Review – focussing on the completed thought in this leftmost column. | Test your recall by hiding this column. |
1 Students’ work in the ____ ____ is intertwined with their work on counting and cardinality, and with the meanings and properties of addition, subtraction, multiplication, and division. | base-ten system |
2 Work in the base-ten system relies on these meanings and ____ , but also contributes to deepening students’ understanding of them. | properties |
3 Position | |
4 The base-ten system is a remarkably efficient and uniform system for systematically representing ____ ____. | all numbers |
5 Using only the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, every number can be represented as a ____ ____ ____, where each digit represents a value that depends on its place in the string. | string of digits |
6 The relationship between ____ ____ by the places in the base-ten system is the same for whole numbers and decimals: the value represented by each place is always 10 times the value represented by the place to its immediate right. | values represented |
6a The relationship between values represented by the places in the base-ten system is the same for ____ ____ and decimals: the value represented by each place is always 10 times the value represented by the place to its immediate right. | whole numbers |
6b The relationship between values represented by the places in the base-ten system is the same for whole numbers and ____: the value represented by each place is always 10 times the value represented by the place to its immediate right. | decimals |
6c The relationship between values represented by the places in the base-ten system is the same for whole numbers and decimals: the value represented by each place is always ____ times the value represented by the place to its immediate right. | 10 |
7 In other words, moving one place to the ____, the value of the place is multiplied by 10. | left, |
8 In other words, moving one place to the left, the value of the place is ____ by 10. | multiplied |
9 Because of this uniformity, ____ ____ for computations within the base-ten system for whole numbers extend to decimals | standard algorithms |
10 Base-ten units | |
11 Each ____ of a base-ten numeral represents a base-ten unit: ones, tens, tenths, hundreds, hundredths, etc. |
place |
12 The ____ in the place represents 0 to 9 of those units. | digit |
13 Because ____ ____ ____ make a unit of the next highest value, only ten digits are needed to represent any quantity in base ten. | ten like units |
14 The basic unit is a ____ (represented by the right-most place for whole numbers). | one |
15 In learning about whole numbers, children learn that ten ones compose a new kind of unit called a ____ . | ten |
16 They understand ____ ____ numbers as composed of tens and ones, and use this understanding in computations, decomposing 1 ten into 10 ones and composing a ten from 10 ones. | two-digit |
17 The power of the base-ten system is in repeated bundling by ____: 10 tens make a unit called a “____”. | ten, hundred |
18 Repeating this process of creating new units by bundling in ____ ____ ____ creates units called thousand, ten thousand, hundred thousand | groups of ten |
19 In learning about decimals, children ____ a one into 10 equal-sized smaller units, each of which is a tenth. | partition |
18 Each base-ten unit can be understood in terms of any other ____ ____. | base-ten unit |
20 For example, ____ ____ can be viewed as a tenth of a thousand, 10 tens, 100 ones, or 1.000 tenths. | one hundred |
21 ____ for operations in base ten draw on such relationships among the base ten units. | Algorithms |
22 Computations | |
23 Standard algorithms for base-ten computations with the four operations rely on ____ numbers written in base-ten notation into base-ten units. | decomposing |
24 The properties of operations then allow any multi-digit computation to be reduced to a collection of ____ computations. | single-digit |
25 These single-digit computations sometimes require the ____ ____ ____ of a base-ten unit. | composition or decomposition |
31 Strategies and algorithms | |
32 In learning about ____, children partition a one into ____ ^{ }equal-sized smaller units, each of which is a ____. | decimals, 10, tenth |
33 Each base-ten unit can be understood in terms of any other ____ ____. | base-ten unit |
34 For example, ____ hundred can be viewed as a tenth of a thousand,10 tens, 100 ones, or1000 tenths. | one |
35 Algorithms for operations in base ten draw on such ____ among the base-ten units. | relationships |
36 Standard algorithms for base-ten computations with the ____ ____ rely on decomposing numbers written in base-ten notation into base-ten units. | four operations |
37 The properties of operations then allow any ____ computation to be reduced to a collection of single-digit computations. | multi-digit |
38 These single-digit computations sometimes require the ____ or decomposition of a base-ten unit. | composition |
39 Beginning in ____, the requisite [computation] abilities develop gradually over the grades. | Kindergarten |
40 Experience with addition and subtraction made within ____ is a Grade 1 standardand fluency is a Grade 2 standard. | 20 |
41 Computations within 20 that “____ ____,” such as 9 + 8 or 13 – 7 , are especially relevant to NBT because they afford the development of the Level 3 make-a-ten strategies for addition and subtraction described in the OA Progression. | cross 10 |
42 From the NBT perspective, _____ strategies are (implicitly) the ﬁrst instances of composing or decomposing a base-ten unit. |
make-a-ten |
43 Such strategies are a foundation for understanding in Grade 1 that addition may require ____ a tenand in Grade 2 that subtraction may require ____ a ten. | composing, decomposing |
44 The Standards distinguish ____ from algorithms. | strategies |
45 For example, students use strategies for addition and subtraction in Grades K-3, but are expected to ﬂuently ____ ____ ____ whole numbers using standard algorithms by the end of Grade 4. | add and subtract |
46 Use of the ____ ____ can be viewed as the culmination of a long progression of reasoning about quantities, the base-ten system, and the properties of operations. | standard algorithms |
47 This progression distinguishes between two types of computational strategies: ____ strategies and ____ methods. | special, general |
48 For ample, a special strategy for computing 398 + 17 is to ____ 17 as 2 + 15 and evaluate (398 + 2) + 15. | decompose |
49 Special strategies either cannot be ____ to all numbers represented in the base-ten system or require considerable modiﬁcation in to do so. | extended |
51 A more readily generalizable method of computing 398 + 17 is to ____ like base-ten units. | combine |
50 General methods extend to all numbers represented in the base-ten system. A ____ method is not necessarily efficient. | general |
51 For example, ____ on by ones is a general method that can be easily modiﬁed for use with ﬁnite decimals. | counting |
52 General methods based on ____ ____, however, are more efficient and can be viewed as closely connected with standard algorithms. | place value |
53 Mathematical Practices | |
51 Both general methods and special strategies are opportunities to ____ competencies relevant to the NBT standards. | develop |
52 Use and discussion of both types of strategies offer opportunities for developing ____ with place value and properties of operations, and to use these in justifying the correctness of computations (MP 3) | fluency |
53 Special strategies may be advantageous in situations that require ____ computation, but less so when uniformity is useful. | quick |
54 Thus, they offer opportunities to raise the topic of ____ appropriate tools strategically (MP 5) | using |
55 Standard algorithms can be viewed as expressions of regularity in ____ ____ (MP.8) used in general methods based on place value. | repeated reasoning |
56 ____ ____ and recordings of computations, whether with strategies or standard algorithms, afford opportunities for students to contextualize, probing into the referents for the symbols involved (MP.2). | Numerical expressions |
57 Representations such as bundled objects or math drawings (e.g., drawings of hundreds, tens, and ones) and diagrams (e.g., simplified renderings of arrays or area models) afford the mathematical practice of explaining ____ among different representations (MP.1). | correspondences |
58 Drawings, diagrams, and numerical recordings may raise questions related to ____ (MP.6), e.g., does that 1represent 1 one or 1 ten? | precision |
59 This progression gives examples of ____ that can be used to connect _____ with quantities and to connect numerical representations with combination, composition ,and decomposition of base-ten units as students work towards computational fluency. | representations, numerals |