3.NBT.A.2 Standards fr Mathematical Practice 1. Make sense f prblems and persevere in slving them. 2. Reasn abstractly and quantitatively. 3. Cnstruct viable arguments and critique the reasning f thers. 4. Mdel with mathematics. 5. Use apprpriate tls strategically. 6. Attend t precisin. 7. Lk fr and make use f structure. 8. Lk fr and express regularity in repeated reasning. Standards in bld are specifically targeted within instructinal materials. 3.OA.A Represent and slve prblems invlving multiplicatin and divisin. 3.OA.A.3 Use multiplicatin and divisin within 100 t slve wrd prblems in situatins invlving equal grups, arrays, and measurement quantities, e.g., by using drawings and equatins with a symbl fr the unknwn number t represent the prblem. 3.OA.A.4 Determine the unknwn whle number in a multiplicatin r divisin equatin relating three whle numbers. Fr example, determine the unknwn number that makes the equatin true in each f the equatins 8? = 48, 5 = _ 3, 6 6 =? Dmains: Operatins and Algebraic Thinking Clusters: Clusters utlined in bld shuld drive the learning fr this perid f instructin. 3.OA.B Understand prperties f multiplicatin and the relatinship between multiplicatin and divisin. Standards: 3.OA.B.5 Apply prperties f peratins as strategies t multiply and divide. Examples: If 6 4 = 24 is knwn, then 4 6 = 24 is als knwn. (Cmmutative prperty f multiplicatin.) 3 5 2 can be fund by 3 5 = 15, then 15 2 = 30, r by 5 2 = 10, then 3 10 = 30. (Assciative prperty f multiplicatin.) Knwing that 8 5 = 40 and 8 2 = 16, ne can find 8 7 as 8 (5 + 2) = (8 5) + (8 2) = 40 + 16 = 56. (Distributive prperty.) 3.OA.B.6 Understand divisin as an unknwnfactr prblem. Fr example, find 32 8 by finding the number that makes 32 when multiplied by 8. 3.OA.D Slve prblems invlving the fur peratins, and identify and explain patterns in arithmetic. 3.OA.D.8 Slve tw-step wrd prblems using the fur peratins. Represent these prblems using equatins with a letter standing fr the unknwn quantity. Assess the reasnableness f answers using mental cmputatin and estimatin strategies including runding. 3.OA.D.9 Identify arithmetic patterns (including patterns in the additin table r multiplicatin table), and explain them using prperties f peratins. Fr example, bserve that 4 times a number is always even, and explain why 4 times a number can be decmpsed int tw equal addends. Rev 8/2016 Prperty f MPS Page 1 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 Fundatinal Learning 2.OA.C 3.OA.A Key Student Understandings Students will understand hw t use the Distributive Prperty t find larger prducts by breaking apart the prduct int the sum f tw smaller knwn multiplicatin facts. Students will understand hw t use the Assciative and Cmmutative Prperties t multiply with three factrs. Students will understand the Zer and Identity Prperties f Multiplicatin. Students will understand the inverse relatinship between multiplicatin and divisin. Assessments Future Learning 3.MD.C.7 4.NBT.B.5-6 4.OA.A.2 4.OA.B.4 4.NF.B.4 Frmative Assessment Strategies Evidence fr Standards-Based Grading Cmmn Miscnceptins/Challenges 3.OA.B Understand prperties f multiplicatin and the relatinship between multiplicatin and divisin. When using the Distributive Prperty: Students might think that by breaking apart an array they are changing the ttal number f items in the array. Use cncrete mdels t have students put the array back tgether again t see that the ttal did nt change. Students blend multiplicatin algrithms and additin algrithms. Use cncrete mdels t have students break apart arrays and write separate multiplicatin equatins. Then have them add the tw. Use a template that they can fill in ( x ) + ( x ). When using the Assciative Prperty: Students may want t use each factr mre than nce. Remind students the Assciative Prperty is abut the gruping factrs t make calculatins easier. Ask questins such as, Which numbers shuld yu multiply first? Have students put parenthesis arund thse numbers. Then multiply by the next factr. When using inverse peratins: Students may cnfuse factrs and prducts. When slving divisin prblems such as 14 2, ask questins such as 2 times what number is 14? r Hw many grups f 2 are in 14? 3.OA.A Represent and slve prblems invlving multiplicatin and divisin. Students think in terms f individual things rather than in grups. A helpful apprach t nurture the understanding f gruping is by the use f rectangular arrays, equal grups drawings, jumps n a number line and tape diagrams as mdels. Students have difficulty identifying infrmatin in a prblem situatin (which number represents the ttal, the number f grups and/r the number f items in a grup). Students need mre experience making explicit cnnectins between their representatins (cncrete mdels r drawings) and determining the number f grups r the number f items in a grup. Rev 8/2016 Prperty f MPS Page 2 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 3.OA.D Slve prblems invlving the fur peratins, and identify and explain patterns in arithmetic. Students disregard quantities and their relatinships when slving multi-step wrd prblems. Mdel and encurage the use f a Think Alud strategy t truly make sense f prblems befre jumping int cmputatin. Have students restate the prblem in their wn wrds. Students can als identify and underline imprtant infrmatin. Students may need carefully cnstructed questins t help guide them in determining what t d, but shuld NEVER be tld what t d. Students misuse estimatin strategies when applying them t slve multi-step prblems. Students slve prblems first and then adjust their answer. Instructinal Practices Dmain: 3.OA Cluster: 3.OA.B Understand prperties f multiplicatin and the relatinship between multiplicatin and divisin. 3.OA.B.5 Apply prperties f peratins as strategies t multiply and divide. Examples: If 6 4 = 24 is knwn, then 4 6 = 24 is als knwn. (Cmmutative prperty f multiplicatin.) 3 5 2 can be fund by 3 5 = 15, then 15 2 = 30, r by 5 2 = 10, then 3 10 = 30. (Assciative prperty f multiplicatin.) Knwing that 8 5 = 40 and 8 2 = 16, ne can find 8 7 as 8 (5 + 2) = (8 5) + (8 2) = 40 + 16 = 56. (Distributive prperty.) This standard references prperties (rules abut hw numbers wrk) f multiplicatin. While students DO NOT need t nt use the frmal terms f these prperties, student must understand that prperties are rules abut hw numbers wrk, and they need t flexibly and fluently apply them in varius situatins. Students represent expressins using varius bjects, pictures, wrds and symbls in rder t develp their understanding f prperties. They multiply by 1 and 0 and divide by 1. They change the rder f numbers t determine that the rder f numbers des nt make a difference in multiplicatin, but des make a difference in divisin. Given three factrs, they investigate changing the rder f hw they multiply the numbers t determine that changing the rder des nt change the prduct. They als decmpse numbers t build fluency with multiplicatin. Students use prperties f peratins t calculate prducts f whle numbers, using increasingly sphisticated strategies based n these prperties t slve multiplicatin and divisin prblems invlving single-digit factrs. By cmparing a variety f slutin strategies, students learn the relatinship between multiplicatin and divisin. Distributive Prperty Students explre this prperty by applying their knwledge f decmpsing numbers as well as their understanding f multiplicatin (grups and bjects). T develp an understanding f the distributive prperty, students need t decmpse the whle int grups. Arrays can be used t develp this understanding. T find the prduct f 3 X 9, students can decmpse 9 int the sum f 4 and 5 and find 3 x (4 + 5). Rev 8/2016 Prperty f MPS Page 3 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 7 x 8 r 7 grups f 8 a. b. 7 x 8 5 x 8 = 40 2 x 8 = + 16 56 c. 7 x 8 7 x 4 = 28 7 x 4 = + 28 56 Splitting arrays can help students understand the distributive prperty. They can use a knwn fact t learn ther facts that may cause difficulty. Fr example, students can split a 6 X 9 array int 6 grups f 5 and 6 grups f 4; then, add the sums f the grups. The 6 grups f 5 is 30 and the 6 grups f 4 is 24. Students can write 6 X 9 as 6 X 5 + 6 X 4. Students are intrduced t the distributive prperty f multiplicatin ver repeated additin/skip-cunting as a strategy fr using prducts they knw t slve prducts they dn t knw. Here are ways that students culd use the distributive prperty t determine the prduct f 7 x 6. [Students are using the distributive prperty, but can refer t this in infrmal language such as breaking numbers apart r decmpsing.] Student 1 Student 2 Student 3 7 x 6 7 x 5 = 35 7 x 1 = 7 35 + 7 = 42 7 x 6 7 x 3 = 21 7 x 3 = 21 21 + 21 = 42 7 x 6 5 x 6 = 30 2 x 6 = 12 30 + 12 = 42 Assciative Prperty The assciative prperty states that the sum r prduct stays the same when the gruping f addends r factrs is changed. Fr example, when a student multiplies 7 x 5 x 2, a student culd rearrange the numbers t first multiply 5 x 2 = 10 and then multiply 10 x 7 = 70. Rev 8/2016 Prperty f MPS Page 4 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 Prperties f Operatins T further develp understanding f prperties related t multiplicatin and divisin, students use different representatins and their understanding f the relatinship between multiplicatin and divisin t determine if the fllwing types f equatins are true r false. 6 x 8 = 8 x 6 (Cmmutative Prperty f Multiplicatin) 0 x 7 = 7 x 0 = 0 (Zer Prperty f Multiplicatin) 1 x 9 = 9 x 1 = 9 (Multiplicative Identity Prperty f 1) 3.OA.B.6 Understand divisin as an unknwn-factr prblem. Fr example, find 32 8 by finding the number that makes 32 when multiplied by 8. This standard refers t the table referenced in 3.OA.A.3 and the varius prblem structures. Since multiplicatin and divisin are inverse peratins, students are expected t slve prblems and explain their prcesses f slving divisin prblems that can als be represented as unknwn factr multiplicatin prblems. Example: A student knws that 2 x 9 = 18. Hw can they use that fact t determine the answer t the fllwing questin: 18 peple are divided int pairs in P.E. class? Hw many pairs are there? Write a divisin equatin and explain yur reasning. Multiplicatin and divisin are inverse peratins; this understanding can be used t find the unknwn. Fact family triangles illustrate the inverse peratins f multiplicatin and divisin by shwing the tw factrs and hw thse factrs relate t the prduct and/r qutient. Example: 3 x 5 = 15 5 x 3 = 15 15 3 = 5 15 5 = 3 15 Create several triangles like the example. Discuss different equatins and the unknwn factr. x r 3 5 Example: Sarah did nt knw the answer t 63 divided by 7. Are any f the fllwing ways an apprpriate strategy fr Sarah t think abut the prblem? Explain why r why nt with a picture r wrds fr each ne. I knw that 7 x 9 = 63, s 63 divided by 7 must be 9. I knw that 7 x 10 = 70. If I take away a grup f 7, that means that I have 7 x 9 = 63. S 63 divided by 7 is 9. I knw that 7x5 is 35. 63 minus 35 is 28. I knw that 7x4 = 28. S if I add 7x5 and 7x4 I get 63. That means that 7x9 is 63, r 63 divided by 7 is 9. Rev 8/2016 Prperty f MPS Page 5 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 Dmain: 3.OA Cluster: 3.OA.A Represent and slve prblems invlving multiplicatin and divisin. 3.OA.A.3 Use multiplicatin and divisin within 100 t slve wrd prblems in situatins invlving equal grups, arrays, and measurement quantities, e.g., by using drawings and equatins with a symbl fr the unknwn number t represent the prblem. The table belw gives examples f a variety f prblem slving cntexts, in which students need t find the prduct, the grup size, r the number f grups. Students shuld be given ample experiences t explre all f the different prblem structures. The easiest prblem structure includes Unknwn Prduct (3 x 6 =? r 18 3 = 6). The mre difficult prblem structures include Grup Size Unknwn (3 x? = 18 r 18 3 = 6) r Number f Grups Unknwn (? x 6 = 18, 18 6 = 3). Rev 8/2016 Prperty f MPS Page 6 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 Explre wrd prblems using multiple representatins (use real-wrld situatins such as candy bxes, egg in cartns, crayns, and cars in a parking lt) t help develp student cnceptualize quantities. Discuss representatins using the fllwing mdels: Equatins: 3 x 4 =?, 4 x 3 =?, 12 4 =? and 12 3 =? Arrays: Equal grups: Bar diagrams (tape diagrams): 20 20 in all 5 grups 4 4 4 4 4 4 in each grup Number lines Sets f cunters, number lines t skip cunt and relate t multiplicatin and arrays/area mdels will aid students in slving prblems invlving multiplicatin and divisin. Allw students t mdel prblems using these tls. They shuld represent the mdel used as a drawing r equatin t find the slutin. This shws multiplicatin using gruping with 3 grups f 5 bjects and can be written as 3 5. Rev 8/2016 Prperty f MPS Page 7 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 Examples f multiplicatin: There are 24 desks in the classrm. If the teacher puts 6 desks in each rw, hw many rws are there? This task can be slved by drawing an array by putting 6 desks in each rw. This is an array mdel: This task can als be slved by drawing pictures f equal grups. 4 grups f 6 equals 24 bjects: A student can als reasn thrugh the prblem mentally r verbally, I knw 6 and 6 are 12. 12 and 12 are 24. Therefre, there are 4 grups f 6 giving a ttal f 24 desks in the classrm. Students in Grade 3 shuld use a variety f pictures, such as stars, bxes, flwers t represent unknwn numbers (variables). Letters are als intrduced t represent unknwns in third grade. Examples f Divisin: There are sme students at recess. The teacher divides the class int 4 lines with 6 students in each line. Write a divisin equatin fr this stry and determine hw many students are in the class. ( 4 = 6. There are 24 students in the class). Determining the number f bjects in each share (partitin mdel f divisin, where the size f the grups is unknwn see mdel at right): The bag has 92 hair clips, and Laura and her three friends want t share them equally. Hw many hair clips will each persn receive? Rev 8/2016 Prperty f MPS Page 8 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 Measurement mdel f divisin (the number f shares/grups is unknwn): Max the mnkey lves bananas. Mlly, his trainer, has 24 bananas. If she gives Max 4 bananas each day, hw many days will the bananas last? Starting Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 24 24-4= 20 20-4= 16 16-4= 12 12-4= 8 8-4= 4 4-4= 0 Slutin: The bananas will last fr 6 days. 3.OA.A.4 Determine the unknwn whle number in a multiplicatin r divisin equatin relating three whle numbers. Fr example, determine the unknwn number that makes the equatin true in each f the equatins 8? = 48, 5 = _ 3, 6 6 =? This standard refers t Glssary page 89, Table 2 (table als included at the end f this dcument fr yur cnvenience) and equatins fr the different types f multiplicatin and divisin prblem structures. The easiest prblem structure includes Unknwn Prduct (3 x 6 =? r 18 3 = 6). The mre difficult prblem structures include Grup Size Unknwn (3 x? = 18 r 18 3 = 6) r Number f Grups Unknwn (? x 6 = 18, 18 6 = 3). The fcus f 3.OA.A.4 extends beynd the traditinal ntin f fact families, by having students explre the inverse relatinship f multiplicatin and divisin. Students extend wrk frm earlier grades with their understanding f the meaning f the equal sign as the same amunt as t interpret an equatin with an unknwn. When given 4 x? = 40, they might think: 4 grups f sme number is the same as 40 4 times sme number is the same as 40 I knw that 4 grups f 10 is 40 s the unknwn number is 10 The missing factr is 10 because 4 times 10 equals 40. Equatins in the frm f a x b = c and c = a x b shuld be used interchangeably, with the unknwn in different psitins. Examples: Slve the equatins belw: 24 =? x 6 72 = 9 Rachel has 3 bags. There are 4 marbles in each bag. Hw many marbles des Rachel have altgether? 3 x 4 = m Rev 8/2016 Prperty f MPS Page 9 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 Dmain: 3.OA Cluster: 3.OA.D Slve prblems invlving the fur peratins, and identify and explain patterns in arithmetic. 3.OA.D.8 Slve tw-step wrd prblems using the fur peratins. Represent these prblems using equatins with a letter standing fr the unknwn quantity. Assess the reasnableness f answers using mental cmputatin and estimatin strategies including runding. Students in Grade 3 begin the step t frmal algebraic language by using a letter fr the unknwn quantity in expressins r equatins fr ne and twstep prblems. The symbls f arithmetic, x r r * fr multiplicatin and r / fr divisin, cntinue t be used in Grades 3, 4, and 5. (Prgressins fr the CCSSM; Operatins and Algebraic Thinking, CCSS Writing Team, May 2011, page 27) This standard refers t tw-step wrd prblems using the fur peratins. The size f the numbers shuld be limited t related Grade 3 standards. Adding and subtracting numbers shuld include numbers within 1,000 (3.NBT.A.2), and multiplying and dividing numbers shuld include single-digit factrs and prducts less than 100 (3.OA.C.7). Example: Mike runs 2 miles a day. His gal is t run 25 miles. After 5 days, hw many miles des Mike have left t run in rder t meet his gal? Write an equatin and find the slutin (2 x 5 + m = 25). In the diagram at right, Carla s bands are shwn using 4 equal-sized bars that represent 4 x 8 r 32 bands. Agustin s bands are directly belw shwing that the number that Agustin has plus 15 equals 32. The diagram can als be drawn like this: 8 8 8 8 15? (Prgressins fr the CCSSM; Operatins and Algebraic Thinking, CCSS Writing Team, May 2011, page 28) Rev 8/2016 Prperty f MPS Page 10 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 3.OA.D.9 Identify arithmetic patterns, and explain the patterns using prperties f peratins. This standard calls fr students t examine arithmetic patterns invlving bth additin and multiplicatin. Arithmetic patterns are patterns that change by the same rate, such as adding the same number. Fr example, the series 2, 4, 6, 8, 10 is an arithmetic pattern that increases by 2 between each term. Dubles (multiples f 2) in a multiplicatin table fall n hrizntal and vertical lines. All the multiples f 5 end in a 0 r 5, while all the multiples f 10 end with 0. Every ther multiple f 5 is a multiple f 10. Even numbers are always divisible by 2. Even numbers can always be decmpsed int 2 equal addends (14=7+7). Multiples f even numbers (2, 4, 6, and 8) are always even numbers. On a multiplicatin chart, the prducts in each rw and clumn increase by the same amunt (skip cunting). On an additin chart, the sums in each rw and clumn increase by the same amunt. When applying the Zer Prperty f Multiplicatin, the answer is always 0. When applying the Cmmutative Prperty f Multiplicatin, the answer is always the same. When applying the Identity Prperty f Multiplicatin, the answer is always the factr paired with the 1 factr. Cnnect patterns t a 100s chart, a number line, r ther visual mdel t help clarify the structure f the pattern. Example: What d yu ntice abut the numbers highlighted in pink in the multiplicatin table? Explain a pattern using prperties f peratins. Student: When yu change the rder f the factrs yu will still get the same prduct. Fr example 6 x 5 = 30 and 5 x 6 = 30. (cmmutative prperty) x 0 1 2 3 4 5 6 7 8 9 10 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 10 2 0 2 4 6 8 10 12 14 16 18 20 3 0 3 6 9 12 15 18 21 24 27 30 4 0 4 8 12 16 20 24 28 32 36 40 5 0 5 10 15 20 25 30 35 40 45 50 6 0 6 12 18 24 30 36 42 48 54 60 7 0 7 14 21 28 35 42 49 56 63 70 8 0 8 16 24 32 40 48 56 64 72 80 9 0 9 18 27 36 45 54 63 72 81 90 10 0 10 20 30 40 50 60 70 80 90 100 Rev 8/2016 Prperty f MPS Page 11 f 12
Weeks 6-10 September/Octber/Nvember envisinmath2.0 Tpics 3-4 Differentiatin 3.OA.B Understand prperties f multiplicatin and the relatinship between multiplicatin and divisin. Allw struggling students t use cncrete mdels/manipulatives t explre and mdel prperties and the inverse relatinship f multiplicatin and divisin. 3.OA.A Represent and slve prblems invlving multiplicatin and divisin. When wrking with different peratins, students shuld engage in applying quantitative reasning strategy (using the structure f the stry t identify the peratin needed in rder t slve the prblem.) Ask students t represent their thinking using different mdels cncrete, pictrial, verbal. Mdify numbers in the prblem t match student need. When wrking with stry prblem situatins, students need t wrk with a variety f cntexts and then cmpare their slutin strategies. Recrding strategies n charts and discussing appraches is a frm f differentiatin. Prvide varius hands n supprts t figure ut prblems. 3.OA.D Slve prblems invlving the fur peratins, and identify and explain patterns in arithmetic. Ask students t write dwn the verbal descriptin f slving the prblem r t represent their thinking using different mdels cncrete, pictrial, verbal t demnstrate hw t recrd a slutin in different ways. Mdify numbers in prblems t match student need. Give students a ne-step prblem and let them slve it, then have them create anther prblem using the answer as a starting pint. Cmbine bth parts int ne prblem. When wrking with different peratins, students shuld engage in applying quantitative reasning strategies (using the structure f the situatin t identify the peratin needed) in rder t slve the prblem. Literacy Cnnectins Academic vcabulary terms Vcabulary Strategies Literacy Strategies The Cmmn Cre Apprach t Differentiating Instructin (engage ny Hw t Implement a Stry f Units, p. 14-20) Linked dcument includes scafflds fr English Language Learners, Students with Disabilities, Belw Level Students, and Abve Level Students. envisinmath2.0 Tpic 3 Pacing Guide Tpic 4 Pacing Guide Resurces Develping Fluency Multiplicatin Thinking Strategies Grade 3 Games t Build Fluency Multi-Digit Additin & Subtractin Resurces Rev 8/2016 Prperty f MPS Page 12 f 12