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3rd grade common core state standards flip book this document is intended to show the connections to the standards of mathematical practices for the content standards and to get detailed information at each level resources used ccss arizona doe ohio doe and north carolina doe this flip book is intended to help teachers understand what each standard means in terms of what students must know and be able to do it provides only a sample of instructional strategies and examples the goal of every teacher should be to guide students in understanding making sense of mathematics construction directions print single-sided on cardstock cut the tabs on each page starting with page 2 cut the bottom off of this top cover to reveal the tabs for the subsequent pages staple or bind the top of all pages to complete your flip book compiled by melisa hancock send feedback to melisa@ksu.edu 1

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mathematical practice standards mp summary of each standard 1 make sense of problems and persevere in solving them in third grade students know that doing mathematics involves solving problems and discussing how they solved them students explain to themselves the meaning of a problem and look for ways to solve it third graders may use concrete objects or pictures to help them conceptualize and solve problems they may check their thinking by asking themselves does this make sense they listen to the strategies of others and will try different approaches they often will use another method to check their answers third graders should recognize that a number represents a specific quantity they connect the quantity to written symbols and create a logical representation of the problem at hand considering both the appropriate units involved and the meaning of quantities 2 reason abstractly and quantitatively 3 construct viable arguments and critique the reasoning of others in third grade students may construct arguments using concrete referents such as objects pictures and drawings they refine their mathematical communication skills as they participate in mathematical discussions involving questions like how did you get that and why is that true they explain their thinking to others and respond to others thinking students experiment with representing problem situations in multiple ways including numbers words mathematical language drawing pictures using objects acting out making a chart list or graph creating equations etc students need opportunities to connect the different representations and explain the connections they should be able to use all of these representations as needed third graders should evaluate their results in the context of the situation and reflect on whether the results make sense third graders consider the available tools including estimation when solving a mathematical problem and decide when certain tools might be helpful for instance they may use graph paper to find all the possible rectangles that have a given perimeter they compile the possibilities into an organized list or a table and determine whether they have all the possible rectangles as third graders develop their mathematical communication skills they try to use clear and precise language in their discussions with others and in their own reasoning they are careful about specifying units of measure and state the meaning of the symbols they choose for instance when figuring out the area of a rectangle they record their answers in square units in third grade students look closely to discover a pattern or structure for instance students use properties of operations as strategies to multiply and divide commutative and distributive properties students in third grade should notice repetitive actions in computation and look for more shortcut methods for example students may use the distributive property as a strategy for using products they know to solve products that they don t know for example if students are asked to find the product of 7 x 8 they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at 40 16 or 56 in addition third graders continually evaluate their work by asking themselves does this make sense 4 model with mathematics 5 use appropriate tools strategically 6 attend to precision 7 look for and make use of structure deductive reasoning 8 look for and express regularity in repeated reasoning inductive reasoning 2 standards for math practice smp

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summary of standards for mathematical practice 1 make sense of problems and persevere in solving them interpret and make meaning of the problem to find a starting point analyze what is given in order to explain to themselves the meaning of the problem plan a solution pathway instead of jumping to a solution monitor their progress and change the approach if necessary see relationships between various representations relate current situations to concepts or skills previously learned and connect mathematical ideas to one another continually ask themselves does this make sense can understand various approaches to solutions questions to develop mathematical thinking how would you describe the problem in your own words how would you describe what you are trying to find what do you notice about what information is given in the problem describe the relationship between the quantities describe what you have already tried what might you change talk me through the steps you ve used to this point what steps in the process are you most confident about what are some other strategies you might try what are some other problems that are similar to this one how might you use one of your previous problems to help you begin how else might you organize represent show 2 reason abstractly and quantitatively make sense of quantities and their relationships decontextualize represent a situation symbolically and manipulate the symbols and contextualize make meaning of the symbols in a problem quantitative relationships understand the meaning of quantities and are flexible in the use of operations and their properties create a logical representation of the problem attends to the meaning of quantities not just how to compute them what do the numbers used in the problem represent what is the relationship of the quantities how is related to what is the relationship between and what does mean to you e.g symbol quantity diagram what properties might we use to find a solution how did you decide in this task that you needed to use could we have used another operation or property to solve this task why or why not 3 construct viable arguments and critique the reasoning of others analyze problems and use stated mathematical assumptions definitions and established results in constructing arguments justify conclusions with mathematical ideas listen to the arguments of others and ask useful questions to determine if an argument makes sense ask clarifying questions or suggest ideas to improve/revise the argument compare two arguments and determine correct or flawed logic what mathematical evidence would support your solution how can we be sure that how could you prove that will it still work if what were you considering when how did you decide to try that strategy how did you test whether your approach worked how did you decide what the problem was asking you to find what was unknown did you try a method that did not work why didn t it work would it ever work why or why not what is the same and what is different about how could you demonstrate a counter-example 4 model with mathematics understand this is a way to reason quantitatively and abstractly able to decontextualize and contextualize apply the mathematics they know to solve everyday problems are able to simplify a complex problem and identify important quantities to look at relationships represent mathematics to describe a situation either with an equation or a diagram and interpret the results of a mathematical situation reflect on whether the results make sense possibly improving/revising the model ask themselves how can i represent this mathematically what number model could you construct to represent the problem what are some ways to represent the quantities what is an equation or expression that matches the diagram number line chart table where did you see one of the quantities in the task in your equation or expression how would it help to create a diagram graph table what are some ways to visually represent what formula might apply in this situation usd 259 learning services 2011

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summary of standards for mathematical practice 5 use appropriate tools strategically use available tools recognizing the strengths and limitations of each use estimation and other mathematical knowledge to detect possible errors identify relevant external mathematical resources to pose and solve problems use technological tools to deepen their understanding of mathematics questions to develop mathematical thinking what mathematical tools could we use to visualize and represent the situation what information do you have what do you know that is not stated in the problem what approach are you considering trying first what estimate did you make for the solution in this situation would it be helpful to use a graph number line ruler diagram calculator manipulative why was it helpful to use what can using a show us that may not in what situations might it be more informative or helpful to use 6 attend to precision communicate precisely with others and try to use clear mathematical language when discussing their reasoning understand the meanings of symbols used in mathematics and can label quantities appropriately express numerical answers with a degree of precision appropriate for the problem context calculate efficiently and accurately what mathematical terms apply in this situation how did you know your solution was reasonable explain how you might show that your solution answers the problem what would be a more efficient strategy how are you showing the meaning of the quantities what symbols or mathematical notations are important in this problem what mathematical language definitions properties can you use to explain how could you test your solution to see if it answers the problem 7 look for and make use of structure apply general mathematical rules to specific situations look for the overall structure and patterns in mathematics see complicated things as single objects or as being composed of several objects what observations do you make about what do you notice when what parts of the problem might you eliminate simplify what patterns do you find in how do you know if something is a pattern what ideas that we have learned before were useful in solving this problem what are some other problems that are similar to this one how does this relate to in what ways does this problem connect to other mathematical concepts 8 look for and express regularity in repeated reasoning see repeated calculations and look for generalizations and shortcuts see the overall process of the problem and still attend to the details understand the broader application of patterns and see the structure in similar situations continually evaluate the reasonableness of their intermediate results explain how this strategy work in other situations is this always true sometimes true or never true how would we prove that what do you notice about what is happening in this situation what would happen if is there a mathematical rule for what predictions or generalizations can this pattern support what mathematical consistencies do you notice usd 259 learning services 2011

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critical areas for mathematics in 3rd grade in grade 3 instructional time should focus on four critical areas 1 developing understanding of multiplication and division and strategies for multiplication and division within 100 2 developing understanding of fractions especially unit fractions fractions with numerator 1 3 developing understanding of the structure of rectangular arrays and of area and 4 describing and analyzing two-dimensional shapes 1 students 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 multiplication is finding an unknown product and division is finding an unknown factor in these situations for equal-sized group situations division can require finding the unknown number of groups or the unknown group size students use properties of operations to calculate products of whole numbers using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors by comparing a variety of solution strategies students learn the relationship between multiplication and division 2 students develop an understanding of fractions beginning with unit fractions students view fractions in general as being built out of unit fractions and they use fractions along with visual fraction models to represent parts of a whole students understand that the size of a fractional part is relative to the size of the whole for example 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts the parts are longer than when the ribbon is divided into 5 equal parts students are able to use fractions to represent numbers equal to less than and greater than one they solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators 3 students recognize area as an attribute of two-dimensional regions they measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps a square with sides of unit length being the standard unit for measuring area students understand that rectangular arrays can be decomposed into identical rows or into identical columns by decomposing rectangles into rectangular arrays of squares students connect area to multiplication and justify using multiplication to determine the area of a rectangle 4 students describe analyze and compare properties of two-dimensional shapes they compare and classify shapes by their sides and angles and connect these with definitions of shapes students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole 3 critical areas

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domain operations and algebraic thinking oa cluster represent and solve problems involving multiplication and division standard 3.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 for example describe a context in which a total number of objects can be expressed as 5 × 7 standards for mathematical practices to be emphasized mp.1 make sense of problems and persevere in solving them mp.4 model with mathematics mp.7 look for and make use of structure connections 3.oa.1-4 this cluster is connected to the third grade critical area of focus #1 developing understanding of multiplication and division and strategies for multiplication and division within 100 connect this domain with understanding properties of multiplication and the relationship between multiplication and division grade 3 oa 5 ­ 6 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 explanations and examples 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 multiplication is seen as groups of and problems such as 5 x 7 refer to 5 groups of 7 however it is important for teachers to understand there are several ways in which we can think of multiplication 1multiplication is often thought of as repeated addition of equal groups while this definition works for some sets of numbers it is not particularly intuitive or meaningful when we think of multiplying 3 by 1/2 for example or 5 by -2 in such cases it may be helpful to widen the idea of grouping to include evaluation of part of a group this concept is related to partitioning which in turn is related to division ex three groups of five students can be read as 3 · 5 or 15 students while half a group of 10 stars can be represented as 1/2 · 10 or 5 stars these are examples of partitioning each one of the three groups of five is part of the group of 15 and the group of 5 stars is part of the group of 10 2a second concept of multiplication is that of rate or price ex if a car travels four hours at 50 miles per hour then it travels a total of 4 · 50 or 200 miles if cds cost eight dollars each then three cds will cost 3 · $8 or $24 3a third concept of multiplication is that of multiplicative comparison ex sara has four cds joanne has three times as many as sara and sylvia has half as many as sara thus joanne has 3 · 4 or 12 cds and sylvia has 1/2 · 4 or 2 cds example for this level 3.oa.1 jim purchased 5 packages of muffins each package contained 3 muffins how many muffins did jim purchase 5 groups of 3 5 x 3 15 describe another situation where there would be 5 groups of 3 or 5 x 3 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 continued next page to further develop this understanding students interpret a problem situation requiring multiplication using pictures objects words numbers and equations then given a multiplication expression e.g 5 x 6 students interpret the expression using a multiplication context see table 2 page 58 they should begin to use the terms factor and product as they describe multiplication 4

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instructional strategies 3.oa.1-4 in grade 2 students found the total number of objects using rectangular arrays such as a 5 x 5 and wrote equations to represent the sum this strategy is a foundation for multiplication because students should make a connection between repeated addition and multiplication students need to experience problem-solving involving equal groups whole unknown or size of group is unknown and multiplicative comparison unknown product group size unknown or number of groups unknown as shown in table 2 of the common core state standards for mathematics page 74 encourage students to solve these problems in different ways to show the same idea and be able to explain their thinking verbally and in written expression allowing students to present several different strategies provides the opportunity for them to compare strategies sets of counters number lines to skip count and relate to multiplication and arrays/area models will aid students in solving problems involving multiplication and division allow students to model problems using these tools students should represent the model used as a drawing or equation to find the solution show a variety of models of multiplication i.e 3 groups of 5 counters can be written as 3 × 5 provide a variety of contexts and tasks so that students will have more opportunity to develop and use thinking strategies to support and reinforce learning of basic multiplication and division facts have students create multiplication problem situations in which they interpret the product of whole numbers as the total number of objects in a group and write as an expression also have students create division-problem situations in which they interpret the quotient of whole numbers as the number of shares students can use known multiplication facts to determine the unknown fact in a multiplication or division problem have them write a multiplication or division equation and the related multiplication or division equation for example to determine the unknown whole number in 27 ÷ 3 students should use knowledge of the related multiplication fact of 3 × 9 27 they should ask themselves questions such as how many 3s are in 27 or 3 times what number is 27 have them justify their thinking with models or drawings instructional resources/tools sets of counters number lines to skip count and relate to multiplication arrays table 2 common multiplication and division situations common core state standards for mathematics 2010 page 58 in this document national council of teachers of mathematics illuminations exploring equal sets this four-part lesson encourages students to explore models for multiplication the inverse of multiplication and representing multiplication facts in equation form national council of teachers of mathematics illuminations all about multiplication in this four-lesson unit students explore several meanings and representation of multiplications and learn about properties of operations for multiplication common misconceptions 3.oa.1-4 students think a symbol or is always the place for the answer this is especially true when the problem is written as 15 ÷ 3 or 15 x 3 students also think that 3 ÷ 15 5 and 15 ÷ 3 5 are the same equations the use of models is essential in helping students eliminate this understanding the use of a symbol to represent a number once cannot be used to represent another number in a different problem/situation presenting students with multiple situations in which they select the symbol and explain what it represents will counter this misconception 5 3.oa.1

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domain operations and algebraic thinking oa cluster represent and solve problems involving multiplication and division standard 3.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 for example describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8 standards for mathematical practices to be emphasized mp.1 make sense of problems and persevere in solving them mp.4 model with mathematics mp.7 look for and make use of structure connections see 3.oa.1 explanations and examples this standard focuses on two distinct models of division partition models and measurement repeated subtraction models partition models focus on the question how many in each group 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 focus on the question how many groups can you make a context or measurement models would be there are 12 cookies on the counter if you put 3 cookies in each bag how many bags will you fill students need to recognize the operation of division in two different types of situations one situation requires determining how many groups and the other situation requires sharing determining how many in each group students should be exposed to appropriate terminology quotient dividend divisor and factor to develop this understanding students interpret a problem situation requiring division using pictures objects words numbers and equations given a division expression e.g 24 ÷ 6 students interpret the expression in contexts that require both interpretations of division see table 2 page 74 students may use interactive whiteboards to create digital models 6 3.oa.2

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domain operations and algebraic thinking oa cluster represent and solve problems involving multiplication and division standard 3.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 see table 2 standards for mathematical practices to be emphasized mp.1 make sense of problems and persevere in solving them mp.4 model with mathematics mp.7 look for and make use of structure connections see 3.oa.1 explanations and examples this standard references various strategies that can be used to solve word problems involving multiplication division students should apply their skills to solve word problems students should use a variety of representations for creating and solving one-step word problems such as if you divide 4 packs of 9 brownies among 6 people how many brownies does each person receive 4 x 9 36 36 ÷ 6 6 table 2 page 58 of this document gives examples of a variety of problem solving contexts in which students need to find the product the group size or the number of groups students should be given ample experiences to explore all of the different problem structures 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 could 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 students 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 continued next page 7

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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 ÷ 4 6 there are 24 students in the class determining the number of objects in each share partitive 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 starting day 1 day 2 day 3 day 4 day 5 day 6 24 24 ­ 4 20 20 ­ 4 16 16 ­ 4 12 12 ­ 4 8 8 ­ 4 4 4 ­ 4 0 solution the bananas will last for 6 days students use a variety of representations for creating and solving one-step word problems i.e numbers words pictures physical objects or equations they use multiplication and division of whole numbers up to 10 x10 students explain their thinking show their work by using at least one representation and verify that their answer is reasonable word problems may be represented in multiple ways equations 3 x 4 4 x 3 12 ÷ 4 and 12 ÷ 3 array equal groups repeated addition 4 4 4 or repeated subtraction three equal jumps forward from 0 on the number line to 12 or three equal jumps backwards from 12 to 0 continued on next page 8

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examples of division problems determining the number of objects in each share partitive division where the size of the groups is unknown o 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 o 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 starting 24 day 1 244 20 day 2 204 16 day 3 164 12 day 4 124 8 day 5 8-4 4 day 6 4-4 0 solution the bananas will last for 6 days students may use interactive whiteboards to show work and justify their thinking 9 3.oa.3

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domain operations and algebraic thinking oa cluster represent and solve problems involving multiplication and division standard 3.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 standards for mathematical practices to be emphasized mp.1 mp.2 mp.6 mp.7 make sense of problems and persevere in solving them reason abstractly and quantitatively attend to precision look for and make use of structure connections see 2.oa.1 explanations and examples this standard refers to table 2 page 58 of this document and equations for the different types of multiplication and division problem structures the easiest problem structure includes unknown product 3 x 6 or 18 ÷ 3 6 the more difficult problem structures include group size unknown 3 x 18 or 18 ÷ 3 6 or number of groups unknown x 6 18 18 ÷ 6 3 the focus of 3.oa.4 goes beyond the traditional notion of fact families by having students explore the inverse relationship of multiplication and division students apply their understanding of the meaning of the equal sign as the same 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 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 questions 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 continued on next page 10

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students apply their understanding of the meaning of the equal sign as the same 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 examples solve the equations below 24 x 6 rachel has 3 bags there are 4 marbles in each bag how many marbles does rachel have altogether 3 x 4 m students may use interactive whiteboards to create digital models to explain and justify their thinking 72÷ 9 11 3.oa.4

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domain operations and algebraic thinking oa cluster understand properties of multiplication and the relationship between multiplication and division standard 3.oa.5 apply properties of operations as strategies to multiply and divide students need not use formal terms for these properties examples if 6 × 4 24 is known then 4 × 6 24 is also known commutative property of multiplication 3 × 5 × 2 can be found by 3 × 5 15 then 15 × 2 30 or by 5 × 2 10 then 3 × 10 30 associative property of multiplication knowing that 8 × 5 40 and 8 × 2 16 one can find 8 × 7 as 8 × 5 2 8 × 5 8 × 2 40 16 56 distributive property standards for mathematical practices to be emphasized mp.1 mp.4 mp.7 mp.8 make sense of problems and persevere in solving them model with mathematics look for and make use of structure look for and express regularity in repeated reasoning connections 3.oa.5-6 this cluster is connected to the third grade critical area of focus #1 developing understanding of multiplication and division and strategies for multiplication and division within 100 this standard references properties of multiplication while students do not need to use the formal terms of these properties student should understand that properties are rules about how numbers work students do need to be flexibley and fluently applying each of them students represent expressions using various objects pictures words and symbols in order to develop their understanding of properties they multiply by 1 and 0 and divide by 1 they change the order of numbers to determine that the order of numbers does not make a difference in multiplication but does make a difference in division given three factors they investigate changing the order of how they multiply the numbers to determine that changing the order does not change the product they also decompose numbers to build fluency with multiplication 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 adding or multiplying numbers for example if a student knows that 5 x 4 20 then they also know that 4 x 5 20 the array below could be described as a 5 x 4 array for 5 columns and 4 rows or a 4 x 5 array for 4 rows and 5 columns there is no fixed way to write the dimensions of an array as rows x columns or columns x rows students should have flexibility in being able to describe both dimensions of an array explanations and examples xxxxxxxxxxxxxxxxxxxx 4x5 or 5x4 xxxxx xxxxx xxxxx xxxxx continued next page 12

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students should be introduced to the distributive property of multiplication over addition as a strategy for using products they know to solve products they don t know students would be using mental math to determine a product 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 student 1 7x6 7 x 5 35 7x1=7 35 7 42 student 2 7x6 7 x 3 21 7 x 3 21 21 21 42 student 3 7x6 5 x 6 30 2 x 6 12 30 12 42 another example of the distributive property helps students determine the products and factors of problems by breaking numbers apart for example for the problem 6 x 5 students can decompose the 6 into a 4 and 2 and reach the answer by multiplying 4 x 5 20 and 2 x 5 =10 and adding the two products 20+10=30 4x5 2x5 to further develop understanding of properties related to multiplication and division students use different representations and their understanding of the relationship between multiplication and division to determine if the following types of equations are true or false · 0 x 7 7 x 0 0 zero property of multiplication · 1 x 9 9 x 1 9 multiplicative identity property of 1 ·3x6=6x3 commutative property ·8÷22÷8 students are only to determine that these are not equal ·2x3x5=6x5 · 10 x 2 5 x 2 x 2 · 2 x 3 x 5 10 x 3 ·1x6>3x0x2 continued next page 13

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