Fluency Trajectory for Multiplication and Division

Developed by:Danielle Seabold Kalamazoo RESAMatthew Mayer Kalamazoo RESA

Developed using the following resources:http://ime.math.arizona.edu/progressions/http://www.ncpublicschools.org/acre/standards/common-core-tools/#unmathhttp://alex.state.al.us/ccrs/node/76https://www.turnonccmath.nethttp://www.corestandards.org/Mathhttp://katm.org/wp/?page_id=91

Equipartitioning

Overview of Equipartitioning:

Equipartitioning is a strand of the standards that builds from students' experiences with fair sharing to the creation of equal-sized groups or parts from evenly divisible collections or wholes. There is a framework for it in the standards but because it can be a critical foundation on which to build division, multiplication, ratio and fractions and draws on standards from both geometry and number, it is presented as a separate strand here.

EquipartitioningStandard Explanation Concrete Representational Abstract

2.G.2: Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

2.G.2: calls for students to partition a rectangle into squares (or square-like regions) and then determine the total number of squares. The idea of equipartitioning a rectangle is extended to involve area or arrays Problems involving arrays help to convince students of the commutative property of multiplication, a x b = b x a.

Build-it Draw-it Write-itPartition a rectangle (object) into rows and columns of same-size squares and count to find the total number of them.

Partition a rectangle by drawing in rows and columns of the same-size squares and count them to find the total number of them.

Standard Explanation Concrete Representational Abstract

2.OA.4: Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

2.OA.4: calls for students to use rectangular arrays to work with repeated addition. This is a building block for multiplication in 3rd Grade. Students should explore this concept with concrete objects (e.g., counters, bears, square tiles, etc.) as well as pictorial representations on grid paper or other drawings. Based on the commutative property of addition, students can add either the rows or the columns and still arrive at the same solution.

Build-it Draw-it Write-itUse addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns, draw a picture of the array.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation.

Overview for Multiplication:

Students enter division and multiplication with their experience in equipartitioning. From this conceptual development,a.      students understand that fair sharing of evenly divisible collections produces equal-sized groups,b.       the whole collection is n times as large as one person’s share.

 As the Structural Overview illustrates, this Learning Trajectory first highlights the models of division and multiplication (Grades 2-3) that support students’ understanding of these two concepts. It then explores the different problem types, the properties and the strategies that may be used (Grades 3-4) reaching to the examination of factors and multiples at Grades 4-6. Recent research suggests that division may be taught first as fair sharing and then the context is reversed to introduce multiplication. In this trajectory, both ideas are moved forward gradually to yield the two inverse operations but, unlike in the past where multiplication is taught first and division is taught later, the operations are linked from the outset. Adding to these, this Trajectory also extends division and multiplication problems to involve multi-digit whole numbers. 

Second GradeUnderstanding and Relating Multiplication and Division Operations 

Standard Explanation Concrete Representational Abstract2.OA.3: Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

2.OA.3: calls for students to apply their work with doubles addition facts to the concept of odd or even numbers. Students should have ample experiences exploring the concept that if a number can be decomposed (broken apart) into two equal addends (e.g., 10 = 5 +5), then that number (10 in this case) is an even number. Students should explore this concept with concrete objects (e.g., counters, place value cubes, etc.) before moving towards pictorial representations such as circles or arrays.

Build-it Draw-it Write-itDetermine whether a group of objects has an odd or even number of members by pairing objects by 2s.

Determine whether a group of objects has an odd or even number of members by counting them by 2s.

Determine whether a group of objects has an odd or even number of members, write an equation to express an even number as a sum of two equal addends.

Third GradeSection 1: Understanding and Relating Multiplication and Division Operations 

Standard Explanation Concrete Representational Abstract3.OA.2: Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

This standard focuses on two distinct models of division: partition models and measurement (repeated subtraction) models. Partition models provide students with a total number and the number of groups. These models focus on the question, “How many objects are in each group so that the groups are equal?” A context for partition models would be: There are 12 cookies on the counter. If you are sharing the cookies equally among three bags, how many cookies will go in each bag? Measurement (repeated subtraction) models provide students with a total number and the number of objects in each group. These models focus on the question, “How many equal groups can you make?”

Build-it Draw-it Write-itInterpret whole-number quotients of whole numbers by using objects portioning objects into groups.

Interpret whole-number quotients of whole numbers by directly modeling the portioning with drawings.

Interpret whole-number quotients of whole numbers by using written methods.

There are 12 cookies on the counter. If you put 3 cookies in each bag, how many bags will you fill?

Standard Explanation Concrete Representational Abstract3.OA.1: Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.

This standard interpret products of whole numbers. Students recognize multiplication as a means to determine the total number of objects when there are a specific number of groups with the same number of objects in each group. Multiplication requires students to think in terms of groups of things rather than individual things. Students learn that the multiplication symbol ”x” means ― groups of and problems such as 5 x 7 refer to 5 groups of 7. The use of a symbol for an unknown is foundational for letter variables in Grade 4 when representing problems using equations with a letter standing for the unknown quantity (Grade 4 OA 2 and OA 3).

Build-it Draw-it Write-itInterpret products of whole numbers by using objects.

Interpret products of whole numbers by drawing the situation.

Interpret products of whole numbers by writing down the written method.

Standard Explanation Concrete Representational Abstract3.MD.7a: Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths

Students can learn how to multiply length measurements to find the area of a rectangular region. But, in order that they make sense of these quantities, they must first learn to interpret measurement of rectangular regions as a multiplicative relationship of the number of square units in a row and the number of rows. This relies on the development of spatial structuring. To build from spatial structuring to understanding the number of area-units as the product of number of units in a row and number of rows, students might draw rectangular arrays of squares and learn to determine the number of squares in each row with increasingly sophisticated strategies, such as skip-counting the number in each row and eventually multiplying the number in each row by the number of rows. They learn to partition a rectangle into identical squares by anticipating the final structure and forming the array by drawing line segments to form rows and columns. They use skip counting and multiplication to determine the number of squares in the array.

Build-it Draw-it Write-itFind the area of a rectangle with whole-number side lengths by tiling it.

Find the area of a rectangle with whole-number side lengths by drawing in the tiles.

Students should tile rectangle then multiply the side lengths to show it is the same. To find the area one could count the squares or multiply 3 x 4 = 12

Many activities that involve seeing and making arrays of squares to form a rectangle might be needed to build robust conceptions of a rectangular area structured into squares. Students should understand and explain why multiplying the side lengths of a rectangle yields the same measurement of area as counting the number of tiles (with the same unit length) that fill the rectangle’s interior For example, students might explain that one length tells how many unit squares in a row and the other length tells how many rows there are.

Standard Explanation Concrete Representational Abstract3.MD.7b: Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

Students should solve real world and mathematical problems

Build-it Draw-it Write-itMultiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems by tiling it.

Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems by using a drawing.

Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems by using a written method.

Example:Drew wants to tile the bathroom floor using 1 foot tiles. How many square foot tiles will he need?

This standard extends students‘ work with the distributive property. For example, in the picture below the area of a 7 x 6 figure can be determined by finding the area of a 5 x 6 and 2 x 6 and adding the two sums.

Students tile areas of rectangles, determine the area, record the length and width of the rectangle, investigate the patterns in the numbers, and discover that the area is the length times the width.

Standard Explanation Concrete Representational Abstract3.MD.7c: Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

This standard extends students’ work with the distributive property. Using concrete objects or drawings students build competence with composition and decomposition of shapes, spatial structuring, and addition of area measurements, students learn to investigate arithmetic properties using area models. . Students also learn to understand and explain that the area of a rectangular region of, for example, 12 length-units by 5 length-units can be found either by multiplying 12 x 5, or by adding two products, e.g., 10 x 5 and 2 x 5, illustrating the distributive property.

Build-it Draw-it Write-itUse tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c.

Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c by drawing a model.

Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c by using a written method to represent the distributive propery.

Example: Joe and John made a poster that was 4ft. by 3ft. Melisa and Barb made a poster that was 4ft. by 2ft. They placed their posters on the wall side-by-side so that that there was no space between them. How much area will the two posters cover? Students use pictures, words, and numbers to explain their understanding of the distributive property in this context.

Standard Explanation Concrete Representational Abstract3.OA.3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

This standard references various problem solving context and strategies that students are expected to use while solving word problems involving multiplication & division. Students should use a variety of representations for creating and solving one-step word problems.

Build-it Draw-it Write-itUse multiplication and division within 100 by using objects.

Use multiplication and division within 100 by using drawings.

Use multiplication and division within 100 by using equations.

Examples of multiplication:There are 24 desks in the classroom. If the teacher puts 6 desks in each row, how many rows are there? This task can be solved by drawing an array by putting 6 desks in each row. This is an array model:

This task can also be solved by drawing pictures of equal groups. 4 groups of 6 equals 24 objects:

A student can also reason through the problem mentally or verbally, “I know 6 and 6 are 12. 12 and 12 are 24. Therefore, there are 4 groups of 6 giving a total of 24 desks in the classroom.” A number line could also be used to show equal jumps. Students in third grade should use a variety of pictures, such as stars, boxes, flowers to represent unknown numbers (variables). Letters are also introduced to represent unknowns in third grade.

Examples of Division:There are some students at recess. The teacher divides the class into 4 lines with 6 students in each line. Write a division equation for this story and determine how many students are in the class (□ ÷ 6 = 4, there are 24 students in the class). Determin ing the number of objects in each share (partition model of division, where the size of the groups is unknown): Example: The bag has 92 hair clips, and Laura and her three friends want to share them equally. How many hair clips will each person receive?

Determining the number of shares (measurement division, where the number of groups is unknown) Example: Max the monkey loves bananas. Molly, his trainer, has 24 bananas. If she gives Max 4 bananas each day, how many days will the bananas last?

Standard Explanation Concrete Representational Abstract3.OA.4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?

The focus of 3.OA.4 extend beyond the traditional notion of fact families, by having students explore the inverse relationship of multiplication and division. Students extend work from eliar grades with their understanding of the meaning of the equal sign as “the same amount as” to interpret an equation with an unknown. When given 4 x ? = 40, they might think: • 4 groups of some number is the same as 40 • 4 times some number is the same as 40 • I know that 4 groups of 10 is 40 so the unknown number is 10 • The missing factor is 10 because 4 times 10 equals 40. Equations in the form of a x b = c and c = a x b should be used interchangeably, with the unknown in different positions.

Build-it Draw-it Write-itDetermine the unknown whole number in a multiplication or division equation relating three whole numbers by directly modeling with objects.

Determine the unknown whole number in a multiplication or division equation relating three whole numbers by using a drawing.

Determine the unknown whole number in a multiplication or division equation relating three whole numbers by using a written method or number strategy.

Example:Solve the equations below: 24 = ? x 6 72 ÷ = 9 Rachel has 3 bags. There are 4 marbles in each bag. How many marbles does Rachel have altogether? 3 x 4 = m This standard is strongly connected to 3.AO.3 when students solve problems and determine unknowns in equations. Students should also experience creating story problems for given equations. When crafting story problems, they should carefully consider the question(s) to be asked and answered to write an appropriate equation. Students may approach the same story problem differently and write either a multiplication equation or division equation.

Standard Explanation Concrete Representational Abstract3.OA.6: Understand division as an unknown-factor problem.

Since multiplication and division are inverse operations, students are expected to solve problems and explain their processes of solving division problems that can also be represented as unknown factor in multiplication problems. Multiplication and division are inverse operations and that understanding can be used to find the unknown. Fact family triangles demonstrate the inverse operations of multiplication and division by showing the two factors and how those factors relate to the product and/or quotient.

Build-it Draw-it Write-itUnderstand division as an unknown-factor problem by modeling with objects.

Understand division as an unknown-factor problem by drawing a picture.

Understand division as an unknown-factor problem by using a written method.

Students use their understanding of the meaning of the equal sign as “the same as” to interpret an equation with an unknown. When given 32 ÷ □ = 4, students may think: 4 groups of some number is the same as 32 4 times some number is the same as 32 I know that 4 groups of 8 is 32 so the unknown number is 8 The missing factor is 8 because 4 times 8 is 32. Equations in the form of a ÷ b = c and c = a ÷ b need to be used interchangeably, with the unknown in different positions.

Multiplication and Division Problem Types, Properties, and Strategies Standard Explanation Concrete Representational Abstract

3.OA.5: Apply properties of operations as strategies to multiply and divide. •Commutative property of multiplication•Associative property of multiplication•Distributive property

This standard references properties (rules about how numbers work) of multiplication. This extends past previous expectations, in which students were asked to identify properties. While students DO NOT need to not use the formal terms of these properties, student must understand that properties are rules about how numbers work, and they need to be flexibly and fluently applying each of them in various situtations. Students represent expressions using various objects, pictures, words and symbols in order to develop their understanding of properties.

Build-it Draw-it Write-itApply properties of operations as strategies to multiply and divide using objects.

Apply properties of operations as strategies to multiply and divide using drawings.

Apply properties of operations as strategies to multiply and divide using a written method.

The associative property states that the sum or product stays the same when the grouping of addends or factors is changed. For example, when a student multiplies 7 x 5 x 2, a student could rearrange the numbers to first multiply 5 x 2 = 10 and then multiply 10 x 7 = 70.

The commutative property (order property) states that the order of numbers does not matter when you are adding or multiplying numbers. For example, if a student knows that 5 x 4 = 20, then they also know that 4 x 5 = 20.

Students are introduced to the distributive property of multiplication over addition as a strategy for using products they know to solve products they don’t know. Here are ways that students could use the distributive property to determine the product of 7 x 6. Again, students should use the distributive property, but can refer to this in informal language such as “breaking numbers apart”.

Standard Explanation Concrete Representational Abstract

3.OA.7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

This standard uses the word fluently, which means accuracy, efficiency (using a reasonable amount of steps and time), and flexibility (using strategies such as the distributive property). “Know from memory” should not focus only on timed tests and repetitive practice, but ample experiences working with manipulatives, pictures, arrays, word problems, and numbers to internalize the basic facts (up to 9 x 9). By studying patterns and relationships in multiplication facts and relating multiplication and division, students build a foundation for fluency with multiplication and division facts. Students demonstrate fluency with multiplication facts through 10 and the related division facts. Multiplying and dividing fluently refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.

Build-it Draw-it Write-itFluently multiply and divide within 100 by using objects.

Fluently multiply and divide within 100 by using drawings.

Fluently multiply and divide within 100 using properties of operations or another written method.

Strategies students may use to attain fluency include: • Multiplication by zeros and ones • Doubles (2s facts), Doubling twice (4s), Doubling three times (8s) • Tens facts (relating to place value, 5 x 10 is 5 tens or 50) • Five facts (half of tens) • Skip counting (counting groups of __ and knowing how many groups have been counted) • Square numbers (ex: 3 x 3) • Nines (10 groups less one group, e.g., 9 x 3 is 10 groups of 3 minus one group of 3) • Decomposing into known facts (6 x 7 is 6 x 6 plus one more group of 6) • Turn-around facts (Commutative Property) • Fact families (Ex: 6 x 4 = 24; 24 ÷ 6 = 4; 24 ÷ 4 = 6; 4 x 6 = 24) • Missing factors Students should have exposure to multiplication and division problems presented in both vertical and horizontal forms.

Standard Explanation Concrete Representational Abstract3.NBT.3: Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations

This standard extends students’ work in multiplication by having them apply their understanding of place value. This standard expects that students go beyond tricks that hinder understanding such as “just adding zeros” and explain and reason about their products.

Build-it Draw-it Write-itMultiply one-digit whole numbers by multiples of 10 in the range 10–90 using objects based on place value and properties of operations strategies

Multiply one-digit whole numbers by multiples of 10 in the range 10–90 using drawings based on place value and properties of operations strategies.

Multiply one-digit whole numbers by multiples of 10 in the range 10–90 using strategies based on place value and properties of operations.

Fourth GradeUnderstanding and Relating Multiplication and Division Operations 

Standard Explanation Concrete Representational Abstract4.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.

A multiplicative comparison is a situation in which one quantity is multiplied by a specified number to get another quantity (e.g., “a is n times as much as b”). Students should be able to identify and verbalize which quantity is being multiplied and which number tells how many times. Students should be given opportunities to write and identify equations and statements for multiplicative comparisons.

Build-it Draw-it Write-itInterpret a multiplication equation as a comparison by using objects to represent the equation.

Interpret a multiplication equation as a comparison by using drawings to represent the equation.

Interpret a multiplication equation as a comparison by writing the statement or using a written method to represent the equation. See below.

Example: 5 x 8 = 40. Sally is five years old. Her mom is eight times older. How old is Sally’s Mom? 5 x 5 = 25 Sally has five times as many pencils as Mary. If Sally has 5 pencils, how many does Mary have?

Standard Explanation Concrete Representational Abstract4.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.

This standard calls for students to translate comparative situations into equations with an unknown and solve. Students need many opportunities to solve contextual problems. In a multiplicative comparison, the underling question is what amount would be added to one quantity in order to result in the other. In a multiplicative comparison, the underlying question is what factor would multiply one quantity in order to result in the other.

Build-it Draw-it Write-itMultiply or divide to solve word problems involving multiplicative comparison by using objects to represent the problem.

Multiply or divide to solve word problems involving multiplicative comparison by using drawings to represent the problem.

Multiply or divide to solve word problems involving multiplicative comparison by using drawings and equations with a symbol for the unknown number to represent the problem.

Examples: Unknown Product: A blue scarf costs $3. A red scarf costs 6 times as much. How much does the red scarf cost? (3 x 6 = p). Group Size Unknown: A book costs $18. That is 3 times more than a DVD. How much does a DVD cost? (18 ÷ p = 3 or 3 x p = 18).

Number of Groups Unknown: A red scarf costs $18. A blue scarf costs $6. How many times as much does the red scarf cost compared to the blue scarf? (18 ÷ 6 = p or 6 x p = 18).

Factors and Multiples Standard Explanation Concrete Representational Abstract

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.Students distinguish the terms “factors” and “multiples” in multiplication.

This standard requires students to demonstrate understanding of factors and multiples of whole numbers.Prime vs. Composite: •A prime number is a number greater than 1 that has only 2 factors, 1 and itself. •Composite numbers have more than 2 factors. Students investigate whether numbers are prime or composite byFactor Pairs:Students should understand the process of finding factor pairs so they can do this for any number 1 -100.Multiples:Multiples can be thought of as the result of skip counting by each of the factors. When skip counting, students should be able to identify the number of factors counted.

Build-it Draw-it Write-itFind all factor pairs for a whole number in the range 1 – 100 using objects.

Find all factor pairs for a whole number in the range 1 – 100 using drawings.

Find all factor pairs for a whole number in the range 1 – 100 using a written method.

Definitions of prime and composite numbers should not be provided, but determined after many strategies have been used in finding all possible factors of a number. Provide students with counters to find the factors of numbers. Have them find ways to separate the counters into equal subsets. For example, have them find several factors of 10, 14, 25 or 32, and write multiplication expressions for the numbers. Another way to find the factor of a number is to use arrays from square tiles or drawn on grid papers. Have students build rectangles that have the given number of squares. For example if you have 16 squares:

The idea that a product of any two whole numbers is a common multiple of those two numbers is a difficult concept to understand. For example. 5 x 8 is 40; the table below shows the multiples of each factor.

Ask students what they notice about the number 40 in each set of multiples; 40 is the 8th multiple of 5, and the 5th multiple of 8. Knowing how to determine factors and multiples is the foundation for finding common multiples and factors in Grade 6. Writing multiplication expressions for numbers with several factors and for numbers with a few factors will help students in making conjectures about the numbers. Students need to look for commonalities among the numbers.

Multiplication and Division Problems Involving Multi-digit Whole Numbers Standard Explanation Concrete Representational Abstract

4.MD.3: Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

Based on work in third grade students learn to consider perimeter and area of rectangles. By repeatedly reasoning about how to calculate areas and perimeters of rectangles, students can come to see area and perimeter formulas as summaries of all such calculations.

Build-it Draw-it Write-itApply the area and perimeter formulas for rectangles in real world and mathematical problems using objects or models.

Apply the area and perimeter formulas for rectangles in real world and mathematical problems using drawings.

Apply the area and perimeter formulas for rectangles in real world and mathematical problems by applying written methods.

Example: Mr. Rutherford is covering the miniature golf course with an artificial grass. How many 1-foot squares of carpet will he need to cover the entire course?

Students learn to apply these understandings and formulas to the solution of real-world and mathematical problems. Example: A rectangular garden has as an area of 80 square feet. It is 5 feet wide. How long is the garden? Here, specifying the area and the width creates an unknown factor problem. Similarly, students could solve perimeter problems that give the perimeter and the length of one side and ask the length of the adjacent side.

In fourth grade and beyond, the mental visual images for perimeter and area from third grade can support students in problem solving with these concepts. When engaging in the mathematical practice of reasoning abstractly and quantitatively in work with area and perimeter, students think of the situation and perhaps make a drawing. Then they recreate the “formula” with specific numbers and one unknown number as a situation equation for this particular numerical situation. “Apply the formula” does not mean write down a memorized formula and put in known values because in fourth grade students do not evaluate expressions (they begin this type of work in Grade 6). In fourth grade, working with perimeter and area of rectangles is still grounded in specific visualizations and numbers. These numbers can now be any of the numbers used in fourth grade (for addition and subtraction for perimeter and for multiplication and division for area). By repeatedly reasoning about constructing situation equations for perimeter and area involving specific numbers and an unknown number, students will build a foundation for applying area, perimeter, and other formulas by substituting specific values for the variables in later grades.

Standard Explanation Concrete Representational Abstract4.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.

Students who develop flexibility in breaking numbers apart have a better understanding of the importance of place value and the distributive property in multi-digit multiplication. Students use base ten blocks, area models, partitioning, compensation strategies, etc. when multiplying whole numbers and use words and diagrams to explain their thinking. They use the terms factor and product when communicating their reasoning. Multiple strategies enable students to develop fluency with multiplication and transfer that understanding to division. Use of the standard algorithm for multiplication is an expectation in the 5th grade.

Build-it Draw-it Write-itMultiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using objects like base ten blocks.

Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using illustrate and explain the calculation by using rectangular arrays, and/or area models.

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. Explain the calculation by using equations.

Example: To illustrate 154 x 6 students use base 10 blocks or use drawings to show 154 six times. Seeing 154 six times will lead them to understand the distributive property, 154 x 6 = (100 + 50 + 4) x 6 = (100 x 6) + (50 X 6) + (4 X 6) = 600 + 300 + 24 = 924. The area model below shows the partial products. 14 x 16 = 224

Standard Explanation Concrete Representational Abstract4.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.

In fourth grade, students build on their third grade work with division within 100. Students need opportunities to develop their understandings by using problems in and out of context. This standard calls for students to explore division through various strategies.

Build-it Draw-it Write-itFind whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using objects.

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using Illustrate and explain the calculation by using rectangular arrays, and/or area models.

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. Explain the calculation by using equations.

Eamples: A 4th grade teacher bought 4 new pencil boxes. She has 260 pencils. She wants to put the pencils in the boxes so that each box has the same number of pencils. How many pencils will there be in each box? • Using Base 10 Blocks: Students build 260 with base 10 blocks and distribute them into 4 equal groups. Some students may need to trade the 2 hundreds for tens but others may easily recognize that 200 divided by 4 is 50. • Using Place Value: 260 ÷ 4 = (200 ÷ 4) + (60 ÷ 4) • Using Multiplication: 4 x 50 = 200, 4 x 10 = 40, 4 x 5 = 20; 50 + 10 + 5 = 65; so 260 ÷ 4 = 65

Example: • 150 ÷ 6 Using an Open Array or Area Model After developing an understanding of using arrays to divide, students begin to use a more abstract model for division. This model connects to a recording process that will be formalized in the 5th grade.

Standard Explanation Concrete Representational Abstract4.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.

The focus in this standard is to have students use and discuss various strategies. It refers to estimation strategies, including using compatible numbers (numbers that sum to 10 or 100) or rounding. Problems should be structured so that all acceptable estimation strategies will arrive at a reasonable answer. Students need many opportunities solving multistep story problems.

Build-it Draw-it Write-itSolve multistep word problems posed with whole numbers and having whole-number answers using the four operations using objects.

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations using drawings.

Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations using a written method.

Estimation skills include identifying when estimation is appropriate, determining the level of accuracy needed, selecting the appropriate method of estimation, and verifying solutions or determining the reasonableness of situations using various estimation strategies. Estimation strategies include, but are not limited to: • front-end estimation with adjusting (using the highest place value and estimating from the front end, making adjustments to the estimate by taking into account the remaining amounts), • clustering around an average (when the values are close together an average value is selected and multiplied by the number of values to determine an estimate), • rounding and adjusting (students round down or round up and then adjust their estimate depending on how much the rounding affected the original values), • using friendly or compatible numbers such as factors (students seek to fit numbers together - e.g., rounding to factors and grouping numbers together that have round sums like 100 or 1000), • using benchmark numbers that are easy to compute (students select close whole numbers for fractions or decimals to determine an estimate). • This standard references interpreting remainders. Remainders should be put into context for interpretation. Ways to address remainders: • Remain as a left over • Partitioned into fractions or decimals • Discarded leaving only the whole number answer • Increase the whole number answer up one • Round to the nearest whole number for an approximate result

Fifth GradeMultiplication and Division Problems Involving Multi-digit Whole Numbers 

Standard Explanation Concrete Representational Abstract5.MD.5a: Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

These standards involve finding the volume of right rectangular prisms. Students should have experiences to describe and reason about why the formula is true. Specifically, that they are covering the bottom of a right rectangular prism (length x width) with multiple layers (height). Therefore, the formula (length x width x height) is an extension of the formula for the area of a rectangle.

Build-it Draw-it Write-itFind the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes.

Find the volume of a right rectangular prism with whole-number side lengths by drawing it with unit cubes,

5.MD.5b: Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

Build-it Draw-it Write-itApply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths using objects.

Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths using drawings.

Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths.

Examples: When given 24 cubes, students make as many rectangular prisms as possible with a volume of 24 cubic units. Students build the prisms and record possible dimensions.

Standard Explanation Concrete Representational Abstract5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm.

This standard refers to fluency which means students select and use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators. They work flexibly with basic number combinations and use visual models, benchmarks, and equivalent forms. They are accurate and efficient (use a reasonable amount of steps), and flexible (use strategies such as the distributive property or breaking numbers apart (decomposing and recomposing) also using strategies according to the numbers in the problem. While the standard algorithm is mentioned, alternative strategies are also appropriate to help students develop conceptual understanding. The size of the numbers should NOT exceed a three-digit factor by a two-digit factor. Students can continue to use these different strategies as long as they are efficient, but must also understand and be able to use the standard algorithm. In applying the standard algorithm, students recognize the importance of place value.

Build-it Draw-it Write-itFluently multiply multi-digit whole numbers using the standard algorithm using objects.

Fluently multiply multi-digit whole numbers using the standard algorithm using drawings.

Fluently multiply multi-digit whole numbers using the standard algorithm using a written method.

Example: 123 x 34. When students apply the standard algorithm, they, decompose 34 into 30 + 4. Then they multiply 123 by 4, the value of the number in the ones place, and then multiply 123 by 30, the value of the 3 in the tens place, and add the two products.Examples of alternative strategies: There are 225 dozen cookies in the bakery. How many cookies are there? Draw a area model for 225 x 12…. 200 x 10, 200 x 2, 20 x 10, 20 x 2, 5 x 10, 5 x 2

Standard Explanation Concrete Representational Abstract5.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.

This standard references various strategies for division. Division problems can include remainders. Even though this standard leads more towards computation, the connection to story contexts is critical. Make sure students are exposed to problems where the divisor is the number of groups and where the divisor is the size of the groups. In fourth grade, students’ experiences with division were limited to dividing by one-digit divisors. This standard extends students’ prior experiences with strategies, illustrations, and explanations. When the two-digit divisor is a ―familiar‖ number, a student might decompose the dividend using place value.

Build-it Draw-it Write-itFind whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using objects.

Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors. Illustrate and explain the calculation by using rectangular arrays, and/or area models.

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. Explain the calculation by using equations.

Example: There are 1,716 students participating in Field Day. They are put into teams of 16 for the competition. How many teams get created? If you have left over students, what do you do with them?

Standard Explanation Concrete Representational Abstract5.NBT.7: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

This standard builds on the work from fourth grade where students are introduced to decimals and compare them. In fifth grade, students begin adding, subtracting, multiplying and dividing decimals. This work should focus on concrete models and pictorial representations, rather than relying solely on the algorithm. The use of symbolic notations involves having students record the answers to computations, but this work should not be done without models or pictures. This standard includes students’ reasoning and explanations of how they use models, pictures, and strategies. Before students are asked to give exact answers, they should estimate answers based on their understanding of operations and the value of the numbers.

Build-it Draw-it Write-itAdd, subtract, multiply, and divide decimals to hundredths, using concrete models and explain the reasoning used.

Add, subtract, multiply, and divide decimals to hundredths, using drawings and explain the reasoning used.

Add, subtract, multiply, and divide decimals to hundredths, using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Sixth GradeFactors and Multiples 

Standard Explanation Concrete Representational Abstract6.NS.4: Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

In elementary school, students identified primes, composites and factor pairs (4.OA.4). In 6th grade students will find the greatest common factor of two whole numbers less than or equal to 100.

Build-it Draw-it Write-itFind the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12

Multiplication and Division Problems Involving Multi-digit Whole Numbers Standard Explanation Concrete Representational Abstract

6.NS.2: Fluently divide multi-digit numbers using the standard algorithm.

In the elementary grades, students were introduced to division through concrete models and various strategies to develop an understanding of this mathematical operation (limited to 4-digit numbers divided by 2-digit numbers). In 6th grade, students become fluent in the use of the standard division algorithm, continuing to use their understanding of place value to describe what they are doing. Place value has been a major emphasis in the elementary standards. This standard is the end of this progression to address students’ understanding of place value.

Build-it Draw-it Write-itFluently divide multi-digit numbers using the standard algorithm.

Standard Explanation Concrete Representational Abstract

6.NS.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

Procedural fluency is defined by the Common Core as “skill in carrying out procedures flexibly, accurately, efficiently and appropriately”. In 4th and 5th grades, students added and subtracted decimals. Multiplication and division of decimals were introduced in 5th grade (decimals to the hundredth place). At the elementary level, these operations were based on concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. In 6th grade, students become fluent in the use of the standard algorithms of each of these operations. The use of estimation strategies supports student understanding of decimal operations.

Build-it Draw-it Write-itFluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

Properties of operationsHere a, b, and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, and the complex number system.

Associative property of multiplication a•(b•c) = (a•b)•cCommutative property of multiplication a•b = b•a

Multiplicative identity property of 1 a • 1 = 1 • a = aMultiplicative property of 0 a • 0 = 0 • a = 0

Distributive property of multiplication over addition a•(b + c) = a•b + a•c

Types of multiplication and division problems chart: