Number Puzzles Standards by tyndale

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									Unit: Number Puzzles and Multiple Towers
Standard                                                            Description
Number
"Big Idea" Standards
5.N.8       Apply the number theory concepts of common factor, common multiple, and divisibility rules for 2, 3, 5, and 10 to the solution of
            problems. Demonstrate an understanding of the concepts of prime and composite numbers.


5.N.9      Solve problems involving multiplication and division of whole numbers, and multiplication of positive fractions with whole
           numbers.
5.N.12     Accurately and efficiently add and subtract whole numbers and positive decimals. Multiply and divide (using double-digit
           divisors) whole numbers. Multiply positive decimals with whole numbers.

Supported Standards
5.N.1     Demonstrate an understanding of (positive integer) powers of ten.
5.N.2      Demonstrate an understanding of place value through millions and thousandths.
5.N.3      Represent and compare large (millions) and small (thousandths) positive numbers in various forms, such as expanded notation without
           exponents, e.g., 9724 = 9 x 1000 + 7 x 100 + 2 x 10 + 4.

5.N.7      Compare and order whole numbers, positive fractions, positive mixed numbers, positive decimals, and percents.


5.N.14     Estimate sums and differences of whole numbers, positive fractions, and positive decimals. Estimate products of whole numbers and
           products of positive decimals with whole numbers. Use a variety of strategies and judge the reasonableness of the answer.


5.M.1      Apply the concepts of perimeter and area to the solution of problems involving triangles and rectangles. Apply formulas where appropriate.


Additional Ten Minute Math Standards
5.P.3
                                      Number Puzzles and Multiple Towers -- September - beginning of October
Inv. Title & MA Stan. Content Objectives                                 Key Vocabulary Sample Language                                 Pacing
Pages                                                                                     Objectives
Inv. 1:     5.N.8        Students will use arrays to model multiplication.         multiplication,       Listening/Speaking:          7 one-hour sessions
            5.N.12                                                                 division, factor,     Students will understand and
Finding     5.M.1        Students will determine whether one number is a factor array, product,          use key vocabulary during
Factors and 5.N.1        or multiple of another.                                   multiple,             discussion.
Prime       5.N.2                                                                  dimensions,
Factors     5.P.3        Students will identify prime, square, even, and odd       even number,          Students will ask and
                         numbers.                                                  odd number,           respond to questions to
TE 24-64                                                                           prime number,         clarify information.
                         Students will use multiplication facts to find equivalent composite
                         multiplication combinations.                              number,               Writing:
                                                                                   square number,        Students will write short
                         Students will use multiplication facts for numbers        square root,          accounts about classroom
                         related by place value.                                   prime                 topics.
                                                                                   factorization
                         Students will find all the ways to multiply whole         common factor,        Students will write about
                         numbers for a given product.                              common                steps to solve a problem,
                                                                                   multiple,             placing events in a logical
                         Students will use properties and relationships of         divisible, solution   order.
                         numbers to solve problems.

                         Students will determine the prime factorization of a
                         number.

Teachers' Notes: --Ten-Minute Math is Quick Images: Seeing Numbers. Students should write as many multiplication equations to describe the
dot arrangements as they can, using two or more factors. This activity supports standards 5.N.8, 5.N.12, and 5.P.3. --During Session 1.1, Activity 3, have
students generalize for all numbers that would fit the "10 tiles wide" clue. This will reinforce divisibility rules for multiples of 10, part of standard 5.N.8. --At the
end of Session 1.2, allow struggling learners to build or draw arrays for further practice while advanced learners find additional prime numbers greater than 50. --
Homework for Session 1.3 involves practice with basic multiplication facts. For students who continue to exhibit difficulty with memorizing these facts, try a
daily five-minute flash card warm-up and personalized flash cards for daily studying at home. --When students finish solving number puzzles quickly, they can
write their own puzzles to give to classmates to solve. --Numbers factored by students can also vary, giving simpler numbers to struggling students, and asking
for longer multiplication combinations from advanced learners.
Prime factorization addressed in Session 1.7 goes beyond fifth grade standard, addressing the sixth grade number theory standard. This activity could be
reserved for students demonstrating mastery of other factoring activities. --Common multiples, common factors, and divisibility rules from standard 5.N.8 not
addressed directly as part of investigation. Please use complimentary materials to address these concepts.
Inv. Title & MA Stan. Content Objectives                                           Key Vocabulary Sample Language                      Pacing
Pages                                                                                             Objectives


Inv. 2:       5.N.12     Students will solve 2-digit by 2-digit multiplication     distributive        Listening/Speaking:             7 one-hour sessions
              5.N.9      problems.                                                 property,           Students will retell steps of a
Multiplicatio 5.M.1                                                                expanded            process in logical order.
n Strategies 5.N.14      Students will create story problems represented by a      notation,
              5.N.7      multiplication and division expressions.                  standard            Students will compare and
TE 66-107 5.N.2                                                                    notation,           contrast strategies used to
              5.N.3      Students will represent multiplication and division       parenthesis,        solve multiplication problems
              5.N.8      problems with pictures or diagrams.                       associative         orally.
                                                                                   property, less
                         Students will use arrays to model multiplication.         than, greater       Writing:
                                                                                   than, value,        Students will draw and label
                         Students will multiply fluently by multiples of 10.       estimate,           or write story problems which
                                                                                   approximate,        describe multiplication and
                         Students will compare multiplication problems to          rounding            division situations.
                         determine which product is greater.
                                                                                                       Students will write short
                         Students will estimate the product of two numbers.                            explanations of how a
                                                                                                       problems was solved.
                         Students will decompose multiplication problems
                         efficiently.                                                                   Students will organize work
                                                                                                        and use clear and concise
                                                                                                        notation to show how
                                                                                                        multiplication problems have
                                                                                                        been solved.
Teachers' Notes: --Ten-Minute Math becomes Number Puzzles, which continues to reinforce 5.N.8. --To get a feel for the fluency students are
developing for solving multiplication problems, meet with small groups of students to observe them. Discuss misconceptions and provide specific
feedback about progress/mistakes students in each group make. --If students refer to multiplying by 10, or by multiples of 10, as adding a zero,
please correct their phraseology so they address this as attaching a zero. When you add a zero to any number, the value of the original number is unchanged.
--During 2.3-2.6, adjust numbers as necessary so that students are challenged at appropriate level of understanding. --For students struggling with cluster
problems, model an appropriate first step, to start them in the right direction. If students are efficiently finding answers, challenge them to find another way to
solve each problem, using other strategies such as doubling and halving.

During sessions 2.4 -2.6, it is not necessary for all students to study cluster problems and starter problems. In addition to repeated addition, it is essential that
students have at least one other strategy (i.e., area model, lattice method, partial products, cluster problems, U.S. standard algorithm) to be able to multiply 2-
digit numbers efficiently and with understanding (make a connection between the two strategies).
Inv. Title & MA Stan. Content Objectives                                           Key Vocabulary Sample Language                      Pacing
Pages                                                                                             Objectives

Inv. 3:      5.N.9      Students will represent multiplication and division        dividend, divisor, Listening/Speaking:              8 one-hour sessions
             5.N.12     problems with pictures or diagrams.                        quotient,          Students will compare and
Division     5.P.3                                                                 remainder          contrast strategies used to
Strategies   5.N.2      Students will create story problems represented by a                          solve division problems
             5.N.8      multiplication and division expressions.                                      orally.
TE 110-152
                        Students will use knowledge of multiples of 10 to solve                      Writing:
                        division problems.                                                           Students will organize work
                                                                                                     and use clear and concise
                        Students will use and interpret notation that represents                     notation to show how division
                        division.                                                                    problems have been solved.

                        Students will relate division and multiplication                             Students will choose a
                        notations.                                                                   division problem and record
                                                                                                     their strategy for solving that
                        Students will solve division problems with 2-digit                           problem in a short paragraph.
                        divisors.

                        Students will make sense of remainders when solving
                        division problems.

                        Students will solve a division problem by breaking the
                        dividend into parts.

                        Students will compare division problems to determine
                        which quotient is greater.
Teacher Notes: --Ten Minute Math during Investigation 3 will consist of some Quick Images: Seeing Numbers and some Number Puzzles. During Seeing
Numbers, students should write multiplication and division equations that represent the dot arrangements they see. For the Number Puzzles, the Random Clue
variation can be played, in which clues are combined and drawn randomly. This will lead to a greater number of inconclusive results for which students can
modify clues and generate a result. --If students experience difficulty connecting the multiple towers in session 3.2 to multiplication and division, they can write
corresponding expressions next to each multiple. In session 3.4, it is not necessary for students to know all suggested strategies, as long as they are efficient
with one of the strategies and can connect division to multiplication. During class discussions, be sure to emphasize the partial quotients (big seven) strategy
as this will reinforce the U.S. standard algorithm when we reach Unit 7 (How Many People How Many Teams?). --During Session 3.6, Division Compare can be
differentiated, so that struggling students practice division of 2-digit numbers by 1-digit numbers, and proficient students can practice playing without multiples
of ten.
Unit: Thousands of Miles, Thousands of Seats
Standard Number                                                                Description
"Big Idea" Standards
5.N.2              Demonstrate an understanding of place value through millions and thousandths
5.N.3              Represent and compare large (millions) and small (thousandths) positive numbers in various forms, such as expanded
                   notation without exponents, e.g., 9724 = 9 x 1000 + 7 x 100 + 2 x 10 + 4.
5.N.7              Compare and order whole numbers, positive fractions, positive mixed numbers, positive decimals, and percents
5.N.9              Solve problems involving multiplication and division of whole numbers, and multiplication of positive fractions with
                   whole numbers.
5.N.11             Demonstrate an understanding of the inverse relationship of addition and subtraction, and use that understanding to
                   simplify computation and solve problems.
5.N.12             Accurately and efficiently add and subtract whole numbers and positive decimals. Multiply and divide (using double-
                   digit divisors) whole numbers. Multiply positive decimals with whole numbers.
Supported Standards
5.N.1              Demonstrate an understanding of (positive integer) powers of ten
5.N.6              Find and position whole numbers, positive fractions, positive mixed numbers, and positive decimals on a number line
Additional Ten Minute Math Standards
5.N.14
                   Estimate sums and differences of whole numbers, positive fractions, and positive decimals. Estimate products of whole numbers
                   and products of positive decimals with whole numbers. Use a variety of strategies and judge the reasonableness of the answer.
                                                      Thousands of Miles, Thousands of Seats -- November
Inv. Title &MA        Content Objectives                                       Key Vocabulary Sample Language                     Assessment         Pacing
Pages       Stan.                                                                                Objectives                       Suggestions
Inv. 1:     5.N.1     Students will read, write, and sequence numbers to       one, ten,         Listening/Speaking:              observation of     5 one-hour
            5.N.2     10,000 and 100,000.                                      hundred,          Students will appropriately      student ability to sessions
Using Place 5.N.3                                                              thousand, ten     name place values though the     read, write, and
Values      5.N.6     Students will understand the place-value relationships thousand,           millions place.                  order numbers on
            5.N.7     between 10, 100, 1,000, and 10,000.                      hundred                                            10,000 chart (see
TE 24-54    5.N.9                                                              thousand,         Students will listen to and      checklist, M3)
            5.N.11    Students will add and subtract multiples of 100 and      million, billion, understand stories involving
            5.N.12    1,000.                                                   trillion, period, subtraction in order to find a   Related
            5.N.14                                                             base, exponent, solution to the problem.           Problems, SAB
                      Students will find the difference between a number a     power of                                           13-14
                      10,000.                                                                    Writing:
                                                                                                 Students will write brief        journal entries
                      Students will find combinations of 3-digit numbers that                    accounts of how problems         about strategies
                      have a sum of 1,000.                                                       were solved.                     used when
                                                                                                                                  playing Close to
                      Students will use visual representations to explain and                                                     1000.
                      justify solutions to subtraction problems.

                      Students will solve addition and subtraction problems
                      with large numbers by focusing on the place value of
                      the digits.

                      Students will find the difference between a number and
                      10,000.

                      Students will identify place values through one trillion.

Teacher Notes: --Ten Minute math for this investigation is Practicing Place Value and Estimation ad Number Sense: Closest Estimate. Practicing Place
Value supports standards 5.N.2, 5.N.3, and 5.N.12. Closest Estimate supports 5.N.2, 5.N.12, and 5.N.14. --For students who struggle with subtraction, small
group sessions to identify misconceptions would be beneficial. --For students who easily play Close to 1000, introduce a variation with positive and negative
numbers, so that all students are challenged. --When doing Related Problems during Session1.4, adjust numbers as necessary so that all students can gain
flexibility with the number manipulation.
Inv. Title &   MA       Content Objectives                                      Key Vocabulary Sample Language                    Assessment          Pacing
Pages          Stan.                                                                              Objectives                      Suggestions
Inv. 2:        5.N.9    Students will solve whole-number addition and           subtraction,      Listening/Speaking:             observation of      5 one-hour
               5.N.11   subtraction problems efficiently.                       addition,         Students will describe and      subtraction         sessions
Studying       5.N.12                                                           algorithm,        compare subtraction             strategies used
Subtraction    5.N.6    Students will analyze and use different subtraction     addend,           strategies by focusing on how   on SAB 20-21
               5.N.2    strategies.                                             difference,       each strategy starts.
TE 56-84       5.N.3                                                            minus, sum, total                                 Starter Problems,
               5.N.14   Students will understand the meaning of the steps and                     Students will orally defend     SAB 31-32
                        notation of the US algorithm for subtraction.                             how differences represented
                                                                                                  by two subtraction              Distances from
                                                                                                  expressions are related.        Chicago, SAB 39-
                                                                                                                                  40
                                                                                                 Writing:
                                                                                                 Students will organize work     Subtraction
                                                                                                 and use clear and concise       Problems, M15
                                                                                                 notation for recording addition
                                                                                                 and subtraction strategies.


Teacher Notes: --Continue both Practicing Place Value and Estimation ad Number Sense: Closest Estimate during Ten Minute Math time throughout this
investigation. --Students who are strong in one subtraction strategy should be encouraged to try other strategies. The strength in understanding multiple
strategies extends far beyond proficiency in subtraction. It helps to develop strong number sense and deeper understanding of the operation of subtraction.
Conversely, if students are struggling with subtraction, allow them to focus on becoming proficient in one strategy. --When teaching the US algorithm for
subtraction during Session 2.4, try to refer to place value manipulation as regrouping instead of borrowing. This will help connections between addition and
subtraction to be more meaningfully built, as when one "carries" in addition, this too is regrouping. Common language between the operations will help
enhance their inverse relationship.
Inv. Title &   MA       Content Objectives                                      Key Vocabulary Sample Language                     Assessment         Pacing
Pages          Stan.                                                                           Objectives                          Suggestions
Inv. 3:        5.N.2    Students will solve addition and subtraction problems                  Listening/Speaking:                 observations of    5 one-hour
               5.N.7    with large numbers by focusing on the place value of                   Students will retell steps used     students playing   sessions
Adding and     5.N.9    the digits.                                                            to solve a problem in a logical     Close to 7,500.
Subtracting    5.N.11                                                                          order.
Large          5.N.12   Students will add and subtract multiples of 100 and                                                        Rock On!
Numbers        5.N.3    1,000.                                                                    Reading:                         Problems, SAB
               5.N.14                                                                             Students will read and           59-62
TE 86-112               Students will solve multi-step story problems.                            interpret story problems in
                                                                                                  order to find its solution.      End-of-Unit
                        Students will solve whole-number addition and                                                              Assessment, M20
                        subtraction problems efficiently.                                         Writing:
                                                                                                  Students will write short
                        Students will read, write, and sequence numbers to                        stories related to subtraction
                        10,000 and 100,000.                                                       problems which can be solved
                                                                                                  by classmates.

Teacher Notes: --Continue both Practicing Place Value and Estimation ad Number Sense: Closest Estimate during Ten Minute Math time throughout this
investigation. --When the division fact assessment in given during session 3.1, allow students to retake until mastery. If students do poorly on the assessment
and receive a poor mark, that does not help them build the fluency necessary for proficient computation. Retaking for a better mark (even if the assessment is
changed slightly) will promote important practice with the basic facts. --For students who are strong with subtraction, they could find actual stadium capacities
through research during session 3.2. This extension would also provide valuable time the teacher could use for small group intervention for student still
struggling with subtraction. --If students do not use clear and concise notation on their assessments, ask them to orally explain their work to you. This will
enable you to determine if the answer was legitimately found or not. If the answer is incorrect, the mini-conference will help you determine if simple
subtraction errors were made or if the student has weak concepts of subtraction.
Unit: What's That Portion?
Standard Number                                                           Description
"Big Idea" Standards
5.N.4              Demonstrate an understanding of fractions as a ratio of whole numbers, as parts of unit wholes, as parts of a
                   collection, and as locations on the number line.
5.N.5              Identify and determine common equivalent fractions (with denominators 2, 4, 5, 10) and mixed numbers (with
                   denominators 2, 4, 5, 10), decimals, and percents (through one hundred percent), e.g., 3/4 = 0.75 = 75%.
5.N.7              Compare and order whole numbers, positive fractions, positive mixed numbers, positive decimals, and percents
5.N.13             Accurately and efficiently add and subtract positive fractions and mixed numbers with like denominators and with
                   unlike denominators (2, 4, 5, 10 only); multiply positive fractions with whole numbers. Simplify fractions in cases when
                   both the numerator and the denominator have 2, 3, 4, 5, or 10 as a common factor.
Supported Standards
5.N.6              Find and position whole numbers, positive fractions, positive mixed numbers, and positive decimals on a number line
5.M.1              Apply the concepts of perimeter and area to the solution of problems involving triangles and rectangles. Apply formulas where
                   appropriate.
5.D.2              Construct and interpret line plots, line graphs, and bar graphs. Interpret and label circle graphs.
Additional Ten Minute Math Standards
5.N.12             Accurately and efficiently add and subtract whole numbers and positive decimals. Multiply and divide (using double-digit divisors)
                   whole numbers. Multiply positive decimals with whole numbers.
5.N.14             Estimate sums and differences of whole numbers, positive fractions, and positive decimals. Estimate products of whole numbers
                   and products of positive decimals with whole numbers. Use a variety of strategies and judge the reasonableness of the answer.
                                                          What's That Portion? -- December - Mid-January
Inv. Title & MA       Content Objectives                                          Key             Sample Language                 Assessment         Pacing
Pages        Stan.                                                                Vocabulary      Objectives                      Suggestions
Inv. 1:      5.N.4    Students will interpret everyday uses of fractions,         fraction,       Listening/Speaking:             observations of    5 one-hour
             5.N.5    decimals, and percents.                                     decimal,        Students will use appropriate   how students       sessions
Using        5.M.1                                                                percent,        pronunciation of fraction       color 10 by 10
Percents     5.N.12   Students will find fractional parts of a whole or of a      equivalent,     names.                          grids to determine
and          5.N.14   group.                                                      numerator,                                      equivalency
Fractions                                                                         denominator,    Reading:
                      Students will find a percentage of a group.                 fraction bar,   Students will read and          School Days,
TE 20-52                                                                          whole number, recognize fractions in            SAB 15-16 (see
                      Students will find a percentage of a rectangular area.      ratio           numerical and word form.        checklist M11)

                      Students will identify fraction and percent equivalents.

                      Students will interpret the meaning of the numerator
                      and denominator of a fraction.

                      Students will use equivalent fractions and percents to
                      solve problems.

Teacher Notes: --Ten Minute Math during this investigation is Estimation and Number Sense, which supports standards 5.N.12 and 5.N.14, and Guess My
Rule, which supports standard 5.N.4, for parts of a collection. By using fractions in simplest form, one can also supports equivalency in 5.N.5. --Students
will need to be shown fractions written as ratios in accordance with 5.N.4. --Decimal equivalency (5.N.5) not addressed at this time. --When solving multiple
word problems such as that during Session1.1, try summarizing the lesson by making strategy posters and having students do a carousel to view the work of
others. Each group can do one problem and the goal becomes showing as many ways to solve the problem as possible. --During Session 1.3 and 1.4,
percent grids go beyond 5.N.5, which requires equivalency of fractions with denominators 2, 4, 5, and 10. Have students do these fractions first and offer
denominators 3, 6, and 8 as challenge for students who progress quickly.
Inv. Title & MA      Content Objectives                                          Key             Sample Language                 Assessment          Pacing
Pages        Stan.                                                               Vocabulary      Objectives                      Suggestions

Inv. 2:   5.N.4      Students will identify fraction and percent equivalents.    simplest form   Listening/Speaking:             observations of    6 one-hour
          5.N.5                                                                                  Students will connect new       strategies used by sessions
Comparing 5.N.6      Students will order fractions and justify their order by                    learning from class             students to order
and       5.N.7      reasoning about equivalencies.                                              discussions to prior            and compare
Ordering  5.N.12                                                                                 knowledge.                      fractions
Fractions 5.N.14     Students will compare fractions and percents to the
                     landmarks of 0, 1/2, and 1.                                                 Writing:                        Fraction and
TE 54-87                                                                                         Students will provide written   Percent
                     Students will find and compare fractional parts and                         proof explaining why a given    Problems, SAB
                     percents of a whole or a group.                                             fraction is greater than        27-28
                                                                                                 another.
                     Students will compare fractional parts of different-sized                                                   Using Fractions
                     wholes.                                                                     Students will write short       and Percents,
                                                                                                 accounts describing how a       M20
                     Students will use equivalent fractions and percents to                      given problem was solved.
                     solve problems.

Teacher Notes: --Ten Minute Math during this investigation continues to be Estimation and Number Sense and Guess My Rule. --Decimals not
addressed at this time. --For students struggling with simplest form, a list of fractions to be placed on the percent equivalent strip can be provided. --
Encourage students to develop more strategies for comparing fractions than just illustrations. When two fractions have values that are close to one another,
it becomes increasingly difficult to determine which is greater. Also, as students will explore when working with different-sized wholes, if the drawings aren't
perfect, then the size comparison is null.
Inv. Title & MA      Content Objectives                                        Key                 Sample Language               Assessment         Pacing
Pages        Stan.                                                             Vocabulary          Objectives                    Suggestions

Inv. 3:     5.N.4    Students will find fractional parts of the rotation around simplify, mixed    Listening/Speaking:           Adding and        10 one-
            5.N.5    a circle.                                                  number, fraction   Students will explain their   Subtracting       hour
Adding and 5.N.6                                                                greater than one   thinking process for adding   Fractions, SAB 46 sessions
Subtracting 5.N.7    Students will add fractions by using a rotation model.     whole              fractions with unlike         (see checklist,
Fractions   5.N.13                                                                                 denominators.                 M24)
            5.D.2    Students will add and subtract fractions by reasoning
TE 90-144 5.N.12     about equivalencies.                                                                                        Fraction
            5.N.14                                                                                                               Problems, SAB
                     Students will represent fractions on a number line.                                                         60-62

                     Students will use equivalencies to place fractions on a                                                     End-of-Unit
                     set of number lines.                                                                                        Assessment, M32-
                                                                                                                                 M33
                     Students will compare fractions on a number line.

                     Students will order mixed numbers and fractions
                     greater than 1.

                     Students will find combinations of fractions with sums
                     between 0 and 2.

                     Students will add and subtract fractions by using a
                     number line.

Teacher Notes: --Ten Minute Math during this investigation continues to be Estimation and Number Sense and Guess My Rule. --Although some
problems address multiplication of whole numbers and fractions, i.e.: 3/8 of 64, this portion of 5.N.13 is not explicitly taught. When students solve such
problems, summarizing various strategies will be important so that this part of the standard is more directly addressed. --Meet with small groups to observe
how students go about adding fractions. As they gain familiarity with the clock face, remove that scaffold and have students who are ready find ways to add
fractions without the clock face. These students can also play variations, i.e., continuing to a sum of 2, subtraction from 2, using a whole number cube too
and going for a goal of 5, etc. For students struggling with the concept of adding fractions, allow them more time to play Roll Around the Clock before
introducing them to Fraction Tracks.
Unit: Measuring Polygons
Standard Number                                                                 Description
"Big Idea" Standards
5.G.1              Identify, describe, and compare special types of triangles (isosceles, equilateral, right) and quadrilaterals (square,
                   rectangle, parallelogram, rhombus, trapezoid), e.g., recognize that all equilateral triangles are isosceles, but not all
                   isosceles triangles are equilateral.
5.G.3              Identify relationships among points and lines, e.g., intersecting, parallel, perpendicular.
5.M.1              Apply the concepts of perimeter and area to the solution of problems involving triangles and rectangles. Apply formulas
                   where appropriate.
5.M.2              Identify, measure, describe, classify, and draw various angles. Draw triangles given two sides and the angle between
                   them, or given two angles and the side between them, e.g., draw a triangle with one right angle and two sides
                   congruent.
5.M.5              Find the sum of the measures of the interior angles in triangles by measuring the angles, and without measuring the
                   angles.
Additional Ten Minute Math Standards
5.G.5              Describe and perform transformations on two-dimensional shapes, e.g., translations, rotations, and reflections
5.G.6              Identify and describe line symmetry in two-dimensional shapes, including shapes that have multiple lines of symmetry.
5.P.1              Analyze and determine the rules for extending symbolic, arithmetic, and geometric patterns and progressions, e.g., ABBCCC; 1,
                   5, 9, 13…; 3, 9, 27…
5.D.2              Construct and interpret line plots, line graphs, and bar graphs. Interpret and label circle graphs.
                                                       Measuring Polygons -- Mid-January - Mid-February
Inv. Title & MA      Content Objectives                             Key Vocabulary             Sample Language                         Assessment      Pacing
Pages        Stan.                                                                             Objectives                              Suggestions

Inv. 1:     5.G.1    Students will identify the attributes of        polygon, right angle, acute     Listening/Speaking:              observe how       7 one-hour
            5.G.3    polygons.                                       angle, obtuse angle,            Students will classify           comfortable       sessions
Polygons: 5.M.5                                                      degree, equilateral,            descriptive geometry terms       students are with
Names,      5.M.2    Students will describe triangles by the sizes scalene triangle, isosceles       into groupings of their choice.  quadrilaterals
Angles, and 5.G.6    of their angles and the lengths of their sides. triangle, attribute, property                                    having many
Properties 5.P.1                                                     quadrilateral, parallel,        Students will orally justify why names
                     Students will use attributes to describe and trapezoid, parallelogram,          they have grouped the
TE 20-64             compare quadrilaterals.                         rectangle, rhombus,             vocabulary words in the way Some Figures
                                                                     square, perpendicular,          they have chosen.                Have Many
                     Students will define a regular polygon as a intersect, congruent,                                                Names, SAB 8-10
                     polygon with all sides and all angles equal. regular, hexagon,                  Writing:
                                                                     heptagon, octagon,              Students will briefly justify in Angles in the
                     Students will use known angles to find the      pentagon, nonagon,              writing how the measure of a Power Polygons,
                     measures of other angles.                       decagon, dodecagon,             polygon's angles was             SAB 17-19
                                                                     interior angle, exterior        determined.
                                                                     angle, complimentary                                             Quadrilaterals
                                                                     angles, supplementary                                            and Angles, M17
                                                                     angles, kite, protractor,
                                                                     straight angle, reflexive
                                                                     angle, chevron


Teacher Notes: --Ten Minute Math is Quick Images: 2-D, which supports 5.G.1, 5.G.3, 5.G.5, 5.G.6, and 5.P.1, depending on strategy used to visualize
images. --Introduce students to the protractor as a tool to check angle estimates. --Once students are comfortable with angle measures, try a warm up
activity in which students are given an angles and two side measurements. Have them draw the resulting triangle. Did everyone draw the same triangle?
This activity addresses the remaining parts of 5.M.2.
Inv. Title & MA      Content Objectives                               Key Vocabulary            Sample Language                  Assessment        Pacing
Pages        Stan.                                                                              Objectives                       Suggestions

Inv. 2:     5.M.1    Students will compare the perimeters and         perimeter, area, one-      Reading:                        journal entries   6 one-hour
            5.G.1    areas of rectangles when the dimensions          dimensional, two-          Students will read and          about area-       sessions
Finding     5.G.3    are changed.                                     dimensional, square units, understand previously-learned   perimeter
Perimeter 5.G.5                                                                                  geometry vocabulary.            relationships
and Area of 5.G.6    Students will use numerical and geometric
Related     5.P.1    patterns to describe how the perimeters and                                Writing:                         A Sequence of
Rectangles 5.D.2     areas of rectangles change when the                                        Students will appropriately      Rectangles, SAB
                     dimensions change.                                                         group ideas for a brief          35-36
TE 66-104                                                                                       comparison writing
                     Students will create different rectangles with                             assignment.                      Open response
                     the same area but different perimeters.                                                                     about changing
                                                                                                                                 area, but not
                     Students will understand square units as a                                                                  perimeter (a
                     unit of measure.                                                                                            former MCAS
                                                                                                                                 question)
                     Students will create rectangles with the
                     same perimeter but different areas.                                                                          Perimeter and
                                                                                                                                  Area of
                                                                                                                                  Rectangles, M19
Teacher Notes: --Ten Minute Math continues with Quick Images: 2-D, but Quick Survey is also introduced. Quick Survey supports bar graphs and line
plots from 5.D.2, but not line graphs or pie charts. --Although this investigation deals with area and perimeter, it does not address finding the area of a
triangle. Please use complimentary materials to teach this important area formula. --Once students understand perimeter and area conceptually it is
appropriate to show them the standard formulas, as they will be seen on standardized test fact sheets. --During Session 2.4, allow students to use color tiles
or grid paper to help them visualize changing perimeter but not area. --During Session 2.5, students who are ready for a challenge can be asked to
determine under what conditions the area will be the greatest, when the perimeter is constant.
Inv. Title & MA      Content Objectives                              Key Vocabulary                Sample Language                  Assessment          Pacing
Pages        Stan.                                                                                 Objectives                       Suggestions

Inv. 3:       5.G.1 Students will recognize and build similar         similar                                                                             5 one-hour
              5.G.3   figures.                                                                                                                            sessions
Similarity    5.M.1
              5.M.2 Students will examine the relationship
TE 106-135 5.P.5 among angles, lines lengths, and areas of
              5.G.5 similar polygons.
              5.G.6
              5.P.1
              5.D.2
Teacher Notes: -- During investigation 3, both Quick Images: 2-D and Quick Survey are the Ten Minute Math activities. --Similarity goes beyond the fifth
grade geometry standards. This time could be used for working with geometry standards not addressed in the fifth grade curriculum (symmetry, translations,
coordinate plane, etc.). Complimentary materials would be necessary in this case. Completing this investigation could be beneficial though, since it
provides further practice with angles, line length, and area. It also brings in proportional relationships, an important topic in algebra. If the similarity
investigation is omitted, then only questions 1 and 2 should be used on the End-of-Unit assessment.
Unit: Prisms and Pyramids
Standard Number                                                               Description
"Big Idea" Standards
5.M.4              Find volumes and surface areas of rectangular prisms.
5.G.2              Identify, describe, and compare special types of three-dimensional shapes (cubes, prisms, spheres, pyramids) based on
                   their properties, such as edges and faces.
Supported Standards
5.N.8              Apply the number theory concepts of common factor, common multiple, and divisibility rules for 2, 3, 5, and 10 to the solution of
                   problems. Demonstrate an understanding of the concepts of prime and composite numbers.
5.M.3              Solve problems involving simple unit conversions within a system of measurement
Additional Ten Minute Math Standards
5.N.2              Demonstrate an understanding of place value through millions and thousandths
                   Represent and compare large (millions) and small (thousandths) positive numbers in various forms, such as expanded notation
5.N.3              without exponents, e.g., 9724 = 9 x 1000 + 7 x 100 + 2 x 10 + 4.
                   Accurately and efficiently add and subtract whole numbers and positive decimals. Multiply and divide (using double-digit divisors)
5.N.12             whole numbers. Multiply positive decimals with whole numbers.

                   Estimate sums and differences of whole numbers, positive fractions, and positive decimals. Estimate products of whole numbers
5.N.14             and products of positive decimals with whole numbers. Use a variety of strategies and judge the reasonableness of the answer.
                                                                 Prisms and Pyramids -- October
Inv. Title & MA      Content Objectives                                       Key Vocabulary Sample Language                         Assessment           Pacing
Pages        Stan.                                                                              Objectives                           Suggestions

Inv. 1:     5.M.4    Students will determine the number of cubes that will fit volume, cube,         Listening/Speaking:             Student Activity     7 one-hour
            5.N.8    into a box made by a given pattern.                       rectangular prism,    Students will rephrase          Book p.3             sessions
Finding the 5.N.14                                                             cubic units, three-   thoughts and ideas stated by
Volume of 5.N.12     Students will develop a strategy for determining the      dimensional, solid    classmates.                     observations of
Boxes       5.N.2    volume of rectangular prisms.                                                                                   student ability to
            5.N.3                                                                                    Students will use key           apply strategy for
TE 20 - 60           Students will design boxes that hold a given number of                          vocabulary appropriately        finding volume of
                     cubes.                                                                          during discussions.             any prism

                     Students will find the volume of rectangular prisms.                            Reading/Writing:                Student Activity
                                                                                                     Students will record their      Book, p.19
                     Students will consider how the dimensions of a box                              strategy for determining the
                     change when the volume is changed.                                              volume of any box in writing. Finding the Volume
                                                                                                                                   of rectangular
                     Students will organize packages to fit in rectangular                           Students will read and answer Prisms M19-
                     boxes.                                                                          measurement questions,        M20
                                                                                                     using knowledge of volume to
                     Students will design boxes that can be completely filled                        support their answers.
                     with several different-shaped rectangular packages.

Teacher Notes: --Both Quick Images: 3-D and Estimation and Number Sense are introduced as Ten Minute Math during this first Investigation. Quick
Images supports 5.M.4, as students must realize the number of cubic units in the 3-D shape in order to rebuild it. Estimation and Number Sense support
5.N.14, 5.N.12, 5.N.2, and 5.N.3 as students can use a variety of strategies to determine their estimate. Place value rounding, number decomposition, and
calculation are methods which could be shared by students with the rest of the class in order to reinforce all these standards. --During Session 1.1, as students
fill boxes with the unit cubes, have them label their answers with cubic units, since the exact measurement is 3/4-inch cubed. Not requiring a label now, may
yield a lack of volume labels later. --Once volume strategies are solidified, after Session 1.2, it is appropriate to introduce the standard volume formulas, in full
words and with letters only, i.e.: L x W x H =V. Students should see these formulas, as they will see them later on assessment fact sheets. --Differently-Sized
packages in Session 1.4 goes beyond the 5.M.4 standard. This could be reserved for students demonstrating proficiency with determining volume, while other
students continue to build their
understanding about volume. Students demonstrating proficiency could also investigate why doubling all dimensions results in a volume that
increases eight-fold.
Inv. Title & MA      Content Objectives                                        Key Vocabulary Sample Language                     Assessment           Pacing
Pages        Stan.                                                                            Objectives                          Suggestions


Inv. 2:     5.M.4    Students will determine the volume, in cubic              length, width,     Listening/Speaking:             observation of     4 one-hour
            5.M.3    centimeters, of a small prism.                            height             Students will summarize         student strategies sessions
Using       5.N.14                                                                                measurement strategies and      for measuring
Standard    5.N.12   Students will construct units of volume.                                     defend their outcomes.          volume of
Cubic Units 5.N.2                                                                                                                 unmarked box (fill
            5.N.3    Students will choose an appropriate unit of volume to                        Writing:                        or measure sides)
TE 62 - 86           measure a large space.                                                       Students will write clear and
                                                                                                  concise plans they will follow Measuring Volume
                     Students will find the volume of a large space using                         to measure the volume of the in Cubic
                     cubic meters.                                                                classroom.                     Centimeters, M24
                                                                                                                                 (see rubric M25)




Teacher Notes: --Continue Estimation and Number Sense as the Ten-Minute Math during Investigation 2. Since the fifth grade estimation standard
addresses all operations but division, you may choose to try another operation when division estimation is introduced. --During Session, 2.1, encourage
multiple strategies such as filling or measuring dimensions. Have students share these strategies aloud and verify their accuracy. --When assessing student
ability to calculate volume, be sure ask for strategies or clear and concise work, as students could make errors in using a ruler or multiplying when they have a
clear understanding of volume.
Inv. Title & MA      Content Objectives                                        Key Vocabulary Sample Language                      Assessment         Pacing
Pages        Stan.                                                                            Objectives                           Suggestions


Inv. 3:     5.G.2    Students will compare volumes of different-shaped         pyramid, cylinder,   Listening/Speaking:            Student Activity   5 one-hour
            5.M.4    containers.                                               cone, base,          Students will orally compare   Book p.45          sessions
Volume      5.N.14                                                             height, apex,        attributes of 3-D geometric
Relationshi 5.N.12   Students will build geometric solids.                     sphere, face,        shapes, using content          End of Unit
ps Among 5.N.2                                                                 edge, vertex or      vocabulary when appropriate.   Assessment, M35-
Solids      5.N.3    Students will find volume relationships between solids    vertices, net,                                      M36
                     with the same base and height.                            surface area         Writing:
TE 88 - 111                                                                                         Students will write about a
                     Students will build a prism with three times the volume                        method they could use to
                     of a given pyramid.                                                            compare the volume of two
                                                                                                    solids.
                     Students will demonstrate the 3:1 relationship between
                     rectangular prisms and pyramids with the same base
                     and height.

                     Students will find the volume, in cubic centimeters, of
                     prisms, pyramids, cylinders, and cones.
Teacher Notes: --Continue both Quick Images: 3-D and Estimation and Number Sense as Ten Minute Math during Investigation 3. --Although this unit
addresses standards 5.G.2 and 5.M.4, these standards are not addressed entirely. It will be important to find complimentary materials in order to address the
standards completely. For 5.G.2, students will need to work with spheres in addition to the other 3-D shapes. They will also need to spend time during Session
3.1, after the shapes are made from the net designs, to describe and talk about similarities and differences between the shapes made (such as faces, edges,
and vertices). For 5.M.4, students will need experience with surface area.
Unit: Decimal Grids and Number Lines
Standard Number                                                            Description
"Big Idea" Standards
5.N.2              Demonstrate an understanding of place value through millions and thousandths
5.N.3              Represent and compare large (millions) and small (thousandths) positive numbers in various forms, such as expanded
                   notation without exponents, e.g., 9724 = 9 x 1000 + 7 x 100 + 2 x 10 + 4.
5.N.5              Identify and determine common equivalent fractions (with denominators 2, 4, 5, 10) and mixed numbers (with
                   denominators 2, 4, 5, 10), decimals, and percents (through one hundred percent), e.g., 3/4 = 0.75 = 75%.
5.N.7              Compare and order whole numbers, positive fractions, positive mixed numbers, positive decimals, and percents

5.N.12             Accurately and efficiently add and subtract whole numbers and positive decimals. Multiply and divide (using double-
                   digit divisors) whole numbers. Multiply positive decimals with whole numbers.
Supported Standards
5.N.4              Demonstrate an understanding of fractions as a ratio of whole numbers, as parts of unit wholes, as parts of a collection, and as
                   locations on the number line.
5.N.6              Find and position whole numbers, positive fractions, positive mixed numbers, and positive decimals on a number line
5.M.1              Apply the concepts of perimeter and area to the solution of problems involving triangles and rectangles. Apply formulas where
                   appropriate.
Additional Ten Minute Math Standards
5.N.14             Estimate sums and differences of whole numbers, positive fractions, and positive decimals. Estimate products of whole numbers
                   and products of positive decimals with whole numbers. Use a variety of strategies and judge the reasonableness of the answer.
                                                                  Decimals on Grids and Number Lines -- Mid-February - Mid-March
Inv. Title & MA Stan. Content Objectives                                                                       Key             Sample Language                    Assessment        Pacing
Pages                                                                                                          Vocabulary      Objectives                         Suggestions

Inv. 1:    5.N.2       Students will identify everyday uses of fractions and decimals.                          decimal point,    Listening/Speaking:             Decimals on       10 one-hour
           5.N.3                                                                                                tenths,           Students will properly          Hundredths and    sessions
Understand 5.N.5       Students will represent decimal fractions as part of an area.                            hundredths,       pronounce place value terms     Thousandths
ing and    5.N.6                                                                                                thousandths,      in order to maintain meaning.   Grids, SAB 9-14
Comparing 5.N.7        Students will read and write decimals in the tenths, hundredths, and thousandths.        ten-thousandths
Decimals   5.N.4                                                                                                                  Reading:                      Ordering
           5.N.12      Students will identify decimal, fraction, and percent equivalencies.                                       Students will identify the    Decimals, SAB 17
TE 20-80   5.N.14                                                                                                                 morpheme -th, which gives
           5.M.1       Students will represent decimals by using a number line.                                                   some place value words their
                                                                                                                                  meaning.                      Decimal Problems,
                       Students will order decimals and justify their reasoning by connecting to work done in                                                   SAB 23-25 (see
                       class.                                                                                                     Writing:                      checklist, M18)
                                                                                                                                  Students will record patterns
                       Students will compare decimals to the landmarks 0, 1/2, and 1.                                             noticed on the division table Comparing and
                                                                                                                                  in their journals.            Ordering
                       Students will interpret fractions as division.                                                                                           Decimals, M20




Teacher Notes: --Ten Minute Math during this investigation is Practicing Place Value, which supports standards 5.N.2, 5.N.3, and 5.N.12, and Estimation and Number Sense, which supports
5.N.12 and 5.N.14. --During Session 1.1, using a context for shading the grids may help provide meaning to the activity. On SAB 2-4, students will have the context that a garden is shaded.
Using this context to introduce the shading would be beneficial. Students can even color vegetation according to color described, i.e.: tomatoes would be red. This can then be carried over to
adding the decimal in investigation 2. --Session 1.3 brings students to work involving ten-thousandths, which reaches beyond the 5.N.2 standard. If students need additional practice with simpler
decimals, allow only those students who are ready to work with ten-thousandths and their equivalencies. --Students who are having trouble ordering the decimals, can continue to use decimal
grids for shading, and could place a smaller amount of decimals onto the number lines. It will also be important to present students with some whole numbers and mixed whole number-decimals.

Ordering such a set of numbers will require students look for and understand the decimal point. --On SAB p.31, students should record the total number of games played for each team so that
the winning percentage can be calculated as a fraction with total number of games played as the denominator. --During Session 1.8, students can fill in the Fraction-to-Decimal Division Table
where the denominator is 2, 4, 5, or 10, first in accordance with 5.N.5. By looking at descending patterns, students can then predict what the rest of the table will look like filled in.
Inv. Title & MA Stan. Content Objectives                                                                           Key              Sample Language                 Assessment          Pacing
Pages                                                                                                              Vocabulary       Objectives                      Suggestions

Inv. 2:      5.N.2      Students will estimate sums of decimal numbers.                                                              Listening/Speaking:             Adding Decimals, 8 one-hour
             5.N.3                                                                                                                   Students will restate the       SAB 49-50           sessions
Adding       5.N.12     Students will visual representations to add tenths, hundredths, and thousandths.                             strategies classmates use to
Decimals     5.N.14                                                                                                                  add decimals.                   End-of-Unit
                        Students will add decimals to the thousandths through reasoning about place value.                                                           Assessment, M27-
TE 82-119                                                                                                                            Reading:                        28
                                                                                                                                     Students will read and
                                                                                                                                     comprehend story problems
                                                                                                                                     in order to find appropriate
                                                                                                                                     solutions.
Teacher Notes: --Practicing Place Value is the Ten Minute Math to be used during this investigation. Although Estimation and Number Sense is not recommended as the Ten Minute Math
during this session, 5.N.14 does require estimation involving decimal computation. This could be used as a few Ten Minute Math sessions during the investigation or as a warm-up activity. --
Although the purpose of this investigation is not computational fluency, standard 5.N.12 requires fluency for addition, subtraction, and multiplication of decimals (with whole numbers). It will be
important to extend practice as necessary and to complement activities in the investigation with those involving other operations. Division of decimals is not an expectation in fifth grade. --
Students who easily play Fill Two, can be encouraged to play with the with thousandths cards on thousandths grids, or play Fill to a higher number.
Unit: How Many People? How Many Teams?
Standard Number                                                                Description

"Big Idea" Standards
5.N.8              Apply the number theory concepts of common factor, common multiple, and divisibility rules for 2, 3, 5, and 10 to the
                   solution of problems. Demonstrate an understanding of the concepts of prime and composite numbers.
5.N.9            Solve problems involving multiplication and division of whole numbers, and multiplication of positive fractions with
                 whole numbers.
5.N.12           Accurately and efficiently add and subtract whole numbers and positive decimals. Multiply and divide (using double-
                 digit divisors) whole numbers. Multiply positive decimals with whole numbers.
Supported Standards
5.P.3

5.N.14             Estimate sums and differences of whole numbers, positive fractions, and positive decimals. Estimate products of whole numbers
                   and products of positive decimals with whole numbers. Use a variety of strategies and judge the reasonableness of the answer.

Additional Ten Minute Math Standards
5.N.13             Accurately and efficiently add and subtract positive fractions and mixed numbers with like denominators and with unlike
                   denominators (2, 4, 5, 10 only); multiply positive fractions with whole numbers. Simplify fractions in cases when both the
                   numerator and the denominator have 2, 3, 4, 5, or 10 as a common factor.
                                               How Many People? How Many Teams? -- Mid-March - End of April
Inv. Title & MA        Content Objectives                                 Key             Sample Language                             Assessment        Pacing
Pages        Stan.                                                        Vocabulary      Objectives                                  Suggestions

Inv. 1:       5.N.9    Students will generate equivalent multiplication            double, halve,    Reading:                         Finding           4 one-hour
              5.N.12   expressions by multiplying one factor and dividing the      triple,           Students will read and           Equivalent        sessions
Equivalenc 5.P.3       other by the same number.                                   expression,       understand story problems in     Expressions for
e in          5.N.13                                                               equation, twice   order to generate equivalent     40 x 32, SAB 7
Multiplicatio 5.N.14   Students will create visual representations that support                      expressions related to the       (see checklist
n and                  explanations about equivalent expressions.                                    problem.                         M15)
Division
                       Students will compare equivalent multiplication                               Writing:
TE 24-44               expressions to equivalent division expressions.                               Students will write short
                                                                                                     explanations of how two
                       Students will create visual representations of equivalent                     multiplication expressions are
                       division expressions.                                                         related, using evidence from
                                                                                                     the classwork to support their
                       Students will generate equivalent division expressions.                       answer.

                                                                                             Students will write short
                                                                                             stories about equivalent
                                                                                             expressions.
Teacher Notes: --Ten Minute Math for this Investigation is Estimation and Number Sense: Closest Estimate. The estimation problems in this
investigation support 5.N.13 and 5.N.14. --During Session 1.3, students who are comfortable with doubling, halving, tripling, and thirding can work with
larger numbers. They could also be encouraged to work with numbers that once divided into parts are no longer whole numbers. This would help support
multiplication of whole numbers and decimals (5.N.12).
Inv. Title & MA      Content Objectives                                       Key              Sample Language                Assessment         Pacing
Pages        Stan.                                                            Vocabulary       Objectives                     Suggestions

Inv. 2:       5.N.9 Students will solve multi-digit multiplication problems                    Listening/Speaking:            observations       4 one-hour
              5.N.12 fluently.                                                                 Students will orally share     from small group sessions
Reviewing 5.N.14                                                                               strategies used for solving    conferences of
Multiplicatio 5.N.8 Students will compare strategies used to solve multi-                      multiplication problems.       strategies used to
n Strategies         digit multiplication problems.                                                                           multiply multi-
                                                                                               Writing:                       digit numbers
TE 46-64             Students will estimate answers to multiplication and                      Students will order work
                     division problems.                                                        appropriately and use clear    Solving
                                                                                               and concise notation to show   Multiplication
                     Students will understand the US algorithm for                             how a problem was solved.      Problems, SAB
                     multiplication.                                                                                          25

                                                                                                                               253 x 46, M25
Teacher Notes: --Ten Minute Math for this Investigation is Number Puzzles, which supports 5.N.8, and Estimation and Number Sense: Closest
Estimate, which supports 5.N.12 and 5.N.14. --During investigations where computation is taught, try meeting with like-ability groups, where strategies can
be discussed and numbers used can be similar. Struggling students can continue to work with strategies such as clustering or partial product, while more
proficient students should be encouraged to use the US standard algorithm.
Inv. Title & MA      Content Objectives                                        Key           Sample Language                 Assessment        Pacing
Pages        Stan.                                                             Vocabulary    Objectives                      Suggestions


Inv. 3:      5.N.9 Students will represent a division problem with a picture                 Listening/Speaking:            Division: How Did 7 one-hour
             5.N.12 or diagram.                                                              Students will use appropriate I Solve It?, SAB sessions
Division     5.N.8                                                                           language structures for        47-48
Strategies   5.N.14 Students will compare strategies used to solve division                  comparing division strategies.
and                 problems.                                                                                               701 divided by
Notation                                                                                     Reading/Writing:               27, M29
                     Students will solve division problems with a 2-digit                    Students will read and write
TE 66-98             divisor fluently.                                                       story problems which can be
                                                                                             solved using division.
                     Students will solve multi-step word problems.
                                                                                             Students will use clear and
                                                                                             concise notation when solving
                                                                                             division problems.

Teacher Notes: --For Ten Minute Math, continue Number Puzzles and Estimation and Number Sense: Closest Estimate. --As with multiplication, meet
with small groups to work on computation strategies. Allow struggling students to use a method such as clustering or partial quotient ("the big 7") while
encouraging more proficient students to learn the US standard algorithm for division. Struggling students can learn one strategy really well, while more
proficient students explore more strategies and look for more comparisons.
Inv. Title & MA       Content Objectives                                            Key          Sample Language                  Assessment        Pacing
Pages        Stan.                                                                  Vocabulary   Objectives                       Suggestions
Inv. 4:      5.N.9    Students will solve multi-step word problems.                 operation    Listening/Speaking:              Multiplying and   5 one-hour
             5.N.12                                                                              Students will share answers to   Dividing Large    sessions
Using the             Students will solve multi-digit multiplication problems                    problems solved and justify      Numbers, SAB
Operations            fluently.                                                                  how that answer was found.       64-65

TE 100-119            Students will solve division problems with a 2-digit                                                        End-of-Unit
                      divisor fluently.                                                                                           Assessment,
                                                                                                                                  M31-32
                      Students will use all operations fluently in order to solve
                      problems.

                   Students will compare strategies used to solve
                   problems.
Teacher Notes: --For Ten Minute Math, continue Number Puzzles and Estimation and Number Sense: Closest Estimate.
Unit: Growth Patterns
Standard Number                                                          Description
"Big Idea" Standards
5.P.2
5.P.4              Represent real situations and mathematical relationships with concrete models, tables, graphs, and rules in words and
                   with symbols, e.g., input-output tables.
5.P.5              Solve problems involving proportional relationships using concrete models, tables, graphs, and paper-pencil methods.

5.P.6              Interpret graphs that represent the relationship between two variables in everyday situations.
Supported Standards
5.G.4              Using ordered pairs of whole numbers (including zero), graph, locate, and identify points, and describe paths on the Cartesian
                   coordinate plane
5.D.2              Construct and interpret line plots, line graphs, and bar graphs. Interpret and label circle graphs.
5.M.1              Apply the concepts of perimeter and area to the solution of problems involving triangles and rectangles. Apply formulas where
                   appropriate.
5.M.3              Solve problems involving simple unit conversions within a system of measurement
Additional Ten Minute Math Standards
5.N.2              Demonstrate an understanding of place value through millions and thousandths
5.N.3              Represent and compare large (millions) and small (thousandths) positive numbers in various forms, such as expanded notation
                   without exponents, e.g., 9724 = 9 x 1000 + 7 x 100 + 2 x 10 + 4.
5.N.12             Accurately and efficiently add and subtract whole numbers and positive decimals. Multiply and divide (using double-digit divisors)
                   whole numbers. Multiply positive decimals with whole numbers.
5.N.14             Estimate sums and differences of whole numbers, positive fractions, and positive decimals. Estimate products of whole numbers
                   and products of positive decimals with whole numbers. Use a variety of strategies and judge the reasonableness of the answer.
                                                          Growth Patterns -- End of April - Mid-May
Inv. Title & MA      Content Objectives                                       Key               Sample Language                      Assessment          Pacing
Pages        Stan.                                                            Vocabulary        Objectives                           Suggestions

Inv. 1:    5.G.4     Students will plot points on coordinate grids to             rate of change,     Listening/Speaking:            The Krink, the      5 one-hour
           5.D.2     represent situations in which one quantity is changing       growth, steady,     Students will communicate      Trifoot, and the    sessions
Height and 5.P.4     in relation to another.                                      rapid, constant,    about the shape of graphs      Water Weasel,
Growth     5.P.5                                                                  slope, graph,       using common vocabulary        Saab 15-16
           5.P.6     Students will use tables to represent the relationship       axis, table,        generated onto a class list.
TE 24-60   5.M.3     between two quantities.                                      function,                                           Comparing
           5.N.2                                                                  increase,           Writing:                        Animals' Growth,
           5.N.3     Students will identify points in a graph with                decrease,           Students will write brief       M17
           5.N.12    corresponding values in a table and interpret the            coordinates,        stories that describe changes
           5.N.14    information in terms of the situation represented.           origin, point, x-   in rate of growth for a person. Fastwalker, SAB
                                                                                  axis, y-axis, x-                                    18-20
                     Students will describe the slope of graphs or parts of       coordinate, y-
                     graphs in terms of different rates of change.                coordinate

                     Students will describe the relationship of quantities in a
                     situation with a constant rate of change.

                     Students will find the value of one quantity in a
                     situation with a constant rate of change, given the
                     value of the other.

                     Students will compare situations by describing
                     differences in their graphs.

                     Students will describe a situation in which the rate of
                     change is not constant.

                     Students will compare tables, graphs, and situations
                     with constant rates of change with those in which the
                     change is not constant.
Teacher Notes: --Ten Minute Math during this investigation is Estimation and Number Sense, which supports 5.N.12 and 5.N.14, and Practicing Place
Value, which supports 5.N.2, 5.N.3, and 5.N.12. --Although the investigation refers to the slope of lines as steepness, students can still be encouraged to
discuss slope of lines. The steeper the line, the more change between points. This would mean it has a greater slope.
Inv. Title & MA      Content Objectives                                           Key               Sample Language                 Assessment          Pacing
Pages        Stan.                                                                Vocabulary        Objectives                      Suggestions


Inv. 2:     5.P.2    Students will write expressions for finding the value of     rule, variable,   Listening/Speaking:             observation of     8 one-hour
            5.P.4    one quantity in terms of the other in a situation with a     coefficient       Students will orally describe   student ability to sessions
Growing     5.P.5    constant rate of change.                                                       similarities and differences    make
Patterns    5.P.6                                                                                   between graphs they create.     connections
            5.G.4    Students will make rules that relate one variable to the                                                       between hands-
TE 62-114   5.D.2    other in situations with a constant rate of change.                                                            on materials,
            5.M.1                                                                                                                   graphs, tables,
            5.N.2    Students will use symbolic letter notation to represent                                                        and rules (2.3)
            5.N.3    the value of one variable in terms of another variable.
            5.N.12                                                                                                                  oral explanations
            5.N.14   Students will find the value of one quantity in a                                                              of graph
                     situation with a constant rate of change, given the                                                            comparisons
                     value of the other.
                                                                                                                                    The Doubling
                     Students will describe the relationship between two                                                            Penny Jar, SAB
                     quantities in situations with or without constant rates of                                                     50-51
                     change.
                                                                                                                                    Staircase
                     Students will identify points in a graph with                                                                  Towers: Jumps
                     corresponding values in a table and interpret the                                                              of 2, SAB 60-62
                     information in terms of the situation represented.
                                                                                                                                    End of Unit
                     Students will compare situations by describing                                                                 Assessment,
                     differences in their graphs.                                                                                   M22-25

                     Students will describe the slope of graphs or parts of
                     graphs in terms of different rates of change.

                     Students will describe and compare tables, graphs,
                     and situations with constant rates of change with those
                     in which the rate of change is not constant.
Teacher Notes: --Continue Estimation and Number Sense and Practicing Place Value as the Ten Minute Math during this investigation.                      --Students
should be challenged to write rules with variables that would work for any number in these situations.
Unit: Data About Us


5.D.1 Given a set of data, find the median, mean, mode, maximum, minimum, and range, and apply to solutions of problems.


5.D.2 Construct and interpret line plots, line graphs, and bar graphs. Interpret and label circle graphs.

5.N.12 Accurately and efficiently add and subtract whole numbers and positive decimals. Multiply and divide (using double-digit divisors)
whole numbers. Multiply positive decimals with whole numbers.
Invest Title &   MA Std   Content Objective                 Key Vocabulary Language Resources                                         Assessment Suggestions   Pacing
Page No.                                                                        Language objectives for this investigation should
                                                                                include reading, writing, speaking and listening. A
                                                                                sample for this investigation might be:

Invest 1        5.D.1     Students will be able to identify data set,           Students will be able to elaborate on and extend Mathematical Reflection, p. 29 5 days
Looking at Data 5.D.2     maximum, minimum, range,          numerical data,     other students descriptions about data
                          mode, and median (5.D.1)          range,              represented in a line plot (S.3.60)               District Formative available on
Problems                                                                                                                          Galileo:
                                                            maximum,
1.1                                                                             Students will be able to describe how to find the 5.D.1 Median, Mode, and
                          Students will be able to          minimum,
1.2                                                                             median of a set of numerical data using           Range
                          interpret line plots (5.D.2)      mode, median,       appropriate math vocabulary (S.1.28)
1.4                                                         table, line plot,
                                                            bar graph,
pp. 6-29
                                                            typical



Teacher's Notes: Problem 1.3 is not recommended here but may be used for enrichment. In order to align Problem 1.4 with
standard 6.D.2, students should create line plots for the data that is represented in the tables and bar graphs. Problem 1.5
is not recommended since bar graphs have been addressed in earlier grades. However, teachers may use Problem 1.5 with
students who demonstrate a need for review of bar graphs.
Invest Title &   MA Std   Content Objective                  Key Vocabulary Language Resources                                       Assessment Suggestions        Pacing
Page No.                                                                       Language objectives for this investigation should
                                                                               include reading, writing, speaking and listening. A
                                                                               sample for this investigation might be:

Invest 2 Using   5.D.1    Students will be able to           stem, leaves,     Students will be able to use appropriate              Mathematical Reflection, p. 48 3 days
Graphs to        5.D.2    construct stem and leaf plots      stem-and-leaf     mathematical vocabulary and concepts to               #1
Explore Data              for data sets with a large range   plot, key,        describe data represented in a stem and leaf
                                                             interval, modal   plot (S.1.28)
                          (5.D.2)
Problem                                                      interval
2.1

pp 30-48




Teacher's Notes: Students will need practice constructing and interpreting stem-and-leaf plots. Teachers should extend
this problem identifying the mode, median, maximum, minimum, and range of data sets represented in stem-and-leaf plots.
You can find additional items in the ACE questions for Investigation 2 on pp.40-44 and in the CMP2 Additional Practice and
Skills workbook, p. 120.
Problem 2.2 is not recommended since it deals with "back-to-back stem and leaf plots" which are not part of the state
standard. This problem may be appropriate for enrichment.
Problems 2.3 and 2.4 are not recommended in this guide because sufficient work with relationships between two variables
and the first quadrant of the Cartesian coordinate plane is included in the unit "Variables and Patterns".
Invest Title &   MA Std   Content Objective             Key Vocabulary Language Resources                                   Assessment Suggestions         Pacing
Page No.                                                              Language objectives for this investigation should
                                                                      include reading, writing, speaking and listening. A
                                                                      sample for this investigation might be:

Invest 3         5.D.1    Students will be able to        mean        Students will be able to describe the steps in the District formative available on   4 days
                 5.N.12   calculate the mean for a set of             algorithm for calculating mean using appropriate Galileo:
Problems                  data (5.D.1)                                math vocabulary. (S.1.28)                          5.D.1 Mean
3.1
3.3                                                                                                                         End-of-Unit Assessment

pp. 49-63

Teacher's Notes: Calculating the mean provides a chance for teachers to review division algorithms with students who
need some reteaching of this skill. Students who are ready for enrichment can work on problem 3.2.
Additional notes about this unit:

Standard 5.D.2 is only partially covered in this unit since circle graphs are not included in any of these investigations.
However, teachers can use ACE item #12 from page 44 to give students the chance to interpret a circle graph.

								
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