DRAFT Unit 6 Multiplication and Division Application 3 Weeks 1 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
2 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Grade 3 Unit 6 Multiplication and Division Application (3 weeks) UNIT 6 Standards and Learning Targets Daily learning target(s) are to be posted in the room where all students can see the target(s). Teacher should refer to the learning target(s) throughout the lesson to keep students focused on the intended outcome for the day. The learning target(s) should be revisited at the end of the lesson. 3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Learning Targets A I can solve multiplication word problems involving equal groups. Yo puedo solucionar problemas matemáticas escritas de multiplicación que contienen grupos iguales. B I can solve multiplication word problems involving arrays. Yo puedo resolver problemas matemáticas escritas de multiplicación que involucran un conjunto. C I can solve multiplication word problems involving measurement quantities with unknowns in all positions. Yo puedo resolver problemas matemáticas escritas de multiplicación que involucran medidas con números desconocidos en diferentes posiciones. D I can solve division word problems involving equal groups. Yo puedo solucionar problemas matemáticas escritas de división que contienen grupos iguales. E I can solve division word problems involving arrays. Yo puedo resolver problemas matemáticas escritas de división que involucran un conjunto. F I can solve division word problems involving measurement quantities with unknowns in all positions. Yo puedo resolver problemas matemáticas escritas de multiplicación que involucran medidas con números desconocidos en diferentes posiciones. G I can use drawings and equations with a symbol for the unknown number to represent my problem. Yo puedo usar dibujos matemáticos y ecuaciones con símbolos para representar números desconocidos que representan mi problema. 3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations: 8 x? = 48, 15 =? 3, 6 x 6 =? Learning Targets I can determine the unknown number in a multiplication equation relating three whole numbers. H I Puedo determinar el número desconocido en las ecuaciones de multiplicación. I can determine the unknown number in a division equation relating three whole numbers. Puedo determinar el número desconocido en las ecuaciones de división. 3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. Learning Targets J K I can multiply within 100 using multiple strategies. Puedo multiplicarse dentro de 100 utilizando múltiples estrategias. I can divide with 100 using multiple strategies. Puedo dividir en 100 utilizando múltiples estrategias. 3 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
3.OA.8 L M N 3.OA.9 O P Q 3.NBT.3 R S 3.MD.7 T U Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Learning Targets I can solve two-step word problems using the four operations. Yo puedo resolver problemas de dos pasos usando las cuatro operaciones. I can represent a problem using an equation with a letter standing for the unknown quantity. Yo puedo representar un problema usando una ecuación con una letra representando la cantidad desconocida. I can assess the reasonableness of an answer using mental computation and estimation strategies. Yo puedo evaluar si una respuesta es razonable usando cálculos mentales y estrategias de estimación. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. Learning Targets I can identify an arithmetic pattern in an addition table. Puedo identificar los patrones aritméticos en las tablas de suma. I can identify an arithmetic pattern in a multiplication table. Puedo identificar los patrones aritméticos en las tablas de multiplicar. I can explain an arithmetic pattern using the properties of operations. Puedo explicar patrones aritméticos utilizando las propiedades de las operaciones. Multiply one-digit whole numbers by multiples of 10 in the range of 10-90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations. Learning Targets I can multiply one-digit whole numbers by multiples of 10 using strategies based on place value. Yo puedo multiplicar números enteros de un digito por múltiples de 10 usando estrategias basadas en valor posicional. I can multiply one-digit whole numbers by multiples of 10 using strategies based on the properties of operations. Yo puedo multiplicar números enteros de un digito por múltiples de 10 usando estrategias basadas en las propiedades de operación. Relate area to the operations of multiplication and addition. a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. Learning Targets I can show that when I find the area by counting square units I get the same answers as when I multiply the side lengths. Puedo demostrar que cuando encuentro el área contando las unidades cuadradas obtengo la misma respuesta como cuando multiplico las longitudes de los lados. I can solve real world problems by multiplying the side lengths to find area. 4 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
V Yo puedo resolver problemas de la vida real multiplicando la medida de los lados para hallar el área. I can decompose a figure into two non-overlapping rectangles to find area. Yo puedo descomponer figuras en dos rectángulos que no se superponen para hallar el área. 5 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Suggested Manipulatives: Two-color counters Color tiles Unit cubes Base ten blocks MATERIALS Representations: Story Representation Numerical expressions Diagrams Resources: District Curriculum Map Go Math ETA Hand 2Mind Critical Terms: Multiplication/multiplicación Division/división Array/ conjunto / matriz Area/ área Equal groups/ grupos iguales Equal shares/ partes iguales Multiple/ múltiplo Product/ producto Factor/ factor Divisor/ divisor Quotient/ cociente Perimeter/perímetro Fact family/ operaciones con números elementales relacionados(familias) Unknown/ desconocido Strategies/ estrategias Reasonableness/ razonable Mental computation/ matemáticas mental / cálculo mental Operation/ operación Estimation/ estimación Patterns/ patrón VOCABULARY Supplemental Terms: Inverse operation/ operación inversa Distributive Property/ propiedad distributiva Commutative Property/ propiedad conmutativa Zero Property/ propiedad del cero Identity Property/ propiedad de identidad Equation/ ecuación Problem Solving Structures: o o o o o o o Take apart/separar Add to/sumar a Take from/restar a Additive comparison/comparación aditiva Equal groups/grupos iguales Array/ matriz Area Model/modelo de área 6 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
STANDARDS FOR MATHEMATICAL PRACTICE (Practices to be explicitly emphasized are indicated with an *.) *1. Make sense of problems and persevere in solving them. Dan sentido a los problemas y perseveran en su resolución. Students demonstrate their ability to persevere and utilize problem-solving structures to solve multiplication and division problems. 2. Reason abstractly and quantitatively. Razonan de forma abstracta y cuantitativa. Students will reason about the problem-solving structure and employ it to justify and explain their solution. Students will make the connection between quantity and area models of multiplication and division. *3. Construct viable arguments and critique the reasoning of others. Construyen argumentos viables y critican el razonamiento de otros. Students may construct arguments using concrete models, such as objects, pictures, and drawings. They refine their mathematical communication skills as they participate in mathematical discussions that the teacher facilitates by asking questions such as How did you get that? and Why is that true? They explain their thinking to others and respond to others thinking. *4. Model with mathematics. Representación a través de las matemáticas. In this unit, students experiment with representing multiplication and division problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, creating equations, making a chart, list or graph, etc. 5. Use appropriate tools strategically. Utilizan las herramientas apropiadas estratégicamente. Students will use concrete manipulatives to represent multiplication and division situations. 6. Attend to precision. Ponen atención a la precisión Students represent and use clear and precise mathematical language in their discussions with others and in their own reasoning about multiplication and division problem solving. 7. Look for and make use of structure. Reconocen y utilizan estructuras. Students will recognize and utilize properties of operations to evaluate real-world problem-solving situations involving multiplication and division. 8. Look for and express regularity in repeated reasoning. Reconocen y expresan regularidad en el razonamiento repetitivo. Students will observe commonalities within and between multiplication and division, such as using the distributive property. 7 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
GRADE 3 UNIT 6 Multiplication and Division Applications Connections to Previous Learning: Grade 2 students have worked primarily with addition and subtraction situations. They have begun extending the modeling of quantities to equal groups and arrays as a basis for multiplication. Students will model this operation using rectangles partitioned into equivalent squares. Focus of the Unit: In this unit, Grade 3 students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models. It also extends the students work with the distributive property. For example, in the picture below the area of a 7 x 6 figure can be determined by finding the area of a 5 x 6 and 2 x 6 and adding the two sums. Students will continue their discovery of this concept and apply it to the composition and decomposition of shapes. 5 x 6 2 x 6 This unit will further address various problem-solving structures that students are expected to use while solving word problems involving multiplication and division. Students should use a variety of representations for creating and solving one-step word problems. They explain and apply properties of operations as strategies for finding their solutions to problems. In addition, students will solve two-step problems involving all four of the operations. Students will determine the unknown in a multiplication and division equation and understand division as an unknown-factor problem. Patterns within multiplication and division will be identified and students will explain them in more depth. This application unit will focus upon the multiplication and division situations of Equal Groups and Arrays (See Table 3 from the K-5 Operations and Algebraic Thinking Progression Document ) 8 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Table 3 9 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Fluency is also a focus of this unit. By studying patterns and relationships in multiplication facts and relating the operations of multiplication and division, students will build a foundation for fluency with multiplication and division facts. Students will demonstrate fluency with multiplication facts through 10 and the related division facts. Multiplying and dividing fluently refers to the skill of performing these operations accurately (using a reasonable amount of steps and time), flexibly (using strategies such as the distributive property), and efficiently. NOTE: By the end of Grade 3, students will know from memory all products of two one-digit numbers. Connections to Subsequent Learning: Grade 4 students will multiply and divide with multi-digit numbers and within multi-step problem situations including multiplicative comparison. They will also extend multiplication and division concepts into factors and multiples. Patterns that flow from these operations will be generated and analyzed. Finally, Grade 4 students will multiply a fraction by a whole number. Enduring Understandings: Students will understand that Area is additive. Modeling multiplication and division problems based upon their problem-solving structure can help in finding solutions. There is a relationship between area and multiplication. Properties of Operations will assist in problem-solving situations. Metric measurement units are related to place value concepts/multiples of 10. Essential Questions: How can modeling multiplication and division problems help in finding their solutions? What is the relationship between area and multiplication? What are the Properties of Operations? How does metric measurement connect to multiples of 10? 10 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Lesson Sequence 1: Patterns in Multiplication and Division (3.OA.3, 3.OA.9, 3.NBT.3) Cluster: Represent and solve problems involving multiplication and division. 3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Cluster: Solve problems involving the four operations, and identify and explain patterns in arithmetic. 3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends Cluster: Use place value understanding and properties of operations to perform multi-digit arithmetic. A range of algorithms may be used. 3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range of 10-90 (e.g., 9 x 80, 5 x 60) using strategies based on place value and properties of operations. Session 1 Multiply by Multiples of 10 Concrete: Example 1: There are 4 racks in the gym with 20 volleyballs on each rack. How many volleyballs are there all together? Step 1: Students show the multiplication problem using an area model to represent the factors in the problem. Step 2: Students use the distributive property to break apart the 20 into 10 and 10. So, the students would multiply (4 x 10) + (4 x 10) =? The area in the middle is the product of 4 x 20. There are 80 volleyballs in the gym. 11 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Example 2: Chelsea School has seven Third Grade teachers. Each class has 30 students. How many Third Grade students are at Chelsea? Step 1: Students build an area model showing 7 x 30. Step 2: Students multiply 7 x 10 using base 10 blocks to represent the multiplication. They used the distributive property to solve by decomposing the 30 into 10 + 10 + 10. The students solve (7 x 10) + (7 x 10) + (7 x 10) =? to find that 7 x 30 = 210. *Note: The teacher may be writing this on the board as the students do the activity concretely using manipulatives. Representational: Go Math Chapter 5 Lessons 5.3, 5.4, 5.5 Example 1: Dan has a sticker book with 40 pages. Each page has 3 stickers. How many stickers does Dan have? Students build an area model to show 40 x 3 =? Then, the next picture shows the student adding groups of 10. The student multiplied (40 x 1) three times and represented each with a 10 rod. Then, they added 40 + 40 + 40 = 120 to find how many stickers Dan had all together. 12 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Example 2: Lauri s class brought 5 packages of balloons for their science project. There are 20 balloons in each package. How many balloons did the students bring to class? The student used an open number line to add 5 groups of 20. Students take what they know from skip counting by 2 s and apply it to skip counting by 20. Abstract: Go Math Chapter 5 Lessons 5.3, 5.4, 5.5 In this session students are applying what they learned during Unit 5 Multiplication and Division Concepts to find patterns on a table. Abstract: Go Math Chapters 4 and 5 Lessons 4.7, 4.10, 5.1, 5.2 Session 2 Patterns on a Table 13 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Lesson Sequence 2: Area Application (3.MD.7.b,d, 3.OA.3) Cluster: Represent and solve problems involving multiplication and division. 3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Cluster: Geometric measurement: understand concepts of area and relate area to multiplication and to addition. 3.MD.7 Relate area to the operations of multiplication and addition. b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping parts, applying this technique to solve real world problems. Session 1 Area and Perimeter Using Concrete Models Concrete: Example 1 (Same Perimeter/Different Areas): Ethan wants to build a rectangle with a perimeter of 10 units. What could his rectangle look like? What would the area of his rectangle be? Here are two possible student repsonses: Student B: Student A: Student A created a 1 by 4 rectangle with a perimeter of 10 units. Students B also created a rectangle with a perimeter of 10 units (3 by 2), but this student s rectangle has an area of 6 square units. Have a math talk about the various rectangles represented in class and how they are alike and how they are different. Discuss how all of the rectangles have the same perimeter, but have different areas. 14 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Example 2 (Same Perimeter/Different Areas): Logan wants to build a rectangular garden in his backyard with a perimeter of 14 units. Show all the possible ways Logan could build his rectangular garden. What is the area of Logan s garden? Which rectangle would give him the largest garden? Here are two possible rectangles with a perimeter of 14: Students work in partners (or independently) to build representations of Logan s garden using color tiles. Students count the perimeter and area of each rectangle. As they build the rectangles, students should have discussions on what is happening to the rectangles. Have a class discussion on students findings (what is similar and different about their rectangles). Example 3 (Same Area/Different Perimeters): Abigail wants to build a sandbox with an area of 16 square units. Make all the possible ways Abigail could build her sandbox. What would the perimeter of each rectangle be? Two possible rectangles with an area of 16: Students begin with 16 color tiles and build a sandbox. They need to understand that different sized rectangles can represent the same area, but have different perimeters. When students finish, go over the rectangles as a class. There are more rectangles than are pictured here. 15 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Representational: Go Math Chapter 11 Lessons 11.9, 11.10 Example: Kyle says that there is only one way to build a rectangle that has an area of 30 square units. Is he correct? Support your answer with a picture and tell what the perimeter of the rectangle would be. Three possible responses: Session 2 Area and Perimeter Using Drawings and Pictures Students use grid paper to draw rectangles that have an area of 30 square units. They realize that Kyle is not right, there is more than one way to build a rectangle with an area of 30 square units. Have a math talk about how the rectangles are similar and how they are different. Point out that the area stayed the same, but the perimeter changed. 16 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Lesson Sequence 3: Solving Multi-Step Word Problems (3.OA.3, 3.OA.7, 3.OA.8) Cluster: Represent and solve problems involving multiplication and division. 3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. Cluster: Multiply and divide within 100. 3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. The last part of 3.OA.7 states: By the end of Grade 3, know from memory all products of two one-digit numbers. Learning math facts is critical for success in mathematics. It takes time for students to develop quick recall of facts. (However, it is important to note that instructional time should not be used for students to memorize multiplication facts. Instruction should focus on teaching the strategies and giving students ample time to model multiplication and division.) Organizing facts can be extremely helpful for developing fluency. Multiplication and division of related facts should be taught simultaneously. The strategies outlined in this standard support student development. Cluster: Solve problems involving the four operations, and identify and explain patterns in arithmetic. 3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Word problems can involve different problem situations. Often students are presented with problems that represent one or two situations. Because of this, students begin to develop procedural approaches to solving problems. To solve problems, we must understand the problem, select a strategy, and assess if our solution is reasonable. It is critical for students to develop these skills through practice and conversation. As teachers, we must be careful not to highlight or modify a singular approach to solving any one problem. Moreover, research tells us that the use of key words as a strategy for solving problems adds to our students inability to solve problems. Making drawings or representations can be quite helpful for solving problems. Research indicates that using bar diagrams can help students develop understanding of problems and how to solve them. 17 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Session 1 Multi-Step Word Problems Representational: Example 1: Go Math Chapter 7 Lessons 7.10, *7.11 (Lesson 7.11 order of operations is limited to students ability to solving real world multi-step problems) Patricia bought 20 cookies from the store. On the way home she ate two of them. She wanted to share the remaining cookies equally between her and her two siblings. How many cookies would each of the 3 siblings get? Step 1: Students draw 20 circles to represent the 20 cookies Patricia bought at the store. Two of them are crossed off because they represent the ones she ate on the way home. The student wrote the equation of 20 2 = 18 to show how many cookies she had remaining. Step 2: Students may draw three circles to represent the 3 siblings. They distribute the cookies equally to the three circles to represent the siblings getting an equal amount of cookies. Step 3: Students label that each sibling got 6 cookies and then wrote an equation 18 3 = 6 to represent this step of the multistep problem. Each sibling would get 6 of the cookies that Patricia bought at the store. 18 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6
Example 2: Materials and Directions: 1. Put the following story problem up for students to see (chart paper, document camera). Then read it aloud. Vicente is reading a book that has 260 pages. He read 35 pages on Monday night, and 40 pages on Tuesday night. Let p stand for how many pages Vicente has left to read? 2. Give time for the students to solve the problem. 3. Since this standard is looking for mental math or estimation strategies, have students share aloud the strategies they used to solve the problem. 4. Record the student s strategies on the document camera or have the students come up to show their thinking while they explain their strategy. 5. To help all students learn to think aloud and solve problems through mental math or estimation, have another student repeat the strategy given in their own words. 6. Repeat these steps using other two-step problems similar to this one. Considerations: Ask students if they could estimate an answer before trying to solve it. Observe students thinking while solving the problem. Do they have to write down the numbers to add or subtract? Are they able to decompose or think flexibly about the numbers to do the math in their head? Some students may need paper and pencil, but others may not. However, it is important for students to show how they solved a problem. Explanations can be oral or in written form. Teacher notes: The learning targets for this task may include: I can choose the correct operation to perform the first computation, and choose the correct operation to perform the second computation in order to solve two-step word problems. I can write equations using a letter for the unknown number. I can decide if my answers are reasonable using mental math and estimation strategies including rounding. Students who demonstrate complete mastery recognize that they need to add 35 + 40 and then use a strategy such as subtraction to find out how many pages are left (i.e. 260 75). They will apply strategies such as break apart, count up, count back, the open number line or the traditional algorithm. Students who demonstrate partial mastery may solve only one part of the problem. For example, they may solve 35 + 40 or 260 35 or 260 40. Students may also have regrouping errors in all three place values since this problem requires regrouping twice if students choose to use the traditional algorithm to solve 260 75. From https://hcpss.instructure.com/courses/97/pages/3-dot-oa-dot-8-assessment-tasks 19 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 3 Unit 6