Progressions for the Common Core State Standards in ...

1 Progressions for the Common Core State Standards in Mathematics (draft). c The Common Core Standards Writing Team 4 July 2013. Suggested citation: Common Core Standards Writing Team. (2013, July 4). Progressions for the Common Core State Standards in Mathematics (draft). Grades 6 8, The number Sys- tem; High School, number . Tucson, AZ: Institute for Mathematics and Education, University of Arizona. For updates and more information about the Progres- sions, see For discussion of the Progressions and related top- ics, see the Tools for the Common Core blog: http: Draft, 9 July 2013, comment at The number system , 6 8. Overview In Grades 6 8, students build on two important conceptions which have developed throughout K 5, in order to understand the rational numbers as a number system .

2 The first is the representation of whole numbers and fractions as points on the number line, and the second is a firm understanding of the properties of operations on whole numbers and fractions. Representing numbers on the number line In early grades, stu- dents see whole numbers as counting numbers, Later, they also understand whole numbers as corresponding to points on the num- ber line. Just as the 6 on a ruler measures 6 inches from the 0. mark, so the number 6 on the number line measures 6 units from the origin. Interpreting numbers as points on the number line brings fractions into the family as well; fractions are seen as measurements with new units, creating by partitioning the whole number unit into equal pieces.

3 Just as on a ruler we might measure in tenths of an inch, on the number line we have halves, thirds, fifths, sevenths; the number line is a sort of ruler with every denominator. The denom- inators 10, 100, etc. play a special role, partioning the number line into tenths, hundredths, etc., just as a metric ruler is partioned into centimeters and millimeters. Represent whole numbers as lengths from 0 on a num- Starting in Grade 2 students see addition as concatenation of lengths on the number By Grade 4 they are using the ber line diagram with equally spaced points corresponding to the numbers 0, 1, 2, .. , and represent whole- number sums and dif- same model to represent the sum of fractions with the same denom- ferences within 100 on a number line diagram.

4 Inator: 35 75 is seen as putting together a length that is 3 units of one fifth long with a length that is 7 units of one fifth long, making Representing 3. 5 and 7. 5 on the number line 10 units of one fifths in all. Since there are five fifths in 1 (that's 7. what it means to be a fifth), and 10 is 2 fives, we get 35 7. 5 2. 5. Two fractions with di erent denominators are added by representing place a length of 7. next them in terms of a Common unit. 5. to a length of 3. 5. Representing sums as concatenated lengths on the number line 0 1 2. is important because it gives students a way to think about addition 0. 5. 1. 5. 2. 5. 3. 5. 4. 5. 5. 5. 6. 5. 7. 5. 8. 5. 9. 5. 10. 5. that makes sense independently of how numbers are represented symbolically.

5 Although addition calculations may look di erent for numbers represented in base ten and as fractions, addition is the Draft, 9 July 2013, comment at NS, 6 8 3. same operation in each case. Furthermore, the concatenation model Properties of Operations on Rational Numbers of addition extends naturally to negative numbers in Grade 7. Properties of Addition 1. Commutative Property. For any two rational numbers . Properties of operations The number line provides a represen- and , . tation that can be used to building understanding of sums and 2. Associative Property. For any three rational numbers , di erences of rational numbers. However, building understanding and , . of multiplication and division of rational numbers relies on a firm 3.

6 Existence of Identity. The number 0 satisfies understanding of properties of operations. Although students have 0 0. not necessarily been taught formal names for these properties, they 4. Existence of Additive Inverse. For any rational number , there is a number such that 0. have used them repeatedly in elementary school and have been with reasoning with them. The commutative and associative properties Properties of Multiplication of addition and mutiplication have, in particular, been their constant 1. Commutative Property. For any two rational numbers . friends in working with strategies for addition and , and , . The existence of the multiplicative identity (1) and multiplicative 2.

7 Associative Property. For any three rational numbers , and , . inverses start to play important roles as students learn about frac- tions. They might see fraction equivalence as confirming the identity 3. Existence of Identity. The number 1 satisfies 1 1. rule for fractions. In Grade 4 they learn about fraction equivalence 4. Existence of Multiplicative Inverse. For every non-zero rational number , there is a rational number 1 such that 1 1.. The Distributive Property and in Grade 5 they relate this to multiplication by 1 For rational numbers , and , one has .. 1 . Apply properties of operations as strategies to add and thus confirming that the identity rule Apply properties of operations as strategies to multiply and 1.

8 Works for Interpret multiplication as scaling (resizing), by: As another example, the commutative property for multiplication a .. plays an important role in understanding multiplication with frac- b .. and relating the principle of fraction equivalence .. tions. For example, although . to the effect of multiplying by 1. 1 5. 5. 2 2. can be made sense of using previous understandings of whole num- ber multiplication as repeated addition, the other way around, 1 5. 5 . 2 2. seems to come from a di erent source, from the meaning of phrases such as half of and a mysterious acceptance that of must mean multiplication. A more reasoned approach would be to observe that if we want the commutative property to continue to hold, then we must have 1 1 5.

9 5 5 . 2 2 2. Draft, 9 July 2013, comment at NS, 6 8 4. Interpret a fraction as division of the numerator by the de- and that 52 is indeed half of five, as we have understood in Grade nominator ( ). Solve word problems involving division of whole numbers leading to answers in the form of fractions or When students extend their conception of multiplication to in- mixed numbers, , by using visual fraction models or equations clude negative rational numbers, the properties of operations become to represent the problem. crucial. The rule that the product of negative numbers is positive, often seen as mysterious, is the result of extending the properties of operations (particularly the distributive property) to rational num- bers.

10 Draft, 9 July 2013, comment at NS, 6 8 5. Grade 6. As Grade 6 dawns, students have a firm understanding of place value and the properties of operations. On this foundation they are ready to start using the properties of operations as tools of exploration, deploying them confidently to build new understandings of operations with fractions and negative numbers. They are also ready to complete their growing fluency with algorithms for the four operations. Apply and extend previous understandings of multiplication and division to divide fractions by fractions In Grade 6 students con- Interpret and compute quotients of fractions, and solve clude the work with operations on fractions, started in Grade 4, by computing quotients of In Grade 5 students divided word problems involving division of fractions by fractions, , by using visual fraction models and equations to represent the unit fractions by whole numbers and whole numbers by unit frac- problem.