7-8th Grade Suggested Activities

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7-8th Grade Suggested Activities Powered By Docstoc
					  Suggested Teaching and Learning Activities taken from the
  Michigan Curriculum Framework Project


                                        SEVENTH-EIGHTH GRADE


                                    I. Patterns, Relationships and Functions

  1. Students recognize similarities and generalize patterns; use patterns to create models and
  make predictions; describe the nature of patterns and relationships, and construct representations
  of mathematical relationships. (Patterns)


   In the middle years,        Grade                          By engaging in activities such as ...
    students have the          Level
     opportunity to…

1. describe, analyze and        5          recognizing patterns arising
generalize patterns arising                 from everyday situations                          June

in a variety of contexts and                (e.g., geometric patterns in a                1    2       3    4     5

express them in general                     quilt, symmetry patterns in
terms.                                      art and nature, numerical          6    7     8    9      10   11    12


                                            patterns in a calendar) and       13   14    15   16      17   18    19

                                            describing them verbally and
(I.1.M1, 4) Describe,                       symbolically.
                                                                              20   21    22   23      24   25    26

analyze, and generalize                                                       27   28    29   30
algebraic and geometric
patterns




                               5-6         exploring numeric, algebraic and geometric examples of patterns such
                                            as patterns of repeating decimals (numeric), patterns in Pascal’s
                                            triangle (algebraic)


                                               1/11 = .090909...
                                               2/11 = .181818...
                                               3/11 = .272727...
                                                   .
                                                   .
                                                       .

                                           or patterns of change in the area of a square as the length of the side
                                            changes (geometric).




  Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
  1/01 DRAFT                                                                                               7-1
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


 In the middle years,        Grade                          By engaging in activities such as ...
  students have the          Level
   opportunity to…



                                                                                                      1
                            7-8               1/7    =   .142857142857...
                                                                                                  1       1
                                              2/7    =   .285714285714...
                                              3/7    =   .428571428571...                     1       2       1
                                              4/7    =   ???
                                                                                          1       3       3       1
                                                    .
                                                    .                                1        4       64 1
                                                    .                            1        5       10 10 5 1



                            7-8         extending patterns by adding the next few elements or by supplying
                                         missing elements.
                             7-8

                                              T1 = 1, T2 = 3, T3 = 6, T4 = 10… T9 = ?




                            All         creating and describing original patterns; modeling patterns using
                                         physical objects such as counters, attribute pieces, cubes, etc.; and
                                         providing reasons to explain their choices when creating or extending
                                         patterns.


                                        generalizing a rule to describe a
                                         pattern, for example a rule for             S =1
                            7-8                                                       1
                                         forming the next Fibonacci
                                         number or a rule for determining            S =4=1+3
                                                                                      2
                                         the number of degrees in the                S3 = 9 = 1 + 3 + 5
                                         angles of a polygon.
                                                                                     S4 = 16 = 1 + 3 + 5 + 7

                                                                                     S10 = ?




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                                        7-2
 Suggested Teaching and Learning Activities taken from the
 Michigan Curriculum Framework Project


   In the middle years,       Grade                          By engaging in activities such as ...
    students have the         Level
     opportunity to…

2. represent and record        5         using a table to show the number of handshakes when every person in
 patterns in a variety of                 the room shakes hands with everyone else.
 ways including tables,
 charts and graphs, and                  using a graph to show the total distance traveled when moving at a
 translate between various    All         fixed rate of speed.
 representations.
                               6         writing a mathematical expression to describe the total receipts after
 (I.1.M2) Represent and                   selling both adults’ and children’s tickets.
 record patterns in a
 variety of ways and          7-8
                                         drawing a graph to represent a mathematical equation.
 relate between
 representations              7-8        writing an equation to describe a graph.
 data tables
 ordered pairs
 linear algebraic             All        matching a graph to a table of values.
 expressions
 geometric                               organizing data in tables in ways that highlight underlying patterns.
                              6-8
representations
    symmetric patterns
    tessellations




 3. use patterns and their
                                                                                             Tower of Hanoi
generalizations to make and 6-8         • analyzing patterns and using
                                                                                           Number      Number
justify inferences and                    them to predict unknown
                                                                                           of Discs    of Moves
predictions.                              outcomes, for example the
                                          number of diagonals in an                            1            1
(I.1.M3) Make and justify                 n-sided polygon or the tenth term                    2            3
predictions                               in a sequence of numbers.                            3            7
                                                                                               4           15
                                          Tower of Hanoi: Given three posts and disks          •            •
                                          of varying diameter, how many moves are              •            •
                                          needed to transfer n disks from one post to          •            •
                                          another? Only one disk may be moved at a
                                          time, and larger disks may not be placed on
                                          smaller disks.




 Huron County Mathematics Curriculum                                                Seventh-Eighth Grade
 1/01 DRAFT                                                                                                 7-3
     Suggested Teaching and Learning Activities taken from the
     Michigan Curriculum Framework Project


      In the middle years,         Grade                         By engaging in activities such as ...
       students have the           Level
        opportunity to…


                                  All        testing and evaluating conjectures based on observed patterns.

                                             using patterns to solve problems, such as comparing total earnings
                                   6-8        over time to determine which of two wage plans is most beneficial.



                                               Pat is going to make name tags for the science fair participants, and she has
                                               been experimenting to see how many name tags she can cut from a sheet of
                                               paper. How many will she be able to get if she continues folding the paper
                                               as shown? What is happening to the size of the name tags as she continues
                                               to fold? What is a reasonable number of name tags to make from a standard
                                               sheet of paper? Explain your recommendation.




4.   explore and describe                    generating data that exhibit linear
     visual and numeric           5-6         patterns, for example by                         Phone Rate:
     patterns, including linear               investigating the relationship                Minutes Cost
     expressions, near-linear                 between diameters and
     patterns, and symmetric                  circumferences of circles or the                   1        $0.70
     and spatial patterns.                    relationship between the cost of a                 2        $0.85
                                              long distance telephone call and
                                              the length of the call;                            3        $1.00
(I.1.M1, 4) Describe,                                                                            4        $1.15
analyze, and generalize                                                                          5        $1.30
algebraic and geometric
patterns


                                  7-8       • comparing patterns and classifying them as linear (e.g., the
                                              temperature of a liquid being warmed at a steady rate), near linear
                                              (e.g., the cost of postage vs. the weight of the letter), or not linear
                                              (e.g., the area of a square vs. the length of its side, or the number of
                                              cells present in a culture if the cells divide every 24 hours.)



     Huron County Mathematics Curriculum                                              Seventh-Eighth Grade
     1/01 DRAFT                                                                                                    7-4
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


 In the middle years,        Grade                                            By engaging in activities such as ...
  students have the          Level
   opportunity to…


                            7-8        • finding or creating examples to illustrate various types of patterns.




                                                                                  Colored Faces of Unit Cubes
                                                                              #      6      3       2       1          0
                                                 Number of Cubes on an Edge

                                                                              1      1      -       -       -          -
                                                                              2      -      8       -       -          -
                                                                              3      -      8       12      6          1
                                                                              4      -      8       24      24         8
                                                                              5      -      8       36      54         27
                                                                              6      -      ?       ?       ?          ?




Huron County Mathematics Curriculum                                                             Seventh-Eighth Grade
1/01 DRAFT                                                                                                                  7-5
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project




 In the secondary years,     Grade                          By engaging in activities such as …
   students have the          Level
   opportunity to …
5. use patterns and
     generalizations to      5-6          identifying and describing patterns in open-ended situations, such
     solve problems and                    as the patterns found in a multiplication table or in Pascal’s
     explore new                           triangle.
     content.
                                          exploring situations in which looking for patterns and making
(I. 1.M5) Use patterns                     tables, charts and graphs leads to the solutions of interesting
and generalizations to                     problems.
solve real-world
problems
                             7-8
                                                   Locker problem: 1000 lockers, 1000 students …
                                                       S1 opens every locker
                                                       S2 closes (changes) every second locker
                                                       S3 changes every third locker
                                                       S4 changes every fourth locker
                                                       … and so on.
                                                   Which lockers will be open in the end?




Huron County Mathematics Curriculum                                          Seventh-Eighth Grade
1/01 DRAFT                                                                                          7-6
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


In the secondary years,      Grade                          By engaging in activities such as …
  students have the           Level
  opportunity to …

                                         describing the connections among patterns observed in various
                                          contexts, for example, recognizing the triangular numbers when they
                             6-8
                                          occur in seemingly dissimilar situations.




Huron County Mathematics Curriculum                                          Seventh-Eighth Grade
1/01 DRAFT                                                                                          7-7
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                  I. Patterns, Relationships and Functions

2. Students describe the relationships among variables; predict what will happen to
one variable as another variable is changed, analyze natural variation and sources of
variability, and compare patterns of change. (Variability and Change)


  In the middle years,     Grade                        By engaging in activities such as ...
   students have the       Level
    opportunity to…

1. identify and         6-8             • exploring mathematical patterns and physical situations that lead
   describe the nature                    them to encounter variables in a natural way—such as the effect that
   of change; recognize                   varying the amount of light or water has on the growth of a
   change in more                         seedling—and describing how one variable changes in relation to
   abstract and                           another.
   complex situations    5
   and explore                          • charting and graphing the growth patterns of several seedlings and
   different kinds of                     observing the natural variation in height even when the plants have
   change and patterns                    been exposed to identical growing conditions.
   of variation.

(I.2.M1, 2) Recognize
and explore changes
and generalize a rule




                            7-8         • conducting experiments in
                                          which they systematically
                                          manipulate variables, such as
                                          using a balance beam to study
                                          relationships between weights
                                          and their distances from the
                                          fulcrum or projecting
                                          shadows on a screen to study
                                          relationships between shadow
                                          size and the object’s distance
                                          from the screen.




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                         7-8
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,     Grade                                             By engaging in activities such as ...
   students have the       Level
    opportunity to…


                            7-8        • examining graphs, such as a graph of distance vs. time, and
                                         explaining the changes that the graph represents.

                                                                                Trip t o Sault St e. Marie
                                                150
                                             ("The Soo")




                                                Dist ance (miles)




                                                 0
                                              (Home)
                                                                                                                           1 2 noo n
                                                                    7 a.m.
                                                                                            Time ( hours)




Huron County Mathematics Curriculum                                                                 Seventh-Eighth Grade
1/01 DRAFT                                                                                                                         7-9
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project




  In the middle years,      Grade                       By engaging in activities such as ...
   students have the        Level
    opportunity to…

2. connect an initial       7-8         conducting experiments and
   state to a final state                observations and generating
   and generalize a rule                 data to explore relationships
   that describes a                      among dependent and
   pattern of change.                    independent variables in
                                         diverse contexts, for example
((I.2.M1, 2)                             by measuring the length of a
   Recognize and                         spring or rubber band as
explore changes and                      various weights are
generalize a rule point                  suspended from it.




                            6-8         describing what changes and what remains unchanged during
                                         explorations such as transforming geometric shapes using slides,
                                         flips, and turns, or adjusting the class’s test scores by adding 12
                                         points to each person’s score.


                                                                              The polygon on the right has been
                                                                              reflected over the line to form the
                                                                              image on the left. What changed and
                                                                              what remained the same after the
                                                                              reflection?



                            All         playing games of “guess the rule” to develop generalizations about
                                         patterns of change, giving reasons for their guesses, and expressing
                                         the rules verbally and symbolically.




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                          7-10
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,     Grade                         By engaging in activities such as ...
   students have the       Level
    opportunity to…

3. begin to investigate     6-8         using patterns of change to make predictions and to answer questions
   applications in                       such as:
   bivariate data and                     - If the observed pattern continues, what will be the value of the
   linear relationships,                     variable one year from now?
   and explore                            - If the observed pattern has been unchanged, what was the value
   questions of what
   will happen to one                        of the variable one month ago?
   quantity if another
   variable is changed.
                                                Matt shared a secret with his friend, Paul, and made Paul
                                                promise not to tell anyone. But the next day, Paul told two
(I.2.M3) Investigate
                                                people; and the day after that, those two people each told
bivariate data and                              two people. If this pattern continues, how long will it be
linear relationships                            before the entire school (enrollment 1200) knows the secret?
and explore patterns
of variability
variable changes            6-8         conducting experiments to generate data, for example by measuring
                                         and comparing the circumferences and diameters of various round
                                         objects.
                            6-8
                                        using technology, especially spreadsheets and calculators, to generate
                                         data and explore patterns of variability.

                                                   Circle relationships:
                                            radius       diameter         area
                                               1             3.14          3.14
                                               2             6.28         12.56
                                               3             9.42         28.26
                                               4           12.56          50.24
                                               5           15.70          78.50
                                               6           18.84         113.04
                                               7           21.98         153.86
                                               8           25.12         200.96
                                               9           28.26         254.34
                                             10            31.40         314.00
                                             11            34.54         379.94
                                             12            37.68         452.16




Huron County Mathematics Curriculum                                          Seventh-Eighth Grade
1/01 DRAFT                                                                                            7-11
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,     Grade                                                            By engaging in activities such as ...
   students have the       Level
    opportunity to…

4. represent variability    All         expressing patterns of change in several different ways, for example
   or change by                          by constructing a graph to show the total snowfall accumulation
   ordered pairs,                        during the winter or by developing a table to show the number of
   tables, graphs and                    pieces of pizza that result when a pizza is sliced by one, two, three,
   equations.                            etc., cuts.

(I.2.M4) Represent
variability and change                                                                                           M arque tte , MI — Total Snowfall to Date

in a variety of ways                               160
ordered pairs
                                                   140
tables
                                                   120
graphs
equations                                          100
                                          inches




                                                    80

                                                    60                                                                                                                                                                                                     1994-95

                                                    40                                                                                                                                                                                                     Average Year

                                                    20

                                                     0
                                                                                    3-Jan
                                                                                            6-Jan
                                                                                                    9-Jan




                                                                                                                                                                           2-Feb
                                                                                                                                                                                   5-Feb
                                                                                                                                                                                           8-Feb
                                                                                                            12-Jan
                                                                                                                     15-Jan
                                                                                                                              18-Jan
                                                                                                                                       21-Jan
                                                                                                                                                24-Jan
                                                                                                                                                         27-Jan
                                                                                                                                                                  30-Jan




                                                                                                                                                                                                   11-Feb
                                                                                                                                                                                                            14-Feb
                                                                                                                                                                                                                     17-Feb
                                                                                                                                                                                                                              20-Feb
                                                                                                                                                                                                                                       23-Feb
                                                                                                                                                                                                                                                26-Feb
                                                                                                                                                                                                                                                         1-Mar
                                                                                                                                                                                                                                                                  4-Mar
                                                                                                                                                                                                                                                                          7-Mar
                                                         25-Dec
                                                                  28-Dec
                                                                           31-Dec




                                                                                                                                                                                                                                                                                  10-Mar
                                                                                                                                                                                                                                                                                           13-Mar
                                                    (The graph above shows the weather bureau’s reported total-snowfall-to-date
                                                    in Marquette, Michigan, during the 1994-95 winter as well as the “average” or
                                                    “normal” seasonal snowfall. One of the graphs appears nearly linear while the
                                                    other is very irregular. What accounts for this difference in the two graphs?
                                                    What information is concealed by the graph of average snowfall? In what
                                                    ways was 1994-95 “average” and in what ways was it not?)




                           7-8

                                        describing plausible situations that could have produced a given
                                         graph or pattern.




Huron County Mathematics Curriculum                                                                                                                                   Seventh-Eighth Grade
1/01 DRAFT                                                                                                                                                                                                                                                       7-12
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,     Grade                        By engaging in activities such as ...
   students have the       Level
    opportunity to…

5. differentiate            7-8         exploring and classifying examples of relationships, including:
   between functions                      - linear relationships (the relationship between Fahrenheit and
   and relations such as                    Celsius temperatures).
   linear vs. not linear                  - quadratic relationships (the relationship between the area of a
   or continuous vs.
   non-continuous.                          square and the length of its side).
                                          - inverse relationships (the relationship between speed and time
                                            when traveling a fixed distance).
(I.2.M5) Explore and
                                          - patterns of growth or decay (the number of cells present at the
classify examples of
relationships                               end of the month if the cells divide every 48 hours).
linear                                    - non-continuous functions (the cost of postage to mail letters of
quadratic                                   varying weight).
                            7-8
inverse
patterns of growth and                  classifying observed patterns of change as linear or non-linear.
decay                       6-8
non-continuous                          expressing linear relationships graphically, algebraically and
functions                                verbally.



6. continue to explore      6-8         developing and manipulating physical and mathematical models of
   relationships arising                 real phenomena (for example, the period of a swinging pendulum or
   from interesting                      the rebound height of a bouncing ball) and describing the associated
   contexts, and use                     variables and their relationships;
   variables and
   relationships to         All         using a spreadsheet to compare the advantages of investment plans
   solve mathematical                    offering different rates of interest.
   problems.

(I.2.M6) Use
relationships to solve
real-world problems




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-13
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                      II. Geometry and Measurement

1. Students develop spatial sense, use shape as an analytic and descriptive tool, identify
characteristics and define shapes; identify properties and describe relationships among shapes.
(Shape and Shape Relationships)


  In the middle years,      Grade                       By engaging in activities such as …
   students have the        Level
    opportunity to…

1. distinguish among                    exploring diagrams and two- and three-dimensional objects in order
   shapes and                            to:
   differentiate between                  - describe the shapes of the objects.
   examples and non-                      - name common shapes found in the objects.
   examples of shapes                     - identify special characteristics and properties of shapes.
   based on their
   properties; generalize                 - classify objects according to their shapes.
   about shapes of                        - differentiate between examples and non-examples of particular
   graphs and data                           shapes.
   distributions.                         - develop hierarchies of related shapes.

(II.1.M1) Explore
three-dimensional
figures
rectangular prisms
     edges
     faces
     cubes
     spheres

(II.1.M1) Generalize
about shapes of                                                              Identify and describe all the
graphs and data                                                              shapes that you recognize in
distribution                                                                 the diagram to the left.




                            5           going on a geometry walk in the neighborhood and identifying and
                                         recording shapes that are observed.

                             7-8       exploring data sets and giving informal descriptions of the data
                                         distributions, such as symmetric vs. skewed distributions, positive
                                         associations, exponential growth, inverse variations, etc.




Huron County Mathematics Curriculum                                       Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-14
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project




  In the middle years,     Grade                        By engaging in activities such as ...
   students have the       Level
    opportunity to…

2. generalize the           5           using paper folding, geoboards, and other models to generate and test
   characteristics of                    hypotheses about families of shapes and their properties (e.g., the
   shapes and apply                      opposite sides of the rectangles are always equal).
   their generalizations
   to classes of shapes.    6-8         exploring physical models of shapes to determine their symmetries,
                                         and describing and demonstrating the symmetries of two- and three-
(II.1.M2, 3) Explore                     dimensional shapes.
characteristics and
hierarchies of two-                     utilizing computer drawing programs to investigate shapes and shape
and three-dimensional       All
                                         relationships.
shapes
     circle                 7-8         drawing or constructing and naming shapes that satisfy given criteria
     sector
                                         such as “a four-sided figure with one pair of parallel sides.”
     altitude
     n-gon
     polygon
     trapezoid




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-15
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,     Grade                         By engaging in activities such as ...
   students have the       Level
    opportunity to…

3. derive                   All         developing and verifying generalizations about geometric properties
   generalizations                       using informal means such as paper folding, reflecting with a mira,
   about shapes and                      or manipulating shapes with a dynamic computer drawing program.
   apply those
   generalizations to                            The number of faces, vertices, and edges
   develop                                       of a polyhedra is related (v+f-e=2)
   classifications of
   familiar shapes.
                                                 Polyhedra           Vertices     Faces     Edges

(II.1.M2, 3) Explore                             cube                    8          6        12
characteristics and                              octahedron              6          8        12
hierarchies of two-                              tetrahedron             4          4         6
and three-dimensional                            dodecahedron            ?         12         ?
                                                 icosahedron             ?         20         ?
shapes
     circle                 7-8
     sector
     altitude                           developing hierarchies for families of shapes, such as drawing a
     n-gon                               Venn diagram to show the relationships among quadrilaterals,
     polygon                7-8          parallelograms, rectangles, squares, rhombi, trapezoids, etc.
     trapezoid
                                        investigating when a characteristic of a shape is sufficient to define a
(II.1.M4,5 and II.2.M3)                  shape, such as determining whether congruent diagonals guarantees
Construct, combine,                      that a quadrilateral is a square.
dissect, and
transform shapes
     size transformation
     enlargement
     reduction




Huron County Mathematics Curriculum                                             Seventh-Eighth Grade
1/01 DRAFT                                                                                             7-16
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,     Grade                         By engaging in activities such as ...
   students have the       Level
    opportunity to…

4. construct familiar       5-6         constructing shapes using simple materials such as straws,
   shapes using                          toothpicks, pattern blocks, geostrips, or geoboards.
   coordinates or
   appropriate tools
   (including
   technology); sketch
   and draw two- and
   three-dimensional
   shapes.

(II.1.M4,5 and II.2.M3)
Construct, combine,
dissect, and
                                             “How many quadrilaterals can be made on a 3-peg by 3-peg geoboard?”
transform shapes
     size transformation
     enlargement            All         constructing shapes using various tools including Euclidean tools
     reduction                           (compass and straight edge), miras or other reflection devices,
                                         computer drawing tools, coordinate graphs, paper folding, and
                                         tangram pieces.
                            7-8         constructing shapes that conform to given specifications (e.g., make a
                                         trapezoid with two right angles), and determining when it is
                                         impossible to create a certain shape (e.g., make a parallelogram with
                                         only two right angles).


5. combine, dissect         6           combining familiar shapes to produce more complex shapes and
   and transform                         designs.
   shapes.

(II.1.M4,5 and II.2.M3)
Construct, combine,
dissect, and                                                                     Can you combine all seven tangram
transform shapes                                                                 pieces to form a rectangle? a
     size transformation                                                         parallelogram? a triangle? a
     enlargement
     reduction




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                             7-17
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,     Grade                          By engaging in activities such as ...
   students have the       Level
    opportunity to…

                                       A domino is formed when two congruent squares are connected along a side.
                                       Similarly, a triomino is formed by three connected squares. How many distinct
                                       shapes can you find for each of the following?

                                       Name of Shape Number of Connected Squares         Number of Distinct Arrangements
                                        Domino                 2
                                        Triomino               3
                                        Tetromino              4
                                        Pentomino              5




                            7-8         separating complex shapes into simpler component shapes.
                            7-8
                                        predicting and investigating the shapes that result when two objects
                                         intersect, such as when a plane slices through a cube.
                            7-8         determining which shapes will tessellate the plane, and creating
                                         original tessellations.
                             6
                                        identifying shapes found in artistic designs, especially designs from
                                         different cultures.
                             7-8
                                        using various plane and solid shapes to create original designs or
                             7-8         sculptures.

                                        tracing shapes and images of those shapes after sliding, flipping or
                                         turning them.




Huron County Mathematics Curriculum                                             Seventh-Eighth Grade
1/01 DRAFT                                                                                                7-18
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project




  In the middle years,      Grade                       By engaging in activities such as ...
   students have the        Level
    opportunity to…

6. generalize about the     5-6         drawing, tracing, folding, and cutting a variety of shapes and using
   common properties                     them to develop concepts of congruent and similar shapes.
   of similar, congruent,
   parallel and
   perpendicular shapes
   and verify their
   generalizations
   informally.


(II.1.M6) Explore
examples of                                                                Cut the tangram diagram into seven
congruency,                                                                pieces. How do the size and shape of
similarity, parallelism,                                                   the two small triangles compare to the
perpendicularity, and                                                      size and shape of the three larger
angle relationships                                                        triangles?
     angle
     complimentary                                                         Are any of the triangles similar? Are
     corresponding                                                         any of the triangles congruent? Give
     interior                                                              an argument to prove your answer.
     supplementary
     verticaL



                            7-8         exploring physical objects such as globes or models of geometric
                                         solids to develop concepts of perpendicular and parallel lines and
                                         planes.
                            7-8
                                        identifying examples of congruence, similarity, parallelism and
                                         perpendicularity from the environment and from geometric objects.




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                         7-19
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,   Grade                              By engaging in activities such as ...
   students have the     Level
    opportunity to…
7. use shape, shape       7-8              exploring the connections between shape and numbers by
    properties and shape                    investigating figurate numbers (triangular, square...), primes vs.
    relationships to                        composite numbers, the shape of data distributions, the shapes of
    describe the                            graphs, etc.
    physical world and
    to solve problems.    7-8              using isometric dot paper to draw two-dimensional representations of
                                            three-dimensional objects, and constructing three-dimensional
(II.1.M7) Use shape                         models from two-dimensional renderings.
properties and
relationships to solve                                    
real-world problems                                    
                                                          
                                                       
                                                          
                                                       
                                                          
                                                       
                                                          
                                                       
                            5-7
                                           investigating diagrams of shapes as viewed from different angles and
                                            constructing 3-dimensional models to correspond to the 2-
                                            dimensional views.




                            7-8
                                           drawing shapes on a coordinate grid and multiplying the coordinates
                                            of selected points by a constant to produce one-way [e.g., (x,y)  (x,
                                            2y)] and two-way [e.g., (x,y)  (3x, 3y)] distortions.




                           All


                                          conducting open-ended investigations involving shapes, such as
Huron County Mathematics Curriculum                                                         Seventh-Eighth Grade
1/01 DRAFT                                                                                                           7-20
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,     Grade                        By engaging in activities such as ...
   students have the       Level
    opportunity to…
                                         coloring maps or finding all the pentominoes and determining which
                                         pentominoes can be folded to make an "open box."




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-21
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                      II. Geometry and Measurement

2. Students identify locations of objects, location relative to other objects, and the effects of
transformations (e.g., sliding, flipping, turning, enlarging, reducing) on an object. (Position)


  In the middle years,                                      By engaging in activities such as ...
   students have the       Grade
    opportunity to…        Level

1. locate and describe      5-6           locating objects in relation to other objects (e.g., one meter to the
   objects in terms of                     right of the window; two feet above the floor...).
   their position,
   including compass         5-6
                                          describing positions in terms of compass directions.
   directions, Cartesian
   coordinates, latitude     5-6          reading and following maps.
   and longitude, and
   midpoints.
                             5-6          drawing maps and giving verbal directions for locating objects
(II.2.M1) Locate and                       (e.g., how to get to the library from the school).
describe points in the
                             7-8
Cartesian plane                           specifying Cartesian coordinates for points on a grid, and locating
coordinate system                          points given their coordinates.

                                          locating and describing points in three-dimensional space using
                                           positive (x, y, z) coordinates.
                           7-8
                                                               z
                                                                     (x,y,z)


                                                                         y

                            6-8                         x

                                          locating and describing points on a globe using latitude and
                                           longitude.




Huron County Mathematics Curriculum                                            Seventh-Eighth Grade
1/01 DRAFT                                                                                            7-22
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project




  In the middle years,                                  By engaging in activities such as ...
   students have the        Grade
    opportunity to…         Level

2. locate and describe      6           locating and describing points, lines, or physical objects in geometric
   objects in terms of                   terms such as "on the perpendicular bisector" or "at the intersection".
   their orientation and
                            6-8
   relative position,                   using physical objects, graphs or sketches to represent positions such
   including                             as "parallel to," "at the midpoint," or "equidistant from".
   coincident,
   collinear, parallel,     5           using a geometry drawing program to construct objects such as the
   perpendicular;                        midpoint of a segment or the perpendicular to a line.
   differentiate            7-8
   between fixed (e.g.,                 using miras or mirrors to determine whether or not figures or objects
   N-S-E-W) and                          have bilateral symmetry.
   relative (e.g., right-    7-8
   left) orientations;                  using tracings and models to determine whether a figure or object has
   recognize and             5-6         rotational symmetry.
   describe examples
   of bilateral and                     demonstrating that fixed orientations or directions (e.g., two blocks
   rotational symmetry.                  north of my house) yield consistent results, while relative orientations
                            All          (e.g., go three miles to the right) can result in different outcomes.
(II.2.M2) Locate and
describe objects                        examining a set of test scores to determine whether the score
using the terms                          distribution is symmetrical.
coincident, collinear,
parallel,
perpendicular;
describe examples of
bilateral and
rotational symmetry




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                         7-23
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,                                   By engaging in activities such as ...
   students have the        Grade
    opportunity to…         Level

3. describe                             using physical objects, geometric sketches, and computer drawing
   translations,                         tools to explore the effects of translating, reflecting, or rotating a
   reflections, rotations                point or object.
   and dilations using
   the language of                      drawing, both with paper and pencil and with computer drawing
   transformations, and                  tools, the resulting image when given a description of the
   employ                                transformation of a shape or object (e.g., translate 10 centimeters at
   transformations to
   verify congruence of                  an angle of 30° from the horizontal; reflect over the y-axis; rotate 60°
   figures.                              about the vertex).
                                                                                         Z


(II.2.M3) Use
transformations and
verify congruence
dilations
                                                                                Y
                                                                    O




                                                                N


                                                     A
                                                                        M                           X




                                                     Describe the transformation that projected triangle MNO onto
                                                         triangle XYZ.

                                        describing the transformation that produced an observed result when
                                         shown a drawing of a shape and its image.



                                                                                    Describe one motion that will transform:
                                                                                          E to F              B to E
                                                                                          B to D              C to E
                                                                                          A to C              F to A
                                                 A          B               C
                                                                                          B to A              C to F

                                                                                    Describe a combination of two
                                                                                    transformations that will transform:
                                                 D          E           F                  A to C               F to A
                                                                                           D to B               D to C




Huron County Mathematics Curriculum                                             Seventh-Eighth Grade
1/01 DRAFT                                                                                                  7-24
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,                                  By engaging in activities such as ...
   students have the       Grade
    opportunity to…        Level

5. use concepts of          5-6         engaging in treasure hunts that require following a series of
   position, direction                   directions to locate a hidden object.
   and orientation to
   describe the             5-6
                                        drawing treasure maps that lead to selected hiding places.
   physical world and
   to solve problems.       All         giving written or oral instructions to lead another student to a desired
                                         location.
(II.2.M5)  Use
position, direction,        7-8
                                        viewing objects from different locations and drawing and describing
and orientation to
                                         how the perspective changes.
solve real-world            7-8
problems
                                        projecting images of shapes and determining how dilations can be
                                         used to enlarge or reduce the original shape.
                            7-8
                                        constructing three-dimensional models from two-dimensional
                                         drawings, and drawing two-dimensional representations of three-
                                         dimensional objects.




Huron County Mathematics Curriculum                                         Seventh-Eighth Grade
1/01 DRAFT                                                                                         7-25
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                      II. Geometry and Measurement

3. Students compare attributes of two objects or of one object with a standard (unit) and
analyze situations to determine what measurements should be made and to what level of
precision. (Measurement)


  In the middle years,     Grade                        By engaging in activities such as ...
   students have the       Level
    opportunity to…

1. select and use           5-6         measuring everyday objects using:
   appropriate tools;                     - other objects for comparison (e.g., index cards, cubes, string).
   measure objects                        - standard units, both common and metric.
   using standard units                   - common measuring tools (rulers, protractors, thermometers,
   in both the metric
   and common                               scales, graduated cylinders, etc.)
   systems, and                           - rulers that are shorter than the object to be measured (i.e. must be
   measure angles in                        applied more than once) and "broken rulers" whose scales do not
   degrees.                                 begin at zero.
                            All
(II.3.M1, 2) Identify                   engaging in hands-on activities to develop concepts of measurements
and use appropriate                      in various dimensions, such as using string to surround an object
tools and units                          (perimeter), using tiles or paper to cover a surface (area), or using
                                         sand to fill a container (volume).
                            All
                                        measuring common objects using rulers, trundle wheels, balances,
                                         scales, stop watches, clocks, thermometers, graduated cylinders,
                                         protractors, and other appropriate measurement tools.
                            All

                                        reading and interpreting measuring devices such as meters, scales of
                                         various design, calipers, etc.




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-26
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                       By engaging in activities such as ...
   students have the        Level
    opportunity to…

2. identify the attribute   All         identifying the quantity to be measured in a given measuring task and
   to be measured and                    selecting the most appropriate measuring instrument and units of
   select the                            measurement.
   appropriate unit of
   measurement for                      explaining why certain units are preferred over others in a given
   length, mass             All          measurement situation, for example by telling why miles is not a
   (weight), time,                       convenient unit for measuring the length of a table.
   temperature,
   perimeter, area,
   volume, angle.

((II.3.M1, 2) Identify
and use appropriate
tools and units




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-27
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project




  In the middle years,                                    By engaging in activities such as ...
   students have the       Grade
    opportunity to…        Level

3. estimate measures        5           estimating measurements for various objects or conditions in the
   with a specified                      environment, such as the Celsius temperature at which it is
   degree of accuracy                    uncomfortable to go outdoors without a jacket or the metric height of
   and decide if an                      a professional basketball player.
   estimate or a
   measurement is
   “close enough.”                        The following measurements appeared in a report. Which ones are
                                            reasonable and which are not? Explain your choices:
(II.3.M3) Measure with
precision and identify                       •   The runner who took first place in the race ran 60 km/hr
sources of error                             •   To bake the pizza, we preheated the oven to 200  F.
     precision                               •   To bake the pizza, we preheated the oven to 200  C.
                                             •   A dose of cough syrup is 4 cm.
                                             •   Two cubic yards of concrete are needed to pour a slab 13 ft.
                                                 x 12 ft. x 4 in.


                            5-6         developing comparisons of standard units to familiar objects, such as
                                         knowing that one gram is approximately the weight of one paper clip.
                            7-8
                                        deciding on appropriate degrees of precision for measurements and
                                         their estimates in particular contexts.
                            7-8         comparing the magnitude of the error to the quantity being measured
                                         and expressing deviations as a percent of error.
                            7-8
                                        giving examples to illustrate the importance of percent of error, such
                                         as recognizing that an error of one dollar is a significant overcharge
                                         on the purchase of a candy bar, but it is relatively insignificant on the
                            7-8
                                         purchase of an automobile.

                            7-8         identifying sources of error or imprecision in measurements.

                                        using informal geometric explorations to develop measurement
                                         concepts, such as arranging sectors of a circle to approximate the
                                         area, removing and rearranging the vertices of a triangle to find the
                                         sum of the angles, or stacking cardboard wafers to approximate
                                         volume.




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                             7-28
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project




  In the middle years,                                   By engaging in activities such as ...
   students have the       Grade
    opportunity to…        Level

4. interpret               7-8          using geoboards or dot paper to construct similar figures, and
   measurements and                      comparing their linear and area measurements.
   recognize that two
   objects may have        7-8
                                        comparing measurements of sets of objects and drawing conclusions
   the same                              about the dependence or independence of measurements—for
   measurement on one                    example, if two squares have the same area, then they have the same
   attribute (e.g., area),
   but not necessarily                   perimeter (and vice versa); however two triangles can have the same
   on another (e.g.,                     area but different perimeters.
   perimeter).
                                        expressing relationships in descriptive terms (e.g., if the sides of the
(II.3.M4) Find area,                     similar triangles are in the ratio of 2 to 1, then the areas are in the
surface area,                            ratio of 4 to 1).
perimeter, and volume
of objects and        7-8
compare attributes
    surface area
    pi (π)



                                             “When I doubled (tripled) the length of each side of the triangle, I
                                             discovered four (nine) copies of the original figure. The side lengths
                                             doubled (tripled), but the area is four (nine) times as great.”
                           7-8
                                        drawing diagrams or making physical models with geoboards or
                                         other construction materials to illustrate observed relationships and
                                         explain why they are true.
                           7-8
                                       finding the area of a plane shape; cutting the shape into two or more
                                          pieces; rearranging the pieces to form a new shape; and comparing
                                          the area of the result to the area of the original.




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                             7-29
  Suggested Teaching and Learning Activities taken from the
  Michigan Curriculum Framework Project




   In the middle years,                                    By engaging in activities such as ...
    students have the         Grade
     opportunity to…          Level

5. use proportional          6-8           comparing measurements of length, area and volume of similar
reasoning and indirect                      objects and drawing conclusions about the relationships among
measurements to draw                        these variables.
inferences.

(II.3.M5) Use
proportional reasoning
to draw measurement
inferences
     proportion


                                                     A                             B                C              D
                                        E        F


                                            Determine the area of each of the trangram pieces if:


                                                                  Area of:     A       B   C    D       E    F

                                                                                           1

                                                                                                        4

                                                                               4

                                                                                       7

                                                                               1




  Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
  1/01 DRAFT                                                                                                7-30
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project



In the middle years,                                              By engaging in activities such as ...
 students have the          Grade
  opportunity to…           Level

                           6-8               deriving strategies for calculating measurements of one quantity
                                              from measurements of another (for example, area of a rectangle
                                              from the measurements of length and width).



                                                                      B
                                                  A                                        C                          D




                                                                                                                  J
                                                          F

                                                                                           I

                                          E           G                     H
                                        Copy these shapes on your geoboard. For each shape, record the number of
                                      geoboard
                                        pegs that lie on the polygon and determine the area of each polygon. Organize your
                                        data (number of pegs, area) in a table. Examine your findings and see if you can
                                        determine a relationship that tells the area of a geoboard polygon in terms of the
                                        number of pegs.


                                          Make some additional polygons of your own and check to see if your rule holds.
                                          Also try your rule on the following examples:




                                                              R

                                              Q                                                T             U
                                                                                S

                                                                                    W

                                                                  V


                                                                                                         Z
                                                      X               Y




                                          Does your rule still hold? If not, can you modify it so that it applies to these cases,
                                          too?
Huron County Mathematics Curriculum                                                   Seventh-Eighth Grade
1/01 DRAFT                                                                                                         7-31
  Suggested Teaching and Learning Activities taken from the
  Michigan Curriculum Framework Project



   In the middle years,                                       By engaging in activities such as ...
    students have the         Grade
     opportunity to…          Level

                             6-8            using informal approaches with geoboards, graph paper and other
                                             concrete objects to develop and verify basic formulas for area and
                                             volume.
                                             Use your tangram pieces to help you find formulas for the area of a rectangle,   a
                                             triangle, a parallelogram, and a trapezoid. Draw or construct other triangles,
                                             parallelograms, and trapezoids and test your formulas on those examples.




                             6-8
                                             finding strategies to measure indirectly certain quantities that are
                                              very large, very small, or inaccessible, such as measuring the
                                              thickness of a book to estimate the thickness of one page, sampling
                                              the number of words on several pages to estimate the number of
                                              words in a book, or measuring shadows to determine the heights of
                                              tall buildings.
6. apply measurement to                      experimenting with different-size samples of a given material to
                             6-8
   describe the real-world                    observe that, although mass and volume each vary from one sample
   and to solve problems.                     to another, the ratio of mass to volume (i.e., the density) remains
                                              constant.
(II.3.M6) Use
measurement to solve         5               using observed patterns of measurements to predict other
real-world problems                           measurements (such as predicting the height of a seedling from
                                              observed patterns of growth).
                             6
                                             comparing the sizes and distances of planets in the solar system, and
                                              constructing solar-system models that show both sizes and distances
                                              on the same scale.
                             6-8
                                             scaling various measurements, such as figuring how high you could
                                              jump on the moon or on another planet.
                             7-8
                                             exploring the significance of size in living organisms.
                             7-8
  Huron County Mathematics Curriculum                                                Seventh-Eighth Grade
  1/01 DRAFT                                                                                                    7-32
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project



In the middle years,                                    By engaging in activities such as ...
 students have the          Grade
  opportunity to…           Level


                                        researching how law enforcement officers use measurements of skid
                                         marks and other evidence to reconstruct an accident scene.




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-33
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                      III. Data Analysis and Statistics

1. Students collect and explore data, organize data into a useful form and develop skill in
representing and reading data displayed in different formats. (Collection, Organization and
Presentation of Data)


  In the middle years,      Grade                           By engaging in activities such as ...
   students have the
    opportunity to…          Level

1. collect and explore     All            posing questions of interest and importance to them, such as
   data through                            questions about favorite movies or the depletion of natural
   observation,                            resources, and gathering data to help answer those questions.
   measurement,
   surveys, sampling       All           devising and implementing a variety of strategies for collecting data,
   techniques and                         such as measuring, counting, sampling, surveying, etc.
   simulations.
                           All           conducting experiments with random devices (spinners, dice,
(III.1.M1) Collect and
explore data                              computer programs, etc.) to generate and explore data.
surveys
sampling techniques
simulations

2. organize data using All                developing posters, charts, bulletin boards, and other means for
   tables, charts,                         presenting data that they have gathered.
   graphs, spreadsheets
   and data bases.      7-8
                                         selecting examples of different modes of data presentation from
                                          newspapers and magazines and describing the information that they
(III.1.M2, 3) Organize                    contain.
and present data and
explain choice of
representation
     box (whisker) plot
     stem & leaf plot
     circle graph
     outlier
     quartile
     scatter plot




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                           7-34
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project




  In the middle years,                                    By engaging in activities such as ...
   students have the       Grade
    opportunity to…        Level

3. present data using a    All          deciding an appropriate way to present data in various situations and
   variety of                            explaining their choices (for example, showing why a line graph is
   appropriate                           preferred to a bar graph or a pie chart in a given setting).
   representations, and
   explain why one                           “I used a circle graph because I wanted to show parts of the whole.”
   representation is
   preferred over                                     How I Spent My Allowance
   another or how a
   particular
   representation may
   bias the                                                 clothes
   presentation.                                                            save for
                                                                            college


(III.1.M2, 3) Organize                                                         misc.
and present data and                                        movies &
explain choice of                                           tapes      snacks &
representation                                                         food

     box (whisker) plot
     stem & leaf plot
                           6-8
     circle graph                       generating examples of situations in which one would choose a
     outlier                             particular form of presentation, such as examples of data sets that
     quartile                            favor presentation via box plots.
     scatter plot          7-8
(III.1.M3, 4) Collect
                                        discussing the ways in which data are presented in newspaper and
and explore data to                      magazine articles, and identifying questions that can be answered
solve real-world           All           from the data.
problems
                                        using a spreadsheet to generate several different presentations from
                           7-8           the same data set.

                                        comparing and evaluating several different presentations of the same
                           All           data.

                                        giving examples of how the presentation of data can be biased, for
                                         example by changing the scale of a graph.




Huron County Mathematics Curriculum                                               Seventh-Eighth Grade
1/01 DRAFT                                                                                                  7-35
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,                                  By engaging in activities such as ...
   students have the       Grade
    opportunity to…        Level

4. identify what data   All             identifying everyday situations that rely on data in answering
   are needed to                         questions, such as predicting election outcomes or making marketing
   answer a particular                   decisions.
   question or solve a
   given problem, and All               identifying interesting problems and asking open-ended questions,
   design and                            identifying the information needed to answer those questions, and
   implement strategies                  devising and implementing data-collection procedures to arrive at
   to obtain, organize
   and present those                     appropriate conclusions.
   data.

(III.1.M3, 4) Collect
and explore data to
solve real-world
problems




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-36
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                      III. Data Analysis and Statistics

2. Students examine data and describe characteristics of the distribution, relate data to the
situation from which they arose, and use data to answer questions convincingly and
persuasively. (Description and Interpretation)


  In the middle years,      Grade                         By engaging in activities such as ...
   students have the        Level
    opportunity to…

1. critically read data    All           examining presentations of data encountered in everyday situations
from tables, charts or                    (news reports, advertising campaigns, political speeches, sports
graphs and explain the                    statistics, etc.) and raising questions about the data.
source of the data and
                                             A major television network reported the following information on its evening
what the data represent.                     news show:

(III.2.M1) Read data                               Percent of Families with Computers in the Home
from tables, charts,
and graphs
examine the source and                           Whites       27%
explain the meaning of                           Blacks       14%
                                                 Hispanics    13%
data
                                             The anchorman then noted: “Twenty-seven percent of white families have
                                             computers at home—as many as black and Hispanic families together.

                                             Do you agree with the anchorman’s conclusion? Explain.
                           All
                                        • posing questions about a set of data, and answering the questions
                                          posed by other students.




Huron County Mathematics Curriculum                                             Seventh-Eighth Grade
1/01 DRAFT                                                                                                7-37
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                       By engaging in activities such as ...
   students have the        Level
    opportunity to…

2. describe the shape     7-8           discussing the distribution of data using various examples and
   of a data distribution                describing the shapes of those distributions (e.g., rectangular,
   and identify the                      symmetric, bimodal).
   center, the spread,
   correlations, and any 6-8            classifying data sets according to selected properties, such as those
   outliers.                             that are symmetric about their mean.
                           6-8
(III.2.M2) Calculate                    describing the center of a distribution and explaining how the
measures of central
                                         “center” can be interpreted in different ways, such as the mean vs. the
tendency
                                         median.
use most appropriate       7-8
measure of central
tendency                                describing special features of various data distributions, such as their
     mean                                spread (range), cluster points, gaps (discontinuities), maximum and
     median                7-8           minimum values, and outliers.
     mode
     range                              exploring how certain features of a data distribution can be inferred
     outliers                            from visual representations, such as how the length of the box in a
     correlation (+, -,                  box-and-whisker plot conveys information about the spread of the
        none)                            data.


3. draw, explain and       6-8          preparing presentations that use data to convince the class of a
   justify conclusions                   certain conclusion.
   based on data.
                           All
                                        writing persuasive letters in which conclusions are presented and
(III.2.M3) Draw,                         supported by data.
explain, and justify
conclusions                7-8          staging a class debate in which each side bases its arguments on data
                                         it has collected and presented.




Huron County Mathematics Curriculum                                         Seventh-Eighth Grade
1/01 DRAFT                                                                                         7-38
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                       By engaging in activities such as ...
   students have the        Level
    opportunity to…

4. critically question    7-8           reading print media and viewing television presentations of data-
   the sources of data;                  based conclusions and discussing their validity.
   the techniques used
   to collect, organize 7-9             collecting examples of biased presentations and writing rebuttals to
   and present data; the                 expose their biases.
   inferences drawn
   from the data; and
   the possible sources All             examining how the method of collecting data can influence the
   of bias in the data or                outcome, for example by comparing the results of a survey given to a
   their presentation.                   random sample of adults with the results of the same survey given to
                                         parents of preschool children.
                           5-6
(III.2.M4) Critically
question data                           examining how the size of the sample can influence the outcome of
data sources                             the data gathering.
collection techniques      7-8
inferences drawn                        identifying how data are used in advertising, and raising questions
data bias                                about the sources of those data and the conclusions that the
presentation                             advertisers hope the public will draw.




5. formulate questions 5-6              raising questions of interest to the students and implementing
   and problems, and                     procedures for gathering data to help answer those questions.
   gather and interpret
   data to answer those 7-8             identifying issues affecting students in the school, such as a proposed
   questions.                            Student Council project or a change in the dress code, and gathering
                                         data as the basis for a report about student preferences.
(III.2.M5) Use data to
pose and answer real-
world questions




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-39
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                      III. Data Analysis and Statistics

3. Students draw defensible inferences about unknown outcomes; make predictions and
identify the degree of confidence they have in their prediction. (Inference and Prediction)


  In the middle years,      Grade                        By engaging in activities such as ...
   students have the        Level
    opportunity to…

1. make and test            5-6          posing interesting questions and generating testable hypotheses to
   hypotheses.                            help answer those questions, such as:
                                           - What kind of music do students prefer, and which band should
(III.3.M1) Make and                           we hire for the dance?
test hypotheses                            - If a salad bar is added in the school cafeteria, will the students
                                              support it?
                            7-8
                                         generating conjectures about physical situations, such as the amount
                                          a spring stretches as weights are suspended from it, and conducting
                                          experiments to test those hypotheses.


2. design experiments       5-6          conducting experiments that model problem situations, such as using
   to model and solve                     dried beans to model a capture-recapture experiment to estimate the
   problems using                         fish population in a lake.
   sampling,
   simulations, and         7-8          exploring phenomena involving several variables, such as
   controlled                             conjecturing about the factors (weight of the pendulum bob, length of
   investigations.                        the pendulum, height from which the pendulum is released) that
                                          might determine the period of a pendulum, and designing
(III.3.M2) Design
experiments to model
                                          experiments to systematically control and test variables.
and solve problems
sampling
simulations
controlled investigations




Huron County Mathematics Curriculum                                         Seventh-Eighth Grade
1/01 DRAFT                                                                                          7-40
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                       By engaging in activities such as ...
   students have the        Level
    opportunity to…

3. formulate and        5-6             explaining how they arrived at their conclusions after conducting an
   communicate                           investigation.
   arguments and
   conclusions based    5-6
                                        analyzing the conclusions that other students have put forth and
   on data and evaluate                  questioning one another about those conclusions.
   their arguments and
   those of others.
                        7-8             writing and delivering persuasive arguments designed to convince the
(III.3.M3) Develop,                      listener based on the data presented.
explain, and analyze
conclusions




4. make predictions        5-6          looking for patterns in data as a strategy for making predictions.
   and decisions based
   on data, including                   using data collected in experiments to predict unknown results, and
   interpolations and      All
                                         testing those predictions experimentally.
   extrapolations.
                           6-8          analyzing the sources of the data and the methods used to gather
(III.3.M4) Make
                                         them, and deciding how much confidence they can place in the
predictions and
decisions based on
                                         conclusions drawn from those data.
data                       7-8
interpolations                          evaluating the context of a data-gathering situation and determining
extrapolations                           the extent to which conclusions drawn from the data can be extended
                                         to other situations.




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                         7-41
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                       By engaging in activities such as ...
   students have the        Level
    opportunity to…

5. employ              5-6              solving problems using data-analysis strategies, such as predicting
   investigations,                       the “life expectancy” of various boxes of cereal based on a set of
   mathematical                          factors which the students identify as affecting the consumption rate
   models and                            of cereal.
   simulations to make
   inferences and
   predictions to
   answer questions
   and solve problems.

(III.3.M5) Use data and
investigations to
solve real-world        7-8
problems                                conducting simulations to solve problems where actual experiments
                                         are impossible, such as to estimate the fish population of a lake or to
                                         determine the number of packs of gum they can expect to buy in
                                         order to collect an entire set of baseball cards packed with the gum.




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                         7-42
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                      IV. Number Sense and Numeration

1. Students experience counting and measuring activities to develop intuitive sense about
numbers; develop understanding about properties of numbers; understand the need for and
existence of different sets of numbers; and investigate properties of special numbers.
(Concepts and Properties of Numbers)


  In the middle years,      Grade                         By engaging in activities such as ...
   students have the        Level
    opportunity to…

1. develop an            All              using hands-on manipulative materials to develop number concepts
   understanding of                        including concepts of place value, prime and composite numbers,
   integers and rational                   fractions and decimals, equivalence, inverses, common factors and
   numbers, and                            common multiples.
   represent rational
   numbers in both       5-6              using physical models of fractional quantities (e.g., number line,
   fraction and decimal                    fraction circles, fraction bars) to represent rational quantities such as
   form.                                   “fractions close to zero (or 1/2 or 1 …).
(IV.1.M1) Understand       5-6            expressing numbers using a variety of equivalent representations,
and represent
integers and rational                      including:
numbers                                     - fractions, decimals and percents.
                                            - equivalent fractions with common denominators.
                           7-8              - physical or geometric representations of fractions or decimals.

                                          expressing mathematical quantities in various ways, such as
                                                                                                  10·94
                                          recognizing that 5·94 can be viewed as 10·94÷2, as 2            , as (5·90 +
                                          5·4), or as (5·100 - 5·6).




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                                7-43
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                        By engaging in activities such as ...
   students have the        Level
    opportunity to…

2. extend their        7-8              exploring very large and very small numbers, and representing such
   understanding of                      numbers in terms of powers of ten.
   numeration systems
   to include decimal  5-6
                                        constructing concrete representations of various quantities, for
   numeration,                           example models that illustrate how large is a billion.
   scientific
   numeration and non-
   decimal numeration All               exploring selected numeration systems used in other cultures, such as
   systems.                              Egyptian or Mayan numerals, as a basis for understanding the power
                                         and convenience of a place-value system.
(IV.1.M2) Understand
                        7-8             engaging in hands-on activities with multibase blocks, chip trading,
different numeration
systems                                  or similar materials to explore non-decimal numeration systems.
scientific (small       7-8
numbers)                                investigating the use of binary and hexadecimal systems in
non-decimal (other than                  computing languages.
base 10)                All
    scientific notation
                                        using concrete models, such as an 8-hour “clock,” to develop
                                         understanding of modular systems.


3. develop an                7-8        discovering a strategy to illustrate that between any two rational
   understanding of the                  numbers there is always another rational number (density).
   properties of the
   integer and rational      7-8
                                        arranging sets of numbers, including positive and negative integers
   number systems                        and rational numbers, in increasing or decreasing order.
   (e.g., order, density)
   and of the properties
   of special numbers        5          giving examples to illustrate the results of adding or multiplying by
   including 0, 1, π,                    zero or one.
   and the additive and      6
   multiplicative                       measuring circumferences and diameters of round objects to develop
   inverses.                             a concept of π as the ratio c/d.
                             6-8
(IV.1.M3) Understand                    demonstrating the behavior of additive and multiplicative inverses,
properties of integers       7-8         and determining the inverses of specified numbers.
and rational numbers
π                                       developing strategies for mental computation based on the
inverses
                                         distributive property, such as recognizing that
density
    reciprocals                          (3.5 x 9) = (3.5 x 10) - 3.5 = (4 x 9) - (.5 x 9) = 3·9 + .5·9.




Huron County Mathematics Curriculum                                          Seventh-Eighth Grade
1/01 DRAFT                                                                                           7-44
  Suggested Teaching and Learning Activities taken from the
  Michigan Curriculum Framework Project


    In the middle years,      Grade                       By engaging in activities such as ...
     students have the        Level
      opportunity to…

  4. apply their             5-6          constructing numerical expressions to represent problem situations,
     understanding of                      such as writing an expression for the cost of two 75¢ ice-cream cones
     number systems to                     and three 60¢ soft drinks, plus tax.
     model and solve
     mathematical and        6            suggesting real applications that correspond to a mathematical
     applied problems.                     expression, such as describing a situation that might be represented
                                           by (3 x 17) + 5.
(I (IV.1.M4) Use number
  systems to solve real- 7-8
                                          discussing “numerical situations without numbers” such as:
  world problems
                                               Two fractions have a positive sum that is less than one.
                                               What can you tell about the fractions? What can you tell
                                               about the product of the same two fractions?




  Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
  1/01 DRAFT                                                                                        7-45
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                      IV. Number Sense and Numeration

2. Students recognize that numbers are used in different ways such as counting, measuring,
ordering and estimating; understand and produce multiple representations of a number; and
translate among equivalent representations. (Representation and Uses of Numbers)


  In the middle years,      Grade                        By engaging in activities such as …
   students have the        Level
    opportunity to…

1. give geometric        7-8              using tiles or markers to represent properties of numbers, such as
   representations of                      the geometric arrangements that generate the sequences of
   fractions, prime and                    triangular and square numbers, or the result that composite
   composite numbers,                      numbers can be represented by rectangular arrays in more than one
   triangular and square                   way while prime numbers can only be represented as 1-by-n arrays.
   numbers, and other
   number concepts;                       using a number-line model to represent positive and negative
   represent rational    All
                                           integers and rational numbers.
   numbers and integers
   on the number line.

(IV.2.M1) Give
geometric
representations of
numbers
triangular numbers

(IV.2.M1) Represent
integers and rational
numbers on the
number line



5. select appropriate    All              solving appropriate consumer problems involving discounts,
   representations for                     rebates, interest, and taxes.
   numbers, including
   integers and rational All              solving real-life problems involving numbers expressed as
   numbers, in order to                    fractions, decimals, or percents, or written in scientific notation.
   simplify and solve
   problems.

(IV.2.M5) Use
appropriate
representations for
numbers to solve real-
world problems



Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                            7-46
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                      IV. Number Sense and Numeration

3. Students investigate relationships such as equality, inequality, inverses, factors and
multiples; and represent and compare very large and very small numbers. (Number
Relationships)


  In the middle years,      Grade                           By engaging in activities such as ...
   students have the        Level
    opportunity to…

1. compare and order     All              evaluating expressions without calculating, such as:
   integers and rational                    318 + 661 + 976 ? 918 + 599 + 268 [<, >, = or ?]
   numbers using                            68 x 5 ? 70 + 69 + 68 + 67 + 66        [<, >, = or ?]
   relations of equality
   and inequality.       All              ordering fractions by reasoning about their relationship to whole
                                                                                     23                  22
(IV.3.M1) Compare                         numbers, for example recognizing that 24 is greater than 23
and order integers                                                                          1
                                          because, in the first case, the “part not used” (24 ) is less than in the
and rational numbers
(positive and                                           1
                                          second case (23 ).
negative)
                           7-8            locating common irrational numbers on the number line by placing
                                           them between two integers (e.g., √5 is between 2 and 3) or two
                                           rational numbers (e.g., π is between 3.1 and 3.2).
                           All
                                          exploring advertised prices and discounts that involve per cents, such
                                           as determining how much the customer will save on various
                                           purchases or what the discount rate is for an item that is marked
                                           down $10.

                                              A local craft store ran the following advertisement:

                                                                             SALE
                                                                         Your Choice!
                                                Take 50% off                   OR                   Take 25% off

                                                Any One Item                              Your Entire Purchase



                                              How would you decide whether to choose the 50% option or the
                                              25% option? Explain your reasons.




Huron County Mathematics Curriculum                                            Seventh-Eighth Grade
1/01 DRAFT                                                                                             7-47
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                        By engaging in activities such as ...
   students have the        Level
    opportunity to…

2. express numerical       All          exploring common applications of rates, such as baseball batting
   comparisons as                        averages, unit pricing or miles-per-hour, and explaining the meaning
   ratios and rates.                     of such numbers.

(IV.3.M2) Explore          7-8          using manipulatives to model ratios, such as paper cut-outs to model
applications of                          the ratios between corresponding parts of similar polygons;
ratios and rates           7-8
     scale                              exploring simple geometric sequences such as:
                                           3   9    27
                                         1, 2 , 4 , 8    …, and calculating the common ratio of successive
                                         terms.
                           All
                                        working with maps and scale drawings, and constructing their own
                                         scale drawings of selected objects.
                           7-8
                                        graphing linear relationships and explaining how slope represents a
                                         rate of change.




Huron County Mathematics Curriculum                                         Seventh-Eighth Grade
1/01 DRAFT                                                                                         7-48
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                         By engaging in activities such as ...
   students have the        Level
    opportunity to…

3. distinguish between 7-8              developing strategies for determining whether a number is prime, and
   prime and                             using a calculator to carry out the investigation.
   composite numbers;
   identify factors,     7-8
                                        constructing a systematic method of finding all the factors of a given
   multiples, common                     number with the aid of a calculator.
   factors and           5-6
   multiples, and
   relatively prime                     writing factor trees to illustrate the prime decomposition of various
   numbers; and apply                    numbers.
   divisibility tests to
   numbers.
                                        Two students were factoring the number 48. Natalie wrote 48 = 2 x 24, while Jeremy
                                       began with 48 = 6 x 8. If they each develop a factor tree for 48, will
(IV.3.M3) Identify
                                       they get the same set of prime factors in the end?
numerical
relationships                          Can you produce any other factor trees for 48 besides the ones Natalie and Jeremy
relatively prime                       developed? How many? Did you get the same prime factors?

                                       Is what you found out about the prime factors of 48 true for other numbers
                                       as well?

                           All         (Fundamental Theorem of Arithmetic)



                                       exploring common divisibility tests and applying them to selected
                                        numbers.

                                         Goldbach, a famous mathematician, proposed that any even number greater
                           5             than 2 can be written as a sum of two prime numbers (for example, 36 = 5 + 31).
                                         Pick any 10 even numbers greater than 2 and     see if they follow Goldbach’s
                                         conjecture. Can you find an even number greater than 2 that does not obey this
                           7-8               rule

                                       • demonstrating a procedure for finding multiples of a number.

                                       • developing a procedure for finding common factors and common
                                         multiples of two or more numbers with the aid of a calculator.




Huron County Mathematics Curriculum                                              Seventh-Eighth Grade
1/01 DRAFT                                                                                                  7-49
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                       By engaging in activities such as ...
   students have the        Level
    opportunity to…

4. explain the meaning 7-8              exploring the result of repeatedly doubling or halving a quantity, and
   of powers and roots                   comparing that to the result of repeatedly adding or subtracting two.
   of numbers and use
   calculators to      7-8
                                        conducting experiments that illustrate exponential growth or decay,
   compute powers and                    such as measuring the heights of successive bounces of a ball.
   square roots.
                           7-8          graphing the results of experiments involving exponential growth or
(IV.3.M4) Explain and                    decay.
compute powers and
square roots               7-8
                                        investigating the relationship between the length of the side and the
Tools: scientific
                                         area of a square, and determining one of the two values when the
calculators
                                         other is known.



5. apply their             6            using published data from a report on contamination in local streams
   understanding of                      to compare actual conditions to environmental standards for
   number                                acceptable water quality.
   relationships in
   solving problems.       7-8          conducting their own environmental studies of air, soil or water
                                         quality, radon levels in the home, etc., and writing reports or making
(IV.3.M5) Apply                          presentations on their findings.
understanding of
number relationships       5
                                        monitoring their food and vitamin intake for one day and comparing
to solve real-world
problems
                                         their consumption to recommended daily allowances.




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-50
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                  V. Numerical and Algebraic Operations and Analytical Thinking

1. Students understand and use various types of operations (e.g., addition, subtraction,
multiplication, division) to solve problems. (Operations and their Properties)


  In the middle years,      Grade                       By engaging in activities such as ...
   students have the        Level
    opportunity to…

1. use manipulatives       5-6          extending familiar models and manipulatives used for operations
   and diagrams to                       with whole numbers (number line, multibase blocks, chips,
   model operations                      rectangular arrays, etc.) to model operations with fractions, decimals
   and their inverses                    and integers.
   with integers and
   rational numbers
   and relate the
   models to their
   symbolic
   expressions.

(V.1.M1) Use
manipulatives and     7-8               writing mathematical
diagrams to model                        expressions to correspond to
                                                                                                         3
operations involving                     physical models, for example
algebraic expressions                    writing (n+2)·3 to describe.
                                                                                       n           2


                           6-8          drawing pictures to represent mathematical operations, for example a
                                         diagram that illustrates (1/3 + 1/2) or (2/3 · 1/4).




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-51
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,       Grade                      By engaging in activities such as ...
   students have the         Level
    opportunity to…

2. compute with              7-8        developing strategies for estimating computations, for example
   integers, rational                    recognizing that (2.47 x 3.93) is approximately 10 (2.5 · 4).
   numbers and simple
   algebraic                            applying mental estimation to consumer situations, for example
   expressions using         5           reasoning that the bill for 12 assorted candy bars must be less than $6
   mental computation,                   since each one is less than 50¢.
   estimation,
   calculators, and          7-8
   paper-and-pencil;                    developing and practicing strategies for mental computation such as
   explain what they                     combining addends that sum to 10 ($13.30 + $5.95 + $3.70 = $17 +
   are doing and know                    $5.95 = $23 - 5¢) or using related facts (“5+5 =10, so 6+6 would be
   which operations to                   12, and then 6+7 would have to be 13”).
   perform in a given        6
   situation.                           comparing the meaning of expressions like (5 + 7) x 3 and
                                         5 + (7 x 3) and explaining the need for an order-of-operations rule for
(V.1.M2) Perform
                             7-8         5 + 7 x 3.
operations
integers
rational numbers                        developing facility with a calculator, including the use of memory
simple algebraic                         and special keys, and using a calculator to carry out complex
expressions                              procedures.
percents (x, , using
    scientific calculator)

(V.1.M2) Explain
strategies and know
which operation to
perform in a given
situation




Huron County Mathematics Curriculum                                        Seventh-Eighth Grade
1/01 DRAFT                                                                                        7-52
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                       By engaging in activities such as ...
   students have the        Level
    opportunity to…

3. describe the          5              using a number line to model the associative and commutative
   properties of                         properties for addition with rationals and integers.
   operations with
   rationals and         5-6
                                        using an area model to demonstrate the associative and commutative
   integers (e.g.,                       properties for multiplication with rationals and integers.
   closure; associative,
   commutative and
   distributive          7-8            exploring the distributive property using physical models, diagrams
   properties) and give                  and mathematical expressions.
   examples of how
   they use those                                    3 people (2 books + 3 toys)               3(2+3) =3(5)
   properties.                                       (6 books + 9 toys)                        (6 + 9)
                                                     (3 people x (n+2) presents)               3(n + 2)
(V.1.M3) Describe and
apply properties with
                           7-8
rationals and integers
distributive                            using manipulatives and diagrams to show that n/n = 1 (n≠0) and
division by zero                         using that result to find equivalent fractions by multiplying or
                           All           dividing by numeral (n/n).

                                        exploring the relationship between multiplication and division,
                                         including multiplication by zero, and offering reasons to explain why
                                         division by zero is undefined.


4. efficiently and     All              reading and discussing problem situations and explaining what
   accurately apply                      mathematical operations will be needed, and why.
   operations with
   integers, rational  5
                                        predicting the approximate result of a computation before actually
   numbers, and simple                   carrying it out.
   algebraic           6
   expressions in
   solving problems.                    writing examples of problems that would appropriately be solved
                                         with mental computation, paper and pencil, or calculator.
(V.1.M4) Use               7-8
operations to solve                     examining the results of several computations and, without
real-world problems                      calculating, determining which answers are reasonable and which are
                                         not.




Huron County Mathematics Curriculum                                          Seventh-Eighth Grade
1/01 DRAFT                                                                                           7-53
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                  V. Numerical and Algebraic Operations and Analytical Thinking

2. Students analyze problems to determine an appropriate process for solution and use
algebraic notations to model or represent problems. (Algebraic and Analytic Thinking)


  In the middle years,     Grade                            By engaging in activities such as ...
   students have the       Level
    opportunity to…

                          All            writing verbal descriptions of situations that could be represented by
1. read and write                         given algebraic expressions (e.g., 14x + 37y could represent the total
   algebraic                              cost of 14 cards and 37 stamps).
   expressions; develop
   original examples      7-8            writing algebraic expressions and equations that describe selected
   expressed verbally                     situations (e.g., How could you express the total cost of tickets for
   and algebraically;                     the field trip if adult tickets are $5 and children’s tickets are $1.50?).
   simplify expressions 6-8
   and translate between                 translating verbal and algebraic expressions that were written by
   verbal and algebraic                   other students.
   expressions; and
   solve linear equations 7-8            determining solutions for linear equations and inequalities using
   and inequalities.                      informal means including physical models and guess-and-check
                                          strategies.
(V.2.M1) Translate
between verbal and
algebraic expressions

(V.2.M1) Read, write,
and simplify algebraic
expressions

(V.2.M1) Write and
solve linear equations
and inequalities




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                           7-54
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project




  In the middle years,     Grade                            By engaging in activities such as ...
   students have the       Level
    opportunity to…




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                           7-55
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                           By engaging in activities such as ...
   students have the        Level
    opportunity to…

                            6-8          engaging in hands-on explorations with manipulatives (e.g., algebra
2. represent algebraic                    tiles, balance beams) and using them to represent algebraic
   concepts with                          expressions, equations and inequalities.
   geometric models
   (e.g., algebra tiles),
   physical models (e.g.,
   balance beam), tables
   and graphs; and write
   algebraic expressions
   to correspond to the
   multiple
   representations.
                            7-8          constructing tables of values, including tables generated with
                                          computer spreadsheets, to represent algebraic expressions, such as a
(V.2.M2) Use models,                      table of ordered pairs that satisfy the equation m = 3n - 1.
tables, and graphs to
represent algebraic
concepts



                            7-8
                                         identifying a pattern in a table of values and using that pattern to
                                          construct an algebraic expression to represent the given relationship.

                                                 x          1      2       3        4       5
                                                                                                       y =?
                                                 y          2      5       10       17      26

                            7-8
                                         graphing values from a table and explaining how the graph
                                          represents the table and the equation derived from that table (e.g.,
                                          explaining why the graph of entries in a table of circumference vs.
                                          diameter of round objects yields a straight line and how the slope of
                                          that line relates to the value of pi).
                            7-8
                                         demonstrating how variables in an algebraic expression are related
                                          by showing what happens to one variable when another is changed,
                                          for example demonstrating that when the diameter of a circle is
                                          tripled, the circumference is also tripled.




Huron County Mathematics Curriculum                                             Seventh-Eighth Grade
1/01 DRAFT                                                                                              7-56
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project




  In the middle years,     Grade                            By engaging in activities such as ...
   students have the       Level
    opportunity to…

3. solve linear            7-8           using number lines to model solutions to linear equations and
   equalities and                         inequalities in one variable.
   inequalities using
   algebraic and
   geometric methods,
   and use the context
   of the problem to
   interpret and explain                                        5 x < 35
   their solutions.
                           7-8           using geometric representations, including graphs in the Cartesian
(V.2.M3) Graph and
                                          plane, to model solutions of linear equations in two variables.
explain solutions for
linear equations and
inequalities




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                           7-57
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,     Grade                            By engaging in activities such as ...
   students have the       Level
    opportunity to…

4. analyze problems     7-8              discussing how individual students understand and represent
   modeled by linear                      selected problems and comparing such multiple representations;
   functions, determine
   strategies for       All
                                         devising and explaining their own strategies for solving problems.
   solving the
   problems, and        7-8              discussing whether a solution is acceptable in the context of the
   evaluate the
   adequacy of the                        problem (for example, 11.69 is an appropriate solution to 456/39
   solutions in the                       when computing the average cost of 39 items ordered from a
   context of the                         catalog—but when computing the number of 39-passenger busses
   problems.                              needed to transport 456 students, the answer is 12, while the answer
                                          is 11 when determining the number of boxes that can be filled when
(V.2.M4) Choose a                         456 cookies are packaged in boxes of 39).
problem-solving
strategy and decide if
the solution is
appropriate




5. explore problems      5-6             comparing long-distance calling plans from several different carriers
   that reflect the                       and deciding on the most economical plan to buy.
   contemporary uses
   of mathematics in                     using a spread sheet to explore the way a savings account grows
   significant contexts 5-6               given different initial investments and rates of interest, and using the
   and use the power of 7-8               information to plan a personal savings strategy.
   technology and
   algebraic and
   analytic reasoning to
   experience the ways
   mathematics is used
   in society.

(V.2.M5) Use
technology, algebraic
concepts, and
reasoning to solve
real-world problems




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                           7-58
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                               VI. Probability and Discrete Mathematics

1. Students develop an understanding of the notion of certainty and of probability as a
measure of the degree of likelihood that can be assigned to a given event based on the
knowledge available; and, make critical judgments about claims that are made in
probabilistic situations. (Probability)


  In the middle years,      Grade                           By engaging in activities such as ...
   students have the        Level
    opportunity to…

3. conduct experiments 7-8              conducting experiments where items are replaced after they are
   and give examples                     selected compared to experiments where items are not replaced after
   to illustrate the                     they are selected.
   difference between
   dependent and       7-8              investigating whether the occurrence of the first event affects the
   independent events.                   occurrence of the second event.

(VI.1.M3) Conduct
experiments involving
dependent and
independent events
   mutually exclusive
   compound events


4. explain the           7-8            conducting experiments to generate data that lead to empirical
   difference between                    probabilities (such as the probability of rolling “5” on a die, calculated
   probabilities                         experimentally by rolling a die 20 times) and comparing those results
   determined from                       to the corresponding mathematical or theoretical probability.
   experiments or
   chance events         7-8            combining experimental data and using computer simulations to
   (empirical) and                       produce large numbers of trials, and comparing the probabilities
   probabilities derived                 derived from such large samples with those derived from small
   mathematically
   (theoretical), and                    samples and with the theoretical values.
   explain how the
   empirical
   probability changes
   for a large number
   of trials.

(VI.1.M4) Compare
experimental and
theoretical
probabilities




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                           7-59
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                           By engaging in activities such as ...
   students have the        Level
    opportunity to…

5. conduct probability     All          identifying situations in which probability is used to solve real
   experiments and                       problems, such as in actuarial applications or product testing.
   simulations to
   model and solve         7-8
                                        applying probabilities to model and solve real problems such as using
   problems.                             capture-recapture simulations to estimate wildlife populations.
(VI.1.M5) Conduct
                           6-8          exploring the concept of “fair games” by playing and analyzing simple
probability
experiments and
                                         games and by designing original games.
simulations to solve
real-world problems




Huron County Mathematics Curriculum                                           Seventh-Eighth Grade
1/01 DRAFT                                                                                           7-60
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

                                 VI. Probability and Discrete Mathematics

2. Students investigate practical situations such as scheduling, routing,sequencing,
networking, organizing and classifying; and analyze ideas like recurrence relations, induction,
iteration and algorithm design. (Discrete Mathematics)


  In the middle years,      Grade                           By engaging in activities such as …
   students have the        Level
    opportunity to…

1. use manipulatives,  6-8               using physical manipulatives to model simple permutation and
   diagrams, and the                      combination problems.
   fundamental
   theorem of counting 7-8               exploring the fundamental theorem of counting (i.e., if one task can
   to count                               be performed in m ways and the second task can be performed in n
   permutations and                       ways, then the number of ways of performing the two tasks is mn
   combinations.
                                          ways).
(VI.2.M1) Use              7-8
                                         observing patterns in the number of permutations of 2, 3, 4, …
manipulatives,
                                          objects and deriving an expression for the number of possible
diagrams, and
counting principle to                     permutations of any number of objects.
                           7-8
count permutations
and combinations                         giving examples of situations that illustrate the difference between
                                          permutations, where the order of objects is important, and
                                          combinations, where order is not important.


2. use sets and set     All              constructing Venn diagrams to represent mathematical situations
   relationships to                       (for example, various partitions on the set of integers, or diagrams
   explore and solve                      of the relationships among different sets of quadrilaterals).
   simple algebraic and
   geometric problems. All               describing attributes of given attribute blocks
(VI.2.M2) Use sets and
set relationships to
solve geometric and
algebraic problems
Venn diagrams




Huron County Mathematics Curriculum                                          Seventh-Eighth Grade
1/01 DRAFT                                                                                          7-61
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                           By engaging in activities such as …
   students have the        Level
    opportunity to…

3. solve problems          7-8           tracing paths through a network of vertices and edges.
   involving networks,
   for example
   planning delivery
   routes or counting
   paths between
   points.

(VI.2.M3) Solve            7-8           using networks to solve problems such as laying out the most
problems involving                        efficient route for delivering newspapers.
networks




4. explore recurrence      All           analyzing and extending arithmetic and geometric sequences.
relations and iterations
                           7-8           investigating Pascal’s triangle and the Fibonacci sequence.
(VI.2.M4) Explore
recurrence relations
and iterations



5. continue to use      7-8              designing an algorithm to accomplish a task:
   manipulatives and                      - design a flow diagram to “assemble” a sandwich.
   drawings to model                       - use a map to write directions to get from point A to point B.
   the concepts and
   procedures for the   All              creating and solving logic problems.
   standard arithmetic
   algorithms, and
   develop and analyze 7-8               investigating the Euclidean algorithm to find the greatest common
   their own and other                    factor of two numbers.
   students' algorithms
   to accomplish a task
   or solve a
   mathematical
   problem.

(VI.2.M5) Develop and
analyze algorithms to
solve problems




Huron County Mathematics Curriculum                                          Seventh-Eighth Grade
1/01 DRAFT                                                                                          7-62
Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project


  In the middle years,      Grade                           By engaging in activities such as …
   students have the        Level
    opportunity to…

6. use discrete         All              using discrete mathematics concepts to model situations and solve
   mathematics                            problems, such as:
   concepts as                             - developing schedules for school, buses, television, etc.
   described above to                      - creating maps and design delivery routes or rubbish pick-up
   model situations and                       routes.
   solve problems; and
   look for whether or                     - using tree diagrams to investigate genetic characteristics.
   not there is a                          - investigating the concept of “pyramid schemes” or telephone
   solution (existence                        relays.
   problems),                              - investigating graphs like the Konigsberg bridge problem.
   determine how                           - determining possible combinations of clothing, food, friends,
   many solutions there                       activities, etc. and describe results of adding/subtracting items.
   are (counting
   problems), and
   decide upon a best
   solution
   (optimization
   problems).

(VI.2.M6) Use discrete
math to solve
existence, counting,
and optimization real-
world problems




Huron County Mathematics Curriculum                                          Seventh-Eighth Grade
1/01 DRAFT                                                                                          7-63

				
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