# 7-8th Grade Suggested Activities

Document Sample

Suggested Teaching and Learning Activities taken from the
Michigan Curriculum Framework Project

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
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:
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
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

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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 ...
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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.
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).
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

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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
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 ...
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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
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 ...
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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 ...
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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

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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