Draft -Unit 1 Whole Number Computation and Application 8 Weeks 1 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
2 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Grade 5 Unit 1: Whole Number Computation and Applications (8 weeks) UNIT 1 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. 5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. Learning Targets A I can model and explain that the value of a digit changes as you move to the left (10 times more) or to the right (1/10 of) using manipulatives, pictures, and/or language. Puedo modelar y explicar que el valor de un digito cambia al moverse al lado izquierdo (10 veces más) o al lado derecho (1/10 menos) usando manipulativos, dibujos, y/o lenguaje. 5.NBT.2 Explain patterns in the number of zeroes of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. Learning Targets B I can multiply and divide by powers of ten. Yo puedo multiplicar y dividir en potencias de 10. C I can explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of ten. Yo puedo explicar patrones en la posición del punto decimal cuando un decimal es multiplicado o dividido en potencias de 10. D I can use whole number exponents to denote powers of 10. Yo puedo usar exponentes de números enteros para marcar potencias de 10. 5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm. Learning Targets E I can use the standard algorithm to calculate the product of multi-digit whole numbers. Yo puedo usar el algoritmo estándar para calcular el producto de números enteros con múltiples dígitos. 5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Learning Targets F I can divide up to a four-digit by a two-digit number using place value strategies. Yo puedo usar la estrategia de valor posicional para dividir un número de cuatro dígitos por un número de dos dígitos. G I can divide up to a four-digit by a two-digit number using the properties of operations. Yo puedo usar las propiedades de operaciones para dividir un número de cuatro dígitos por un número de dos dígitos. 3 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
H 5.OA.1 I 5.OA.2 J K L I can divide up to a four-digit by a two-digit number using the relationship between multiplication and division. Yo puedo usar la relación entre multiplicación y división para dividir un número de cuatro dígitos por un número de dos dígitos. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. I can write and evaluate numerical expressions. Yo puedo escribir y evaluar expresiones numéricas. Learning Targets Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation add 8 and 7, then multiply by 2 as 2 (8 + 7). Recognize that 3 (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. Learning Targets I can write simple expressions to record calculations and real world problems. Yo puedo escribir expresiones simples para anotar calculaciones y problemas escritos de situaciones del mundo real. I can interpret expressions without evaluating them. Yo puedo interpretar expresiones sin evaluarlas. I can compare expressions without evaluating them. Yo puedo comparar expresiones sin resolverlas. 4 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Suggested Manipulatives: Base-ten Blocks Color Tiles Centimeter Cubes Place Value Discs Place Value Mats Critical Terms: Expression / expresión Parentheses/ paréntesis Brackets/ paréntesis o corchete Braces/ abrazaderos Powers of 10/ potencia de 10 Exponents/exponentes Distributive Property/ Propiedad Distributiva MATERIALS Representations: Number Lines Story Representation Numerical Expressions Matching Games VOCABULARY Supplemental Terms: Dividend/ dividendo Divisor/ divisor Quotient/ cociente Remainder/remanente Array/ conjunto / orden / matriz Area Model/ modelos de área Compatible Numbers/Numeros Compatibles Resources: STANDARDS FOR MATHEMATICAL PRACTICE (Practices to be explicitly emphasized are indicated with an *.) District Curriculum Guide Go Math ETA Hands 2 Mind Base/Base Evaluate/Evaluar Inverse Operations/Operaciones Inversas Period/Periodo Factor/Factor Product/Producto Partial Quotient/ Cociente Parcial 1. Make sense of problems and persevere in solving them (Dan sentido a los problemas y perseveran en su resolución.) Students persevere in solving problems to represent and solve in a range of contexts by selecting appropriate strategies. 2. Reason abstractly and quantitatively. (Razonan de forma abstracta y cuantitativa.) Students reason abstractly and quantitatively when they read word problems and determine the appropriate math strategy and/or operation to solve the problem. After solving the problem, students will determine does this answer make sense in the context of the problem. 3. *Construct viable arguments and critique the reasoning of others. (Construyen argumentos viables y critican el razonamiento de otros.) Students explain calculations using models, properties of operations, and rules that generate patterns when they talk and write about the steps they take to solve problems. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like: How did you get that? Why is that true? Students explain their thinking to others and respond to others thinking. 4. Model with mathematics. (Representación a través de las matemáticas) Students make diagrams and equations to represent the multiplication and division situations. 5. Use appropriate tools strategically. (Utilizan las herramientas apropiadas estratégicamente.) Use manipulatives to model division (e.g. base- ten materials and Cuisenaire Rods) 6. Attend to precision. (Ponen atención a la precisión.) Students are using mathematical language about place value and power of ten appropriately and consistently. 5 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
7. Look for and make use of structure. (Reconocen y utilizan estructuras.) Students will look for the place value structure of numbers to aide in efficient calculation. 8. *Look for and express regularity in repeated reasoning. (Reconocen y expresan regularidad en el razonamiento repetitivo) Students use repeated reasoning to understand algorithms and make generalizations about patterns when multiplying and dividing multi-digit numbers. Students connect place value and their prior work with operations to understand algorithms to fluently multiply multi-digit numbers. Grade 5 Unit 1 Whole Number Computation and Applications Connections to Previous Learning: In fourth grade, students fluently multiply (4-digit by 1-digit, 2-digit by 2-digit) and divide (4-digit by 1-digit) using strategies based on place-value and the properties of operations. 5 th grade students will extend these skills to develop fluency with efficient procedures for multiplying and dividing multi-digit whole numbers. Focus of the Unit: Much of this unit focuses on extending previous skills of multiplication and division to multi-digit whole numbers. Students will fluently multiply multi-digit numbers by using a standard algorithm. Using partial products, including area models and place value strategies is using an algorithm. The work that students do in grade 5, prepares them for the traditional standard algorithm as natural transitions from what they know, understand and are able to apply using partial products. They will use area models as a stepping stone to partial products. Then partial products will help with place value understanding as they learn the column multiplication method. All three of these are recording strategies for the standard algorithm. Students should be exposed to all three methods so that they are adequately prepared to critique the reasoning of others when any of these methods are chosen. Students will divide (whole numbers with up to four-digit dividends and with no more than two-digit divisors) using strategies based on place-value, the properties of operations, and/or the relationship between multiplication and division; they will illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. The fluency standard for division, using the standard algorithm, does not apply until grade 6. At this point in the students learning, they can fluently apply all operations using whole numbers. Now they are ready to begin incorporating the order of operations by using parentheses as grouping symbols when decomposing multi-digit numbers to multiply and divide. They do not study the entire order of operations because focus in this unit is only on the application of the distributive property. Connections to Subsequent Learning: 6 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Fifth grade students experiences with numerical expressions prepare students to work with algebraic expressions in the sixth grade. In sixth grade, students will read, write, and evaluate more complex expressions using variables. Experiences with parentheses, brackets, and braces in grade 5 lead 6 th grade students to learn the order of operations. In grades 6-8, students begin using properties of operations to manipulate algebraic expressions and produce equivalent expressions for different purposes. This builds on the extensive work done in K-5 working with addition, subtraction, multiplication, and division. Enduring Understanding: Students will understand that Order of operations is about communication: Rules and symbols are used to communicate with others the sequence that should be used to simplify an expression. Parentheses, brackets, and braces are used to guide the order of operations when simplifying expressions. A standard algorithm is used to fluently multiply multi-digit whole numbers. A variety of different strategies can be used to divide multi-digit numbers, visual models (rectangular array, equations, and/or area model) and strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Essential Questions: How do parentheses, brackets, and braces affect the way you simplify expressions? How do I use parentheses, brackets, and braces to communicate the sequence for simplifying expressions? How do you multiply multi-digit numbers using a standard algorithm? How do you choose different division strategies to divide multi-digit numbers? 7 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Lesson Sequence 1 Multiplicative Patterns of Powers of Ten (5.NBT.1, 5.NBT.2) Understand the place value system. 5.NBT.1- Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 5. NBT.2- Explain patterns in the number of zeroes of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. Session 1 Place Value Understanding and Powers of Ten MP 7: Look for and make use of structure: Students use a table to find patterns in powers of ten. Words Number Number of Power of Multiplication Expression Zeroes Ten Exponent Millions 1,000,000 6 1 10 10 10 10 10 10 10 6 Sixth power 6 Hundred 100,000 5 1 10 10 10 10 10 10 Thousand Fifth power 5 Ten Thousand 10,000 4 1 10 10 10 10 10 4 Fourth power 4 Thousand 1,000 3 1 10 10 10 10 3 Third power 3 Hundred 100 2 1 10 10 10 2 Second power 2 Ten 10 1 1 10 10 1 First power 1 One 1 0 1 1 10 0 0 Students write expressions with powers of ten and recognize that numbers can be described as ten, a hundred, or a thousand times greater based on the power of ten. Misconception: Adding a Zero When students refer to multiplying by 10 as multiplying numbers and then adding a zero they are not correctly articulating what adding a zero" represents. This should be authentically demonstrated to students so they recognize that adding a zero is not the same as multiplying by 10. As indicated by math practice standard (MP6) it is imperative that students attend to precision in their explanation of the numbers of zeroes in their solution. When you add zero to any number your result is that number (Identity property of addition). Correct student justification: 14,560 ones multiplied by one hundred means I have 14,560 hundreds so I write 1,456,000 because I shifted all of the digits two places to the left. 8 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
In addition, students learn that 500,000 is equivalent to the expression 5 10 5 Number Number of Expression Multiplication Expression Exponent Zeroes 500,000 5 5 10 10 10 10 10 5 5 x 10 5 50,000 4 5 10 10 10 10 4 5 x 10 4 5,000 3 5 10 10 10 3 5 x 10 3 500 2 5 10 10 2 5 x 10 2 50 1 5 10 1 5 x 10 1 5 0 5 1 0 5 x 10 0 MP 8: Look for and express regularity in repeated reasoning. Students are able to make a generalization about multiplying and dividing by powers of ten. Since each factor of 10 shifts the digits one place to the left, multiplying 10 10 10 (which can be recorded in exponential form as 10 3 ) shifts the position of the digits to the left 3 places. Concrete: Students use place value chips to compose and decompose units to illustrate the 10 times larger and 1/10 as large as relationships among whole numbers 9 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Math Practice 8: Look for and express regularity in repeated reasoning First remind students how to make equivalents with place value chips. For example, one million is equal to 10 hundred thousand. Divide 10 hundred thousand by 10. Students can draw ten circles and put a single hundred thousand chip in each circle. Repeat, by asking how to make an equivalent to one hundred thousand. One hundred thousand is equal to 10 ten-thousands. Now divide ten thousand by ten. Students draw ten circles and put a single ten thousand chip in each circle. Continue this sequence until the ones place is reached, emphasizing the unbundling for 10 of the smaller units and then the division. Record the place values and equations (using unit form) on the board being careful to point out the 1 tenth of relationship: 1 million 10 = 1 hundred thousand 1 hundred thousand 10 = 1 ten thousand 1 ten thousand 10 = 1 thousand 1 thousand 10 = 1 hundred 1 hundred 10 = 1 ten 1 ten 10 = 1 one 10 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Students use Place Value Chips to multiply 3 by 10 across the place value chart starting in the ones moving to the thousands. [= As you move to the left on the place value chart you are multiplying by ten (10 times larger) and then regrouping into the next higher place value. 11 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Representational: Bridge the concrete (C) to representational (R) by drawing the place value chips. What happens if we start with 3 million and divide by ten? Students model drawing chips using the same pattern and process of the previous segment. The students draw ten circles and put 3 ten thousand disks in each circle. Students continue the pattern to further their conceptual understanding of the ten times greater and the one tenth as large relationship. Activity: Students use base ten block representations on a place value mat to help them understand that each time they multiply by 10 the digits shift to the left one place value. We want students to understand that a bundle of 10 tens is the same as 1 hundred, or 10 thousands is the same as 1 ten-thousand. A bundle of ten in one place value is the same as 1 in the place value to the right. Key Unit Rod Flat Example 1: 3 10 = 3 groups of ten ones which is 30 ones which is 3 tens 30 10 = 3 groups of 10 tens or 3 hundreds Example 2: Example 3: Place value chips (500 10) = (5 100) 10 = 5 (10 100) 12 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Bridge from Representational to Abstract: Example 1: Students write digits in the place value chart and multiply by ten to shift to the left. For example, students notice that multiplying 8 by 1,000 shifts the position of the digits to the left three places, changing the digits relationships to the decimal point and producing a product with a value that is 10 10 10 as large (8000.0) Students should justify their work reasoning about place value (see example below). Incorporate the use of exponents to represent powers of ten (5.NBT.2). For example, students can write 10 x 10 x 10 as 10 3 Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones 8 0. 8 0 0 0. 8 0 0 0 0... Tenths MATH PRACTICE 3: Construct Viable Arguments and Critique the Reasoning of Others. Students should justify their reasoning using place value by saying I know that 80 times 1 hundred is 80 hundreds and 80 hundreds is equivalent to 8 thousand. I know that 8 thousands is 100 times greater than 8 tens. I know that 8 thousands times 10 is 80 thousands and is equivalent to 8 ten thousands. I know that 80 thousands is 10 times greater than 8 thousands. Math Practice 2: Reason Abstractly and Quantitatively Each time I multiply by ten the digit 8 in the above example becomes ten times greater than in the previous number. Students will be able to transfer this idea to real-world situations. Real World Example: A microscope has a setting that magnifies an object so that it appears 1000 times as large when viewed through the eyepiece. If an ant has a width of 2 mm, what would the width appear to be in millimeters through the microscope? Explain how you know. Students can write an equation 2 10 3 = 2000 mm or use a place value chart to solve the problem. 13 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Bride from Representational to Abstract Example 1 Continued: Students write digits in the place value chart and divide by ten to move the digit one place value to the right. For example, students notice that dividing 80 by 1,000 moves the position of the digit three place values to the right, changing the digits relationships to the decimal point. Students should see the problems as 8,000 10, 8,000 100, and 8,000 1,000 or 80 10 1, 80 10 2, 80 10 3 (5.NBT.2). Avoid tricks like 10 4 means you add 4 zeros to the number because adding zero to any number yields the same number (Additive identity property). This trick creates a misconception and students lack a conceptual understanding of the standard. Millions Hundred Ten Thousands Hundreds Tens Ones Thousands Thousands. Tenths 8 0 0 0. 8 0 0. 8 0. 8. Example 2: Multiply Mixed Units by 10, 100, 1000 78 x 10 1, 78 x 10 2, 78 x 10 3 Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones 7 7 7. Tenths 8. 0 8 0. 0 8 0 0. 0 7 8 0 0 0. 0 14 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Divide Mixed Units by 10, 100, 1000 89000 10 1, 89000 10 2, 89000 10 3 Millions Hundred Thousands Ten Thousands 8 9 Thousands Hundreds Tens Ones. Tenths 0 0 0. 0 8 9 0 0. 0 8 9 0. 0 8 9. 0 MATH PRACTICE 7: Look for and make use of structure Lead students to discuss how the digits shift as a result of their change in value by isolating one digit, such as the 3, and comparing its value in each product. Students should be able to compare the 8 in any of the numbers above. For example, the 8 in 7800 is 100 times greater than the 8 in 78 because 8 100 = 800. Give students opportunities to practice and discuss what they are doing when multiplying by 10, 100, or 1,000 a. 345 10 = 3450 Thousands Hundreds Tens Ones Tenths Hundredths Thousandths 3 4 5 3 4 5 b. 345 100 = Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Each digit in the product is 10 times greater than the original factor so the value of the product is 10 times greater than the original factor. When I multiply by 10, I shift all of the digits one place to the left. Each digit in the product is 100 times greater than the original factor so the value of the product is 100 times greater than original factor. When I multiply by 100, I shift all of the digits two places to the left. 15 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
c. 345 1,000 = Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Each digit in the product is 1000 times greater than the original factor so the value of the product is 1000 time greater than the original factor. When I multiply by 1000, I shift all of the digits three places to the left. Give students opportunities to practice and discuss what they are doing when dividing by 10, 100, or 1,000. a. 34500 10 = 345 Thousands Hundreds Tens Ones Tenths Hundredths Thousandths 3 4 5 0 0 3 4 5 0 0 Each digit in the quotient is 1 or 0.1 the value of the original digit in 10 the dividend so the quotient is 1 or 0.1 the value or the original 10 dividend. b. 34500 100 = Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Each digit in the quotient is 1 or 0.01 the value of the original digit in 100 the dividend so the quotient is 1 or 0.01 the value of the original 100 dividend. c. 34500 1,000 = Thousands Hundreds Tens Ones Tenths Hundredths Thousandths dividend. d. Explain how and why the value of the 4 changed in the quotients in (a), (b), and (c). Each digit in the quotient is the dividend so the number is The focus for this explanation should show the connection between the digit and the change in the value of the digit. The value of every digit in the quotient is one tenth, one hundredth or one thousandth of the dividend. See the notes for each above. 1 1000 1 1000 or 0.001 the value of the original digit in or 0.001 the value of the original 16 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Abstract: Go Math Chapter 1 Lesson 1.1, 1.2, 1.4, 1.5 Find the products. a. 14,560 10 = b. 14,560 100 = c. 14,560 1,000 = Explain how you decided on the number of zeros in the products for a, b, and c. The focus for this explanation should show the connection between the digit and the change in the value of the digit. The value of every digit in the quotient is one tenth, one hundredth or one thousandth of the dividend. See previous pages for examples. a. 15278 10 = b. 15278 100 = c. 15278 1,000 = 17 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Lesson Sequence 2 Multiplying Multi-digit Numbers (5.NBT.5, 5.OA.1, 5.OA.2) Perform operations with multi-digit whole numbers and with decimals to hundredths. 5. NBT.5- Fluently multiply multi-digit whole numbers using the standard algorithm. Write and interpret numerical expressions. 5. OA.1- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 5. OA.2- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation add 8 and 7, then multiply by 2 as 2 (8 + 7). Recognize that 3 (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. Abstract Go Math Chapter 1 Lessons 1.3, 1.10, 1.11, 1.12* *5.OA.1 does not require nested parentheses, brackets, and braces Session 1 Properties BIG IDEA: Multi-digit whole numbers and multiples of 10 can be multiplied using place value patterns, and the distributive and associative properties. These strategies create a foundation for understanding the standard multiplication algorithm. ASSOCIATIVE PROPERTIES Abstract Example: Students write simple expressions with grouping symbols to organize their thinking. 40 5 40 50 40 500 40 5000 We can write 40 5 as (4 X 10) X 5. If we write the numerical expression as (4 X 5) X 10, is it equivalent to the other expression? Why or why not? The commutative property of multiplication allows me to write the factors in any order. The associative property of multiplication allows us to change the way we group the factors. We can write 40 X 50 as (4 X 10) X (5 X 10). We can use the commutative and associative properties of multiplication to write an equivalent expression (4 X 5) X (10 X 10). Is there another equivalent expression? (4 X 5) X 10 2 18 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
2-digit multiplied by 1-digit: Session 2 Multiply by 1-digit Situation Model Strategy Equal Groups Unknown Product Base Ten Blocks 2 digit times 1 digit STORY PROBLEM There are 23 tables set up in the gym for the awards ceremony. Ms. Gilmore wants to put 8 chairs at each table. How many chairs do we need for all the tables to have 8 chairs? CONCRETE REPRESENTATIONAL ABSTRACT The students use base ten blocks to model the problem situation. The teacher models the work with equations that the students have modeled with the blocks. (8x10)+(8x10)+(8x3) 8 tens + 8 tens + 2 tens + 4 ones 8 tens + 2 tens + 8 tens + 4 ones 10 tens + 8 tens + 4 ones 1 hundred + 8 tens + 4 ones 184 We need 184 chairs for 23 tables. Bridge students to representational drawings by having them draw pictures of the work they modeled with the base ten blocks. Students transition to the area model Students connect the representational drawings to the partial products algorithm. We need 184 chairs for 23 tables. Students connect partial products to the standard algorithm. We need 184 chairs for 23 tables. Students use standard algorithm. We need 184 chairs for 23 tables. (8x10)+(8x10)+(8x3) 8 tens + 8 tens + 2 tens + 4 ones 8 tens + 2 tens + 8 tens + 4 ones 10 tens + 8 tens + 4 ones 1 hundred + 8 tens + 4 ones 184 We need 184 chairs for 23 tables. 19 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
3-digit multiplied by 1-digit Example: A tank of oil can fill up 324 motorcycles. If there are 3 tanks of oil, how many motorcycles can be filled up? Patterns and the Distributive Property Concrete bridged to the Representational: 324 X 3 Students are given multiple opportunities to work with place value manipulatives (chips). When the students are working with their place value manipulatives, the teacher is formatively assessing how the students are placing their chips on the mats and how their explanation of what they are doing mirrors the task. Representational: Area Model Students bridge from the concrete to create a representational illustration of what they modeled. When students are able to explain this bridging activity, they are increasing their conceptual understanding of multiplication. The following is an example of how students Bridge to an Area Model Representation of the expression, 324 X 3 Students can use area models with place value strategies and distributive property to decompose numbers to find products of multi-digit whole numbers. 300 20 4 900 + 60 + 12= 972 3 900 60 12 20 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Situation Model Strategy Equal Groups Unknown Product Measurement Example 3 digit times 1 digit STORY PROBLEM Melanie ran 6 miles per day for 124 days last year. How many total miles did she run last year? CONCRETE REPRESENTATIONAL ABSTRACT The students model the problem with place value chips. The teacher writes the equations to match the student work. (6x100)+(6x20)+(6x4) 6 hundreds + 12 tens + 24 ones 6 hundreds + 1 hundred + 2 tens + 2 tens + 4 ones 7 hundreds + 4 tens + 4 ones Students can use an area model. The students begin to label their drawings with equations. (6x100)+(6x20)+(6x4) 6 hundreds + 12 tens + 24 ones 600 + 120 + 24 = 744 miles Students connect area models to partial product algorithm Students connect the partial products algorithm to the standard algorithm. Students use the standard algorithm to model the problem situation. Melanie ran 744 miles last year. Melanie ran 744 miles last year. Representational Bridge to Abstract Go Math Chapter 1 Lesson 1.8 21 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
4-digit multiplied by 1-digit Example: A barrel of water can fill up 1,243 cups. If there are 4 barrels of water, how many cups can be filled up? Concrete: Students use place value chips/base ten blocks to model multiplication problems. Students put the model for the first factor on the place value mat. Students place the first factor on the mat four times. Students write an expression to match what they have placed on the mat. Students evaluate the expression. (1000 X 4) + (200 X 4) + (40 X 4) + (3 X 4) = 4000 + 800 + 160 + 12 = 4972 22 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Representational: To bridge from the concrete to the representational, students use the concrete model to create their representational drawing. They should not be Situation Model Strategy Equal Groups Unknown Product 4 digit times 1 digit STORY PROBLEM There are 2,135 seats in the stadium. They sold tickets for all the seats Saturday and Sunday for the last two weeks. How many tickets did they sell in all? CONCRETE REPRESENTATIONAL ABSTRACT The students can model the problem with place value chips. The teacher will use equations to model the work the students are doing with the place value chips. The students will model the problem with area models. Students connect the area model to partial products. The students will model the problem with partial products and then the standard algorithm. 23 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Situation Model Strategy STORY PROBLEM CONCRETE REPRESENTATIONAL ABSTRACT Compare Unknown Product 4 digit by 1 digit Mike scored 2,498 points on the Pac Man video game. His brother scored twice as many points. How many points did Mike s brother get on the video game? Students can model the problem situations using place value chips. The teacher can model the problem situation with equations to connect to the work students are doing with the place value chips. Use the distributive property. (2x2000)+(2x400)+(2x90)+(2x8) =4000+800+180+16 Use the associative property to regroup 4000+900+90+6 =4,996 MIke s brother scored 4,996 points playing the Pac Man game. The students can model the problem situation using an area model. Mike s brother scored 4,996 points playing the Pac Man game. The students can use partial products to model the problem situation. The students can model the problem situation using partial products and then the standard algorithm. Mike s brother scored 4,996 points playing the Pac Man game. The students can model the problem situation using the standard algorithm. MIke s brother scored 4,996 points playing the Pac Man game. Mike s brother scored 4,996 points playing the Pac Man game. Abstract Go Math Chapter 2 Lesson 1.6 24 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Abstract Go Math Chapter 1 Lessons 1.7*, 1.9 * Students will need more practice to reach the expectation of 5.NBT.5 which requires fluently multiplying multi-digit whole numbers using the standard algorithm. Situation Model Strategy Arrays/ Area Unknown Product Base Ten Blocks 2 digit times 2 digit STORY PROBLEM The students are setting up the gym for the spring concert. They are making 25 rows of 12 chairs. How many chairs did the students use in all? CONCRETE REPRESENTATIONAL ABSTRACT The students model the problem situation using base ten blocks. The teacher will model the student work with equations. (10x10)+(10x10)+(5x10)+(10 x 2) +(10x2)+(5x2) =100 + 100 + 50 + 20 + 20+10 =200 + 100 =300chairs The students used 300 chairs. Session 3 Multiply by 2-digit The students can model the problem situation with a drawing. The students used 300 chairs The students are labeling their models and writing equations. Students bridge to the partial product algorithm. Students bridge the partial products algorithm to the standard algorithm. The students used 300 chairs. Students use the standard algorithm to model the problem situation. The students used 300 chairs The students used 300 chairs. 25 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Situation Model Strategy Equal Groups Unknown Product Base Ten Blocks STORY PROBLEM CONCRETE REPRESENTATIONAL ABSTRACT The manager at Office Depot is putting boxes of paper clips on shelves. There are 21 boxes of paperclips. Each box of paperclips contains 34 paperclips. What is the total number of paperclips in all of the boxes of paperclips? The student can model the problem situation using base ten blocks. The students can begin drawing pictures of the concrete materials to model the problem situation. Students can model the problem situation with partial products and then the standard algorithm. 2 digit times 2 digit The teacher models the problem situation using equations that models the work students did with the blocks. 2134 (20x30)+(1x30)+(20x4)+(1x4) We can use different properties to help us solve this problem. =(2x3x10x10)+30+(2x4x10)+4 =(6x100)+30+(8x10)+4 =600+30+80+4 =600+100+10+4 =714 There are 714 paperclips in all. The students can label the drawings to help them find the total. The student can model the problem situation using an area model. 26 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Situation Model Strategy Compare Unknown Product 2 digit by 2 digit Base Ten Blocks STORY PROBLEM Puppy Bits dog food company shipped 48 cases of dog food to PetCo last week. The company shipped 12 times as many cases to Dog Mart. How many cases of dog food did Puppy Bits ship to Dog Mart? CONCRETE REPRESENTATIONAL ABSTRACT The students can model the problem situation with base ten blocks. 48 12 =? The teacher will use equations to model the work students do with base ten blocks. (10x40)+(10x8)+(2x40)+(2x8) =400 + 80 + 80+16 =400 +160 +16 =500 + 70 + 6 =576cases of dog food. The students can use an area model. The students can use partial products to model the problem situation. The students can use partial products and then the standard algorithm to model the problem situation. The students use the standard algorithm to model the problem situation. 27 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
3-digit multiply by 2-digit Example: An acre of farmland can yield 324 pounds of peas a season. If Uncle Lee used 18 acres of farmland for planting peas, how many pounds of peas can be harvested? 18 324 MP 3: Construct viable arguments and critique the reasoning of others. Students should be able to explain how the area model represents multiplication of multi-digit numbers using distributive property, associative and commutative properties, and place value in their explanations. Students should be able to identify errors in the models and explanations of others. Student A was asked to draw an area model for the expression 324 80. The student drew the area model pictured above. Student B says the area model is incorrect. Students should be able to identify that the numbers outside the area model are additive. The side is 300 + 20 + 4 or 324 and along the top is 8 + 10 or 18. Not 80. 28 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Connecting the Area Model and Partial Products to the standard algorithm Students start to write the standard algorithm next to the area model drawing. Students find the sum of the partial products. Using the Communitive property, the order of the factors can be rearranged to reflect the process of the standard algorithm. Students should have the understanding that the area model is flexible and can be used how they choose. Students may be working at various levels of mastery with the algorithm. 29 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
APPLICATION: MP 2: Reason abstractly and quantitatively. Do not wait until students are working with the standard algorithm to introduce real-world application problems involving multiplication. Students will decontextualize the problem to set up and solve with one of the above strategies. Students will have to reflect on whether or not the product makes sense in context of the situation. Situation Model Strategy Arrays/ Area Unknown Product STORY PROBLEM CONCRETE REPRESENTATIONAL ABSTRACT Jimena has a rectangular table that is 125 feet by 13 feet. What is the area of the top of Jimena s table? The students can use an area model to model the problem situation. The students can use partial products and then the standard algorithm to model the problem situation. 3 digit times 2 digit The students can use partial products to model the problem situation. The students can use the standard algorithm to model the problem situation. 30 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Situation Model Strategy Compare Unknown Product 3 digit by 2 digit STORY PROBLEM CONCRETE REPRESENTATIONAL ABSTRACT A manufacturing company shipped out 437 cases of ballpoint pens. The company shipped out 15 times more cases of gel pens. How many cases of gel pens did the company ship out.? The students can use an area model. The students can use partial products and then the standard algorithm to model the problem situation. The students can use an area model and partial products. The students use the standard algorithm to model the problem situation. 31 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Example : A container can hold 122 tennis balls. If there are 4 such containers, how many tennis balls can be held in the containers? 3-digit by 1-digit: Concrete (Array): Representational (Array to Area Model): When moving from drawing a representation of the place value to chips to creating an Area Model, make sure to discuss that the area (partitioning) of the box should match the magnitude of the number. For example, 4x100 should be a larger area of the box then 4 x 2. Example: An acre of farmland can harvest 1,122 pounds of soybeans. If there are 3 acres of farmlands, how many pounds of soybeans can be harvested? 4-digit by 1-digit: Concrete (Array): Representational (Array to Area Model): When moving from drawing a representation of the place value using chips to creating an Area Model make sure to discuss that the area (partitioning) of the box should match the magnitude of the number. For example, 3x1000 should be a larger area of the box then 3 x 20. 32 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
** The magnitude of the problems for some the examples above are too large to model with concrete materials. Use these examples when the students have moved beyond the need for concrete examples. Addressing Misconception: Place Value Misconceptions can make algorithms difficult for students to understand. 1. Student under generalizes results of multiplication by powers of 10 and does not understand the shifting of the digits to higher place values is like multiplying by powers of 10. For example, when asked to solve a problem like,? x 25=2500, the student either divides or cannot respond. 2. The student under generalizes the result of multiplication by powers of 10. To find products like 4 x 30=120or 40 x 30=1200, he or she must work it out using a long method of computation. For example, Remediation suggestions: Continue to work on patterns with powers of ten. Use Number Talks for mental math strategies. Problem Solving Misconceptions about Multiplication 3. Student has oversimplified during the learning process so that she recognizes some multiplications situations as multiplication and fails to classify others appropriately. For example, student recognizes groups of problems as multiplication but does not know how to solve scale, rate, or other types of multiplication situations. 4. Student knows how to multiply but does not know when to multiply other than when he is told to do so or it is written as a multiplication problem for computation. For example, the student cannot explain why he should multiply or connect multiplication using concrete materials. 33 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Remediation suggestions: Be sure you are exposing the students to a variety of problem situations for multiplication. (See page 37 for all problem situations) Practice reasoning about what kind of math is found in different problem situations. Provide examples of problem situations that are multiplication and others that are not multiplicative but uses similar wording (Some students use keywords that can work for both addition and subtraction). Go back to concrete materials to help student develop vocabulary about multiplication. Conceptual Misconceptions 5. The student does not understand the distributive property and does not know how to apply it to simplify the work of multiplication. For example, the student is fluent with basic facts but cannot multiply 12 x 8 or 23 x 6 6. The student can state and give examples of properties of multiplication but does not know how to apply them to simplifying computation. For example, a. The student multiplies 3 x 14 with ease but struggles to find the product of 14 x 3. b. The student labors to find the product of 12 x 15 because he does not realize that he could instead perform the equivalent but much easier computation, 6 x 30 c. The student is fluent with basic facts but cannot multiply 7 x 25 Remediation suggestions: Use Number Talks to build multiplication fluency Review strategies for solving multiplication problems Re-teach properties of multiplication in contextual situations Create an environment where students share out solution strategies and make connections between the different strategies. Algorithm Misconceptions 7. The student misapplies the procedure for multiplying multi-digit numbers by ignoring place value. a. Student multiplies correctly by ones digit but ignores the fact that the 3 in the tens place means 30 so 30 x 60=1800 34 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
b. Student multiplies each digit as if it represented a number of ones and/or Ignores place value completely. 8. The student misapplies the procedures for regrouping as follows; a. The first step (multiplying by ones) is done correctly but the same numbers are used for regrouping again when multiplying by 10s whether it is appropriate or not. 9. The student over generalizes the procedure learned for addition and applies it to multi-digit multiplication inappropriately. 35 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
10. The student generalizes what she learned about single digit multiplication and applies it to multi-digit multiplication by multiplying each column as a separate single-digit multiplication problem. Remediation suggestions: When students make mistakes with standard algorithm, move back one step in the CRA progression and see if the student can identify his or her own mistake. Keep moving back one step until you find where the students is proficient. Move forward again through the progression reviewing the bridged connections between each step. 36 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
37 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Lesson Sequence 3 Dividing Multi-digit Numbers (5.NBT.6, 5.OA.1, 5.OA.2) Perform operations with multi-digit whole numbers and with decimals to hundredths. 5. NBT.6- Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Write and interpret numerical expressions. 5. OA.1- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 5. OA.2- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation add 8 and 7, then multiply by 2 as 2 (8 + 7). Recognize that 3 (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. There are two distinct interpretations for division. Although the quotients are the same, the approaches are different. Partitive Division (Group Size Unknown): 15 apples were placed equally into 3 bags. How many apples were in each bag? Measurement Division (Number of Groups Unknown): 15 apples were put in bags with 3 apples in each bag. How many bags were needed? Session 1 Divide by 1-digit Abstract Go Math Chapter 2 Lessons 2.1*, 2.2** * 5.NBT.6 requires students to use strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. When using this lesson have students use one of the listed strategies in 5.NBT.6 to solve the problems. ** Students will need additional practice to meet the expectations of 5.NBT.6 which requires students to use strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. 38 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
2-digit divided by 1-digit Situation STORY Model PROBLEM Strategy Equal Groups Group Size Unknown How many in each group? Division Jeremiah had 24 cookies shared equally in 3 bags. How many cookies were there in each bag? CONCRETE REPRESENTATIONAL ABSTRACT The students can use base ten blocks to model the problem situation. The students can draw pictures of the models with base ten blocks. Student shows the three bags as the divisor and starts to distribute or count out the cookies up to 24 cookies. The student can reason about division through its relationship with multiplication. Twenty-four shared into three equal groups is equal to what number? 24 3 =? Three groups of what number are equal to 24? 3? = 24 Base Ten Blocks Two digit divided by one digit Connect the factor track that was used to model multiplication problem situations. What do we know? We know the total number of cookies. We know the number of groups (3 bags). We put the three groups on the side and now the student should fit the blocks (cookies) into each bag by distribution. Once the student is done with the distribution, he or she can count how many are in each bag or row. The number in each row is the quotient. The student can explain the division problem by identifying if they know the number of groups or the size of the groups. When computing basic division it does not matter if it is the group or the number of groups because multiplication is commutative. However, in problem situations being able to identify the number of groups versus the number in each group can have effect if the correct answer is given. The student may have to make trades in order to distribute. The student checks the work by reasoning about multiplication. For example, when providing the solution to this problem, the student should say that there are 8 cookies in each bag. If they do not understand the difference between what is in each group and the number of groups they may say that we need 8 bags. There are 8 cookies in each bag. Misconceptions about the meaning of the solution becomes more complex when there are remainders so it is best to make sure we understand the problem from the start. 39 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Situation Model Strategy Arrays Group Size Unknown How many in each group? Division STORY PROBLEM CONCRETE REPRESENTATIONAL ABSTRACT Kimmy is planning to sell bouncy balls at the school fun fair. She has 78 bouncy balls and 4 baskets to put them in. If she wants to put the same number of bouncy balls in each basket, how many balls should she put it each basket? The students can model the problem situation using base ten blocks. The students can model the problem situation using an area model. Thought Bubble Base Ten Blocks Two digit divided by one digit with remainders The student puts out 78 for the dividend because that is the total number of bouncy balls. The student puts out 4 to show the number of baskets. Students can distribute ten bouncy balls (1 rod) to each basket. The student can see that there are only 3 rods so they need to make trades. The students should be able to interpret the remainder. There are 19 bouncy balls in each of the baskets but 2 are left over. 40 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Situation Model Strategy Equal Groups Number of Groups Unknown How many groups? Division STORY PROBLEM There are 45 people coming to the 5th grade Fall Festival Presentation. The teacher wants to put 9 chairs in each row. How many rows of chairs will there be? CONCRETE REPRESENTATIONAL ABSTRACT The students can model the problem situation with base ten blocks. The students can draw pictures of the models they made with base ten blocks. What do I know? I know that there should be 9 chairs in each row so that is my divisor. The student can reason about division through its relationship with multiplication. Forty-five with 9 chairs in each row equals how many rows? 45 9 =? How many rows of 9 chairs equals 45 chairs.? 9 = 45 Two digit divided by one digit The students know that they need 45 chairs and that there will be 9 chairs in each row. The number of chairs in one row is the divisor. The total number of chairs is the dividend. We have to make trades again. to solve the problem. I need to make there be 9 chairs in each column until I get to 45 total chairs. The student can explain the division problem by identifying if they know the number of groups or the size of the groups. When computing basic division it does not matter if it is the group or the number of groups because multiplication is commutative. However, in problem situations being able to identify the number of groups versus the number in each group can effect if the correct answer is given. How many rows of chairs is the quotient. The students find the quotient to be 5 because that is the number of rows with 9 chairs in each. For example, when providing the solution to this problem,the students should say that there will be 5 rows of chairs, not 5 chairs in each row. Misconceptions about the meaning of the solution becomes more complex when there are remainders so it is best to make sure we understand the problem from the start. The students made 5 rows with 9 chairs in each row. The students made 5 rows with 9 chairs in each row. 41 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
3-digit Divided by 1-digit Situation STORY Model PROBLEM Strategy Equal groups Group Size Unknown How many in each group? Division Measurement Three digit divided by one digit Jay cuts 8 blocks of wood from a board that was 112 inches long. How long were each of the blocks of wood? CONCRETE REPRESENTATIONAL ABSTRACT The students can use base ten blocks to model the problem situation. The students put out blocks that show the total number of inches for the board and the divisor that shows the number of blocks Jay cut from the board. Trades will be made to determine how long each block was in inches. The students can draw pictures to model the problem situation. The student shows that there are 8 blocks of wood. When drawing the student count up to the total number while drawing and then uses subtraction. I can see that the blocks have to be greater than 10 inches long. So I will pass out 1 inch more to each block until I use up all the inches. Students can reason about this problem by estimating and then checking. If each block is 10 inches long then I am using only 80 inches so they need to be longer than 10 inches. If I subtract the 80 inches from 112 inches, I have 32 extra inches to divide up to each of the blocks. If I know that 32 divided by 8 is four my job is easy. If not I have to continue to check different amounts. A student may make a table. Students might start using partial quotients and take out part and then find the remainder. It will take the students time to become proficient at making the correct trades for these types of problems. Students will start to notice, for example, that if the divisor is less than 10 and there is 100 or more they will have to trade the 100 for tens. The student continues to subtract 8 groups of 1 inch until there are 0 inches left. Any way you do it, each block is 14 inches long. The blocks are 14 inches long. The blocks are 14 inches long. 42 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Situation Model Strategy Equal groups Number of Groups Unknown How many groups? Division STORY PROBLEM There are 128 fluid ounces in a gallon of juice. If one serving of juice is 4 fluid ounces, how many servings of juice are there in the gallon? CONCRETE REPRESENTATIONAL ABSTRACT The students can use base ten blocks to model the problem situation. The students can draw pictures of the models they have created for the problem situation. The students can uses repeated subtraction to model the problem situation. Some students will have to start with subtracting 4 ounces for each serving. Count with them as they subtract, one serving, two servings, three servings Is there another way to do this? I hope so! Measurement Three digit divided by one digit The student puts out the total number of ounces as the dividend and shows the 4 fluid ounces as the divisor. The student sees that there will need to be trades to find out the number of servings that are 4 ounces each. The student asks how many ounces of juice do I use if I give out 10 servings that are 4 ounces each? I will use 40 ounces of juice. Student continues to pass out servings. But what happens when the amount of juice I have left is less than 40 ounces? I switch to units. The divisor represents a 4 ounce serving so each of the 10 rods represents 10 servings that are 4 ounces. How many four ounce servings are there in 128 fluid ounces of juice? How many 4 ounce servings are there is 8 ounces of juice? There are 2 servings. What happens if we subtract in groups of 4 ounces.? There are a total of 32 four ounce servings in 128 ounces of juice. There are 32 servings in 128 fluid ounces. Or you could say that we can serve 32 people if we gave them each 4 fluid ounces of juice. I can give out 32 servings of juice. 43 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
4-Digit Divided by 1-digit Situation Model Strategy Equal Groups Group Size Unknown How many in each group? Division Comparison Place Value Chips Four Digit Divided by One Digit STORY PROBLEM The balance of Jeffery s bank account is 3 times greater than Shelly s bank account. Jeffery has $1,218 in his bank account. How much money does Shelly have? CONCRETE REPRESENTATIONAL ABSTRACT The students can model the problem situation using place value chips. The students trade the 1,000 for ten hundreds. The students trade the one ten for ten ones. The students can model the problem situation using an area model and partial quotient. Students make think bubble sof multiplication facts to help guide their thinking. Encourage students to use multiplicative patterns with powers of ten. The students can use mental math by decomposing the dividend into numbers divisible by 3. 1218 can be decomposed into 1200+18 Students can write equations to model the work that they are doing. 3 x?=1200 3 x 400=1200 3 x?=18 3 x 6=18 3406=1218 Shelly has $406 in her bank account. Now distribute the dollars to each of the three groups. Shelly has $406 in her bank account. 44 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Session 2 Divide by 2-digit CONCRETE: Example: 10 exercise pads cost $240. How many dollars does each pad cost? Students use place value disks or base ten blocks to model division problems. Students model the dividend of the problem. Two ways to look at the problem: For a partitive division problem the divisor represents the number of groups and the quotient will represent how many are placed in each group. For measurement division the divisor represents the number of items in each group and the quotient will represent the number of groups. Adding context to the situation will help students see the difference. Partitive: Students trade chips so that they can arrange them into groups of ten to find the quotient (number in each group) is 24. Measurement: Students arrange chips so there is 10 in each group to find the quotient (number of groups) to be 24. There are twenty tens chips and four groups of 10 ones chips so 20 + 4 = 24 groups of 10. 45 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
PLACE VALUE PATTERNS MP 7: Look for and make use of structure. Abstract: There are some problems that are too big to model with chips, but students can learn to use what they know about multiplicative patterns of place value. For example, students can look at powers of ten from the first lesson of this unit. The focus should be on the pattern for the number of zeroes in the dividend and divisor. 180 10 180 10 1 18 tens 1 ten 18 1800 100 1800 10 2 18 hundreds 1 hundred 18 18000 1000 18000 10 3 18 thousands 1 thousand 18 Students learn to use this pattern to use mental strategies to simplify the problem. For example, students can decompose the divisor and use brackets to write numerical expressions and what they know about order of operations to use place value strategies. 18000 600 (18000 100) 6 (18000 10 2 ) 6 18 6 = 3 3500 500 (3500 100) 5 (3500 10 2 ) 5 35 5 = 7 81000 9000 (81000 1000) 9 (81000 10 3 ) 9 81 9 = 9 USING ESTIMATION Students can use the place value patterns and structure to estimate quotients. For example, given the problem 602 29, the students can round to 600 30 and write the equivalent expression (600 10) 3. Another estimation strategy requires students to round the divisor and then list multiples of the divisor. For example, given 149 71, the students first round the 71 to 70. Usually students round the 149 to 150 but with this strategy they list the multiples of 70 and round the dividend to the closest multiple of 70. Multiples of 70: 70, 140, 210 140 is the closest so our estimation becomes 140 70 = 2. (Students can combine the two strategies by writing the equivalent expression, (140 10) 7 if needed. Learning these estimation and place value strategies will help students with the following concrete examples for solving division problems: 46 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
AREA MODEL with BASE TEN BLOCKS Students use base ten blocks and a factor mat to model division. Example: 336 passengers went into 24 buses equally. How many passengers were there on each bus? 336 24 Students arrange the divisor along the outside of the mat as one dimension or side length of the area model. Students get the blocks that model the dividend of the problem. The next step is to arrange the blocks so they form a rectangle with the one side length equal to the divisor along the one side of the mat. The student may find that they cannot arrange the blocks into a rectangle. Ask students if there is a way to trade the blocks so they can move them in different ways. For example, trade a 100 flat for 10 rods or trade 1 rod for 10 units and try again. If students are trading blocks in for other blocks make sure they end with the correct value. Students can now look at the side length across the top or the bottom of the rectangle they created to find the quotient. In this case, the students need 1 rod and four units along the edge. The quotient (missing factor) is 14. 47 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Situation Model Strategy Equal groups Number of Groups Unknown How many groups? Division STORY PROBLEM Mr. McDonald s chickens laid 288 eggs. He put a dozen eggs in each carton. How many cartons did Mr. McDonald fill? CONCRETE REPRESENTATIONAL ABSTRACT The students can model the problem situation with base ten blocks. The students can model the problem situation using an area model The students can use any multiple of 12. Example 1: The students can use partial quotients to model the problem situation. 3 digit by 2 digit The dividend is the number of eggs the chicken laid. The divisor is a dozen. A dozen equals 12 eggs. Example 2: The students create a rectangle with one side equal to the divisor 12. Give the students time to figure out how to arrange the blocks. Example 3: The students find the quotient by completing the model. Connect to multiplication. 48 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
Situation Model Strategy STORY PROBLEM CONCRETE REPRESENTATIONAL ABSTRACT Equal groups Group Size Unknown How many in each group? Division Base Ten Blocks Three Digit Divided by Two Digit Josie had 368 beads. She used all the beads to make 23 necklaces. How many beads does each necklace have? The students can use base ten blocks to model the problem situation. Students put out 368 to represent the dividend and 23 to represent the divisor. For this example, ask the students if they think the quotient will be greater than or less than 20. We want the students to be able to reason that 20 x 20=400and there is only 300 so it will be less than 20. So we need to make some trades. 368-230=138 Let s try half of 10. 5 x 10 =50 and 5 x 10 =50 that will be 100 more beads from the start. 138-115=23 The students can draw pictures to model the problem situation with area model. Above you see one example of how students can find the quotient or the number of beads on each necklace. The thinking can be, if I put 10 beads on a necklace I would use 230 beads. So there are more than 10 beads, If I put 5 more beads on the necklace, then I will have used 345 beads, so there are 23 more beads. I can put one more bead on each necklace. Now I have used all of the beads. Each necklace has 16 beads. The students can use partial products to model the problem situation. Students are really guessing about the number of beads that are on each necklace. Students should be able to use educated guesses based on multiples of 23. Look for students to use group of 10 or 100 and then 5 or 50 to identify those that are developing fluency. There is only one more group of 23 left. The quotient is 16. 49 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1
4-Digit Divided by 2-digit Situation STORY Model PROBLEM Strategy 4 digit by 2 digit Taco Hut served 1,232 tacos in 4 weeks. If they served the same number of tacos each day, how many tacos did they serve in one day. CONCRETE REPRESENTATIONAL ABSTRACT The student can model the problem situation with base ten blocks. The student can model the problem situation with an area model. Example 1: The student can model the problem situation using the partial quotient. Student A: The student put out 1,232 to represent the dividend or the total number of tacos. They put out 28 as the divisor because there are 28 days in 4 weeks. Example 2: The student uses multiples of 28.. Use multiples of 28 to help decide how to make the rectangle. Student B: Number Talk: Student talk about how the models are the same and how they are different. 50 Joliet Public Schools District 86 DRAFT Curriculum Guide 2017-2018, Grade 5, Unit 1