Approximate Time Frame: 8-9 weeks Connections to Previous ...ioer.ilsharedlearning.org/ContentDocs/bc2cc184-41bf-464b-a363-11a… · the base-ten system to non-whole numbers. Using

Grade 4 Unit 1 Multiplication and Division Concepts

3/29/2014 5:39:19 PM Adapted from UbD Framework Page 1

Approximate Time Frame: 8-9 weeks Connections to Previous Learning: In Grade 3, students fluently add and subtract within 1,000, and they also fluently multiply and divide within 100. They 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. 4th grade students will extend these skills to develop fluency with efficient procedures for multiplying whole numbers and apply their understanding of models for multiplication and division.

Focus of the Unit: Much of this unit focuses on extending previous skills to multi-digit whole numbers. Students will read, write, and compare multi-digit whole numbers up to a million, conceptually understanding that a digit in one place represents 10 times what it represents in the place to its right. Students will also 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. They will illustrate and explain their calculations by using equations, rectangular arrays, and/or area models. Students will develop an understanding of multiplication as multiplicative comparisons (35 is 5 times as much as 7.) They will apply their understanding of multiplication comparison, multiplication skills, and division skills while estimating and problem-solving. Students extend the idea of decomposition to multiplication and learn to use the terms multiple, factor, prime, and composite.

Connections to Subsequent Learning: In fifth grade, students will extend their understanding of the base-ten system for decimals to thousandths. They will fluently compute products of whole numbers using the standard algorithm. They will continue their fourth grade work on division, extending it to computation of whole number quotients with dividends of up to four digits and two-digit divisors.

Progression Citation: From the K-5, Number and Operations in Base Ten, pp. 12-16, K-5 Operations and Algebraic Thinking, pp. 29-31 At Grade 4, students extend their work in the base-ten system. They use standard algorithms to fluently add and subtract. They use methods based on place value and properties of operations supported by suitable representations to multiply and divide with multi-digit numbers.

Generalize place value understanding for multi-digit whole numbers. In the base-ten system, the value of each place is 10 times the value of the place to the immediate right.4.NBT.1 Because of this, multiplying by 10 yields a product in which each digit of the multiplicand is shifted one place to the left. To read numerals between 1,000 and 1,000,000, students need to understand the role of commas. Each sequence of three digits made by commas is read as hundreds, tens, and ones, followed by the name of the appropriate base-thousand unit (thousand, million, billion, trillion, etc.). Thus, 457,000 is read “four hundred fifty seven thousand.”4.NBT.2 The same methods students used for comparing and rounding numbers in previous grades apply to these numbers, because of the uniformity of the base-ten system.



Decimal notation and fractions - students in Grade 4 work with fractions having denominators 10 and 100.4.NF.5 Because it involves partitioning into 10 equal parts and treating the parts as numbers called one tenth and one hundredth, work with these fractions can be used as preparation to extend the base-ten system to non-whole numbers.

Using the unit fractions 1

10 and

1

100, non-whole numbers like 23

7

10 can be written in an expanded form

that extends the form used with whole numbers: 2 10 3 1 7 1 10 .4.NF.4b As with whole-number expansions in the base-ten system, each unit in this decomposition is ten times the unit to its right. This can be connected with the use of base-ten notation to

represent 2 × 10 + 3 × 1 + 7 ×1

10 as 23.7. Using decimals allows students to apply familiar place

value reasoning to fractional quantities.4.NF.6 The Number and Operations—Fractions Progression discusses decimals to hundredths and comparison of decimals4.NF.7 in more detail. The decimal point is used to signify the location of the ones place, but its location may suggest there should be a “oneths" place to its right in order to create symmetry with respect to the decimal point. However, because one is the basic unit from which the other base ten units are derived, the symmetry occurs instead with respect to the ones place.

Ways of reading decimals aloud vary. Mathematicians and scientists often read 0.15 aloud as “zero point one five" or “point one five." (Decimals smaller than one may be written with or without a zero before the decimal point.) Decimals with many non-zero digits are more easily read aloud in this manner. (For example, the number π, which has infinitely many non-zero digits, begins 3.1415 …..). Other ways to read 0.15 aloud are “1 tenth and 5 hundredths” and “15 hundredths,” just as 1,500 is sometimes read “15 hundred” or “1 thousand, 5 hundred.” Similarly, 150 is read “one hundred and fifty” or “a hundred fifty” and understood as 15 tens, as 10 tens and 5 tens, and as 100 + 50. Just as 15 is understood as 15 ones and as 1 ten and 5 ones in computations with whole numbers, 0.15 is viewed as 15 hundredths and as 1 tenth and 5 hundredths in computations with decimals. It takes time to develop understanding and fluency with the different forms. Layered cards for decimals can help students become fluent with decimal equivalencies such as three tenths is thirty hundredths. Use place value understanding and properties of operations to perform multi-digit arithmetic. At Grade 4, students become fluent with the standard addition and subtraction algorithms.4.NBT.4 As at the beginning of this progression, these algorithms rely on adding or subtracting like base-ten units (ones with ones, tens with tens, hundreds with hundreds, and so on) and composing or decomposing base-ten units as needed (such as composing 10 ones to make 1 ten or

decomposing 1 hundred to make 10 tens). In mathematics, an algorithm is defined by its steps and not by the way those steps are recorded in writing. With this in mind, minor variations in methods of recording standard algorithms are acceptable.



In fourth grade, students compute products of one-digit numbers and multi-digit numbers (up to four digits) and products of two two-digit numbers. They divide multi-digit numbers (up to four digits) by one-digit numbers. As with addition and subtraction, students should use methods they understand and can explain. Visual representations such as area and array diagrams that students draw and connect to equations and other written numerical work are useful for this purpose. By reasoning repeatedly about the connection between math drawings and written numerical work, students can come to see multiplication and division algorithms as abbreviations or summaries of their reasoning about quantities. Students can invent and use fast special strategies while also working towards understanding general methods and the standard algorithm. One component of understanding general methods for multiplication is understanding how to compute products of one-digit numbers and multiples of 10, 100, and 1000. This extends work in Grade 3 on products of one-digit numbers and multiples of 10. We can calculate 6 ×700 by calculating 6 × 7 and then shifting the result to the left two places (by placing two zeros at the end to show that these are hundreds) because 6 groups of 7 hundred is 6 × 7 hundreds, which is 42 hundreds, or 4,200. Students can use this place value reasoning, which can also be supported with diagrams of arrays or areas, as they develop and practice using the patterns in relationships among products such as 6 × 7, 6 × 70, 6 × 700 and 6 × 7000. Products of 5 and even numbers, such as 5 × 4, 5 × 40, 5 × 400, 5 × 4000 and 4 × 5, 4 × 50, 4 × 500, 4 × 5000 might be discussed and practiced separately afterwards because they may seem at first to violate the patterns by having an “extra” 0 that comes from the one-digit product. Another part of understanding general base-ten methods for multi-digit multiplication is understanding the role played by the distributive property. This allows numbers to be decomposed into base-ten units, products of the units to be computed, then combined. By decomposing the factors into like base-ten units and applying the distributive property, multiplication computations are reduced to single-digit multiplications and products of numbers with multiples of 10, of 100, and of 1000. Students can connect diagrams of areas or arrays to numerical work to develop understanding of general base-ten multiplication methods. Computing products of two two-digit numbers requires using the distributive property several times when the factors are decomposed into base-ten units. For example,

36 × 94 = (30 + 6) × (90 + 4) = (30 + 6)90 + (30 + 6) × 4



= 30 × 90 + 6 × 90 + 30 × 4 + 6 × 4 General methods for computing quotients of multi-digit numbers and one-digit numbers rely on the same understandings as for multiplication, but cast in terms of division.4.NBT.6 One component is quotients of multiples of 10, 100, or 1000 and one-digit numbers. For example, 42 ÷ 6 is related to 420 ÷ 6 and 4200 ÷ 6. Students can draw on their work with multiplication and they can also reason that 4200 6 means partitioning 42 hundreds into 6 equal groups, so there are 7 hundreds in each group. Another component of understanding general methods for multi-digit division computation is the idea of decomposing the dividend into like base-ten units and finding the quotient unit by unit, starting with the largest unit and continuing on to smaller units. As with multiplication, this relies on the distributive property. This can be viewed as finding the side length of a rectangle (the divisor is the length of the other side) or as allocating objects (the divisor is the number of groups). See the figures on the next page for examples. Multi-digit division requires working with remainders. In preparation for working with remainders, students can compute sums of a product and a number, such as 4 × 8 + 3. In multi-digit division, students will need to find the greatest multiple less than a given number. For example, when dividing by 6, the greatest multiple of 6 less than 50 is 6 × 8 = 48. Students can think of these “greatest multiples” in terms of putting objects into groups. For example, when 50 objects are shared among 6 groups, the largest whole number of objects that can be put in each group is 8, and 2 objects are left over. (Or when 50 objects are allocated into groups of 6, the largest whole number of groups that can be made is 8, and 2 objects are left over.) The equation 6 × 8 + 2 = 50 (or 8 × 6 + 2 = 50) corresponds with this situation.



Cases involving 0 in division may require special attention.



Desired Outcomes

Standard(s):

Use the four operations with whole numbers to solve problems.

4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

Generalize place value understanding for multi-digit whole numbers.

4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

4.NBT.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.

Gain familiarity with factors and multiples.

4.OA.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

Use place value understanding and properties of operations to perform multi-digit arithmetic.

4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-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.

Use the four operations with whole numbers to solve problems.

4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. 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.



Transfer: Students will apply concepts and procedures for adding, subtracting, multiplying and dividing multi-digit whole numbers to solve real-world and mathematical problems.

Ex: A basketball costs 3 times as much as a tennis ball. If the tennis ball costs $4, how much does a basketball cost?

Ex: The dimensions of a small-size soccer field are 65 yards by 45 yards. What is the area of the soccer field?

Understandings: Students will understand that …

Place value is based on groups of ten and the value of a number is determined by the place of its digits.

Whole numbers are read from left to right using the name of the period; commas are used to separate periods.

A number can be written using its name, standard, or expanded form.

Flexible methods of computation involve grouping numbers in strategic ways.

The distributive property is connected to the area model and/or partial products method of multiplication.

Multiplication and division are inverse operations.

There are three different structures for multiplication and division problems: Area/Arrays, Equal Groups, and Comparison, and the unknown quantity in multiplication and division situations is represented in three ways: Unknown Product, Group Size Unknown, and Number of Groups Unknown.

Some division situations will produce a remainder, but the remainder should 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. When using division to solve word problems, how the remainder is interpreted depends on the problem situation.



Essential Questions:

How does the position of a digit in a number affect its value, and how can the value of digits be used to compare two numbers?

In what ways can numbers be composed and decomposed?

How do I determine the factors of a number?

What is the difference between a prime and composite number?

How are multiplication and division related to each other?

What are different models for multiplication and division?

What are efficient methods for finding products and quotients, and how can place value properties aid computation?

How are dividends, divisors, quotients, and remainders related?

What real-life situations require the use of multiplication or division?

How can a remainder affect the answer in a division word problem?



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Highlighted Mathematical Practices:

1. Make sense of problems and persevere in solving them. Students will solve word problems using all operations. Students will represent word problems using various modalities. Students will demonstrate their perseverance by selecting an effective modality to solve problems.

2. Reason abstractly and quantitatively. Students determine what operations to use to solve word problems and think about the reasonableness of their solutions. Students use reasonable estimates based upon the value of the numbers.

3. Construct viable arguments and critique the reasoning of others. Students demonstrate their ability to construct viable arguments when they talk and write about the steps they take to solve problems. Students restate and respond to each other about their mathematical thought processes.

4. Model with mathematics. Students write equations when recording all steps used to solve multiplication and division problems. They use all modalities (picture, manipulatives, written symbols, real world situations, oral/written language) when representing a situation.

5. Use appropriate tools strategically. Students choose and use the appropriate tools to solve problems. Tools could include: color tiles, base ten blocks, counters, math lines, geoboards, Unifix® cubes, websites, formulas, numerical equations, numerical expressions, diagrams, graph paper, shape diagrams, songs, whiteboards.

6. Attend to precision. Students demonstrate precision in calculation by using their inverse operations to check their work and by using precise vocabulary in their oral and written explanations.

7. Look for and make use of structure. Students recognize and identify patterns existing between addition and multiplication along with division and subtraction. Students use this knowledge when applying strategies to evaluate real-world problems such as area and perimeter. They also use problem solving structures to aid in solving real world problems.

8. Look for and express regularity in repeated reasoning. Students identify and explain patterns within multiplication and division arrays (prime numbers, square numbers, and composite numbers, factors, multiples), determine and communicate whether identified patterns always work, determine and communicate when to apply specific patterns, describe the relationship between multiplication, addition, subtraction, and division, and justify that solutions make sense (ie., using mental computation and/or estimation).



Prerequisite Skills/Concepts:

Students should already be able to…

Add and subtract fluently within 1000

Read and write numbers to 1000 using base-ten numerals, number names, and expanded form

Compare two three-digit numbers

Apply properties to solve problems

Interpret products and quotients of whole numbers

Solve word problems involving x and ÷ with equal groups, arrays, and measurement

Determine unknowns in x and ÷ equations

Understand division as unknown factor problems

Fluently x and ÷ within 100

Multiply 1-digit whole number by multiples of 10

Solve two-step word problems involving +, -, x and ÷

Advanced Skills/Concepts:

Some students may be ready to…

Multiply larger than 2 digit by 2 digit numbers

Divide by multiples of ten

Solve problems involving multi-step equations with two or more operation

Create multi-step problems using multiple structures

Use parentheses to represent and solve multi-step problems

Knowledge: Students will know…

Multiplication equations can show comparisons (4.OA.1)

When to apply single equations or more than one equation using manipulatives, and/or diagrams to represent multiplicative comparison. (4.OA.1)

Skills: Students will be able to …

Translate comparative situations into drawings and equations with a symbol for the unknown and unknowns in all 3 locations. (4.OA.2)

Solve word problems involving multiplicative comparison using drawings and (multiplication or division) equations with a symbol for the unknown number and unknowns in all 3 locations. (4.OA.2)



Verbal statements of multiplicative comparisons can be written as equations with and without variables. (i.e., Sally is five years old. Her mom is eight times older. How old is Sally’s Mom? 5 x 8 = 40) (4.OA.1)

A digit in one place represents ten times what it represents in the place to its right, by using manipulatives, pictures, language, and/or equations to explain their reasoning. (4.NBT.1)

Strategies for multiplying and dividing based on place value, the properties of operations, and/or the relationship between multiplication and division. (4.NBT.5) (4.NBT.6)

Explain the difference between additive comparison and multiplicative comparison using visuals and words. (4.OA.2)

Read and write whole numbers up to a million using standard, word, and expanded form. (4.NBT.2)

Compare two multi-digit (up to a million) numbers. (4.NBT.2)

Use manipulatives, pictures, and language to show the relationship between the numerals and their place value representations in multiple ways. (4.NBT.2)

Identify all factor pairs for any given number 1-100. Recognize that a whole number is a multiple of each of its factors. (4.OA.4)

Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. (4.OA.4)

Determine whether a given whole number in the range 1-100 is prime or composite. (4.OA.4)

Use visuals, symbols and/or language to explain their reasoning. (4.OA.4)

Multiply up to 4-digit by 1-digit numbers and 2-digit by 2-digit numbers. (4.NBT.5)

Use place value manipulatives to represent multiplication calculations. Illustrate and explain the calculation by using written equations, rectangular arrays, and area models. (4.NBT.5)

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors. Illustrate and explain the calculation by using written equations, rectangular arrays, and area models. (4.NBT.6)

Use place value manipulatives to represent division calculations. (4.NBT.6)

Use the relationship between multiplication and division to explain calculations. (4.NBT.6)

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations and represent those problems using equations with a variable standing for the unknown quantity. Interpret remainders when solving multi-step word problems (4.OA.3)

Assess the reasonableness of answers using mental computation and estimation strategies, including rounding. (4.OA.3)



WIDA Standard: English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. English language learners benefit from:

A preview of critical vocabulary terms before instruction.

The use of visuals to make explicit connections between the vocabulary and the content being learned.

Academic Vocabulary:

Critical Terms: Multiplicative comparison Standard Form Written Form Expanded Form Factor Multiple Prime Composite Divisor Dividend Remainder

Supplemental Terms: Array Area Model Equation Product Quotient

Assessments

Pre-Assessments Formative Assessments Summative Assessments Self-Assessments Additive Comparisons Exit Slip Multiplicative Comparison

Problem Cards

Multiplicative Comparisons

Place Value

Factors and Multiples

Multiplication and Division

Place Value Skeleton

Multiplication and Division Self –Assessment Skeleton

Sample Lesson Sequence

1. 4.OA.1 and 4.OA.2 Multiplicative Comparisons (model lesson) 2. 4.NBT.1 & 4.NBT.2 Place value and multiple representations of multi-digit whole numbers 3. 4.OA.4 Factors, multiples, prime numbers and composite numbers 4. 4.NBT.5 and 4.OA.3 Multiplication of whole numbers and multi-step problem solving 5. 4.NBT.6 and 4.OA.3 Division of whole numbers and multi-step problem solving

Documents

Approximate Time Frame: 8-9 weeks Connections to Previous ...ioer.ilsharedlearning.org/ContentDocs/bc2cc184-41bf-464b-a363-11a… · the base-ten system to non-whole numbers. Using