# Landmarks _ Large Numbers

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```					Unit 5: Landmarks Large Numbers                                      September to Early October                                                                     Grade 4
24 Sessions

Inv. # and    MA Std    Content Objective                                             Key Vocabulary               Language               Assessment           Pacing
Name                                                                                                               Objective

Inv. 1: How   4.N.1,    1. Students will read, write, and sequence numbers to         place value, base-ten        This is one            Review p. 14 for the 6 - 1 hour
Much is       4.N.2,    1000.                                                         number system,               example of a           assessment in this    sessions
1000? TE p.   4.N.5                                                                   sequence, addition,          language               unit. Complete all
28 - 56       4.N.7,    2. Students will add and subtract multiples of 10, 100, and   subtraction, multiple,       objective. Please      'Ongoing
4.N.9,    1000.                                                         difference, number line,     consider your          Assessments,
4.N.11                                                                  open number line,            students' needs        Writing
4.N.12,   3. Students will use multiples of 10 and 100 to find the      commutative property,        when writing your      Opportunities,
4.N.16,   difference b/w any 3-digit number and 1000.                   identity property, inverse                          Portfolio
language
4.N.17,                                                                 operation, landmarks,                               Opportunities, & End-
objectives:
4.M.1,    4. Students will represent addition and subtraction on a      rounding, expanded                                  of-Unit
Students will use
4.P.2     number line.                                                  notation, closer to,                                Assessments.'
content specific
between, before, after,                             Portfolio: SAB p. 9-
vocabulary to orally
expression, ones, tens,                             10, M19
explain their
hundreds, thousands
thinking.

Teacher's Notes: Broken Calculator (Ten Minute Math) reinforces 4.P.2, 4.N.9, 4.N.11, and 4.N.12. Inv. 1.1, make connections between landmark whole
numbers and fractional & decimal landmarks, for ex. 25 is 1/4 of 100, 500 is 1/2 of 1000, etc. When assessing students familiarity with the structure of the
number system up to the 100's, use base 10 blocks as needed. Be cognizant of Teacher Note # 5 on p. 31 of the teacher's guide. When students are finding
numbers in their 1000s book make connections to rounding, for example, if a students states 541 is closer to 500 rather than 600, explain that it is a form of
rounding to the nearest 100, and so on. The teacher may want to take this opportunity to reintroduce estimation (4.N.17) as a strategy. SAB p. 3, 11, & 15
reinforce 4.N.11. This is a great time to introduce the place value game "The Greatest." Ask your resource teacher for a copy from the district created math
specialist binder. Standard 4.N.1 expects mastery up to 100,000, challenge & differentiate up to that place value when completing any or all of the sessions.

Focal Points (connection): Number and Operations
CIA                                                                       Lowell Public Schools                                                           Working 3/5/2010
Unit 5: Landmarks Large Numbers                                      September to Early October                                                         Grade 4
24 Sessions

Inv. # and   MA Std    Content Objective                                            Key Vocabulary              Language      Assessment           Pacing
Name                                                                                                            Objective

Inv. 2:      4.N.1,    1. Students will add 3 and 4-digit numbers.                 addition strategies,                       Portfolio: M23       6 - 1 hour
Adding it    4.N.2,                                                                expressions, equivalent,                                        sessions
Up TE p.     4.N.8,    2. Students will identify, describe and compare addition    addition notation,
60 - 93      4.N.9,    strategies by focusing on how each strategy starts.         subtraction notation, U.S.
4.N.10,                                                               standard algorithm, digit,
4.N.12,   expressions are equivalent (e.g., 597 + 375 = 600 + 372)    decompose, open number
4.N.13,                                                               line, flats, rods, units,
4. Students will use story context and representations to   representations, equation,
4.N.14,
4.N.17,
4.M.1,
5. Students will use clear and concise notation to record
4.P.2,

6. Students will understand the meaning of the steps
and notation of the U.S. algorithm for addition.

7. Students will find combinations of 3-digit numbers that

Teacher's Notes: Broken Calculator (Ten-Minute Math) reinforces 4.P.2, 4.N.9. 4.N.11, 4.N.12. Student activity sheet p. 16-17 develop the understanding of
distance and miles for 4.M.1, p. 18 reinforces 4.N.10 & 4.N.17; p. 27 reinforces 4.N.11, 4.N.12; p. 30 reinforces 4.P.2, and p. 32 reinforces 4.N.10, 4.N.11,
4.N.12, & 4.N.13. After playing Close to 1,000 extend the game by having students go back to their recording sheets and write the sums in expanded
notation and in words, additional game sheets with this differentiation can be located by your math resource teacher in the district specialist created
binder. Continue playing "The Greatest" game.

Focal Points (connection): Number and Operations
CIA                                                                     Lowell Public Schools                                                  Working 3/5/2010
Unit 5: Landmarks Large Numbers                                        September to Early October                                                       Grade 4
24 Sessions

Inv. # and     MA Std    Content Objective                                             Key Vocabulary               Language    Assessment         Pacing
Name                                                                                                                Objective

Inv. 3:        4.N.1,    1. Students will read, write, and sequence numbers to         equivalence, place value,                                   5 - 1 hour
Working        4.N.2,    10,000.                                                       digit, multiple, equation,                                  sessions
with           4.N.5,                                                                  base-ten number system,
Numbers to     4.N.8,    2. Students will understand the structure of 10,000 and       power of ten, decompose,
10,000 TE p.   4.N.9,    its equivalence to 1,000 10s, 100 100s, and 10 1000s          ones, tens, hundreds,
100 - 125      4.N.10,                                                                 thousands, ten-thousand,
4.N.11,   3. Students will recognize the place value of digits in       rows, columns, chunk,
4.N.12,                                                                 mental math,
large numbers.
4.N.13,
4.N.14,   4. Students will add and subtract multiples of 10, 100, and
4.N.17,   1000.
4.M.1,
4.M.2,    5. Students will add 3 and 4-digit numbers.
4.P.2
6. Students will find combinations of 3-digit numbers that

Teacher's Notes: Ten-Minute Math Broken Calculator reinforces 4.P.2, 4.N.9. 4.N.11, 4.N.12. During Inv. 3.1, make connections between landmark whole
numbers and fractional and decimal landmarks, for example, 2,500 is 1/4 of 10,000, 5,000 = .5 of 10,000, etc. When students are finding numbers in their
10,000s chart make connections to rounding, for example, if a student states 3,218 is between 3,200 and 3,300 they have rounded to the nearest 100, or
they could round to the nearest 1000, etc. Student Activity sheet p. 42 reinforces 4.N.10, 4.N.11, 4.N.13. During Inv. 3.4, differentiate by having students
convert miles to feet (teacher may use a map of Lowell and simpler problems.) and show the addition another way, this may extend problems up to 5-digit
numbers for 4.N.12 and 4.M.2 simple unit conversion. Remember to use horizontal and vertical notation. Play the differentiated game "Close to 10,000".
Then extend the game by having students go back to their recording sheets and write the sums in expanded notation and in words. Additional game
sheets with this differentiation can be located by your math resource teacher in the district specialist created binder. Continue playing "The Greatest"
game.

Focal Points (connection): Number and Operations
CIA                                                                       Lowell Public Schools                                               Working 3/5/2010
Unit 5: Landmarks Large Numbers                                     September to Early October                                                              Grade 4
24 Sessions

Inv. # and    MA Std    Content Objective                                              Key Vocabulary                Language    Assessment            Pacing
Name                                                                                                                 Objective

Inv. 4:       4.N.1,    1. Students will understand the action of subtraction          subtraction strategies,                   Portfolio: SAB p. 55- 7 - 1 hour
Subtraction   4.N.2,    problems.                                                      subtraction situation/story               56, M27, and Unit     sessions
TE p. 132 -   4.N.5,    2. Students will represent subtraction situations.             problems, open number                     Assessment M29
165           4.N.7,    3. Students will represent addition and subtraction on a       line, decompose/break
4.N.8,    number line.                                                   apart, chunks, landmark,
4.N.9,    4. Students will solve subtraction problems by breaking        difference, expression,
4.N.10,   numbers apart.                                                 take away, minus,
5. Students will find combinations of 3-digit numbers that     difference, positive
4.N.11,
4.N.12,
6. Students will identify, describe, and compare subtraction   numbers, missing
4.N.13,
strategies by focusing on how each strategy starts.            addend, comparison,
4.N.14,
7. Students will develop arguments about how the               borrowing/regrouping,
4.N.16,                                                                  U.S. Standard algorithm,
differences represented by two subtraction expressions are
4.N.17,   related (e.g., 432 - 198 and 432 - 200)                        inverse, equation, farther,
4.M.1,    8. Students will use story context and representations to      plus,
4.M.2,    support explanations about related subtraction expressions.
4.P.2     9. Students will add and subtract 3 and 4 digit numbers.
10. Students will solve multi-step addition and subtraction
problems.
11. Students will combine positive and negative numbers.

Teacher's Notes: Broken Calculator (Ten-Minute Math) reinforces 4.P.2, 4.N.9. 4.N.11, 4.N.12. Encourage students to use the inverse to check their
answers. Standard 4.N.1 expects mastery up to 100,000, challenge and differentiate up to that place value when completing any or all of the sessions in
these investigations. During Inv. 4.5, introducing Which is Farther?, have students convert miles to feet, this will reinforce 4.M.2 & 4.N.12. Student Activity
sheets p. 67 & 68 reinforce 4.N.10 & 4.N.12; p. 69 reinforces 4.N.12; p. 70-71 reinforce 4.N.7 & 4.N.11; p. 73-74 reinforces 4.N.11, 4.N.12, 4.M.1, & 4.M.2.
Differentiate sheets 73-74 by having students convert to tons and ounces and show their solutions using those units of measure.

Focal Points (connection): Number and Operations
CIA                                                                     Lowell Public Schools                                                    Working 3/5/2010
Unit 5: Landmarks & Large Numbers
Standards #                                                                    Description
Exhibit an understanding of the base ten number system by reading, modeling, writing, and interpreting whole numbers to at least 100,000;
4.N.1     demonstrating an understanding of the values of the digits; and comparing and ordering the numbers.

4.N.2     Represent, order, and compare large numbers (to at least 100,000) using various forms, including expanded notation, e.g., 853 = 8 x 100 + 5 x 10 + 3.
4.N.5
Recognize classes (in particular, odds, evens; factors or multiples of a given number; and squares) to which a number may belong, and identify the
4.N.7     numbers in those classes. Use these in the solution of problems
Select, use, and explain various meanings and models of multiplication and division of whole numbers. Understand and use the inverse
4.N.8     relationship between the two operations.
Select, use, and explain the commutative, associative, and identity properties of operations on whole numbers in problem situations, e.g., 37 x 46 = 46
4.N.9     x 37, (5 x 7) x 2 =     5 x (7 x 2).
Select and use appropriate operations (addition, subtraction, multiplication, and division) to solve problems, including those involving
4.N.10    money.
Know multiplication facts through 12 x 12 and related division facts. Use these facts to solve related multiplication problems and compute related
4.N.11    problems, e.g., 3 x 5 is related to 30 x 50, 300 x 5, and 30 x 500
4.N.12    Add and subtract (up to five-digit numbers) and multiply (up to three digits by two digits) accurately and efficiently.
Divide up to a three-digit whole number with a single-digit divisor (with or without remainders) accurately and efficiently. Interpret any
4.N.13    remainders.
Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition and subtraction (up to five-digit
4.N.14    numbers), and multiplication (up to three digits by two digits).
4.N.16    Round whole numbers through 100,000 to the nearest 10, 100, 1000, 10,000, and 100,000.
Select and use a variety of strategies (e.g., front-end, rounding, and regrouping) to estimate quantities, measures, and the results of whole-
4.N.17    number computations up to three-digit whole numbers and amounts of money to \$1000, and to judge the reasonablenes
Demonstrate an understanding of such attributes as length, area, weight, and volume, and select the appropriate type of unit for measuring each
4.M.1     attribute.
4.M.2     Carry out simple unit conversions within a system of measurement, e.g., hours to minutes, cents to dollars, yards to feet or inches, etc.

4.P.2     sentences that use =, <, >).

CIA                                                                            LPS                                                                            3/5/2010
Unit 1: Factors, Multiples, and Arrays                                  Early October - End October                                                                      Grade 4
14 Sessions

Invest Title &   MA Std    Content Objective                                                 Key Vocabulary                    Language                Assessment           Pacing
Page No.                                                                                                                       Objective

Inv #1:          4.N.1,    1. Students will use arrays to model multiplication               multiplication, array/            This is one             Review p. 14 for     5 - 1 hour
Representing     4.N.2,    situations.                                                       rectangular arrangement,          example of a            the assessment in sessions
Multiplication   4.N.7,                                                                      dimensions, 2-dimension,          language                this unit. Complete
with Arrays      4.N.8,    2. Students will break arrays into parts to find the product      package, related numbers,         objective. Please       all 'Ongoing
TE Pages 22-     4.N.9,    represented by the array.                                         strategy, odd/even, multiple/     consider your           Assessments,
47               4.N.10,                                                                     product, skip counting, factor,   students' needs         Writing
4.N.11,   3. Students will find the multiples of a number by skip           factor pairs, multiplication      when writing your       Opportunities, &
4.N.12,   counting.                                                         expression, prime number,         language                Portfolio
composite number, square                                  Opportunities, &
4.P.1,                                                                                                        objectives:
4. Students will use arrays to find factors of 2-digit numbers.   number, multiplication                                    End-of-Unit
4.P.2,                                                                                                        Students will use
combination, row, column,                                 Assessments
4.P.4                                                                                                         content specific
5. Students will identify features of numbers, including prime,   divide, multiplication                                    Portfolio: SAB p. 2,
vocabulary to orally
square, and composite numbers.                                    equation, commutative                                     12, M31
explain their
property, associative             thinking.
6. Students will identify and learn multiplication combinations   property, properties of
not yet known fluently.                                           factors, common factors,
doubling/halving, decompose
7. Students will use known multiplication combinations to         (distributive property)
determine the products of more difficult combinations.

Teacher's Notes: Standards 4.P.2, and 4.P.4 are addressed during Ten-Minute Math. When making arrays, make connections to related numbers as ways to find other
factors for the given multiple, for example, doubling, halving, divisibilty, odd/even. The addition and subtraction in 4.N.12 is practiced and reinforced during Student
Activity Book (SAB) practice, as well as, 4.N.10. During Inv. 1 when looking for patterns in the arrays, finding factor pairs, and classifying numbers, make connections
to the Identity and Commutative Properties to cover 4.N.9. For those students who are ready, distributive property can be introduced during the discussions about
using related multiplication combinations to find more difficult combinations. See page 16 in teacher's guide.

Focal Points (addressed): Number and Operations, Algebra (supported): Measurement
CIA                                                           Lowell Public Schools                                                                Working Document 3/5/2010
Unit 1: Factors, Multiples, and Arrays                               Early October - End October                                                        Grade 4
14 Sessions

Invest Title &   MA Std    Content Objective                                                Key Vocabulary                  Language     Assessment        Pacing
Page No.                                                                                                                    Objective

Inv #2:          4.N.1,    1. Students will use arrays to model multiplication              product, multiple, factor,                   Portfolio: M51    5 - 1 hour
Multiplication   4.N.2,    situations.                                                      "groups of," arrangement,                                      sessions
Combinations     4.N.7,                                                                     equation, image, related
TE Pages 54 -    4.N.8,    2. Students will identify and learn multiplication combinations  numbers, odd, even, prime,
83               4.N.9,    not yet known fluently.                                          composite, doubling, halving,
4.N.10,                                                                    divisibilty, multiplication,
4.N.11    3. Students will use known multiplication combinations to        pattern, expression,
4.N.12    determine the products of more difficult combinations.           equations, skip count,
4.P.2,                                                                     division, strategies,
4.P.3,    4. Students will use determine whether one number is a factor or associative property,
4.P.4     multiple of another.                                             commutative property,
distributive property

Teacher's Notes: Standards 4.N.8 (in addition to the Investigation), 4.P.2, and 4.P.4 are addressed during Ten-Minute Math. During Inv. 2.1, for students who are ready
the associative property, 4.N.9, can be emphasized when discussing related numbers, for example, (2 x 2) x 12 = 48 or 2 x (2 x 12) = 48. During the discussion of
strategies for difficult multiplication combinations (p.62-63) and finding factors and multiples (p. 69-70), ask questions that make connections to related numbers, for
ex. odd, even, prime, composite, divisibility rules, doubling, and halving, and make connections to the inverse operation. The addition and subtraction in 4.N.12 is
practiced and reinforced during Student Activity Book practice, as well as 4.N.10. Standard 4.P.3 can be introduced during the missing factor p. 70 in Inv. 2.3. For
those students who are ready, distributive property can be introduced during the discussions about using related multiplication combinations to find more difficult
combinations. See page 16 in teacher's guide. During Inv. 2 when looking for patterns in the arrays, finding factor pairs, and classifying numbers, make connections
to the identity property to cover 4.N.9.

Focal Points (addressed): Number and Operations, Algebra (supported): Measurement
CIA                                                           Lowell Public Schools                                                  Working Document 3/5/2010
Unit 1: Factors, Multiples, and Arrays                                      Early October - End October                                                          Grade 4
14 Sessions

Invest Title &   MA Std    Content Objective                                                      Key Vocabulary                   Language     Assessment          Pacing
Page No.                                                                                                                           Objective

Inv #3:          4.N.1,    1. Students will use arrays to model multiplication                    factors, multiples/product,                   Portfolio: SAB p.   4- 1 hour
Finding          4.N.2,    situations.                                                            skip counting, doubling and                   33-34, Unit         sessions
Factors          4.N.7,                                                                           halving, patterns, square                     Assessment M55-
4.N.8,    2. Students will use arrays to find factors of 2-digit numbers.        numbers, landmark, estimate,                  M56
TE Pages 88-     4.N.9,                                                                           expressions, equations,
114              4.N.10,   3. Students will find the multiples of a number by skip                factor pairs, related numbers,
4.N.11,   counting.                                                              conjecture, strategies,
4.N.12,                                                                          divide/division, evenly,
4.P.3     4. Students will identify the factors of a given number.               arrays, representations,
commutative property,
5. Students will identify all the factors of 100.                      rectangle, number string,
Identity property
6. Students will use knowledge of the factors of 100 to find
factors of multiples of 100.

7.Students will use known multiplication combinations to find
related multiplication combinations for a given product (e.g., if 4
x 50 = 200, then 8 x 25 = 200).

8. Students will use representations to show that a factor of a
number is also a factor of its multiples (e.g., if 25 is a factor of
100, then 25 is also a factor of 300).

9. Students will use the known multiplication combinations to
determine the product of more difficult combinations.

Teacher Notes: The addition and subtraction in 4.N.12 is practiced and reinforced during Student Activity Book practice, as well as, 4.N.10. Standard 4.P.3 can be
introduced during the missing factor p. 70 in Inv. 2.3. For those students who are ready, distributive property can be introduced during the discussions about using
related multiplication combinations to find more difficult combinations. See p. 16 in teacher's guide. During Inv. 3 when identifying all the factors of a 100 and
multiples of 100 make connections to the Identity property.

Focal Points (addressed): Number and Operations, Algebra (supported): Measurement
CIA                                                           Lowell Public Schools                                                         Working Document 3/5/2010
Unit 1: Factors, Multiples, & Arrays
Standards #                                              Description
4.N.1      Exhibit an understanding of the base ten number system by reading, modeling, writing, and
interpreting whole numbers to at least 100,000; demonstrating an understanding of the values

4.N.2
Represent, order, and compare large numbers (to at least 100,000) using various forms, including

4.N.7
Recognize classes (in particular, odds, evens; factors or multiples of a given number; and squares) to
which a number may belong, and identify the numbers in those classes. Use these in the solution of

4.N.8
Select, use, and explain various meanings and models of multiplication and division of whole

4.N.9      Select, use, and explain the commutative, associative, and identity properties of operations on whole

4.N.10      Select and use appropriate operations (addition, subtraction, multiplication, and division) to

4.N.11      Know multiplication facts through 12 x 12 and related division facts. Use these facts to solve related
multiplication problems and compute related problems, e.g., 3 x 5 is related to 30 x 50, 300 x 5, and 30

4.N.12      Add and subtract (up to five-digit numbers) and multiply (up to three digits by two digits) accurately and

4.P.1      Create, describe, extend, and explain symbolic (geometric) and numeric patterns, including

4.P.2

4.P.3
4.P.4      Use pictures, models, tables, charts, graphs, words, number sentences, and mathematical
notations to interpret mathematical relationships.

CIA                                                        LPS                                                         3/5/2010
Unit 4: Size, Shape, Symmetry                                        November - Early December                                                                          Grade 4
20 Sessions

Inv. # and Name MA Std     Content Objective                                       Key Vocabulary                    Language Objective        Assessment         Pacing

Inv. 1: Linear   4.M.1,    1. Students will review the lengths of units of         area, volume, perimeter, linear   This is one example       Review p. 14 for   5 - 1 hour
Measurement      4.M.2,    measure (inches, feet, yards, cm, meters).              measurement, inch, foot, yard,    of a language             the assessment     sessions
TE p. 22 - 49    4.M.4,                                                            centimeter, meter, benchmark,     objective. Please         in this unit.
4.M.5,    2. Students will use U.S. Standard and Metric           millimeter, standard system,      consider your             Complete all
4.N.1,    units to accurately measure length.                     metric, kilometer, estimate,      students' needs           'Ongoing
4.N.2,                                                            perimeter, unit, length,          when writing your         Assessments,
4.N.3,    3. Students will estimate lengths based on              expression, equation, hand        language objectives:      Writing
4.N.4,                                                            span                              Students will use         Opportunities, &
common units (cm, in., ft. yd. m.).
4.N.5,                                                                                              content specific          Portfolio
4.N.6,    4. Students will determine when estimates and                                             vocabulary to orally      Opportunities, &
4.N.11,   exact measurements are needed.                                                            explain their thinking.   End-of-Unit
4.N.12,                                                                                                                       Assessments.
4.N.13,   5. Students will find perimeter using standard units.                                                               Portfolio: M13
4.N.14,
4.P.2     6. Students will recognize and explain possible
sources of measurement error.

7. Students will compare different paths that have
the same length.

Teacher's Notes: Broken Calculator, reinforces 4.P.2, 4.N.1, 4.N.2, 4.N.12, 4.N.14. SAB p. 3 & p. 5 reinforce 4.N.11 & 4.N.12. Students should practice doing
simple conversions, 4.M.2. When recording measurements have students record t. same measurement in different ways, e.g. 2 1/4 meters = 2 meters 25 cm =
2.25 meters and 3 1/2 feet = 3 feet 6 inches = 3.5 ft. These differentiations connect to 4.N.3, 4.N.4, 4.N.5, 4.N.6. When exploring Measuring Perimeter make
connections & comparisons b/w area & perimeter. Differentiate by having students find area of a desk, etc. Conference w/ your Computer Lab teacher about
creating a cheat sheet for the LogoPath Activity. See page 151-153. "Measuring Lengths" (SAB p. 7) can be differentiated by having students complete add'l
columns: Simple Unit Conversions, e.g. metric & standard; Record t. same measurement in different ways; Find Area. SAB p. 9 reinforces 4.N.13. Inv. 1.4
Mapping 100 Feet can also be Mapping 100 Yards. Discuss t. relationship. Inv. 1 (measurement) can be supported by Specialists (Art, Gym, Science or
Computer teachers). If this is the case, begin t. unit on Inv. 2. Also, Inv. 4 (symmetry) can be supported in Art.

CIA                                                                       Lowell Public Schools                                                   Working Document 3/5/2010
Unit 4: Size, Shape, Symmetry                                        November - Early December                                                                   Grade 4
20 Sessions

Inv. # and Name MA Std      Content Objective                                    Key Vocabulary                      Language Objective   Assessment       Pacing

Inv. 2:       4.G.1,        1. Students will define polygons as closed figures   polygon, line segment, end                               Portfolio: M21   5 - 1 hour
Polygons of   4.G.2,        with line segments as sides, and vertices.           point, vertex/vertices,                                                   sessions
Many Types TE 4.G.3,                                                             orientation, side, attribute,
p. 54 - 83    4.G.4,        2. Students will classify polygons by attributes     angle, trapezoid, equilateral
4.G.5,        including number of sides, length of sides, and      triangle, scalene triangle,
4.G.9,        size of angles.                                      isosceles triangle, acute
4.N.8,                                                             triangle, obtuse triangle,
4.N.11,       3. Students will combine polygons to make new        equiangular triangle,
4.N.12        polygons.                                            parallelogram, hexagon, prefix,
right angle, parallel, acute,
4. Students will recognize number of sides as a      obtuse, perpendicular,
descriptor of various polygons.                      intersecting, benchmark,
5. Students will develop vocabulary to describe      power polygon set, ,2-D,"side
length", regular polygon,
nonregular, concave, convex,
'gon', 'poly',
6. Students will understand the relationship b/w
squares and rectangles.

Teacher's Notes: Inv. 2.2, Making Polygons, have students record each # (at least 3 different ex.s of t. # of polygons) on separate pieces of paper. In explore &
summary, ask students to count & label t. sides, names, triangles, quadrilaterals, regular vs. nonregular, etc. on each sheet. Differentiate by having students
examine & identify which of their polygons are concave and convex. SAB p. 20, "Names for Polygons" can be summarized using a teacher created
transparency or chart paper posted for each prefix. Students can go around recording all t. terms they found. For those prefixes that are few, challenge
students to search in dictionaries or thesauruses, etc. Summarize the new terms and how they relate to the meaning of the prefix and root words. When
playing "Guess My Rule" with all the shape cards make connections to angle terms, for ex. if a student describes an angle > right angle= obtuse, or < 90
degrees = acute. Use benchmarks at this point to determine angle type. Also, encourage students to go beyond just naming a three-sided polygon as a
triangle, encourage them to use the specific triangle names. Do the same w/ quadrilaterals. SAB p. 25 reinforces 4.N.8, 4.N.11, 4.N.12.

CIA                                                                       Lowell Public Schools                                              Working Document 3/5/2010
Unit 4: Size, Shape, Symmetry                                         November - Early December                                                                       Grade 4
20 Sessions

Inv. # and Name MA Std      Content Objective                                      Key Vocabulary                       Language Objective   Assessment        Pacing

Inv. 3:           4.G.4,    1. Students will identify a right angle as 90 degrees. angle, degree, right angle,                               Portfolio: SAB p. 3 - 1 hour
Measuring         4.M.5,                                                           equilateral triangle, acute angle,                        41-43             sessions
Angles TE p. 88 - 4.N.1,    2. Students will measure acute angles by relating      obtuse angle, benchmark, turn
108               4.N.2,    them to 90 degrees.
4.N.3,
4.N.7,    3. Students will use known angles to find the
4.N.11,   measure of other angles.
4.N.12,
4.N.14,
4.P.1,
4.P.2

Teacher's Notes: Ten-Minute Math Broken Calculator reinforces standards 4.P.2, 4.N.1, 4.N.2, 4.N.12, 4.N.14. When making right angles and deciding how many
degrees make connections to halving, doubling, and fractional numbers, e.g. 45 x 2 = 90, 45 is half of 90, etc. (4.N.3). On Student Activity Book p. 39 -43 & 49,
encourage students to record polygon names and angle names. Encourage students to write an equation for Building Angles p. 42 - 43. During the discussion
of Smallest Angle, Biggest Angle make connections to benchmarks 180 (half a turn) and 360 ( full turn) degrees and that each unit in the turn is 1 degree.
Student Activity Sheets p. 47 & 48 reinforces 4.P.1, 4.N.7, 4.N.11, 4.N.12.

CIA                                                                         Lowell Public Schools                                               Working Document 3/5/2010
Unit 4: Size, Shape, Symmetry                                            November - Early December                                                                  Grade 4
20 Sessions

Inv. # and Name MA Std      Content Objective                                         Key Vocabulary                  Language Objective   Assessment        Pacing

Inv. 4: Finding 4.M.1,      1. Students will make designs with mirror               symmetry, symmetrical, mirror                          Portfolio: SAB p. 7 - 1 hour
Area TE p. 112 - 4.M.4,     symmetry.                                               symmetry, area, square unit,                           70-72, Unit       sessions
148              4.M.5,                                                             pentagon, irregular polygon,                           Assessment M27-
4.G.2,     2. Students will find the area of symmetrical designs. decompose, rectangle, triangle,                         28
4.G.8,                                                             hexagon, equilateral triangle,
4.G.9,     3. Students will understand that the larger the unit of trapezoid,
4.N.1,     area, the smaller the number of units needed to         parallelogram/rhombus, line of
4.N.2,     measure the area.                                       symmetry, standard unit,
4.N.3,                                                             nonstandard unit,
4.N.8,     4. Students will divide irregular polygons into two     congruent/congruence,
4.N.11,    shapes that have equal area.                            geoboard, array, rows, columns,
4.N.12,                                                            length, width
4.N.14,    5. Students will find the area of polygons by
4.P.2      decomposing shapes.

6. Students will find the area of polygons using
square units.

7. Students will find the area of rectangles.

8. Students will find the area of triangles in relation
to the area of rectangles.

Teacher's Notes: Student Activity Sheet p. 53, p. 65 reinforces 4.N.11 & 4.N.12. Make connections to fractions when comparing units of measure during Inv.
4.2, 4.4, 4.5 (4.N.3). Student Activity Sheet p. 60 reinforces 4.N.8, 4.N.11, 4.N.12. To review and reinforce the polygon names, encourage students to name the
crazy cake and geoboard figures appropriately, e.g. nonregular hexagon. Ten-Minute Math Broken Calculator reinforces standards 4.P.2, 4.N.1, 4.N.2, 4.N.12,
4.N.14.

CIA                                                                          Lowell Public Schools                                            Working Document 3/5/2010
Unit 4: Size, Shape, and Symmetry
Standards #                                              Description
Exhibit an understanding of the base ten number system by reading, modeling, writing, and
interpreting whole numbers to at least 100,000; demonstrating an understanding of the values of
4.N.1
Represent, order, and compare large numbers (to at least 100,000) using various forms, including
4.N.2
Demonstrate an understanding of fractions as parts of unit wholes, as parts of a collection, and
4.N.3

Select, use, and explain models to relate common fractions and mixed numbers (1/2, 1/3, 1/4, 1/5, 1/6,
4.N.4
Identify and generate equivalent forms of common decimals and fractions less than one whole
4.N.5
Exhibit an understanding of the base ten number system by reading, naming, and writing decimals
4.N.6
Recognize classes (in particular, odds, evens; factors or multiples of a given number; and squares) to
which a number may belong, and identify the numbers in those classes. Use these in the solution of
4.N.7

Select, use, and explain various meanings and models of multiplication and division of whole
4.N.8
Know multiplication facts through 12 x 12 and related division facts. Use these facts to solve related
multiplication problems and compute related problems, e.g., 3 x 5 is related to 30 x 50, 300 x 5, and 30 x
4.N.11
Add and subtract (up to five-digit numbers) and multiply (up to three digits by two digits) accurately and
4.N.12
Divide up to a three-digit whole number with a single-digit divisor (with or without remainders)
4.N.13

CIA                                                          LPS                                                             3/5/2010
Unit 4: Size, Shape, and Symmetry

Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for
4.N.14
Demonstrate an understanding of such attributes as length, area, weight, and volume, and select the
4.M.1
Carry out simple unit conversions within a system of measurement, e.g., hours to minutes, cents to
4.M.2
Estimate and find area and perimeter of a rectangle, triangle, or irregular shape using diagrams, models,
4.M.4
Identify and use appropriate metric and English units and tools (e.g., ruler, angle ruler, graduated
(e.g., number of sides, faces, corners, right angles, diagonals, and symmetry) of two- and three-
4.M.5    dimensional geometric shapes.
Compare and analyze attributes and other features (e.g., number of sides, faces, corners, right angles,
4.G.1

Describe, model, draw, compare, and classify two- and three-dimensional shapes, e.g., circles,
4.G.2
4.G.3
4.G.4
4.G.5
4.G.8
Predict and validate the results of partitioning, folding, and combining two- and three-dimensional
4.G.9    shapes.
Create, describe, extend, and explain symbolic (geometric) and numeric patterns, including multiplication
4.P.1

4.P.2

CIA                                                        LPS                                                        3/5/2010
Unit 3: Multple Towers and Division Stories                       Early December - Middle January                                                                     Grade 4
20 Sessions

Inv. # and Name        MA Std    Content Objective                            Key Vocabulary                  Language Objective          Assessment           Pacing
Resources:

Inv. 1: Breaking       4.N.2,    1. Students will develop strategies for     multiplication, array, rows,     This is one example of      Review p. 14 for      5 - 1 hour
Apart Multiplication   4.N.7,    multiplying that involve breaking apart     columns, factor, area,           a language objective.       the assessment in sessions
Problems TE p. 28 -    4.N.8,    numbers.                                    product/multiple, equation,      Please consider your        this unit. Complete
54                     4.N.9,                                                break apart/decompose,           students' needs when        all 'Ongoing
4,N.10,   2. Students will represent a multiplication expanded notation, partial       writing your language       Assessments,
4.N.11,   problem with pictures, diagrams, or models. products/subproduct,             objectives: Students        Writing
4.N.12,                                               associative property,            will use content specific   Opportunities, &
4.M.1,    3. Students will review multiplication      distributive property, length,   vocabulary to orally        Portfolio
4.M.2,    combinations to 12 x 12.                    width                            explain their thinking.     Opportunities, &
4.P.2                                                                                                              End-of-Unit
4. Students will use arrays to model                                                                     Assessments.
multiplication.                                                                                          Portfolio: SAB p. 12-
13, M43

Teacher's Notes: Quick Images (Ten-Minute Math) targets standards 4.N.8, 4.N.9, 4.N.11, 4.P.2. Read the Algebraic Connections on p. 16 of this teacher's
guide regarding the distributive property. Try to make connections that the product/multiple equals the area of the array (4.M.1 & 4.M.2). When creating
Small Arrays for Big Arrays, model how to record the dimensions on the outside of the array, especially if students are posting them on additional sheets of
paper or on the board, record the partial products inside each array. For example, 8 x 9 = (4 x 9) + (4 x 9), so the dimension recorded on the outside of the
array should read 4 + 4. Encourage students to use parentheses when writing their equations for Small Array/Big Array (Inv. 1.3). It clarifies order of
operation for them. Ten-Minute Math, Counting Around the Class, Inv. 1.4, reinforces 4.N.11, 4.P.1, 4.P.2.

Focal Points (addressed): Number and Operations, Algebra (connected): Geometry
CIA                                                                    LPS                                                                     Working Document3/5/2010
Unit 3: Multple Towers and Division Stories                   Early December - Middle January                                                                Grade 4
20 Sessions

Inv. # and Name       MA Std    Content Objective                             Key Vocabulary                 Language Objective   Assessment           Pacing
Resources:

Inv. 2: Division TE   4.N.2,    1. Students will develop strategies for       division, divide, remainder,                        Portfolio: SAB p. 25- 6 - 1 hour
p. 60 - 91            4.N.7,    multiplying that involve breaking apart       left over, divisor/ factor,                         26, M46               sessions
4.N.8,    numbers.                                      dimension,
4.N.10,   2. Students will solve division story         dividend/product/multiple,
4.N.11,   problems.                                     story problem, standard
4.N.12,   3. Students will use and interpret division   notation, represent,
4.N.13,   notation.                                     multiplication combinations,
4.M.1,    4. Students will solve division problems by   rows, sharing, grouping,
4.M.2,    making groups of the divisor.                 quotient, equation, inverse
4.P.1,    5. Students will use known multiplication
4.P.2,    combinations to solve division
4.P.3     problems.
6. Students will make sense of
remainders in terms of the problem
context.
7. Students will represent a division
problem with pictures, diagrams, or
models.
8. Students will create a story problem to
represent a division expression.

Teacher's Notes: Counting Around the Class (Ten-Minute Math) reinforces 4.N.11, 4.P.1, 4.P.2. As you go through the division story problems always try to
identify with students if it is a sharing or grouping division problem. Identify with students how divisor is synonymous with factor and vice versa, as well
as, dividend is to product is to multiple. You may use the nmemonic that the standard notation for division (i.e. the house), is an adaptation of the array for
that problem. Ten-Minute Math, Quick Images, targets standards 4.N.8, 4.N.9, 4.N.11, 4.P.2. Encourage students to use a circle or square to write their
equations for the "Missing Factors/Quotient." Continue to make the connection that the product/multiple equals the area of the array (4.M.1 & 4.M.2).

Focal Points (addressed): Number and Operations, Algebra (connected): Geometry
CIA                                                                    LPS                                                             Working Document3/5/2010
Unit 3: Multple Towers and Division Stories                       Early December - Middle January                                                              Grade 4
20 Sessions

Inv. # and Name      MA Std     Content Objective                                Key Vocabulary                Language Objective   Assessment           Pacing
Resources:

Inv. 3: Multiplying 4.N.1,      1. Students will review multiplication           multiple, factor, mulitply,                        Portfolio: SAB p. 37- 4 - 1 hour
Tens TE p. 94 - 120 4.N.7,      combinations to 12 x 12.                         sequence, visual                                   38, M48-49, SAB p. sessions
4.N.8,      2. Students will multiply multiples of 10.       representation,                                    42-43, M50
4.N.11,     3. Students will understand the effect of        mutliplication
4.N.12,     multiplying by a multiple of 10 (e.g.,           situations/story problems,
4.N.14,     describing the relationship b/w 3 x 4 & 3 x      multiple towers, landmarks
4.M.5,      40).                                             multiples, equation,
4.P.1,      4 Students will find multiples of 2-digit        expression
4.P.2       numbers.
5. Students will describe a sequence of
multiples in order to predict other multiples.
6. Students will represent a
multiplication problem with pictures,
diagrams, or models.
7. Students will compare a visual
representation of multiplication
situations.

Notes: Counting Around Class reinforces 4N11, 4P1, 4P2. Connect multiple towers to mathematical expressions, ex. ask How many multiples have we
counted? Multiple 27 is expressed 3 x 9 or 9 multiples of 3 have been counted. Inv. 3.1, when looking at list of multiples of 3 look for patterns, ex. the sum
of digits in the multiples of 3 is always 3, 6, or 9. Therefore, the number is a multiple of 3 or divisible by 3. Apply this same thinking to other numbers when
counting around the room & record multiples. Apply thinking of looking for patterns when creating multiple towers. Touch on measurement (4.M.5) when
students are creating their own multiple towers, measure evenly sized sections. SAB p. 39 & 40 reinforce 4N8 & 4N11. Inv. 3.2 the idea of "Adding a Zero" is
discussed. Use the term "ATTACHING A ZERO" instead, adding a zero will not change the sum. For ex. 4 x 4 = 24 & 40 x 6 = 240, but 24 + 0 is not 240. We
want students to see pattern of attaching zero! Encourage students to make drawings, use grid paper, etc. to show what happens to arrays as we multiply
by 10s. As students become fluent with 2-digit x 1-digit multiplication differentiate & solve 2 x 2 & 3 x 2 multiplication.

Focal Points (addressed): Number and Operations, Algebra (connected): Geometry
CIA                                                                    LPS                                                               Working Document3/5/2010
Unit 3: Multple Towers and Division Stories                    Early December - Middle January                                                                Grade 4
20 Sessions

Inv. # and Name      MA Std    Content Objective                              Key Vocabulary                  Language Objective   Assessment           Pacing
Resources:

Inv. 4: Strategies   4.N.1,    1. Students will determine the effect on the   double, halve, factor,                               Portfolio: SAB p. 57- 5 - 1 hour
for Multiplication   4.N.2,    product when the factor is doubled or          product/multiple, strategies,                        58, Unit              sessions
TE p. 124 - 150      4.N.8,    halved.                                        cluster, expanded notation,                          Assessment M51-
4.N.9,                                                   partial                                              52
4.N.10,   2. Students will represent a                   products/subproducts,
4.N.11,   multiplication or division problem with        equivalent expression,
4.N.12,   pictures, diagrams, or models.                 inverse operation, array, big
4.P.1,                                                   & small array, cluster
4.P.2,    3. Students will develop strategies for        problems, length, width,
multiplying that involve breaking apart        dimensions, area, equation,
numbers.                                       divide, distributive property

4. Students will use arrays to model
multiplication.

Teacher's Notes: Quick Images (Ten-Minute Math) targets standards 4.N.8, 4.N.9, 4.N.11, 4.P.2. For students, who are still struggling seeing the arrays, you
may try drawing in the hash marks along the length and width during, Inv.4.2. Record the products inside the array. Continue to make connections to area.
As students become fluent with 2-digit by 1-digit multiplication differentiate by challenging them to try 2 by 2 and 3 by 2 multiplication. Students who
master doubling and halving may try to experiment with tripling and quadrupling. Ten-Minute Math, Counting Around the Class, reinforces 4.N.11, 4.P.1,
4.P.2. Students who are flexible with using and explaining their strategies for multiplication and division should be exposed to the standard algorithms.
Students compare the standard algorithm for multiplication with partial products/distributive property. If students are not ready for the standard algorithms
they will have additional opportunities to master them during Unit 8: How Many Packages? How Many Groups?

Focal Points (addressed): Number and Operations, Algebra (connected): Geometry
CIA                                                                    LPS                                                              Working Document3/5/2010
Unit 3: Multiple Towers & Division Stories
Standards #                                                        Description
Exhibit an understanding of the base ten number system by reading, modeling, writing, and interpreting whole
numbers to at least 100,000; demonstrating an understanding of the values of the digits; and comparing and
4.N.1      ordering the numbers.
Represent, order, and compare large numbers (to at least 100,000) using various forms, including expanded notation,
4.N.2
Recognize classes (in particular, odds, evens; factors or multiples of a given number; and squares) to which a number
4.N.7
Select, use, and explain various meanings and models of multiplication and division of whole numbers.
4.N.8
Select, use, and explain the commutative, associative, and identity properties of operations on whole numbers in problem
4.N.9
Select and use appropriate operations (addition, subtraction, multiplication, and division) to solve problems,
4.N.10
Know multiplication facts through 12 x 12 and related division facts. Use these facts to solve related multiplication
4.N.11

4.N.12
Divide up to a three-digit whole number with a single-digit divisor (with or without remainders) accurately and
4.N.13
Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition and
4.N.14      subtraction (up to five-digit numbers), and multiplication (up to three digits by two digits).
Demonstrate an understanding of such attributes as length, area, weight, and volume, and select the appropriate type of
4.M.1
Carry out simple unit conversions within a system of measurement, e.g., hours to minutes, cents to dollars, yards to feet
4.M.2      or inches, etc.

Identify and use appropriate metric and English units and tools (e.g., ruler, angle ruler, graduated (e.g., number
4.M.5      of sides, faces, corners, right angles, diagonals, and symmetry) of two- and three-dimensional geometric shapes.
Create, describe, extend, and explain symbolic (geometric) and numeric patterns, including multiplication patterns like 3,
4.P.1

4.P.2
4.P.3

CIA                                                                LPS                                                                      3/5/2010
Unit 6: Fraction Cards Decimal Squares                                Mid January - Mid February                                                                     Grade 4
20 Sessions

Inv. # and MA Std Content Objective                                                  Key Vocabulary              Language Objectives         Assessment       Pacing
Name

Inv. 1:      4.N.1,    1. Students will find fractional parts of a rectangular       fraction, denominator,      This is one example of      Review p. 14 for 7 - 1 hour
Parts of     4.N.2,    area.                                                         numerator, rectangular      a language objective.       the assessment in sessions
Rectangle    4.N.3,    2. Students will interpret the meaning of the numerator       area, half, fourths,        Please consider your        this unit.
TE p. 20 -   4.N.4,    and denominator of a fraction.                                thirds, sixths, eighths,    students' needs when        Complete all
62           4.N.5,    3. Student will identify relationships b/w unit fractions     unit fraction, multiple,    writing your language       'Ongoing
4.N.7,    when one denominator is a multiple of another (e.g.           equivalent fraction,        objectives: Students        Assessments,
4.N.8,    halves and fourths, thirds and sixths).                       fraction notation, whole,   will use content specific   Writing
sum, estimate, square       vocabulary to orally        Opportunities, &
4.N.12,   4. Students will identify equivalent fractions.
unit, divide/division,      explain their thinking.     Portfolio
4.N.18,   5. Students will find fractional parts of a group (e.g. of
congruent, halving,                                     Opportunities, &
4.M.1,    objects, people, etc.)
doubling, groups of,                                    End-of-Unit
4.M.4,    6. Student will write, read, and apply fraction notation.
dimensions, length,                                     Assessments.
4.P.2     7. Students will compare the same fractional parts of
width, row, columns,                                    Portfolio: M12,
different sized wholes.                                       combinations, greater                                   SAB p. 19-21
8. Students will use representations to add fractions that    than >, less than <,
sum to one.                                                   equal to =
9. Students will add fractions with the same and related
denominators (e.g., halves, fourths, and eights; thirds and
sixths)
10. Students will estimate sums of fractions.

Teacher's Notes: Practicing Place Value reinforces 4N1, 4N2, 4N12, 4P2. While discussing 1/2s & 1/4s show notation w/ symbols & words, for ex. 1/4 of a 4
x 6 rectangle is 6 or 1/4 of 24 = 6 or 1/4 x 24 = 6 or 24/4 = 6, etc. This will reinforce 4N8. Continue this throughout unit. Be aware that the same area ≠
congruent figures. Use Math Note # 3 p. 27 in guide. Inv. 1.1 p. 29, record 1/4 + 3/4 = 4/4 = 1, continue this throughout the unit. SAB p. 2, p. 8, p. 17
reinforce 4.N.12. Standard 4N4 calls for mastery of the following fractions: 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12, 1 1/2. Find these fractional parts of wholes
when appropriate, for ex. when finding fractions of the 5 x 12 rectangle, 8ths would not work. Continue this throughout the unit. SAB p. 15, differentiate by
having students write an equivalent equation, for example 1/2 of 24 = 24/2 = 1/2 x 24. Be aware of common misconception when adding fractions with
unlike denominators (ex. 1/2 + 2/6 ¹ 3/8). Use models to clarify why this doesn't work. During Inv. 1.7, More or Less Than 1? encourage students to write
inequalities using the > & < signs when deciding, for ex. 1/2 + 3/4 > 1, etc., this will continue to extend 4P2.

Focal Points (addressed): Number and Operations
CIA                                                                       Lowell Public Schools                                                 Working Document 3/5/2010
Unit 6: Fraction Cards Decimal Squares                               Mid January - Mid February                                                              Grade 4
20 Sessions

Inv. # and MA Std Content Objective                                                Key Vocabulary             Language Objectives   Assessment        Pacing
Name

Inv. 2:      4.N.3,    1. Students will interpret the meaning of the numerator     numerator,                                       Portfolio: M23    6 - 1 hour
Ordering     4.N.4,    and denominator of the fraction.                            denominator, fraction,                                             sessions
Fractions    4.N.5,                                                                rectangular area,
TE p. 68 -   4.N.10,   2. Students will represent fractions greater than one.      equivalent fractions,
97           4.N.12,                                                               landmarks, number
4.N.18,   3. Students will find fractional parts of a rectangular     line, fraction cards,
4.M.1,    area.                                                       greater than, less than,
4.M.2,                                                                "strategies for
comparing fractions",
4.M.4,    4. Students will identify equivalent fractions.
greater than, less than,
4.P.2,
equal to, conjecture,
4.D.1,    5. Students will order fractions and justify their order
mixed number
4.D.2,    through reasoning about fraction equivalencies and
4.D.3     relationships.

6. Students will compare fractions to the landmarks 0,
1/2, 1, 2.
7. Students will represent fractions using a number line.

Teacher's Notes: Quick Survey reinforces 4D1, 4D2, & 4D3. In Inv. 2.1, Fractions > 1, encourage students to write inequalities using the > & < signs, for ex.
1/2 + 3/6 + 4/8 > 1, etc. This will cont. to extend 4P2. ALERT! SAB P. 26, FOR THIS UNIT IS PRINTED INCORRECTLY, IT SHOULD SHOW 1 WHOLE
RECTANGLE w 3/4 SHADED OF ANOTHER. GIVE STUDENTS A COPY OF t. TEACHER'S MASTER P. 19 OR HAVE STUDENTS MAKE t. CORRECTION IN
THEIR BOOKS. SAB p.28, p.34, reinforces 4N12, differentiation may be done by having students do simple conversions (4.M.2). When comparing fractions
for ordering, create a list of student strategies, for ex., "One piece missing" compare t. size of t. missing piece. The smaller t. piece, t. larger the fraction.
With "like denominators" compare numerators, etc. Begin this strategy chart w/ Missing Piece & it leave posted to add to in the future. When playing
"Capture Fractions", Students can differentiate by recording the expressions & equations using the > & < symbols, for ex. 2/3 < 1 3/4, etc. On SAB p. 38,
encourage students to use the > and < symbols as opposed to just circling. SAB p. 40 reinforces 4.N.10 & 4.N.12.

Focal Points (addressed): Number and Operations
CIA                                                                      Lowell Public Schools                                          Working Document 3/5/2010
Unit 6: Fraction Cards Decimal Squares                                  Mid January - Mid February                                                           Grade 4
20 Sessions

Inv. # and MA Std Content Objective                                                       Key Vocabulary      Language Objectives   Assessment        Pacing
Name

Inv. 3:       4.N.1,    1. Students will identify every day uses of fractions and decimal, fraction, 10ths,                         Portfolio: SAB p. 7 - 1 hour
Working       4.N.3,    decimals.                                                 100ths, area,                                     57-58, Unit       sessions
with          4.N.4,                                                              equivalence, estimate,                            Assessment M31
Decimals      4.N.5,    2. Students will read and write 10ths and 100ths.         sums, multiples, mixed
TE p. 104 -   4.N.6,                                                              number, decimal
137           4.N.7,    3. Students will represent 10ths and 100ths as part of an notation, equations
4.N.10,   area.
4.N.12,
4.N.14,   4. Students will order decimals and justify their order through
4.N.17,   reasoning about representations and meanings of the
4.N.18,   numbers.
4.P.2,
4.M.1,    5. Students will identify decimal and fraction
4.M.4     equivalence.

6. Students will use representations to combine 10ths
and 100ths.

7. Students will estimate sums of decimal numbers.

8. Students will add decimal numbers that are multiples of
0.1 and 0.25 (e.g., 2.3 + 3.25)

Teacher's Notes: Practicing Place Value (Ten-Minute Math) reinforces 4.N.1, 4.N.6, and 4.N.12. SAB p. 46 , p. 49 reinforce 4.N.12. During Inv. 3.2, "Decimal
Compare" differentiate by having students record the fraction notation for each decimal card. During Inv. 3.3, "Fill Two", differentiate by having students
record the equivalent fraction equations for example, 0.45 + 0.1 + 0.35 = 0.9, similarly, 45/100 + 1/10 + 35/100 = 9/10, this extends to 4.N.18. With SAB p. 55
& 59 encourage students to use the greater than, less than, or equal to symbols as opposed to circling the greater decimal. SAB p. 60 reinforces 4.N.12,
4.N.17, 4.N.10.

Focal Points (addressed): Number and Operations
CIA                                                                         Lowell Public Schools                                      Working Document 3/5/2010
Unit 6: Fraction Cards & Decimal Squares
Standards #                                                                     Description
Exhibit an understanding of the base ten number system by reading, modeling, writing, and interpreting whole numbers to at least 100,000;
4.N.1      demonstrating an understanding of the values of the digits; and comparing and ordering the numbers.

4.N.2      Represent, order, and compare large numbers (to at least 100,000) using various forms, including expanded notation, e.g., 853 = 8 x 100 + 5 x 10 + 3.
Demonstrate an understanding of fractions as parts of unit wholes, as parts of a collection, and as locations on the number line
4.N.3
Select, use, and explain models to relate common fractions and mixed numbers (1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12, and 11/2), find equivalent
4.N.4      fractions, mixed numbers, and decimals, and order fractions.

4.N.5       Identify and generate equivalent forms of common decimals and fractions less than one whole (halves, quarters, fifths, and tenths).
4.N.6      Exhibit an understanding of the base ten number system by reading, naming, and writing decimals between 0 and 1 up to the hundredths.
Recognize classes (in particular, odds, evens; factors or multiples of a given number; and squares) to which a number may belong, and identify the
4.N.7      numbers in those classes. Use these in the solution of problems.
Select, use, and explain various meanings and models of multiplication and division of whole numbers. Understand and use the inverse
4.N.8      relationship between the two operations.

4.N10       Select and use appropriate operations (addition, subtraction, multiplication, and division) to solve problems, including those involving money.
4.N.12      Add and subtract (up to five-digit numbers) and multiply (up to three digits by two digits) accurately and efficiently.
Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition and subtraction (up to five-digit
4.N.14      numbers), and multiplication (up to three digits by two digits).

Select and use a variety of strategies (e.g., front-end, rounding, and regrouping) to estimate quantities, measures, and the results of whole-
4.N.17      number computations up to three-digit whole numbers and amounts of money to \$1000, and to judge the reasonablenes
4.N.18      Use concrete objects and visual models to add and subtract common fractions.
Demonstrate an understanding of such attributes as length, area, weight, and volume, and select the appropriate type of unit for measuring each
4.M.1       attribute.
4.M.2       Carry out simple unit conversions within a system of measurement, e.g., hours to minutes, cents to dollars, yards to feet or inches, etc.
4.M.4       Estimate and find area and perimeter of a rectangle, triangle, or irregular shape using diagrams, models, and grids or by measuring.
4.D.1       Collect and organize data using observations, measurements, surveys, or experiments, and identify appropriate ways to display the data.
4.D.2       Match a representation of a data set such as lists, tables, or graphs (including circle graphs) with the actual set of data.
Construct, draw conclusions, and make predictions from various representations of data sets, including tables, bar graphs, pictographs, line
4.D.3      graphs, line plots, and tallies.

4.P.2      sentences that use =, <, >).

CIA                                                                         LPS                                                                                    3/5/2010
Unit 2: Describing The Shape of the Data                             End February - Middle March                                                                           Grade 4
17 Sessions

Invest Title & MA Std   Content Objective                                                             Key Vocabulary            Language               Assessment    Pacing
Page No.                                                                                                                        Objective

Invest #1    4.D.1,     1. Students will organize ordered, numerical data to describe a data          data, data set,           This is one            Review p. 14 5 - 1 hour
Landmarks    4.D.2,     set.                                                                          numerical data, list,     example of a           for the        sessions
in the Data  4.D.3,     2. Students will desribe the shape of the data set; where the data are        spread,                   language               assessment in
4.M.2,     spread out or concentrated, what the highest and lowest values are,           repesentation, bar        objective. Please      this unit.
TE pages 22- 4.M.5,     what the range is, and what the outliers are.                                 graph, line plot,         consider your          Complete all
50-          4.N.2      3. Students will describe what values are typical or atypical in a data       range, median,            students' needs        'Ongoing
4.N.11     set.                                                                          mode, outlier, typical,   when writing           Assessments,
4.N.12     4. Students will use a line plot to represent ordered numerical data.         atypical,                 your language          Writing
4.P.2                                                                                    interpret/analyze,                               Opportunities,
5. Students will determine the range of a data set.                                                     objectives:
order, organize,                                 & Portfolio
6. Students will use U.S. standard units to measure lengths longer than                                 Students will use
shape of data,                                   Opportunities,
the measuring tool.                                                                                     content specific
standard unit, length,                           & End-of-Unit
7. Students will record and keep track of data.                                                         vocabulary to
expression, equation,                            Assessments
8. Students will represent 2 sets of data in order to compare them.                                     orally explain their
landmark, frequency                              Portfolio: SAB
9. Students will consider how well a data representation communicates to an                             thinking.
p. 7, M7-8
audience.
10. Students will describe and interpret data that compare two groups.
11. Students will develop arguements based on data.
12. Students will find the median of a data set.
13. Students will use medians to compare groups.
14. Students will consider what information a median does and does
not provide.

Notes: 10-Minute Math covers 4P2 & 4N12 with writing, solving equations, addition, and subtraction. When having students describe the shape of date they should
use correct terminology, as opposed to referring to "bumps, clumps, & holes", mathematicians prefer "mean, median, mode, outlier, range, frequency, and typical."
SAB p. 1 targets 4N11. Prior to start of Inv. 1.2, consider a mini-lesson on proper use of measuring tools. Another form of differentiation may be to use
measurements to the nearest whole in. or 1/2 in. through the height data collection. Inv. 1.2 & 1.3 students who are ready may convert inches to feet (4M2) as a form
of differentiation. In session 1.3, please consider the sensitivity around the HW "How Many Cavities?" You may substitute w/ a more appropriate question since
you will need data for the assessment in Inv. 1.5 or use the one provided in the M pages. When comparing heights, Inv. 1.4, have students represent data in more
than one way as a form of differentiation. Daily practice p 8 & 9 in SAB cover 4N11 & 4P2. SAB p. 29 reinforces 4N11, 4N12.

Focal Points (supported): Number and Operations, Algebra (connected): Data Analysis
Lowell Public Schools                                                                Working Document [Date]
Unit 2: Describing The Shape of the Data                            End February - Middle March                                                                Grade 4
17 Sessions

Invest Title & MA Std   Content Objective                                                             Key Vocabulary          Language    Assessment     Pacing
Page No.                                                                                                                      Objective

Invest #2    4.D.1,     1. Students will develop and revise a survey question.                        survey, numerical                   Portfolio: SAB 7 - 1 hour
Using Data   4.D.2,     2. Students will record and keep track of a set of data.                      data, , categorical                 p.34-35        sessions
To Compare   4.D.3      3. Students will represent two sets of data in order to compare them.         data, line plot, bar
4.P.2,     4. Students will consider how well a data representation communicates to an   graph, conclusion,
TE pages 56- 4.N.11,    audience.                                                                     value, data sets,
90           4.N.12,    5. Students will compare two sets of data by using the shape and              represent, shape,
6. Students will draw conclusions based on data.                              range, outliers,
7. Students will develop arguments based on data.                             expression, equation,
8. Students will use a line plot to represent ordered numerical data.         frequency, typical
9. Students will use medians to compare groups.
10. Students will consider what information a median does or does not
provide.
11. Students will describe the shape of a set of data; where the data are
spread out or concentrated, what the highest and lowest values are,
what the range is, and what the outliers are.

Teacher's Notes: Ten Minute Math covers 4.P.2 and 4.N.12 with writing, solving equations, addition, and subtraction. When students are playing Broken calculator,
make connections to expanded notation (4.N.2). Students may represent their data in more than one way as a form of differentiation and may write several
statements describing and comparing their data in detail. Teachers need to be attentive to the various typs of survey questions student create for their assessment,
e.g. students may develop numerical questions around measurement. Be aware of the measurement standards that survey question connects with. SAB p. 18, 19
supports 4.N.11. Page 22 of the daily practice in the SAB reinforces 4.N.13; p. 23, 24, 36 reinforces 4.N.11.

Focal Points (supported): Number and Operations, Algebra (connected): Data Analysis
Lowell Public Schools                                                   Working Document [Date]
Unit 2: Describing The Shape of the Data                                 End February - Middle March                                                                  Grade 4
17 Sessions

Invest Title & MA Std   Content Objective                                                                   Key Vocabulary           Language    Assessment     Pacing
Page No.                                                                                                                             Objective

Invest #3    4.D.1,     1. Students will associate the word probability with how likely will is to occur.   probability, likely,                 Portfolio: SAB 5 - 1 hour
Finding &    4.D.2,                                                                                         likelihood, unlikely,                p.49-50,Unit   sessions
Comparing    4.D.3,     2. Students will arrange events along a line representing the range of              event, certain,                      Assessment
Probabilities4.D.4,     certain to impossible.                                                              definate, impossible,                M18-21
4.D.6,                                                                                         possible, range,
TE pages 94- 4.N.2,     3. Students will use numbers from 0-1 as measures of probability.                   trials, experimental
119          4.N.3,                                                                                         probability, shape,
4.N.7,     4. Students will associate verbal descriptions of probability with                  spread, data, predict,
4.N.11,    numeric descriptions.                                                               maybe, chance,
4.N.12,                                                                                        often, not often
4.P.2      5. Students will compare the expected probability of an event with the
actual results of repeated trials of that event.

6. Students will compare two sets of data by using the shape and

Teacher's Notes: Ten Minute Math covers 4.P.2 and 4.N.12 with writing, solving equations, addition, and subtraction. When students are playing Broken Calculator,
make connections to expanded notation (4.N.2). SAB. pgs. 39, 46, 48, 51 reinforces 4.N.11, p. 48 also reinforces 4.N.7.

Focal Points (supported): Number and Operations, Algebra (connected): Data Analysis
Lowell Public Schools                                                          Working Document [Date]
Unit 2: Describing the Shape of the Data
Standards #                                                         Description

Represent, order, and compare large numbers (to at least 100,000) using various forms, including expanded notation, e.g.,
4.N.2
Demonstrate an understanding of fractions as parts of unit wholes, as parts of a collection, and as locations on
4.N.3      the number line.
Recognize classes (in particular, odds, evens; factors or multiples of a given number; and squares) to which a number may
4.N.7
Know multiplication facts through 12 x 12 and related division facts. Use these facts to solve related multiplication problems
4.N.11

4.N.12
Divide up to a three-digit whole number with a single-digit divisor (with or without remainders) accurately and
4.N.13      efficiently. Interpret any remainders.
Carry out simple unit conversions within a system of measurement, e.g., hours to minutes, cents to dollars, yards to feet or
4.M.2      inches, etc

Identify and use appropriate metric and English units and tools (e.g., ruler, angle ruler, graduated (e.g., number of
4.M.5      sides, faces, corners, right angles, diagonals, and symmetry) of two- and three-dimensional geometric shapes.
Collect and organize data using observations, measurements, surveys, or experiments, and identify appropriate ways to
4.D.1      display the data

4.D.2
Construct, draw conclusions, and make predictions from various representations of data sets, including tables,
4.D.3
Represent the possible outcomes for a simple probability situation, e.g., the probability of drawing a red marble from a bag
4.D.4      containing three red marbles and four green marbles.
Classify outcomes as certain, likely, unlikely, or impossible by designing and conducting experiments using concrete objects
4.D.6      such as counters, number cubes, spinners, or coins.

4.P.2      inequalities (mathematical sentences that use =, <, >).

CIA                                                                           LPS                                                                    3/5/2010
Unit 9: Penny Jars Plant Growth                                              Middle March - Middle April                                                                       Grade 4
15 Sessions

Inv. # and MA Std            Content Objective                                                      Key Vocabulary         Language            Assessment       Pacing
Name                                                                                                                       Objectives

Inv. 1:      4.D.1, 4.D.2,   1. Students will interpret the points and shapes of a graph in terms   axis/axes,             This is one         Review p. 16     2 - 1 hour
Looking at   4.D.3,          of the situation the graph represents.                                 horizontal, vertical   example of a        for the          sessions
Graphs       4.N.7,                                                                                 graph, points,         language            assessment in
TE p. 24 -   4.N.11,         2. Students will find the difference b/w two values on a line graph.   shape, difference,     objective.          this unit.
39           4.N.12,                                                                                value, quantity,       Please consider     Complete all
4.N14,          3. Students will discriminate b/w features of a graph that represent   highest, lowest,       your students'      'Ongoing
4.N.17,         quantity and those that represent changes in quantity.                 range, constant,       needs when          Assessments,
interpret, zero        language            Opportunities,
speed, slanted,                            Portfolio
objectives:
increase,                                  Opportunities, &
Students will use
content specific
vocabulary to
slope, steep slope,
orally explain
jagged
their thinking.

Teacher's Notes: PRIOR TO STARTING INV. 1, PLANT LIMA BEANS YOU WILL NEED FOR MEASURING & COLLECTING DATA FOR USE DURING INV. 3
(SEE TEACHER'S GUIDE P. 13-15). Ten-Minute Math, Quick Survey, reinforces 4.D1, 4.D.2, & 4.D.3. During Inv. 1.1, to extend to standard 4.D.2, have
students create a table for each city to show the temperatures for each day. Student Activity Sheet p. 11, reinforces 4.N.7 & 4.N.11. To reduce unit, review all
sessions (especially as connected to the Unit Assessment). Some can be combined or condensed.

Focal Points (supported): Numbers and Operations, Algebra
CIA                                                                              Lowell Public Schools                                                        Working Document 3/5/2010
Unit 9: Penny Jars Plant Growth                                                  Middle March - Middle April                                                              Grade 4
15 Sessions

Inv. # and MA Std           Content Objective                                                               Key Vocabulary   Language     Assessment        Pacing
Name                                                                                                                         Objectives

Inv. 2:     4.D.1, 4.D.2,   1. Students will identify pts. in a graph w/ corresponding values in a                                        Portfolio: SAB p. 8 - 1 hour
Penny       4.D.3, 4.P.2,   table & interpret t. numerical info.                                                                          21-22, p.35-37, sessions
Jars and    4.P.4, 4.P.6,   2. Students will plot points on a coordinate grid to represent a                                              M28-30
Towers      4.N.1, 4.N.2,   situation in which one quantity is changing in relation to another.
TE p 46 -   4.N.7,          3. Students will compare situations by describing the differences in
102         4.N.10,         their graphs.
4.N.13,         4. Students will describe the relative steepness of graphs or parts of
4.N.14,         graphs in terms of different rates of change.
4.N.16,         5. Students will use tables to represent the relationship b/w two
4.N.17          quantities in a situation of constant change & describe their
relationship.
6. Students will interpret #s in a table in terms of t. situation they
represent.
7. Students will find t. value of one quantity in a situation of
constant change, given t. value of t. other.
8. Students will represent a situation of constant change.
9. Students will write an arithmetic expression using symbolic letter
notation to represent and find t. value of one quantity in terms of t. other.
10. Students will make rules that relate one variable to another.

Notes: Closest Estimate (Ten Minute Math) addresses 4N1, 4N2, 4N11, 4N16, 4N17. Inv. 2.1, Share Representations, connects to order of operations when
writing expressions, ex. 4 + (6 x 9). SAB p. 14, 29, 49, 55 reinforces 4N10, 4N14. Inv. 2.2, Start with 2 & Add 3, have students express rules in oral, written, &
symbolic expression. Continue this throughout the unit. When creating tables consider the zero/starting round. It will come up in Inv.2.4. SAB p. 19, 48
reinforces 4N13. 10-Min. Math Quick Survey reinforces 4D1, 4D2, 4D3. Inv. 2.3, Round 20, record the most efficient arithmetic expression in the calculation
column, refer to it as an arithmetic expression, connect to order of operations. In this unit introduce the term method, use synonmously with strategy. SAB
p. 24 reinforces 4N14. During Inv. 2.4, Matching Tables & Graphs, emphasize the reasonableness of connecting the pts, refer to T.N. #1 on pg. 70. Inv.2.5,
Single & Double Tower, consider extending the rule to floors 20 & 100 .
For students who struggle with keeping track of the number of windows, a suggestion is to insert an extra column for writing the arithmetic expressions,
just like the calculation column in Round 20. SAB p.38 reinforces 4N17, 4N14.

Focal Points (supported): Numbers and Operations, Algebra
CIA                                                                                  Lowell Public Schools                                               Working Document 3/5/2010
Unit 9: Penny Jars Plant Growth                                              Middle March - Middle April                                                                  Grade 4
15 Sessions

Inv. # and MA Std            Content Objective                                                        Key Vocabulary         Language     Assessment        Pacing
Name                                                                                                                         Objectives

Inv. 3:      4.P.4,          1. Students will identify points in a graph with corresponding values    points, graph,                      Portfolio: SAB p. 5 - 1 hour
Collecting   4.P.6,          in a table and interpret the numerical information in terms of the       corresponding                       60, Unit          sessions
and          4.D.1, 4.D.2,   situation the graph represents.                                          value, table,                       Assessment
Analyzing    4.D.3,          2. Students will plot points on a coordinate grid to represent a         numerical                           M37-39
Measurem     4.M.5,          situation in which one quantity is changing in relation to another.      information, plot,
ents TE      4.M.2,          3. Students will describe the relative steepness of graphs or parts of   points, coordinate
106 - 131    4.N.1,          graphs in terms of different rates of change.                            graph, quantity,
4.N.2,          4. Students will compare situations by describing the differences in     change, relative
4.N.8,          their graphs.                                                            steepness, rate of
4.N.10,                                                                                  change, difference,
5. Students will interpret the points and shapes of a graph in terms
4.N.11,                                                                                  relationship,
of the situation the graph represents
4.N.12,                                                                                  constant change,
6. Students will compare tables, graphs, and situations of constant
4.N.13,                                                                                  arithmetic
change with those of non-constant change.
expression,
4.N.14,         7. Students will use tables to represent the relationship b/w two        plotting, x-axis, y-
4.N.15,         quantities in a situation of constant change.                            axis, horizontal
4.N.16,         8. Students will write an arithmetic expression for finding the value    axis, vertical axis.
4.N.17          of one quantity in terms of the other in a situation of constant         relative steepness,
9. Students will measure in centimeters.                                 increase

Teacher's Notes: Ten-Minute Math, Quick Survey, reinforces 4.D1, 4.D.2, & 4.D.3. Ten-Minute Math, Closest Estimate, reinforces 4.N.1, 4.N.2, 4.N.11, 4.N.16,
4.N.17. Prior to Inv. 3.1, review metric system of measurement, make connection to the base-ten number system, fraction and decimal notation, and
strategies for measuring accurately. Differentiate by having students record heights in mm and cm, when appropriate, to reinforce 4.M.2 and to record the
measurements to the nearest half and whole. Differentiate by having students represent plant growth data using different representations, e.g. bar graphs.
Student Activity Sheets p. 61-61, 65 reinforce 4.N.10, 4.N.11, 4.N.12,4.N.14. Student Activity Sheets p.64 & 72 reinforce 4.N.8, 4.N.11, 4.N.12, 4.N.13, 4.N.14,
4.N.15. Add a calculation/expression column to the sheet Removing Pennies from a Penny Jar for students who struggle seeing the change everytime.

Focal Points (supported): Numbers and Operations, Algebra
CIA                                                                              Lowell Public Schools                                                   Working Document 3/5/2010
Unit 9: Penny Jars & Plant Growth
Standards #                                                                       Description
Exhibit an understanding of the base ten number system by reading, modeling, writing, and interpreting whole numbers to at least 100,000;
4.N.1    demonstrating an understanding of the values of the digits; and comparing and ordering the numbers.
Represent, order, and compare large numbers (to at least 100,000) using various forms, including expanded notation, e.g., 853 = 8 x 100 + 5 x 10 + 3.
4.N.2
Recognize classes (in particular, odds, evens; factors or multiples of a given number; and squares) to which a number may belong, and identify the
4.N.7    numbers in those classes. Use these in the solution of problems.
Select, use, and explain various meanings and models of multiplication and division of whole numbers. Understand and use the inverse
4.N.8    relationship between the two operations.
Select and use appropriate operations (addition, subtraction, multiplication, and division) to solve problems, including those involving
4.N.10    money.
Know multiplication facts through 12 x 12 and related division facts. Use these facts to solve related multiplication problems and compute related
4.N.11    problems, e.g., 3 x 5 is related to 30 x 50, 300 x 5, and 30 x 500.
4.N.12    Add and subtract (up to five-digit numbers) and multiply (up to three digits by two digits) accurately and efficiently.
Divide up to a three-digit whole number with a single-digit divisor (with or without remainders) accurately and efficiently. Interpret any
4.N.13    remainders.
Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition and subtraction (up to five-digit
4.N.14    numbers), and multiplication (up to three digits by two digits).
Demonstrate in the classroom an understanding of and the ability to use the conventional algorithm for division of up to a three-digit whole number
4.N.15    with a single-digit divisor (with or without remainders).
4.N.16   Round whole numbers through 100,000 to the nearest 10, 100, 1000, 10,000, and 100,000.
Select and use a variety of strategies (e.g., front-end, rounding, and regrouping) to estimate quantities, measures, and the results of whole-
4.N.17    number computations up to three-digit whole numbers and amounts of money to \$1000, and to judge the reasonablenes
4.D.1    Collect and organize data using observations, measurements, surveys, or experiments, and identify appropriate ways to display the data.
4.D.2    Match a representation of a data set such as lists, tables, or graphs (including circle graphs) with the actual set of data.
Construct, draw conclusions, and make predictions from various representations of data sets, including tables, bar graphs, pictographs, line
4.D.3     graphs, line plots, and tallies.
4.M.2     Carry out simple unit conversions within a system of measurement, e.g., hours to minutes, cents to dollars, yards to feet or inches, etc.
Identify and use appropriate metric and English units and tools (e.g., ruler, angle ruler, graduated (e.g., number of sides, faces, corners,
4.M.5     right angles, diagonals, and symmetry) of two- and three-dimensional geometric shapes.

4.P.2    (mathematical sentences that use =, <, >).
Use pictures, models, tables, charts, graphs, words, number sentences, and mathematical notations to interpret mathematical relationships.
4.P.4
4.P.6    Determine how change in one variable relates to a change in a second variable, e.g., input-output tables.

CIA                                                                            LPS                                                                           3/5/2010
Unit 8: How Many Packages? How Many Groups?                                   End April - Middle May                                                                          Grade 4
16 Sessions

Inv. # and       MA Std    Content Objective                                                 Key Vocabulary               Language objectives Assessment             Pacing
Name

Inv. 1:          4.N.1,                                                                estimation, multiplication,
1. Students will estimate solutions to 2-digit multiplication                                  This is one example       Review p. 14 for 5 - 1 hour
Multiplication   4.N.2     problems.                                                   landmark, multiple, factor,        of a language             the assessment in sessions
with 2-Digit     4.N.7,                                                                product/solution,                  objective. Please         this unit.
Numbers          4.N.8,    2. Students will multiply by multiples of 10.               subproduct/partial product,        consider your             Complete all
TE p. 24 - 58    4.N.11,                                                               expression, cluster, division,     students' needs           'Ongoing
4.N.12,   3. Students will solve 2-digit multiplication problems by   array, 'groups of', estimate,      when writing your         Assessments,
4.N.14,   breaking a problem into smaller parts and combining the sub over-estimate, under-estimate,     language objectives:      Writing
4.N.16,   products.                                                   magnitude,strategy, break          Students will use         Opportunities, &
4.N.17                                                                apart/decompose, expanded          content specific          Portfolio
notation, distributive property,   vocabulary to orally      Opportunities, &
4. Students will use story problems represented by a
chunk, arithmetic, big array,      explain their thinking.   End-of-Unit
multiplication expression to keep track of parts of the
smaller array                                                Assessments.
problem.
Portfolio: SAB p.
5-6, 14-15
5. Students will represent a multiplication or division problem
with pictures, or diagrams, including arrays, and pictures of
groups.

Teacher's Notes: Ten-Minute Math, Closest Estimate, reinforces 4.N.1, 4.N.2, 4.N.11, 4.N.16, 4.N.17. StandardS 4.N.12 and 4.N.14 expect students to master 3 by 2 digit
multiplication. Differentiate 'Closest Estimate' to include these types of problems. SAB p. 2 and 16 reinforce 4.N.12 & 4.N.14. When solving 2 by 2 digit multiplication make
connections to the distributive property, order of operations, and array models to help studenSt make sense of the partial products. When students are drawing arrays for
their multiplication problems encourage those who are struggling to use their grid paper as well as drawing the abstract representations. We want students to be able to
draw the abstract models without all the square units. SAB p. 7 reinforces 4.N.8, 4.N.11, 4.N.12, 4.N.14, 4.N.17. SAB p. 10 reinforces 4.N.12 & 4.N.14. Ten-Minute Math,
Counting Around the Class, reinforces standards 4.N.11, 4.P.1, 4.P.2. To reduce unit, review all sessions (especially as connected to the Unit Assessment). Some can be
combined or condensed.

Focal Points (addressed): Number and Operations, Algebra
CIA                                                                            Lowell Public Schools                                                    Working Document 3/5/2010
Unit 8: How Many Packages? How Many Groups?                                 End April - Middle May                                                                            Grade 4
16 Sessions

Inv. # and       MA Std    Content Objective                                                 Key Vocabulary                     Language objectives Assessment       Pacing
Name

Inv. 2:          4.N.1,    1. Students will solve 2-digit multiplication problems by         multiplication, landmark,                             Portfolio: SAB p. 5 - 1 hour
Strategies for   4.N.2,    changing one factor to create an easier problem.                  multiple, factor,                                     71-72,M19         sessions
Multiplication   4.N.7,    2. Students will use a story problem represented by a             product/solution,
TE p. 62 - 84    4.N.8,    multiplication expression to keep track of parts of the           subproduct/partial product,
4.N.9,    problem.                                                          expression, cluster, division,
4.N.10,   3. Students will represent a multiplication or division problem   array, 'groups of', subtract,
4.N.11,   with pictures or diagrams, including arrays and pictures of       rounding, distributive property,
4.N.12,   groups.                                                           related number/problems,
4.N.14,                                                                     cluster
4. Students will solve 2-digit multiplication problems by
4.N.16,   breaking a problem into smaller parts and combining the sub
4.N.17,   products.
4.P.1,    5. Students will multiply multiples of 10.
4.P.2,
4.P.4

Teacher's Notes: Ten-Minute Math, Closest Estimate, reinforces 4.N.1, 4.N.2, 4.N.11, 4.N.16, 4.N.17. SAB p. 21 reinforces 4.P.4, 4.N.10, 4.N.11. SAB p. 28 reinforces 4.N.8,
4.N.9, 4.N.12, & 4.N.14. When solving 2 by 2 digit multiplication make connections to the distributive property, order of operations, and array models to help students make
sense of the partial products. When students are drawing arrays for their multiplication problems encourage those who are struggling to use use their grid paper as well as
drawing the abstract representations. We want students to be able to draw the abstract models without all the square units. SAB p. 29 reinforces 4.N.7, 4.N.11, 4.N.12, &
4.N.14. SAB p. 30 reinforces 4.N.8 & 4.N.11. Ten-Minute Math, Counting Around the Class, reinforces 4.N.11, 4.P.1, 4.P.2. SAB p. 33 reinforces 4.N.11 & 4.N.12.

Focal Points (addressed): Number and Operations, Algebra
CIA                                                                          Lowell Public Schools                                                     Working Document 3/5/2010
Unit 8: How Many Packages? How Many Groups?                                 End April - Middle May                                                                        Grade 4
16 Sessions

Inv. # and    MA Std      Content Objective                                                 Key Vocabulary              Language objectives Assessment           Pacing
Name

Inv. 3:       4.N.1,      1. Students will solve 2-digit multiplication problems by   divisor/factor, quotient,                                Portfolio: SAB p. 6 - 1 hour
Solving       4.N.2,      breaking a problem into smaller parts and combining the sub dividend, division,                                      44-45, Unit       sessions
Division      4.N.7,      products.                                                   multiplication, 'groups of',                             Assessment M21-
Problems TE   4.N.8,                                                                  remainder, left over,                                    22
p. 88 - 111   4.N.9,      2. Students will solve 2-digit multiplication problems by   expression, multiples,
4.N.11,     changing one factor to create an easier problem.            estimate, landmark number,
4.N.12,                                                                 extra, strategies, doubling,
4.N.13,     3. Students will solve division problems by breaking the    halving, rounding, standard
4.N.14,                                                                 notation, inverse
problem into parts.
4.N.15,
4.N.16,     4. Students will use multiples of 10 to solve division
4.N.17,     problems.
4.P.1,
4.P.2       5. Students will use relationships b/w multiplication and
division to solve division problems.

6. Students will represent a multiplication or division problem
with pictures or diagrams including arrays and pictures of
groups.

7. Students will use a story problem represented by a
multiplication or division expression to keep track of parts of
the problem.
Teacher's Notes: Ten-Minute Math, Closest Estimate, reinforces 4.N.1, 4.N.2, 4.N.11, 4.N.16, 4.N.17. SAB p. 39 and 49 reinforce 4.N.12, 4.N.16, & 4.N.17. SAB p. 46 reinforces
4.N.1 & 4.N.12. Differentiate by having students record remainders using fraction notation. SAB p. 47 reinforces 4.N.7, 4.N.11, & 4.N.12. SAB p. 48 reinforces 4.N.8, 4.N.13,
4.N.12, & 4.N.15. Ten-Minute Math, Counting Around the Class, reinforces 4.N.11, 4.P.1, 4.P.2. SAB p. 33 reinforces 4.N.11 & 4.N.12. SAB p. 53 reinforces 4.N.12, 4.N.14,
&4.P.2. Investigation 3 goes beyond our standard when teaching division of 3 digit by 2 digit. 4N13 calls for mastery fro 3 digit by 1 digit. Please be flexible when teaching
this investigation and emphasize the 3 by 1 digit division and use the 3 by 2 digit division for differentiation.

Focal Points (addressed): Number and Operations, Algebra
CIA                                                                          Lowell Public Schools                                                 Working Document 3/5/2010
Unit 8: How Many Packages? How Many Groups?
Standards #                                                                  Description
Exhibit an understanding of the base ten number system by reading, modeling, writing, and interpreting whole numbers to at
4.N.1      least 100,000; demonstrating an understanding of the values of the digits; and comparing and ordering the numbers.
Represent, order, and compare large numbers (to at least 100,000) using various forms, including expanded notation, e.g., 853 = 8 x
4.N.2      100 + 5 x 10 + 3.
Recognize classes (in particular, odds, evens; factors or multiples of a given number; and squares) to which a number may belong,
4.N.7      and identify the numbers in those classes. Use these in the solution of problems.
Select, use, and explain various meanings and models of multiplication and division of whole numbers. Understand and use
4.N.8      the inverse relationship between the two operations.
Select, use, and explain the commutative, associative, and identity properties of operations on whole numbers in problem situations,
4.N.9      e.g., 37 x 46 = 46 x 37, (5 x 7) x 2 =    5 x (7 x 2).
Select and use appropriate operations (addition, subtraction, multiplication, and division) to solve problems, including those
4.N.10      involving money.
Know multiplication facts through 12 x 12 and related division facts. Use these facts to solve related multiplication problems and
4.N.11      compute related problems, e.g., 3 x 5 is related to 30 x 50, 300 x 5, and 30 x 500.
Add and subtract (up to five-digit numbers) and multiply (up to three digits by two digits) accurately and efficiently.
4.N.12
Divide up to a three-digit whole number with a single-digit divisor (with or without remainders) accurately and efficiently.
4.N.13      Interpret any remainders.
Demonstrate in the classroom an understanding of and the ability to use the conventional algorithms for addition and subtraction (up
4.N.14      to five-digit numbers), and multiplication (up to three digits by two digits).
Demonstrate in the classroom an understanding of and the ability to use the conventional algorithm for division of up to a three-digit
4.N.15      whole number with a single-digit divisor (with or without remainders).
4.N.16      Round whole numbers through 100,000 to the nearest 10, 100, 1000, 10,000, and 100,000.
Select and use a variety of strategies (e.g., front-end, rounding, and regrouping) to estimate quantities, measures, and the
results of whole-number computations up to three-digit whole numbers and amounts of money to \$1000, and to judge the
4.N.17      reasonablenes
Create, describe, extend, and explain symbolic (geometric) and numeric patterns, including multiplication patterns like 3, 30, 300,
4.P.1      3000, ….

4.P.2      inequalities (mathematical sentences that use =, <, >).
Use pictures, models, tables, charts, graphs, words, number sentences, and mathematical notations to interpret mathematical
4.P.4      relationships.

CIA                                                                  LPS                                                                             3/5/2010
Unit 7: Moving Between Solids Silhouttes                                Middle May - June                                                                        Grade 4
14 Sessions

Inv. # and       MA Std     Content Objective                                    Key Vocabulary                  Language                  Assessment       Pacing
Name                                                                                                             Objectives

Inv. 1:           4.G.1,    1. Students will describe attributes of geometric    prism, cube, cylinder, solid,   This is one example       Review p. 14 for 4 - 1 hour
Geometric         4.G.2,    solids.                                              vertex/vertices, edge, face,    of a language             the assessment sessions
Solids TE p. 20 - 4.G.3,                                                         geometric, attributes,          objective. Please         in this unit.
43                4.G.9,    2. Students will name geometric solids.              silhouttes, 2-D, 3-D,           consider your             Complete all
4.N.1,                                                         triangular, circular, shape,    students' needs           'Ongoing
4.N.3,    3. Students will understand how 3-D solids project   visualize, figure,              when writing your         Assessments,
4.N.4,    silhouettes with 2-D shapes (e.g., how a cone can    perspective, rectangle,         language                  Writing
4.N.6,    produce triangular and circular silhouettes.)        rectangular prism, triangle,                              Opportunities,
objectives:
4.N.10,                                                        octagonal prism,                                          Portfolio
Students will use
4.N.11,   4. Students will decompose images of 3-D shapes      hemisphere, cone,                                         Opportunities, &
content specific
4.N.12,   and then recombine them to make a given structure.   hexagonal prism, square                                   End-of-Unit
vocabulary to orally
4.N.13,                                                        prism, sphere, arrangement                                Assessments.
explain their thinking.
4.N.18,   5. Students will visualize what 3-D figures look                                                               Portfolio: SAB p.
4.P.2     like from different perspectives.                                                                              5, M14

Teacher's Notes: Ten-Minute Math reinforces place value, standards 4.N.1, 4.N.6, 4.N.12. SAB p. 2 reinforces 4.N.6 & 4.P.2. SAB p. 3,
reinforces 4.N.13, 4.N.3, 4.N.10, 4.N.11. SAB p. 8 relates to standards 4.N.3 and 4.N.4 SAB p. 14 reinforces standards 4.N.10, 4.N.18; p. 15
reinforces 4.N.4 & 4.P.2

Focal Points (supported): Number and Operations, Algebra (connected): Geometry
CIA                                                           Lowell Public Schools                                                         Working Document 3/5/2010
Unit 7: Moving Between Solids Silhouttes                                   Middle May - June                                                             Grade 4
14 Sessions

Inv. # and        MA Std    Content Objective                                        Key Vocabulary                  Language     Assessment         Pacing
Name                                                                                                                 Objectives

Inv. 2: Make      4.G.1,    1. Students will recognize how components of 3-D volume, fill-up, cube, mental                        Portfolio: SAB p. 5 - 1 hour
and Visualizing   4.G.2,    cube buildings come together to form a whole            model, visualize, silhouette,                 30, M16-17        sessions
Cube Bldgs. TE    4.G.3,    building.                                               perspective, faces, edge,
p. 48 - 72        4.G.7,                                                            vertex/vertices, layer, area,
4.G.9,    2. Students will find the volume of cube buildings. square units, cubic units,
4.M.1,                                                            top/bird's eye view, front,
4.N.1,    3. Students will draw silhouettes of 3-D cube buildings right side, flip (reflection),
4.N.3,    from different perspectives.                            turn (rotation), slide
4.N.4,                                                            (translation)
4.N.5,    4. Students will interpret different silhouettes of an
4.N.6,    object, both to form a mental model and to build a
4.N.10,   whole object.
4.N.11,
4.N.12    5. Students visualize what 3-D figures look like
4.N.13,   from different perspectives.
4.N.18

Teacher's Notes: Ten-Minute Math reinforces place value, standards 4.N.1, 4.N.6, 4.N.12. SAB p. 19 reinforces 4.N.13, 4.N.3, 4.N.10, 4.N.11; p. 24 reinforces
4.N.18; and p. 25, 31, and 32 reinforce standards 4.N.4 & 4.N.6. During Inv. 2.2, p. 57, include the terms from standard 4.G.7, when describing the
configurations. Standard 4.M.1 area can be extended for those students who are ready by asking them to find surface area of their buildings, this is
another form of differentiation. Standard 4.N.5 is reinforced with Student Activity sheet p. 34.

Focal Points (supported): Number and Operations, Algebra (connected): Geometry
CIA                                                           Lowell Public Schools                                                Working Document 3/5/2010
Unit 7: Moving Between Solids Silhouttes                                Middle May - June                                                             Grade 4
14 Sessions

Inv. # and      MA Std     Content Objective                                         Key Vocabulary                Language     Assessment        Pacing
Name                                                                                                               Objectives

Inv. 3:         4.G.1,     1. Students will find the number of cubes (volume) rectangular prism, volume,                        Portfolio: SAB p. 5 - 1 hour
Understanding   4.G.2,     that will fit into the box made by a given pattern.     cubes, box, congruent,                       39-42, 45, Unit   sessions
Volume TE p.    4.M.1,                                                             pattern/net, layer, strategy,                Assessment M23-
76 - 96         4.M.4,     2. Students design patterns for boxes that hold a       decompose, dimensions,                       25
4.N.1,     given number of cubes (volume).                         silhoutte, 3-D, perspective,
4.N.3,                                                             length, width/depth,
4.N.4,     3. Students will see that cubes filling a rectangular height/tall, congruent,
4.N.6,     prism can be decomposed into congruent layers. top/bird's eye view, front,
4.N.10,                                                            right side, box pattern/net,
4.N.12,    4. Students will develop a strategy for determining the rows, array,
4.N.18,    volume of rectangular prisms.                           algorithm/notation (l x w x
4.P.1,                                                             h), doubling
4.P.2      5. Students will double the number of cubes for a
given box and consider how that changes the
dimensions of the original box.

6. Students will draw silhouettes of 3-D cube buildings
from different perspectives.

7. Students will find the volume of cube buildings.

Teacher's Notes: Ten-Minute Math in Inv. 3.1 reinforces place value (standards 4.N.1, 4.N.6, 4.N.12). SAB p. 36, reinforces 4.N.18.; p. 37, 43 reinforces
standards 4.N.6 and 4.P.2. Ten-Minute Math in Inv. 3.2 reinforces place value (standards 4.N.6, 4.N.12). When completing the activity "Doubling the Number
of Cubes" Student Activity Sheet p. 47, consider your students who are ready for this and those who are not, also, consider recording the doubling results
on a chart so that students can easily follow the change or pattern, 4.P.1. Student Activity Sheet p. 49 reinforces standards 4.N.3, 4.N.4 & 4.N.10.

Focal Points (supported): Number and Operations, Algebra (connected): Geometry
CIA                                                           Lowell Public Schools                                              Working Document 3/5/2010
Unit 7: Moving Between Solids & Silhouttes
Standards #                                                                  Description
Exhibit an understanding of the base ten number system by reading, modeling, writing, and interpreting whole numbers to at least
4.N.1      100,000; demonstrating an understanding of the values of the digits; and comparing and ordering the numbers.
4.N.3      Demonstrate an understanding of fractions as parts of unit wholes, as parts of a collection, and as locations on the number line.
Select, use, and explain models to relate common fractions and mixed numbers (1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12, and 11/2), find
4.N.4      equivalent fractions, mixed numbers, and decimals, and order fractions.
Identify and generate equivalent forms of common decimals and fractions less than one whole (halves, quarters, fifths, and tenths).
4.N.5
Exhibit an understanding of the base ten number system by reading, naming, and writing decimals between 0 and 1 up to the hundredths.
4.N.6
Select and use appropriate operations (addition, subtraction, multiplication, and division) to solve problems, including those
4.N.10      involving money.
Know multiplication facts through 12 x 12 and related division facts. Use these facts to solve related multiplication problems and compute
4.N.11      related problems, e.g., 3 x 5 is related to 30 x 50, 300 x 5, and 30 x 500.
4.N.12      Add and subtract (up to five-digit numbers) and multiply (up to three digits by two digits) accurately and efficiently.
Divide up to a three-digit whole number with a single-digit divisor (with or without remainders) accurately and efficiently.
4.N.13      Interpret any remainders.
4.N.18      Use concrete objects and visual models to add and subtract common fractions.
Demonstrate an understanding of such attributes as length, area, weight, and volume, and select the appropriate type of unit for measuring
4.M.1       each attribute.
Estimate and find area and perimeter of a rectangle, triangle, or irregular shape using diagrams, models, and grids or by measuring.
4.M.4
Compare and analyze attributes and other features (e.g., number of sides, faces, corners, right angles, diagonals, and symmetry) of two- and
4.G.1      three-dimensional geometric shapes.
Describe, model, draw, compare, and classify two- and three-dimensional shapes, e.g., circles, polygons—especially triangles and
4.G.3      Recognize similar figures.
Describe and apply techniques such as reflections (flips), rotations (turns), and translations (slides) for determining if two shapes are
4.G.7       congruent.
4.G.9      Predict and validate the results of partitioning, folding, and combining two- and three-dimensional shapes.
Create, describe, extend, and explain symbolic (geometric) and numeric patterns, including multiplication patterns like 3, 30, 300, 3000, ….
4.P.1

4.P.2      (mathematical sentences that use =, <, >).

CIA                                                                        LPS                                                                            3/5/2010

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