Students use their knowledge of multiplication and division with whole numbers and with fractions to multiply and divide with decimals, and apply this understanding to the context of measurement conversion.
In Unit 6, students use their procedural knowledge of multiplication and division with whole numbers, combined with their newly acquired understanding of multiplication and division with fractions, to multiply and divide with decimals, reasoning about the placement of the decimal point. They then apply this to the context of word problems, including those involving measurement conversion.
In Grade 4, students were first introduced to decimal notation for fractions and reasoned about their size (4.NF.5—7). Then, in the first unit in Grade 5, students developed a deeper understanding of decimals as an extension of our place value system, understanding that the relationships of adjacent units apply to decimal numbers, as well (5.NBT.1), and using that understanding to compare, round, and represent decimals in various forms (5.NBT.2—4). Next, students learned to multiply and divide with whole numbers in Unit 2 (5.NBT.5—6), skills upon which decimal computations will rely. In Unit 3, students explored the other two operations with decimals not addressed in this unit: addition and subtraction (5.NBT.7). In Unit 5, students learned to multiply and divide with fractions, including relating fractions to the operation of division; multiplying a fraction by a fraction, including mixed numbers; and dividing a unit fraction by a whole number and vice versa (5.NF.3—7), which will help them make sense of analogous cases of decimal multiplication and division. Thus, this unit is dependent on a lot of prior learning, both in Grade 4 and Grade 5.
This unit starts with multiplying a decimal by a single-digit whole number, then multiplying a decimal by a multi-digit whole number, and finally multiplying a decimal by another decimal. Then, students progress to dividing a decimal by a single-digit whole number, then dividing a decimal by a two-digit whole number, and finally solving cases involving decimal divisors. Throughout these topics, students use the same methods to compute decimal products and quotients as they did for whole-number products and quotients, but they must reason about the placement of the decimal point. It is only in the last lesson of each topic that students generalize the pattern of the placement of the decimal point. The various lines of reasoning, and their advantages and disadvantages, can be read on pages 19 and 20 of the NBT Progression linked in the “Unit-Specific Intellectual Preparation” section. Students also solve myriad word problems as well as write and solve expressions involving decimals as a way to support the major work (5.OA.1, 5.OA.2). Finally, the unit closes with students learning to convert among different-sized customary measurement units within a given measurement system and solve word problems that use those conversions (5.MD.1), which extends the work from Grade 4 of converting from a larger unit of measurement to a smaller one in Grade 4 (4.MD.1—2). As noted in the Progressions, “this is an excellent opportunity to reinforce notions of place value for whole numbers and decimals, and the connection between fractions and decimals (e.g., meters can be expressed as 2.5 meters or 250 centimeters)” (GM Progression, p. 26), as well as computations with these types of numbers (5.NBT.7, 5.NF), thus connecting the work of unit conversion with major work of the grade.
Reasoning about the placement of the decimal point affords students many opportunities to engage in mathematical practice, such as constructing viable arguments and critiquing the reasoning of others (MP.3) and looking for and expressing regularity in repeated reasoning (MP.8). For example, “students can summarize the results of their reasoning as specific numerical patterns and then as one general overall pattern such as ‘the number of decimal places in the product is the sum of the number of decimal places in each factor’” (NBT Progression, p. 20).
In Grade 6, students will become fluent with all decimals computations that they’ve developed in Grade 5 (6.NS.3). In Grade 7, students will also learn that every fraction can be represented with a decimal that either terminates or repeats. Then in Grade 8, students learn that terminating and repeating decimals are rational numbers and that there are numbers that are irrational whose decimal expansion does not repeat. Then, students use the work they start in this unit in Grade 8 in the context of scientific notation. Thus, this unit has many interesting connections and applications for many years to come.
Pacing: 27 instructional days (24 lessons, 2 flex days, 1 assessment day)
This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.
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conversion factor
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Intellectual Prep for All Units
Unit-specific Intellectual Prep
5.NBT.B.7
Multiply a single-digit whole number by a decimal in cases that involve basic facts. Estimate the product of a single-digit whole number with a decimal by rounding numbers to their largest place value.
5.NBT.B.7
Multiply a single-digit whole number by a decimal.
5.NBT.B.7
Construct viable arguments and critique the reasoning of others regarding the placement of the decimal point in computations that involve multiplying a single-digit whole number by a decimal.
5.NBT.B.7
Multiply a multi-digit whole number by a decimal.
5.NBT.B.7
Multiply a decimal by a decimal in cases that involve basic facts. Estimate the product of two decimals by rounding numbers to their largest place value.
5.NBT.B.7
Multiply a decimal by a decimal.
5.NBT.B.7
Multiply with decimals in all cases, reasoning about the placement of the decimal point.
5.NBT.B.7
Divide a decimal by a single-digit whole number in cases that involve basic facts. Estimate quotients with single-digit divisors by rounding numbers to compatible numbers.
5.NBT.B.7
Divide a decimal by a single-digit whole number.
5.NBT.B.7
Divide a decimal by a single-digit whole number that requires decomposition in the smallest place value.
5.NBT.B.7
Divide a decimal by a two-digit whole number.
5.NBT.B.7
Divide a decimal by a two-digit whole number that requires decomposition in the smallest place value.
5.NBT.B.7
Divide a whole number or a decimal by 1 tenth or 1 hundredth.
5.NBT.B.7
Divide a whole number or a decimal by a decimal in cases that involve basic facts. Estimate quotients with decimal divisors by rounding numbers to compatible numbers.
5.NBT.B.7
Divide a whole number or a decimal by a decimal.
5.NBT.B.7
Divide with decimals in all cases, reasoning about the placement of the decimal point.
5.NBT.B.7
Solve real-world problems that require interpretation of the remainder involving all possible cases, including further decomposition into a decimal.
5.NBT.B.7
Solve real-world problems involving multiplication and division with decimals.
5.OA.A.1
5.OA.A.2
5.NBT.B.7
Write and evaluate numerical expressions involving multiplication and division with decimals.
5.MD.A.1
Express measurements in a whole number of larger units or mixed units in terms of a smaller unit.
5.MD.A.1
Express measurements in a fractional or decimal number of larger units in terms of a smaller unit.
5.MD.A.1
Express measurements in a whole number of smaller units in terms of a larger unit.
5.MD.A.1
Express measurements in a decimal or fractional number of smaller units in terms of a larger unit.
5.MD.A.1
Solve real-world problems involving measurement conversions.
Key: Major Cluster Supporting Cluster Additional Cluster
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