Carroll County Public Schools Elementary Mathematics Instructional Guide (5 th Grade) Unit #4 : Division of Whole Numbers and Measurement


 Rose Harvey
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1 Background Information and Research By fifth grade, students should understand that division can mean equal sharing or partitioning of equal groups or arrays. They should also understand that it is the same as repeated subtraction, and since it s the inverse of multiplication, the quotient can be thought of as a missing factor. In fourth grade, students divided 4digit dividends by 1 digit divisors. They also used contexts to interpret the meaning of remainders. Division is extended to 2digit divisors in fifth grade, but fluency of the traditional algorithm is not expected until sixth grade. Division models and strategies that have been used in previous grade levels, such as arrays, number lines, and partial quotients, should continue to be used in fifth grade as students deepen their conceptual understanding of this division. There are two common types of division problems. The first type is called repeated subtraction. You know the total amount and the amount in each group and are looking for the number of groups. For example, how many packages of 6 ping pong balls can be made from 42 ping pong balls? You know that there are 7 groups of 6 in 42, so you know 7 packages can be made. The second type is called sharing division. You know the total amount and the number of groups and you need to find out how many are in each group. For example, Jody has 42 apples. She puts them into 6 equally sized groups. How many are in each group? There are 7 in each group. Division strategies in Grade 5 involve breaking the dividend apart into like baseten units and applying the distributive property to find the quotient place by place, starting from the highest place. (Division can also be viewed as finding an unknown factor: the dividend is the product, the divisor is the known factor, and the quotient is the unknown factor.) Students continue their fourth grade work on division, extending it to computation of whole number quotients with dividends of up to four digits and twodigit divisors. Estimation becomes relevant when extending to twodigit divisors. Even if students round appropriately, the resulting estimate may need to be adjusted. Teaching StudentCentered Mathematics (Grades 35) pg Focus in Grade 5 Pages
2 The chart below highlights the key understandings of this cluster along with important questions that teachers should pose to promote these understandings. The chart also includes key vocabulary that should be modeled by teachers and used by students to show precision of language when communicating mathematically. Enduring Understandings Essential Questions Key Vocabulary There are two common situations where division may be used: fair sharing (given the total amount and the number of equal groups, determine how many/much in each group) and measurement (given the total amount and the amount in a group, determine how many groups of the same size can be created). Some division situations will produce a remainder, but the remainder will always be less than the divisor. If the remainder is greater than the divisor, that means at least one more can be given to each group (fair sharing) or at least one more group of the given size (the dividend) may be created. The quotient remains unchanged when both the dividend and the divisor are multiplied or divided by the same number. The properties of multiplication and division help us solve computation problems easily and provide reasoning for choices we make in problem solving. How can estimating help us when solving division problems? What strategies can we use to efficiently solve division problems? How can I effectively explain my mathematical thinking and reasoning to others? How can I effectively critique the reasoning of others? How can identifying patterns help determine multiple solutions? Dividend Divisor Quotient Remainder Properties 2
3 Throughout this cluster, students will develop their use of the 8 Mathematical Practices while learning the instructional standards. Specific connections to this cluster and instructional strategies are provided in the following chart. Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them Cluster Connections and Instructional Strategies Students solve problems by applying their understanding of operations with whole numbers, including the order of operations. Students seek the meaning of a problem and look for efficient ways to solve it. 2. Reason abstractly and quantitatively Students demonstrate abstract reasoning to connect quantities to written symbols and create a logical representation of the problem at hand. Students write simple expressions that record calculations with numbers and represent numbers using place value concepts. 3. Construct viable arguments and critique the reasoning of others Students construct arguments using concrete referents, such as objects, pictures, and drawings. They explain calculations based upon models and properties of operations and rules that generate patterns. They explain their thinking to others and respond to others thinking. 4. Model with mathematics Students use base ten blocks, drawings, and equations to represent place value and powers of ten. They interpret expressions and connect them to representations. 5. Use appropriate tools strategically Students select and use tools such as estimation, graph paper, and place value charts to solve problems with whole number operations. 6. Attend to precision Students use clear and precise language (math talk) in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to expressions, place value, and powers of ten 7. Look for and make use of structure Students use properties of operations as strategies to add, subtract, multiply, and divide with whole numbers. They explore and use patterns to evaluate expressions. Students utilize patterns in place value and powers of ten and relate them to graphical representations of them. 8. Look for and express regularity in repeated reasoning Students use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and properties of operations to fluently perform operations. 3
4 Maryland College and CareerReady Standards Convert like measurement units within a given measurement system. 5.MD.1. Convert among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, real world problems. SMP.2 Reason abstractly and quantitatively SMP.3 Construct viable arguments and critique the reasoning of others SMP.5 Use appropriate tools strategically SMP.6 Attend to precision Sizing Up Measurement pp. 152, , , Instructional Strategies and Resource Support Students worked with both metric and customary units of length in second grade. In third grade, students work with metric units of mass and liquid volume. In fourth grade, students work with both systems and begin conversions within systems in length, mass and volume using whole numbers. Early in the year, students worked on converting measurements in the metric system. During this unit, students will work with converting customary units of measure and time. Students present solutions to multistep problems in the form of valid chain of reasoning, using symbols such as equal signs appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as 1+4=5+7=12, even if the final answer is correct), or identify or describe errors in solutions to multstep problems and present corrected solutions. (SMP.3, 5,6) Students should apply their understanding of converting within the metric system to converting within the customary system. Students should see the pattern and make the generalization that: To change from a larger unit to a smaller unit, multiply by the appropriate number. To change from a smaller unit to a larger unit, divide by the appropriate number. Students should have securely held knowledge from 4 th grade and will not be available on the reference sheet: 1 foot=12 inches, 1 yard=3 feet, 1 day =24 hours Sample Formatives Write <, >, = to compare measurements. 4
5 Maryland College and CareerReady Standards Convert like measurement units within a given measurement system. 5.MD.1. Convert among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, real world problems. SMP.2 Reason abstractly and quantitatively SMP.3 Construct viable arguments and critique the reasoning of others SMP.5 Use appropriate tools strategically SMP.6 Attend to precision Instructional Strategies and Resource Support Sample Formatives 5
6 Perform operations with multidigit whole numbers and with decimals to hundredths. Maryland College and CareerReady Standards 5.NBT.6 Find wholenumber quotients of whole numbers with up to fourdigit dividends and twodigit 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. *5.NBT.6 is a milestone along the way to reaching fluency with the standard algorithm. In this unit 5.NBT.6 will focus on operations with whole numbers only. Operations with decimals will be introduced in March and will be finalized in May and June, but should be practiced throughout the year to provide opportunities for students to develop proficiency with these operations. SMP.3 Construct viable arguments and critique the reasoning of others SMP.5 Use appropriate tools strategically SMP.6 Attend to precision SMP.8 Look for and express regularity in repeated reasoning Instructional Strategies and Resource Support Developing fluency with multidigit division takes three years because students first develop and explain the approach of the standard algorithm with visual models for dividing by onedigit numbers in Grade 4 and then extend the approach in Grade 5 to dividing by 2digit numbers where the difficulties of estimating complicate division. In Grade 6 the standard algorithm is used fluently for one and twodigit divisors. Teaching StudentCentered Mathematics (Grades 35) pg. 65 *The Broken Division Key  Activity 2.26 pg *Figure 4.17 pg Focus in Grade 5 Pages 926 See Focus in Grade 5 book for complete explanation on using rectangular arrays and area models as well as partial quotient methods. See Excerpt from NCSM JournalFall/Winter below for an explanation of methods used that will teach the concept of division with multidigit whole numbers and lead to the standard algorithm. When students use the traditional algorithm, they often treat each digit in the dividend separately and do not look at the value of the entire number. Encourage the students to estimate prior to dividing, this helps them see what a reasonable quotient will be. When students show an understanding of the concept, they will naturally work toward fluency with the standard algorithm. Fifth grade students should begin working with the standard algorithm, but are not expected to be fluent until 6 th grade. Students look for regularity in their work with multiplication and division use their understanding of the structure (MP.8) to make sense of their solutions and understand the approaches of other students (MP.3) *On Summative Unit and MidYear, students will be allowed to solve using any method. Students will also need to explain their thinking and/or the thinking of others. They will base arithmetic explanations/reasoning on diagrams, connecting diagrams to a symbolic method.(smp.3, SMP.5, SMP.6) Sample Formative Assessments Use the area model to find the quotient. Show your thinking. Jeremy divided this array of 336 stars into the groups below. What division equation has he represented? What is the division sentence for the visual model? Savannah bought a used car for $9,900. She needs to make 36 equal payments Use the diagram to complete the equation and show your thinking. 6
7 Excerpt from NCSM JournalFall/Winter Array and area models can support understanding of strategies for division. Figure 7 shows Method A including both the recording of the numerical calculations and the area model that corresponds to those calculations as shown in the NBT Progression document. The full multiplier of the divisor at each step in Method A is written above the dividend so that students can see the place value and make a clear connection to the place values in the area model. The full product is written at each step, and the amount of the dividend not yet used is also written in full. After experience connecting it to a drawing, Method A can also be undertaken without a drawing as a standard algorithm when calculating quotients that are whole numbers or decimals. Another written method for this standard algorithm is also shown as Method B in Figure 7. In this method, the zeros are not written in the multipliers or in the partial products within the problem. Digits are brought down within the problem one at a time. Method B makes it more difficult to make the connections to the meaning of the computation, and for that reason, we included conceptual language to communicate the underlying meaning of each step of the calculation. But this method does show clearly the singledigit calculations that are used, and these singledigit calculations are in their placevalue locations, as indicated in Method A. Method B becomes important when a quotient has many places, for example, when explaining why decimal expansions of fractions eventually repeat (8.NS.1). Some students may be able to understand and explain Method B right away, and many students can move to it from strategies like Method A, because the move is small. There are also other written methods for the standard algorithm that are variations in between the two shown in Methods A and B. For instance, you could write the zeros on the top but not within the subtractions, or vice versa. Students might develop and use any of these variations. 7
8 Method A has a further advantage when dividing by 2 digit numbers because it can involve the use of reasonable estimates. The example shown in the NBT Progression document is depicted in Figure 8 and demonstrates how an estimate of the quotient can be used and then adjusted. Underestimates can be repaired in any place by subtracting another partial product for that place (and adding another rectangle to the area model when first using models). Such a simple repair is an acceptable written method for the standard algorithm. However, some introductions to division allow students to dramatically underestimate the quotient, and as a result, they then have many partial products (e.g., multipliers of ) that are used to adjust the estimate. Many extra multipliers and partial products are not consistent with the fluency expectations of the standard algorithm, so students need to be encouraged to be brave and use a multiplier as close as possible to the largest multiplier, for the sake of efficiency. With Method B students cannot continue on from a low estimate. They need to be exact, which often means erasing their underestimate or overestimate. Of course, a student could leave the product and difference they already found, cross out their low multiplier, increase it by 1 or 2, and take away that partial product as another step as in Method A. (See the second variation in Figure 8). Note that overestimates are still best fixed by erasing and trying a lower partial quotient because repairs are difficult to carry out correctly. 8
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