Investigations math example

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Source: TERC
    11/29/12, v3 ! TERC, 2012 1 Math Content by Strand 1   Number and Operations with Whole Numbers Multiplication and Division Grade 3 In Grade 3, students investigate the properties of multiplication and division, including the inverse relationship between these two operations, and develop strategies for solving multiplication and division  problems. Their work focuses on developing the idea that multiplication involves some  number of equal-sized groups ,   and that division also involves equal groups. Students are introduced to arrays—rectangular arrangements of objects in rows and columns—to help them develop visual images that support their understanding of multiplication. They use these rectangular arrays to represent the relationship between a product and its factors. Students determine, describe, and compare sets of multiples, noticing their characteristics and relationships, and use these to investigate important ideas about how multiplication works. 1   This document applies to the 2nd edition of Investigations   (2008, 2012). See for changes when implementing Investigations and the Common Core Standards  .    11/29/12, v2 ! TERC, 2012 2 They learn the multiplication combinations with products up to 50. Students solve division situations that involve  sharing  , (“Divide 35 pennies among 5 people equally. How many pennies are in each share?”) and those that involve  grouping   (“How many groups of 5  pennies can I make if I have 35 pennies?”). Sharing: Divide 35 pennies among 5 people equally. How many pennies are in each share? Grouping: How many groups of 5 pennies can I make if I have 35 pennies? Students use their knowledge of the relationship between division and multiplication by reasoning in ways like the following: “I know that five 5s is 25, and two more 5s make 35, so I have 7 groups of 5.” Students are also introduced to two forms of division notation –– 35 ÷ 5 and 5 35 )  –– and learn how to interpret these numbers and symbols in terms of the meaning and actions of division. The Algebra Connections page in the curriculum unit that focuses on multiplication and division shows how students are applying the commutative and distributive properties of multiplication as they solve  problems. It also highlights students’ application of the inverse relationship between multiplication and division.     4 x 5 = 5 x 4 = Start with _____________  7 x 6 = 6 x 7 = Start with _____________   11/29/12, v2 ! TERC, 2012 3 Emphases Whole Number Operations   ã   Understanding the meaning of multiplication ã   Reasoning about numbers and their factors and multiples ã   Understanding and working with an array model of multiplication ã   Developing strategies for division based on understanding the inverse relationship between multiplication and division  Computational Fluency   ã   Learning the multiplication combinations with products to 50 fluently Benchmarks ã   Demonstrate an understanding of multiplication and division as involving groups of equal groups ã   Solve multiplication combinations and related division problems using skip counting or known multiplication combinations ã   Interpret and use multiplication and division notation ã   Demonstrate fluency with the multiplication combinations with products up to 50 (by the end of Grade 3) Grade 4 In Grade 4, three of the four curriculum units on number and operations with whole numbers focus on multiplication and division. This major component of students’ work centers on reasoning about numbers and their factors and multiples, using models, representations, and story contexts to help them visualize and solve multiplication and division problems; and understanding the relationship between multiplication and division.  11/29/12, v2 ! TERC, 2012 4 Students learn the multiplication combinations (facts) to 12 x 12 so that they can use these fluently to solve both multiplication and division problems. They develop strategies for solving multiplication and division problems based on looking at the problem as a whole, thinking about the relationships of the numbers in the problem, and choosing an approach they can carry out easily and accurately, often  breaking the numbers apart or changing the numbers in some way. Visualizing how multiplication works is critical in applying the distributive property to solve problems and in keeping track of parts of the problem. Learning to multiply by multiples of 10 is also a key component of this work. Examples of Multiplication Strategies Breaking numbers apart by addition 48 x 42 = 48 x 42 = 40 x 40 = 1,600 48 x 40 = 1,920 40 x 2 = 80 48 x 2 = 96 8 x 40 = 320 1,920 + 96 = 2,016 8 x 2 = 16 1,600 + 80 + 320 + 16 = 2,016 Students interpret and solve division problems, both in story contexts and numerical contexts. They work with both grouping and sharing situations, and consider how to make sense of a remainder within the context of the problem. They use the inverse relationship between multiplication and division to solve division problems, including those related to the multiplication combinations to 12 x 12 (the division “facts”), and problems in which 3-digit numbers are divided by 1-digit and small 2-digit divisors.
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