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Developing Computational Proficiency with Multiplication and Division Students learn to add before they learn to multiply. Therefore, early attempts to multiply may be done using repeated addition connected to a groups of equal quantities scenario. Since many students will not be very accurate or speedy when it comes to repeated addition of numbers other than 2s, 5s, or 10s, it is necessary to provide explicit instruction in a variety of other computational strategies. Sequence for Teaching Multiplication Facts and Strategies Though there is no one correct way to sequence teaching of the multiplication facts, most researchers and educators agree that 2s, 5s, and 10s come first, then 1s and 0s. Researcher Susan O’Connell suggests 2, 10, 5, 1, 0, 3, 4, 6, 9, 8, 7 as the sequence for learning multiplication facts. In its May 2015 Vol. 21, No. 9 publication, Teaching Children Mathematics, the National Council of Teachers of Mathematics (NCTM) suggests a similar sequence, but adds square products as foundational facts to be learned early. See Table 1. Table 1: Sequence and Strategies for teaching multiplication facts. Foundational Facts 2s, 5s, 10s (begin these late in grade 2) 0s, 1s, multiplication squares (2x2, 3x3..) Use story problems, arrays, skip counting, and patterns on a hundreds chart, and a multiplication table to learn these facts. Derived Fact Strategies Adding or subtracting a group Halving and doubling Using a square product Decomposing a factor Start with a nearby 2s, 5s, or 10s fact, then subtract (or add) the group. Example, I don’t know 9 x 6, so I think “10 x 6 = 60” and subtract one group of 6 to get 54. Look for an even factor. Find the fact for half of that factor, then double it. Example, I don’t know 6 x 8, so I think “3 x 8 = 24” and double that to get 48. Look for a nearby square. Find that fact then add on or subtract off the extra group. Example, I don’t know 7 x 6. I use 6 x 6 = 36 and add one more 6 to get 42. Partition one of the factors into a convenient sum of known facts, find the known facts and combine the products. Example, I don’t know 7 x 6. I break the 7 into 2 and 5, because I know 2 x 6 and 5 x 6. Then I add 12 and 30 to get 42. Note: There are many interpretations of 3 x 2. In this document the supporting examples will reference it as 3 groups of 2. EduGAINS Mathematics K-12 Page 1 of 10 Knowing the foundational facts, along with properties and strategies, allows students to reason the facts for 3, 4, 6, 7, 8, and 9. What follows is designed to support the teaching and learning of foundational facts and computational strategies. Multiplication: Establish the Foundational Facts and Strategies Establishing 2s, 5s, and 10s Effective ways to establish mastery of the 2s, 5s, and 10s foundational facts include frequent opportunities involving: • connecting products of 2 to skip counting by 2, products of 5 to skip counting by 5, products of 10 to skip counting by 10 and make use of the Commutative Property of Multiplication Example: 5 x 3 may not be automatic when thought of as 5 groups of 3, however, thinking of it as 3 groups of 5 may be determined by skip counting 5, 10, 15. See page 5 for more detail on the Commutative Property of Multiplication. • repeated addition on a number line • observing patterns in a hundreds chart • recitation during games, jumping rope, counting objects, etc. Establishing 1s and 0s a)Multiplying by 1 Once students are comfortable with interpreting 2 x 4 as 2 groups of 4, 5 x 3 as 5 groups of 3, etc., they can be led to interpret 1 x 2 as 1 group of 2, and 1 x 5 as 1 group of 5. After students have worked with enough examples, support students in explicitly understanding multiplying any number by 1 yields a product equal to the original number (i.e., a number keeps its original identity when multiplied by 1). Hence, 1 is called the Multiplicative Identity. b) Multiplying by Zero Students can think of 0 x 3 = 0 or 3 x 0 as “no groups of 3” as well as “addition of 3 zeros” to generate products of 0. They may even extend the product pattern. Example: 4x3 3x3 2x3 1x3 0x3 = 12 =9 =6 =3 =0 Take away 3 Take away 3 Take away 3 Take away 3 c) Dividing by 1 Students who have had experience with the unknowns being written in all three of the possible locations in a x b = c will appreciate that 2 ÷ 1 = ? means the same as 2 = 1 x ?. Their understanding of the multiplicative identity, tells them that ? = 2. Like multiplication by 1, division by 1 also leaves a number unchanged (e.g., 2 ÷ 1 = 2, 99 ÷ 1 = 99). d) Divisions Involving 0 Students can reason about 0 ÷ 2 = ? by asking themselves, “How many groups of 2 yield a total of 0?” This quotative division question is answered by 0 since it takes zero groups of 2 to yield a total of 0. If students ask the partitive division question, “2 groups of how many yield a total of 0?”, they again, answer 0. Similarly, 0 divided by any number other than 0 yields a quotient of 0. EduGAINS Mathematics K-12 Page 2 of 10 What is Partitive and Quotative Divison? Example: 4 x 7 = 28, if 4 represents the number of groups and 7 represents the number of items in each group, then the result of 28 ÷ 4 is the number of items in each group (partitive) and the result of 28 ÷ 7 is the number of groups (quotative) Depending on the context of a problem, you may be finding one or the other. If not given a context, then one should be considering both types of divisions. Some students may be curious about division by zero. Guide them to reason that 2 ÷ 0 = ? can be a partitive division where the quantity per part is unknown. In that case, ask “Zero groups of what quantity yield a total of 2?” Since this question does not define any number, we say that 2 ÷ 0 is undefined. If 2 ÷ 0 is a quotative division, then 2 ÷ 0 = ? asks, “How many groups of 0 are in a total of 2?”. Again, this question does not define any number, so 2 ÷ 0 is undefined. Any number other than 0 can replace 2 in the argument above. A few students may be curious about 0 ÷ 0 = ? By changing this division question to the equivalent multiplication question, 0 = 0 x ?, students can reason that ? can be 1, or ? can be 2, or ? can be 99. In fact, ? can be any number. Since the value for ? cannot be determined without further information 0 ÷ 0 is said to be indeterminate. Establishing the Square Products Effective ways to establish mastery of these foundational facts include frequent opportunities involving: • observing patterns in a hundreds chart • drawing nested squares and observing the pattern in adding to one square to make the next. EduGAINS Mathematics K-12 Page 3 of 10 The following visual summaries show the multiplication facts for 2, 5, 10, 0, and 1, and the facts established using the commutative property. Visual Summaries of Sequence for Teaching Multiplication Facts 2s, 5s, and 10s Examples: 2 x 3, 5 x 4 Using Commutative Property for 2s, 5s, and 10s 0s, 1s, and square products Examples: 0 x 5, 4 x 4 Using Commutative Property for 0s, and 1s 2s, 5s, 10s, 0s, 1s and using the Commutative Property altogether Notice how few facts have yet to be learned once the foundational facts and the Commutative Property have been learned. EduGAINS Mathematics K-12 Page 4 of 10 Multiplication: Naming and Using Properties Commutative Property of Multiplication It is helpful to demonstrate that an expression like 2 x 6 yields the same product as 6 x 2. The fact that you get the same product when you switch the order of the numbers in a multiplication question is called the Commutative Property of Multiplication. Model of 2 x 6 Model of 6 x 2 Using the Commutative Property of Multiplication can reduce the cognitive load for students by taking advantage of multiplication facts already mastered when learning a new multiplication fact. For example, students typically learn certain facts before others (e.g., 2-times, and 3-times before 6-times). The models above convince students that 6 x 2 (a new fact) yields the same product as 2 x 6 (a known fact). Similarly, if the student knows 3 x 6 = 18, they can reason that 6 x 3 = 18. By applying the Commutative Property to known facts, students can reduce the number of new facts they need to learn. It should be noted that the Commutative Property does not apply to division (e.g., 10 ÷ 2 does not have the same value as 2 ÷ 10). Associative Property of Multiplication This property says that you can group numbers in whatever order you want when multiplying. 7x2x5 = (7 x 2) x 5 = 14 x 5 = 70 7 x 2 x 5 Most people find it easier to compute 7 x 10 than 14 x 5. = 7 x (2 x 5) The brackets do not have to be included. They are used here to show = 7 x 10 which numbers are associated first. = 70 Distributive Property of Multiplication Over Addition or Subtraction This property says you can multiply a sum by multiplying each of the addends and then add the partial products. For example, 6 x (3 + 2) is equal to 6 x 3 + 6 x 2. The examples below show how decomposing a number allows us to take advantage of the Distributive Property. Example 1: 7x7 = (5 + 2) x 7 = (5 x 7) + (2 x 7) = 35 + 14 = 49 When learning the fact 7 x 7 = 49, a student may think of 5 groups of 7 and 2 more groups of 7. Example 2: 7 x 23 = 7 x (20 + 3) = (7 x 20) + (7 x 3) = 140 + 21 = 161 This property is especially useful for multiplying larger numbers. Here, a student can think “7 groups of 20 and 7 groups of 3”. For more detail about the Distributive Property and visual justification for using this strategy, see Pictorial Strategy for Multiplying All Types of Numbers on page 9. EduGAINS Mathematics K-12 Page 5 of 10 Multiplication: Connecting Computational Strategies and Properties *Many steps are included for understanding but these steps do not have to be written. Students may use less formal notation. Strategy Examples* Additional Information Skip Counting 6x2 2, 4, 6, 8, 10, 12 Students may use their fingers to keep track of the six 2s as they skip count. Repeated Addition 6x2 =2+2+2+2+2+2 = 12 7 x 2 may seem easier as Commutative Property = 2 x 7 Half then Double Anchor of 2s, 5s, or 10s plus a group Anchor of 2s, 5s, or 10s minus a group Anchor of 2s, 5s, or 10s plus or minus more than one group 6x7 =3x7x2 = 21 x 2 = 42 3x7 = (2 +1) x 7 =2x7+1x7 = 14 + 7 = 21 9x7 = (10 - 1) x 7 = 10 x 7 – 1 x 7 = 70 – 7 = 63 8x7 = (10 – 2) x 7 = 10 x 7 – 2 x 7 = 70 – 14 = 56 EduGAINS Mathematics K-12 As the number of addends increases, the student is less likely to use repeated addition or skip counting. Many students will use skip counting only for 2s, 5s, and 10s, and prefer one of the strategies outlined below for the other facts. It is helpful to support students with understanding the connection between the partial sums 2, 2 + 2, 2 + 2 + 2, ... to the skip count “2, 4, 6, 8, 10, 12”. Switching the order of a multiplication question is sometimes done to determine unknown facts. If a quantity is halved, then doubled, the end result is the same as the starting quantity (e.g., 6 halved is 3, then 3 doubled is 6). This makes strategic use of the Associative Property and decomposes an unknown multiplier. Start with a nearby anchor of a 2s, 5s, or 10s fact, then add or subtract one or more groups. This makes strategic use of the Distributive Property and decomposes an unknown multiplier. Page 6 of 10 Multiplication: Choosing Which Computational Strategy to Use Being flexible in using strategies is a great skill. Many computational strategies depend on breaking larger numbers down to smaller or friendly numbers (decomposing the number), then applying the Distributive or the Associative Property. Which particular strategy a student chooses will depend on which particular facts the student has already mastered. The best strategy for decomposing factors depends on which number facts the student knows. The following chart illustrates that multiple strategies can be used for the same product. Multiple Strategies for the Product 6 x 7 Strategy Anchor of 5 Plus a Group Perfect Square Plus a Group Perfect Square Minus a Group Anchor of 5 Plus Two Groups Half then Double Example 6x7 6x7 6x7 6x7 6x7 Decompose = (5 + 1) x 7 = 6 x (6 + 1) = (7 – 1) x 7 = 6 x (5 + 2) =3x7x2 Use the =5x7+1x7 Distributive or = 35 + 7 Associative = 42 Property 6 is near the Student anchor of 5 Thinking =6x6+6x1 = 36 + 6 = 42 =7x7–1x7 = 49 – 7 = 42 =6x5+6x2 = 30 + 12 = 42 = 21 x 2 = 42 6 x 7 is near the square product 6 x 6 6 x 7 is near the square product 7 x 7 7 is near the anchor of 5 6 is even, so take half, then double Visual summaries of the various decomposing strategies are shown below and illustrate the fact that more than one strategy can be used for some of the facts. If the Commutative Property was used as well, the first and third charts will have more facts filled in. 2s, 5s, or 10s Anchor plus or minus 1 group Square Product Anchor plus or minus 1 group EduGAINS Mathematics K-12 2s 5s, or 10s Anchor plus or minus 2 groups Half then Double Page 7 of 10 Multiplication and Division: Strategies for Larger Numbers Rename Numbers to Multiply and Divide by 10, 100, 1000 Mentally Students learn early that 10 ones can be traded for a ten, 10 tens can be traded for a hundred, and 10 hundreds can be traded for a thousand. Gap Closing J/I Representing and Renaming Whole Numbers Student Book p.18 Using trading of units and flexibility in renaming numbers, students can make sense of multiplication and division by 10, 100, and 1000. 52 tens can be written as 520 52 x 10 can mean 52 tens 630 x 100 can mean 630 hundreds 630 hundreds can be written as 63000 can mean how many hundreds are in 63 000? 63 000 can be thought of as 630 hundreds so the answer is 630 can mean how many tens are in 520? 520 can be thought of as 52 tens so the answer is 52 63 000 ÷ 100 520 ÷ 10 so the answer is 520 so the answer is 63 000 Multiplying Double-Digit Numbers The following strategies are related to strategies shown earlier for single-digit multiplication. Again, several strategies can be used for the same fact. *Many steps are included for understanding but these steps do not have to be written. Students may use less formal notation. Strategy Decompose to Use Known Facts Examples* 14 x 25 = 14 x (20 + 5) = 14 x 20 + 14 x 5 = 280 + 70 = 350 14 x 25 = 14 x 100 ÷ 4 Scaling Up and Scaling Down = 1400 ÷ 4 = 350 EduGAINS Mathematics K-12 Additional Information Use a known multiplier as an anchor then use the distributive property. “Quadruple then Quarter” and “Half then Double” strategies are both based on keeping equivalence while using numbers that are easier to work with mentally. Page 8 of 10 Pictorial Strategy for Multiplying All Types of Numbers The following chart contains examples of how an open partitioned array can be used to find a product by decomposing larger whole numbers, decimals, fractions, and algebraic expressions. Many steps are included for understanding but these steps do not have to be written Whole Numbers Decompose using friendly numbers Decimal Numbers 14 x 25 7.1 x 8.5 =(10 + 4)x(20 + 5) = (7 + 0.1) x (8 + 0.5) Fractions Algebra 2 1/4 x 3 1/2 = (2 + 1/4) (3 + 1/2) (c + 2)(c + 3) 6 + 1 + 3/4 +1/8 = 7 7/8 c² + 3c + 2c + 6 = c² + 5c + 6 Use an open array Determine partial products Add partial products 200 + 80 + 20 + 50 = 350 EduGAINS Mathematics K-12 56 + 3.5 + 0.8 + 0.05 = 60.35 Page 9 of 10 Standard Algorithm Whole Numbers Decimal Numbers Fractions Algebra 2 1/4 x 3 1/2 = 9/4 x 7/2 = 63/8 = 7 7/8 • • • Multiply 4 x 25. Multiply 1 x 25 and move 25 over 1 place value position since it is really 10 x 25 or 250. Add lined up digits. • • • • Multiply 5 x 71 Multiply 8 x 71 and move over one place value position Add lined up digits Put the decimal in by adding the number of decimal places in the question and counting over that many places from the right. EduGAINS Mathematics K-12 • • • • Convert to improper fractions. Multiply numerators and multiply • denominators Simplify fractions if necessary Convert the improper fraction to a mixed fraction if desired or required. • c² + 3c + 2c + 6 = c² + 5c + 6 Multiply using “FOIL” Multiply the F irst terms, O uter terms, I nner terms, L ast terms, Collect like terms. Page 10 of 10