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Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project SEVENTH-EIGHTH GRADE I. Patterns, Relationships and Functions 1. Students recognize similarities and generalize patterns; use patterns to create models and make predictions; describe the nature of patterns and relationships, and construct representations of mathematical relationships. (Patterns) In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 1. describe, analyze and 5 recognizing patterns arising generalize patterns arising from everyday situations June in a variety of contexts and (e.g., geometric patterns in a 1 2 3 4 5 express them in general quilt, symmetry patterns in terms. art and nature, numerical 6 7 8 9 10 11 12 patterns in a calendar) and 13 14 15 16 17 18 19 describing them verbally and (I.1.M1, 4) Describe, symbolically. 20 21 22 23 24 25 26 analyze, and generalize 27 28 29 30 algebraic and geometric patterns 5-6 exploring numeric, algebraic and geometric examples of patterns such as patterns of repeating decimals (numeric), patterns in Pascal’s triangle (algebraic) 1/11 = .090909... 2/11 = .181818... 3/11 = .272727... . . . or patterns of change in the area of a square as the length of the side changes (geometric). Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-1 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 1 7-8 1/7 = .142857142857... 1 1 2/7 = .285714285714... 3/7 = .428571428571... 1 2 1 4/7 = ??? 1 3 3 1 . . 1 4 64 1 . 1 5 10 10 5 1 7-8 extending patterns by adding the next few elements or by supplying missing elements. 7-8 T1 = 1, T2 = 3, T3 = 6, T4 = 10… T9 = ? All creating and describing original patterns; modeling patterns using physical objects such as counters, attribute pieces, cubes, etc.; and providing reasons to explain their choices when creating or extending patterns. generalizing a rule to describe a pattern, for example a rule for S =1 7-8 1 forming the next Fibonacci number or a rule for determining S =4=1+3 2 the number of degrees in the S3 = 9 = 1 + 3 + 5 angles of a polygon. S4 = 16 = 1 + 3 + 5 + 7 S10 = ? Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-2 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 2. represent and record 5 using a table to show the number of handshakes when every person in patterns in a variety of the room shakes hands with everyone else. ways including tables, charts and graphs, and using a graph to show the total distance traveled when moving at a translate between various All fixed rate of speed. representations. 6 writing a mathematical expression to describe the total receipts after (I.1.M2) Represent and selling both adults’ and children’s tickets. record patterns in a variety of ways and 7-8 drawing a graph to represent a mathematical equation. relate between representations 7-8 writing an equation to describe a graph. data tables ordered pairs linear algebraic All matching a graph to a table of values. expressions geometric organizing data in tables in ways that highlight underlying patterns. 6-8 representations symmetric patterns tessellations 3. use patterns and their Tower of Hanoi generalizations to make and 6-8 • analyzing patterns and using Number Number justify inferences and them to predict unknown of Discs of Moves predictions. outcomes, for example the number of diagonals in an 1 1 (I.1.M3) Make and justify n-sided polygon or the tenth term 2 3 predictions in a sequence of numbers. 3 7 4 15 Tower of Hanoi: Given three posts and disks • • of varying diameter, how many moves are • • needed to transfer n disks from one post to • • another? Only one disk may be moved at a time, and larger disks may not be placed on smaller disks. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-3 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… All testing and evaluating conjectures based on observed patterns. using patterns to solve problems, such as comparing total earnings 6-8 over time to determine which of two wage plans is most beneficial. Pat is going to make name tags for the science fair participants, and she has been experimenting to see how many name tags she can cut from a sheet of paper. How many will she be able to get if she continues folding the paper as shown? What is happening to the size of the name tags as she continues to fold? What is a reasonable number of name tags to make from a standard sheet of paper? Explain your recommendation. 4. explore and describe generating data that exhibit linear visual and numeric 5-6 patterns, for example by Phone Rate: patterns, including linear investigating the relationship Minutes Cost expressions, near-linear between diameters and patterns, and symmetric circumferences of circles or the 1 $0.70 and spatial patterns. relationship between the cost of a 2 $0.85 long distance telephone call and the length of the call; 3 $1.00 (I.1.M1, 4) Describe, 4 $1.15 analyze, and generalize 5 $1.30 algebraic and geometric patterns 7-8 • comparing patterns and classifying them as linear (e.g., the temperature of a liquid being warmed at a steady rate), near linear (e.g., the cost of postage vs. the weight of the letter), or not linear (e.g., the area of a square vs. the length of its side, or the number of cells present in a culture if the cells divide every 24 hours.) Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-4 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 7-8 • finding or creating examples to illustrate various types of patterns. Colored Faces of Unit Cubes # 6 3 2 1 0 Number of Cubes on an Edge 1 1 - - - - 2 - 8 - - - 3 - 8 12 6 1 4 - 8 24 24 8 5 - 8 36 54 27 6 - ? ? ? ? Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-5 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the secondary years, Grade By engaging in activities such as … students have the Level opportunity to … 5. use patterns and generalizations to 5-6 identifying and describing patterns in open-ended situations, such solve problems and as the patterns found in a multiplication table or in Pascal’s explore new triangle. content. exploring situations in which looking for patterns and making (I. 1.M5) Use patterns tables, charts and graphs leads to the solutions of interesting and generalizations to problems. solve real-world problems 7-8 Locker problem: 1000 lockers, 1000 students … S1 opens every locker S2 closes (changes) every second locker S3 changes every third locker S4 changes every fourth locker … and so on. Which lockers will be open in the end? Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-6 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the secondary years, Grade By engaging in activities such as … students have the Level opportunity to … describing the connections among patterns observed in various contexts, for example, recognizing the triangular numbers when they 6-8 occur in seemingly dissimilar situations. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-7 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project I. Patterns, Relationships and Functions 2. Students describe the relationships among variables; predict what will happen to one variable as another variable is changed, analyze natural variation and sources of variability, and compare patterns of change. (Variability and Change) In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 1. identify and 6-8 • exploring mathematical patterns and physical situations that lead describe the nature them to encounter variables in a natural way—such as the effect that of change; recognize varying the amount of light or water has on the growth of a change in more seedling—and describing how one variable changes in relation to abstract and another. complex situations 5 and explore • charting and graphing the growth patterns of several seedlings and different kinds of observing the natural variation in height even when the plants have change and patterns been exposed to identical growing conditions. of variation. (I.2.M1, 2) Recognize and explore changes and generalize a rule 7-8 • conducting experiments in which they systematically manipulate variables, such as using a balance beam to study relationships between weights and their distances from the fulcrum or projecting shadows on a screen to study relationships between shadow size and the object’s distance from the screen. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-8 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 7-8 • examining graphs, such as a graph of distance vs. time, and explaining the changes that the graph represents. Trip t o Sault St e. Marie 150 ("The Soo") Dist ance (miles) 0 (Home) 1 2 noo n 7 a.m. Time ( hours) Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-9 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 2. connect an initial 7-8 conducting experiments and state to a final state observations and generating and generalize a rule data to explore relationships that describes a among dependent and pattern of change. independent variables in diverse contexts, for example ((I.2.M1, 2) by measuring the length of a Recognize and spring or rubber band as explore changes and various weights are generalize a rule point suspended from it. 6-8 describing what changes and what remains unchanged during explorations such as transforming geometric shapes using slides, flips, and turns, or adjusting the class’s test scores by adding 12 points to each person’s score. The polygon on the right has been reflected over the line to form the image on the left. What changed and what remained the same after the reflection? All playing games of “guess the rule” to develop generalizations about patterns of change, giving reasons for their guesses, and expressing the rules verbally and symbolically. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-10 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 3. begin to investigate 6-8 using patterns of change to make predictions and to answer questions applications in such as: bivariate data and - If the observed pattern continues, what will be the value of the linear relationships, variable one year from now? and explore - If the observed pattern has been unchanged, what was the value questions of what will happen to one of the variable one month ago? quantity if another variable is changed. Matt shared a secret with his friend, Paul, and made Paul promise not to tell anyone. But the next day, Paul told two (I.2.M3) Investigate people; and the day after that, those two people each told bivariate data and two people. If this pattern continues, how long will it be linear relationships before the entire school (enrollment 1200) knows the secret? and explore patterns of variability variable changes 6-8 conducting experiments to generate data, for example by measuring and comparing the circumferences and diameters of various round objects. 6-8 using technology, especially spreadsheets and calculators, to generate data and explore patterns of variability. Circle relationships: radius diameter area 1 3.14 3.14 2 6.28 12.56 3 9.42 28.26 4 12.56 50.24 5 15.70 78.50 6 18.84 113.04 7 21.98 153.86 8 25.12 200.96 9 28.26 254.34 10 31.40 314.00 11 34.54 379.94 12 37.68 452.16 Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-11 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 4. represent variability All expressing patterns of change in several different ways, for example or change by by constructing a graph to show the total snowfall accumulation ordered pairs, during the winter or by developing a table to show the number of tables, graphs and pieces of pizza that result when a pizza is sliced by one, two, three, equations. etc., cuts. (I.2.M4) Represent variability and change M arque tte , MI — Total Snowfall to Date in a variety of ways 160 ordered pairs 140 tables 120 graphs equations 100 inches 80 60 1994-95 40 Average Year 20 0 3-Jan 6-Jan 9-Jan 2-Feb 5-Feb 8-Feb 12-Jan 15-Jan 18-Jan 21-Jan 24-Jan 27-Jan 30-Jan 11-Feb 14-Feb 17-Feb 20-Feb 23-Feb 26-Feb 1-Mar 4-Mar 7-Mar 25-Dec 28-Dec 31-Dec 10-Mar 13-Mar (The graph above shows the weather bureau’s reported total-snowfall-to-date in Marquette, Michigan, during the 1994-95 winter as well as the “average” or “normal” seasonal snowfall. One of the graphs appears nearly linear while the other is very irregular. What accounts for this difference in the two graphs? What information is concealed by the graph of average snowfall? In what ways was 1994-95 “average” and in what ways was it not?) 7-8 describing plausible situations that could have produced a given graph or pattern. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-12 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 5. differentiate 7-8 exploring and classifying examples of relationships, including: between functions - linear relationships (the relationship between Fahrenheit and and relations such as Celsius temperatures). linear vs. not linear - quadratic relationships (the relationship between the area of a or continuous vs. non-continuous. square and the length of its side). - inverse relationships (the relationship between speed and time when traveling a fixed distance). (I.2.M5) Explore and - patterns of growth or decay (the number of cells present at the classify examples of relationships end of the month if the cells divide every 48 hours). linear - non-continuous functions (the cost of postage to mail letters of quadratic varying weight). 7-8 inverse patterns of growth and classifying observed patterns of change as linear or non-linear. decay 6-8 non-continuous expressing linear relationships graphically, algebraically and functions verbally. 6. continue to explore 6-8 developing and manipulating physical and mathematical models of relationships arising real phenomena (for example, the period of a swinging pendulum or from interesting the rebound height of a bouncing ball) and describing the associated contexts, and use variables and their relationships; variables and relationships to All using a spreadsheet to compare the advantages of investment plans solve mathematical offering different rates of interest. problems. (I.2.M6) Use relationships to solve real-world problems Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-13 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project II. Geometry and Measurement 1. Students develop spatial sense, use shape as an analytic and descriptive tool, identify characteristics and define shapes; identify properties and describe relationships among shapes. (Shape and Shape Relationships) In the middle years, Grade By engaging in activities such as … students have the Level opportunity to… 1. distinguish among exploring diagrams and two- and three-dimensional objects in order shapes and to: differentiate between - describe the shapes of the objects. examples and non- - name common shapes found in the objects. examples of shapes - identify special characteristics and properties of shapes. based on their properties; generalize - classify objects according to their shapes. about shapes of - differentiate between examples and non-examples of particular graphs and data shapes. distributions. - develop hierarchies of related shapes. (II.1.M1) Explore three-dimensional figures rectangular prisms edges faces cubes spheres (II.1.M1) Generalize about shapes of Identify and describe all the graphs and data shapes that you recognize in distribution the diagram to the left. 5 going on a geometry walk in the neighborhood and identifying and recording shapes that are observed. 7-8 exploring data sets and giving informal descriptions of the data distributions, such as symmetric vs. skewed distributions, positive associations, exponential growth, inverse variations, etc. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-14 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 2. generalize the 5 using paper folding, geoboards, and other models to generate and test characteristics of hypotheses about families of shapes and their properties (e.g., the shapes and apply opposite sides of the rectangles are always equal). their generalizations to classes of shapes. 6-8 exploring physical models of shapes to determine their symmetries, and describing and demonstrating the symmetries of two- and three- (II.1.M2, 3) Explore dimensional shapes. characteristics and hierarchies of two- utilizing computer drawing programs to investigate shapes and shape and three-dimensional All relationships. shapes circle 7-8 drawing or constructing and naming shapes that satisfy given criteria sector such as “a four-sided figure with one pair of parallel sides.” altitude n-gon polygon trapezoid Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-15 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 3. derive All developing and verifying generalizations about geometric properties generalizations using informal means such as paper folding, reflecting with a mira, about shapes and or manipulating shapes with a dynamic computer drawing program. apply those generalizations to The number of faces, vertices, and edges develop of a polyhedra is related (v+f-e=2) classifications of familiar shapes. Polyhedra Vertices Faces Edges (II.1.M2, 3) Explore cube 8 6 12 characteristics and octahedron 6 8 12 hierarchies of two- tetrahedron 4 4 6 and three-dimensional dodecahedron ? 12 ? icosahedron ? 20 ? shapes circle 7-8 sector altitude developing hierarchies for families of shapes, such as drawing a n-gon Venn diagram to show the relationships among quadrilaterals, polygon 7-8 parallelograms, rectangles, squares, rhombi, trapezoids, etc. trapezoid investigating when a characteristic of a shape is sufficient to define a (II.1.M4,5 and II.2.M3) shape, such as determining whether congruent diagonals guarantees Construct, combine, that a quadrilateral is a square. dissect, and transform shapes size transformation enlargement reduction Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-16 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 4. construct familiar 5-6 constructing shapes using simple materials such as straws, shapes using toothpicks, pattern blocks, geostrips, or geoboards. coordinates or appropriate tools (including technology); sketch and draw two- and three-dimensional shapes. (II.1.M4,5 and II.2.M3) Construct, combine, dissect, and “How many quadrilaterals can be made on a 3-peg by 3-peg geoboard?” transform shapes size transformation enlargement All constructing shapes using various tools including Euclidean tools reduction (compass and straight edge), miras or other reflection devices, computer drawing tools, coordinate graphs, paper folding, and tangram pieces. 7-8 constructing shapes that conform to given specifications (e.g., make a trapezoid with two right angles), and determining when it is impossible to create a certain shape (e.g., make a parallelogram with only two right angles). 5. combine, dissect 6 combining familiar shapes to produce more complex shapes and and transform designs. shapes. (II.1.M4,5 and II.2.M3) Construct, combine, dissect, and Can you combine all seven tangram transform shapes pieces to form a rectangle? a size transformation parallelogram? a triangle? a enlargement reduction Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-17 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… A domino is formed when two congruent squares are connected along a side. Similarly, a triomino is formed by three connected squares. How many distinct shapes can you find for each of the following? Name of Shape Number of Connected Squares Number of Distinct Arrangements Domino 2 Triomino 3 Tetromino 4 Pentomino 5 7-8 separating complex shapes into simpler component shapes. 7-8 predicting and investigating the shapes that result when two objects intersect, such as when a plane slices through a cube. 7-8 determining which shapes will tessellate the plane, and creating original tessellations. 6 identifying shapes found in artistic designs, especially designs from different cultures. 7-8 using various plane and solid shapes to create original designs or 7-8 sculptures. tracing shapes and images of those shapes after sliding, flipping or turning them. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-18 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 6. generalize about the 5-6 drawing, tracing, folding, and cutting a variety of shapes and using common properties them to develop concepts of congruent and similar shapes. of similar, congruent, parallel and perpendicular shapes and verify their generalizations informally. (II.1.M6) Explore examples of Cut the tangram diagram into seven congruency, pieces. How do the size and shape of similarity, parallelism, the two small triangles compare to the perpendicularity, and size and shape of the three larger angle relationships triangles? angle complimentary Are any of the triangles similar? Are corresponding any of the triangles congruent? Give interior an argument to prove your answer. supplementary verticaL 7-8 exploring physical objects such as globes or models of geometric solids to develop concepts of perpendicular and parallel lines and planes. 7-8 identifying examples of congruence, similarity, parallelism and perpendicularity from the environment and from geometric objects. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-19 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 7. use shape, shape 7-8 exploring the connections between shape and numbers by properties and shape investigating figurate numbers (triangular, square...), primes vs. relationships to composite numbers, the shape of data distributions, the shapes of describe the graphs, etc. physical world and to solve problems. 7-8 using isometric dot paper to draw two-dimensional representations of three-dimensional objects, and constructing three-dimensional (II.1.M7) Use shape models from two-dimensional renderings. properties and relationships to solve real-world problems 5-7 investigating diagrams of shapes as viewed from different angles and constructing 3-dimensional models to correspond to the 2- dimensional views. 7-8 drawing shapes on a coordinate grid and multiplying the coordinates of selected points by a constant to produce one-way [e.g., (x,y) (x, 2y)] and two-way [e.g., (x,y) (3x, 3y)] distortions. All conducting open-ended investigations involving shapes, such as Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-20 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… coloring maps or finding all the pentominoes and determining which pentominoes can be folded to make an "open box." Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-21 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project II. Geometry and Measurement 2. Students identify locations of objects, location relative to other objects, and the effects of transformations (e.g., sliding, flipping, turning, enlarging, reducing) on an object. (Position) In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 1. locate and describe 5-6 locating objects in relation to other objects (e.g., one meter to the objects in terms of right of the window; two feet above the floor...). their position, including compass 5-6 describing positions in terms of compass directions. directions, Cartesian coordinates, latitude 5-6 reading and following maps. and longitude, and midpoints. 5-6 drawing maps and giving verbal directions for locating objects (II.2.M1) Locate and (e.g., how to get to the library from the school). describe points in the 7-8 Cartesian plane specifying Cartesian coordinates for points on a grid, and locating coordinate system points given their coordinates. locating and describing points in three-dimensional space using positive (x, y, z) coordinates. 7-8 z (x,y,z) y 6-8 x locating and describing points on a globe using latitude and longitude. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-22 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 2. locate and describe 6 locating and describing points, lines, or physical objects in geometric objects in terms of terms such as "on the perpendicular bisector" or "at the intersection". their orientation and 6-8 relative position, using physical objects, graphs or sketches to represent positions such including as "parallel to," "at the midpoint," or "equidistant from". coincident, collinear, parallel, 5 using a geometry drawing program to construct objects such as the perpendicular; midpoint of a segment or the perpendicular to a line. differentiate 7-8 between fixed (e.g., using miras or mirrors to determine whether or not figures or objects N-S-E-W) and have bilateral symmetry. relative (e.g., right- 7-8 left) orientations; using tracings and models to determine whether a figure or object has recognize and 5-6 rotational symmetry. describe examples of bilateral and demonstrating that fixed orientations or directions (e.g., two blocks rotational symmetry. north of my house) yield consistent results, while relative orientations All (e.g., go three miles to the right) can result in different outcomes. (II.2.M2) Locate and describe objects examining a set of test scores to determine whether the score using the terms distribution is symmetrical. coincident, collinear, parallel, perpendicular; describe examples of bilateral and rotational symmetry Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-23 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 3. describe using physical objects, geometric sketches, and computer drawing translations, tools to explore the effects of translating, reflecting, or rotating a reflections, rotations point or object. and dilations using the language of drawing, both with paper and pencil and with computer drawing transformations, and tools, the resulting image when given a description of the employ transformation of a shape or object (e.g., translate 10 centimeters at transformations to verify congruence of an angle of 30° from the horizontal; reflect over the y-axis; rotate 60° figures. about the vertex). Z (II.2.M3) Use transformations and verify congruence dilations Y O N A M X Describe the transformation that projected triangle MNO onto triangle XYZ. describing the transformation that produced an observed result when shown a drawing of a shape and its image. Describe one motion that will transform: E to F B to E B to D C to E A to C F to A A B C B to A C to F Describe a combination of two transformations that will transform: D E F A to C F to A D to B D to C Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-24 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 5. use concepts of 5-6 engaging in treasure hunts that require following a series of position, direction directions to locate a hidden object. and orientation to describe the 5-6 drawing treasure maps that lead to selected hiding places. physical world and to solve problems. All giving written or oral instructions to lead another student to a desired location. (II.2.M5) Use position, direction, 7-8 viewing objects from different locations and drawing and describing and orientation to how the perspective changes. solve real-world 7-8 problems projecting images of shapes and determining how dilations can be used to enlarge or reduce the original shape. 7-8 constructing three-dimensional models from two-dimensional drawings, and drawing two-dimensional representations of three- dimensional objects. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-25 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project II. Geometry and Measurement 3. Students compare attributes of two objects or of one object with a standard (unit) and analyze situations to determine what measurements should be made and to what level of precision. (Measurement) In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 1. select and use 5-6 measuring everyday objects using: appropriate tools; - other objects for comparison (e.g., index cards, cubes, string). measure objects - standard units, both common and metric. using standard units - common measuring tools (rulers, protractors, thermometers, in both the metric and common scales, graduated cylinders, etc.) systems, and - rulers that are shorter than the object to be measured (i.e. must be measure angles in applied more than once) and "broken rulers" whose scales do not degrees. begin at zero. All (II.3.M1, 2) Identify engaging in hands-on activities to develop concepts of measurements and use appropriate in various dimensions, such as using string to surround an object tools and units (perimeter), using tiles or paper to cover a surface (area), or using sand to fill a container (volume). All measuring common objects using rulers, trundle wheels, balances, scales, stop watches, clocks, thermometers, graduated cylinders, protractors, and other appropriate measurement tools. All reading and interpreting measuring devices such as meters, scales of various design, calipers, etc. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-26 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 2. identify the attribute All identifying the quantity to be measured in a given measuring task and to be measured and selecting the most appropriate measuring instrument and units of select the measurement. appropriate unit of measurement for explaining why certain units are preferred over others in a given length, mass All measurement situation, for example by telling why miles is not a (weight), time, convenient unit for measuring the length of a table. temperature, perimeter, area, volume, angle. ((II.3.M1, 2) Identify and use appropriate tools and units Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-27 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 3. estimate measures 5 estimating measurements for various objects or conditions in the with a specified environment, such as the Celsius temperature at which it is degree of accuracy uncomfortable to go outdoors without a jacket or the metric height of and decide if an a professional basketball player. estimate or a measurement is “close enough.” The following measurements appeared in a report. Which ones are reasonable and which are not? Explain your choices: (II.3.M3) Measure with precision and identify • The runner who took first place in the race ran 60 km/hr sources of error • To bake the pizza, we preheated the oven to 200 F. precision • To bake the pizza, we preheated the oven to 200 C. • A dose of cough syrup is 4 cm. • Two cubic yards of concrete are needed to pour a slab 13 ft. x 12 ft. x 4 in. 5-6 developing comparisons of standard units to familiar objects, such as knowing that one gram is approximately the weight of one paper clip. 7-8 deciding on appropriate degrees of precision for measurements and their estimates in particular contexts. 7-8 comparing the magnitude of the error to the quantity being measured and expressing deviations as a percent of error. 7-8 giving examples to illustrate the importance of percent of error, such as recognizing that an error of one dollar is a significant overcharge on the purchase of a candy bar, but it is relatively insignificant on the 7-8 purchase of an automobile. 7-8 identifying sources of error or imprecision in measurements. using informal geometric explorations to develop measurement concepts, such as arranging sectors of a circle to approximate the area, removing and rearranging the vertices of a triangle to find the sum of the angles, or stacking cardboard wafers to approximate volume. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-28 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 4. interpret 7-8 using geoboards or dot paper to construct similar figures, and measurements and comparing their linear and area measurements. recognize that two objects may have 7-8 comparing measurements of sets of objects and drawing conclusions the same about the dependence or independence of measurements—for measurement on one example, if two squares have the same area, then they have the same attribute (e.g., area), but not necessarily perimeter (and vice versa); however two triangles can have the same on another (e.g., area but different perimeters. perimeter). expressing relationships in descriptive terms (e.g., if the sides of the (II.3.M4) Find area, similar triangles are in the ratio of 2 to 1, then the areas are in the surface area, ratio of 4 to 1). perimeter, and volume of objects and 7-8 compare attributes surface area pi (π) “When I doubled (tripled) the length of each side of the triangle, I discovered four (nine) copies of the original figure. The side lengths doubled (tripled), but the area is four (nine) times as great.” 7-8 drawing diagrams or making physical models with geoboards or other construction materials to illustrate observed relationships and explain why they are true. 7-8 finding the area of a plane shape; cutting the shape into two or more pieces; rearranging the pieces to form a new shape; and comparing the area of the result to the area of the original. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-29 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 5. use proportional 6-8 comparing measurements of length, area and volume of similar reasoning and indirect objects and drawing conclusions about the relationships among measurements to draw these variables. inferences. (II.3.M5) Use proportional reasoning to draw measurement inferences proportion A B C D E F Determine the area of each of the trangram pieces if: Area of: A B C D E F 1 4 4 7 1 Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-30 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 6-8 deriving strategies for calculating measurements of one quantity from measurements of another (for example, area of a rectangle from the measurements of length and width). B A C D J F I E G H Copy these shapes on your geoboard. For each shape, record the number of geoboard pegs that lie on the polygon and determine the area of each polygon. Organize your data (number of pegs, area) in a table. Examine your findings and see if you can determine a relationship that tells the area of a geoboard polygon in terms of the number of pegs. Make some additional polygons of your own and check to see if your rule holds. Also try your rule on the following examples: R Q T U S W V Z X Y Does your rule still hold? If not, can you modify it so that it applies to these cases, too? Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-31 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 6-8 using informal approaches with geoboards, graph paper and other concrete objects to develop and verify basic formulas for area and volume. Use your tangram pieces to help you find formulas for the area of a rectangle, a triangle, a parallelogram, and a trapezoid. Draw or construct other triangles, parallelograms, and trapezoids and test your formulas on those examples. 6-8 finding strategies to measure indirectly certain quantities that are very large, very small, or inaccessible, such as measuring the thickness of a book to estimate the thickness of one page, sampling the number of words on several pages to estimate the number of words in a book, or measuring shadows to determine the heights of tall buildings. 6. apply measurement to experimenting with different-size samples of a given material to 6-8 describe the real-world observe that, although mass and volume each vary from one sample and to solve problems. to another, the ratio of mass to volume (i.e., the density) remains constant. (II.3.M6) Use measurement to solve 5 using observed patterns of measurements to predict other real-world problems measurements (such as predicting the height of a seedling from observed patterns of growth). 6 comparing the sizes and distances of planets in the solar system, and constructing solar-system models that show both sizes and distances on the same scale. 6-8 scaling various measurements, such as figuring how high you could jump on the moon or on another planet. 7-8 exploring the significance of size in living organisms. 7-8 Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-32 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level researching how law enforcement officers use measurements of skid marks and other evidence to reconstruct an accident scene. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-33 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project III. Data Analysis and Statistics 1. Students collect and explore data, organize data into a useful form and develop skill in representing and reading data displayed in different formats. (Collection, Organization and Presentation of Data) In the middle years, Grade By engaging in activities such as ... students have the opportunity to… Level 1. collect and explore All posing questions of interest and importance to them, such as data through questions about favorite movies or the depletion of natural observation, resources, and gathering data to help answer those questions. measurement, surveys, sampling All devising and implementing a variety of strategies for collecting data, techniques and such as measuring, counting, sampling, surveying, etc. simulations. All conducting experiments with random devices (spinners, dice, (III.1.M1) Collect and explore data computer programs, etc.) to generate and explore data. surveys sampling techniques simulations 2. organize data using All developing posters, charts, bulletin boards, and other means for tables, charts, presenting data that they have gathered. graphs, spreadsheets and data bases. 7-8 selecting examples of different modes of data presentation from newspapers and magazines and describing the information that they (III.1.M2, 3) Organize contain. and present data and explain choice of representation box (whisker) plot stem & leaf plot circle graph outlier quartile scatter plot Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-34 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 3. present data using a All deciding an appropriate way to present data in various situations and variety of explaining their choices (for example, showing why a line graph is appropriate preferred to a bar graph or a pie chart in a given setting). representations, and explain why one “I used a circle graph because I wanted to show parts of the whole.” representation is preferred over How I Spent My Allowance another or how a particular representation may bias the clothes presentation. save for college (III.1.M2, 3) Organize misc. and present data and movies & explain choice of tapes snacks & representation food box (whisker) plot stem & leaf plot 6-8 circle graph generating examples of situations in which one would choose a outlier particular form of presentation, such as examples of data sets that quartile favor presentation via box plots. scatter plot 7-8 (III.1.M3, 4) Collect discussing the ways in which data are presented in newspaper and and explore data to magazine articles, and identifying questions that can be answered solve real-world All from the data. problems using a spreadsheet to generate several different presentations from 7-8 the same data set. comparing and evaluating several different presentations of the same All data. giving examples of how the presentation of data can be biased, for example by changing the scale of a graph. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-35 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, By engaging in activities such as ... students have the Grade opportunity to… Level 4. identify what data All identifying everyday situations that rely on data in answering are needed to questions, such as predicting election outcomes or making marketing answer a particular decisions. question or solve a given problem, and All identifying interesting problems and asking open-ended questions, design and identifying the information needed to answer those questions, and implement strategies devising and implementing data-collection procedures to arrive at to obtain, organize and present those appropriate conclusions. data. (III.1.M3, 4) Collect and explore data to solve real-world problems Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-36 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project III. Data Analysis and Statistics 2. Students examine data and describe characteristics of the distribution, relate data to the situation from which they arose, and use data to answer questions convincingly and persuasively. (Description and Interpretation) In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 1. critically read data All examining presentations of data encountered in everyday situations from tables, charts or (news reports, advertising campaigns, political speeches, sports graphs and explain the statistics, etc.) and raising questions about the data. source of the data and A major television network reported the following information on its evening what the data represent. news show: (III.2.M1) Read data Percent of Families with Computers in the Home from tables, charts, and graphs examine the source and Whites 27% explain the meaning of Blacks 14% Hispanics 13% data The anchorman then noted: “Twenty-seven percent of white families have computers at home—as many as black and Hispanic families together. Do you agree with the anchorman’s conclusion? Explain. All • posing questions about a set of data, and answering the questions posed by other students. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-37 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 2. describe the shape 7-8 discussing the distribution of data using various examples and of a data distribution describing the shapes of those distributions (e.g., rectangular, and identify the symmetric, bimodal). center, the spread, correlations, and any 6-8 classifying data sets according to selected properties, such as those outliers. that are symmetric about their mean. 6-8 (III.2.M2) Calculate describing the center of a distribution and explaining how the measures of central “center” can be interpreted in different ways, such as the mean vs. the tendency median. use most appropriate 7-8 measure of central tendency describing special features of various data distributions, such as their mean spread (range), cluster points, gaps (discontinuities), maximum and median 7-8 minimum values, and outliers. mode range exploring how certain features of a data distribution can be inferred outliers from visual representations, such as how the length of the box in a correlation (+, -, box-and-whisker plot conveys information about the spread of the none) data. 3. draw, explain and 6-8 preparing presentations that use data to convince the class of a justify conclusions certain conclusion. based on data. All writing persuasive letters in which conclusions are presented and (III.2.M3) Draw, supported by data. explain, and justify conclusions 7-8 staging a class debate in which each side bases its arguments on data it has collected and presented. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-38 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 4. critically question 7-8 reading print media and viewing television presentations of data- the sources of data; based conclusions and discussing their validity. the techniques used to collect, organize 7-9 collecting examples of biased presentations and writing rebuttals to and present data; the expose their biases. inferences drawn from the data; and the possible sources All examining how the method of collecting data can influence the of bias in the data or outcome, for example by comparing the results of a survey given to a their presentation. random sample of adults with the results of the same survey given to parents of preschool children. 5-6 (III.2.M4) Critically question data examining how the size of the sample can influence the outcome of data sources the data gathering. collection techniques 7-8 inferences drawn identifying how data are used in advertising, and raising questions data bias about the sources of those data and the conclusions that the presentation advertisers hope the public will draw. 5. formulate questions 5-6 raising questions of interest to the students and implementing and problems, and procedures for gathering data to help answer those questions. gather and interpret data to answer those 7-8 identifying issues affecting students in the school, such as a proposed questions. Student Council project or a change in the dress code, and gathering data as the basis for a report about student preferences. (III.2.M5) Use data to pose and answer real- world questions Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-39 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project III. Data Analysis and Statistics 3. Students draw defensible inferences about unknown outcomes; make predictions and identify the degree of confidence they have in their prediction. (Inference and Prediction) In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 1. make and test 5-6 posing interesting questions and generating testable hypotheses to hypotheses. help answer those questions, such as: - What kind of music do students prefer, and which band should (III.3.M1) Make and we hire for the dance? test hypotheses - If a salad bar is added in the school cafeteria, will the students support it? 7-8 generating conjectures about physical situations, such as the amount a spring stretches as weights are suspended from it, and conducting experiments to test those hypotheses. 2. design experiments 5-6 conducting experiments that model problem situations, such as using to model and solve dried beans to model a capture-recapture experiment to estimate the problems using fish population in a lake. sampling, simulations, and 7-8 exploring phenomena involving several variables, such as controlled conjecturing about the factors (weight of the pendulum bob, length of investigations. the pendulum, height from which the pendulum is released) that might determine the period of a pendulum, and designing (III.3.M2) Design experiments to model experiments to systematically control and test variables. and solve problems sampling simulations controlled investigations Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-40 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 3. formulate and 5-6 explaining how they arrived at their conclusions after conducting an communicate investigation. arguments and conclusions based 5-6 analyzing the conclusions that other students have put forth and on data and evaluate questioning one another about those conclusions. their arguments and those of others. 7-8 writing and delivering persuasive arguments designed to convince the (III.3.M3) Develop, listener based on the data presented. explain, and analyze conclusions 4. make predictions 5-6 looking for patterns in data as a strategy for making predictions. and decisions based on data, including using data collected in experiments to predict unknown results, and interpolations and All testing those predictions experimentally. extrapolations. 6-8 analyzing the sources of the data and the methods used to gather (III.3.M4) Make them, and deciding how much confidence they can place in the predictions and decisions based on conclusions drawn from those data. data 7-8 interpolations evaluating the context of a data-gathering situation and determining extrapolations the extent to which conclusions drawn from the data can be extended to other situations. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-41 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 5. employ 5-6 solving problems using data-analysis strategies, such as predicting investigations, the “life expectancy” of various boxes of cereal based on a set of mathematical factors which the students identify as affecting the consumption rate models and of cereal. simulations to make inferences and predictions to answer questions and solve problems. (III.3.M5) Use data and investigations to solve real-world 7-8 problems conducting simulations to solve problems where actual experiments are impossible, such as to estimate the fish population of a lake or to determine the number of packs of gum they can expect to buy in order to collect an entire set of baseball cards packed with the gum. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-42 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project IV. Number Sense and Numeration 1. Students experience counting and measuring activities to develop intuitive sense about numbers; develop understanding about properties of numbers; understand the need for and existence of different sets of numbers; and investigate properties of special numbers. (Concepts and Properties of Numbers) In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 1. develop an All using hands-on manipulative materials to develop number concepts understanding of including concepts of place value, prime and composite numbers, integers and rational fractions and decimals, equivalence, inverses, common factors and numbers, and common multiples. represent rational numbers in both 5-6 using physical models of fractional quantities (e.g., number line, fraction and decimal fraction circles, fraction bars) to represent rational quantities such as form. “fractions close to zero (or 1/2 or 1 …). (IV.1.M1) Understand 5-6 expressing numbers using a variety of equivalent representations, and represent integers and rational including: numbers - fractions, decimals and percents. - equivalent fractions with common denominators. 7-8 - physical or geometric representations of fractions or decimals. expressing mathematical quantities in various ways, such as 10·94 recognizing that 5·94 can be viewed as 10·94÷2, as 2 , as (5·90 + 5·4), or as (5·100 - 5·6). Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-43 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 2. extend their 7-8 exploring very large and very small numbers, and representing such understanding of numbers in terms of powers of ten. numeration systems to include decimal 5-6 constructing concrete representations of various quantities, for numeration, example models that illustrate how large is a billion. scientific numeration and non- decimal numeration All exploring selected numeration systems used in other cultures, such as systems. Egyptian or Mayan numerals, as a basis for understanding the power and convenience of a place-value system. (IV.1.M2) Understand 7-8 engaging in hands-on activities with multibase blocks, chip trading, different numeration systems or similar materials to explore non-decimal numeration systems. scientific (small 7-8 numbers) investigating the use of binary and hexadecimal systems in non-decimal (other than computing languages. base 10) All scientific notation using concrete models, such as an 8-hour “clock,” to develop understanding of modular systems. 3. develop an 7-8 discovering a strategy to illustrate that between any two rational understanding of the numbers there is always another rational number (density). properties of the integer and rational 7-8 arranging sets of numbers, including positive and negative integers number systems and rational numbers, in increasing or decreasing order. (e.g., order, density) and of the properties of special numbers 5 giving examples to illustrate the results of adding or multiplying by including 0, 1, π, zero or one. and the additive and 6 multiplicative measuring circumferences and diameters of round objects to develop inverses. a concept of π as the ratio c/d. 6-8 (IV.1.M3) Understand demonstrating the behavior of additive and multiplicative inverses, properties of integers 7-8 and determining the inverses of specified numbers. and rational numbers π developing strategies for mental computation based on the inverses distributive property, such as recognizing that density reciprocals (3.5 x 9) = (3.5 x 10) - 3.5 = (4 x 9) - (.5 x 9) = 3·9 + .5·9. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-44 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 4. apply their 5-6 constructing numerical expressions to represent problem situations, understanding of such as writing an expression for the cost of two 75¢ ice-cream cones number systems to and three 60¢ soft drinks, plus tax. model and solve mathematical and 6 suggesting real applications that correspond to a mathematical applied problems. expression, such as describing a situation that might be represented by (3 x 17) + 5. (I (IV.1.M4) Use number systems to solve real- 7-8 discussing “numerical situations without numbers” such as: world problems Two fractions have a positive sum that is less than one. What can you tell about the fractions? What can you tell about the product of the same two fractions? Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-45 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project IV. Number Sense and Numeration 2. Students recognize that numbers are used in different ways such as counting, measuring, ordering and estimating; understand and produce multiple representations of a number; and translate among equivalent representations. (Representation and Uses of Numbers) In the middle years, Grade By engaging in activities such as … students have the Level opportunity to… 1. give geometric 7-8 using tiles or markers to represent properties of numbers, such as representations of the geometric arrangements that generate the sequences of fractions, prime and triangular and square numbers, or the result that composite composite numbers, numbers can be represented by rectangular arrays in more than one triangular and square way while prime numbers can only be represented as 1-by-n arrays. numbers, and other number concepts; using a number-line model to represent positive and negative represent rational All integers and rational numbers. numbers and integers on the number line. (IV.2.M1) Give geometric representations of numbers triangular numbers (IV.2.M1) Represent integers and rational numbers on the number line 5. select appropriate All solving appropriate consumer problems involving discounts, representations for rebates, interest, and taxes. numbers, including integers and rational All solving real-life problems involving numbers expressed as numbers, in order to fractions, decimals, or percents, or written in scientific notation. simplify and solve problems. (IV.2.M5) Use appropriate representations for numbers to solve real- world problems Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-46 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project IV. Number Sense and Numeration 3. Students investigate relationships such as equality, inequality, inverses, factors and multiples; and represent and compare very large and very small numbers. (Number Relationships) In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 1. compare and order All evaluating expressions without calculating, such as: integers and rational 318 + 661 + 976 ? 918 + 599 + 268 [<, >, = or ?] numbers using 68 x 5 ? 70 + 69 + 68 + 67 + 66 [<, >, = or ?] relations of equality and inequality. All ordering fractions by reasoning about their relationship to whole 23 22 (IV.3.M1) Compare numbers, for example recognizing that 24 is greater than 23 and order integers 1 because, in the first case, the “part not used” (24 ) is less than in the and rational numbers (positive and 1 second case (23 ). negative) 7-8 locating common irrational numbers on the number line by placing them between two integers (e.g., √5 is between 2 and 3) or two rational numbers (e.g., π is between 3.1 and 3.2). All exploring advertised prices and discounts that involve per cents, such as determining how much the customer will save on various purchases or what the discount rate is for an item that is marked down $10. A local craft store ran the following advertisement: SALE Your Choice! Take 50% off OR Take 25% off Any One Item Your Entire Purchase How would you decide whether to choose the 50% option or the 25% option? Explain your reasons. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-47 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 2. express numerical All exploring common applications of rates, such as baseball batting comparisons as averages, unit pricing or miles-per-hour, and explaining the meaning ratios and rates. of such numbers. (IV.3.M2) Explore 7-8 using manipulatives to model ratios, such as paper cut-outs to model applications of the ratios between corresponding parts of similar polygons; ratios and rates 7-8 scale exploring simple geometric sequences such as: 3 9 27 1, 2 , 4 , 8 …, and calculating the common ratio of successive terms. All working with maps and scale drawings, and constructing their own scale drawings of selected objects. 7-8 graphing linear relationships and explaining how slope represents a rate of change. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-48 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 3. distinguish between 7-8 developing strategies for determining whether a number is prime, and prime and using a calculator to carry out the investigation. composite numbers; identify factors, 7-8 constructing a systematic method of finding all the factors of a given multiples, common number with the aid of a calculator. factors and 5-6 multiples, and relatively prime writing factor trees to illustrate the prime decomposition of various numbers; and apply numbers. divisibility tests to numbers. Two students were factoring the number 48. Natalie wrote 48 = 2 x 24, while Jeremy began with 48 = 6 x 8. If they each develop a factor tree for 48, will (IV.3.M3) Identify they get the same set of prime factors in the end? numerical relationships Can you produce any other factor trees for 48 besides the ones Natalie and Jeremy relatively prime developed? How many? Did you get the same prime factors? Is what you found out about the prime factors of 48 true for other numbers as well? All (Fundamental Theorem of Arithmetic) exploring common divisibility tests and applying them to selected numbers. Goldbach, a famous mathematician, proposed that any even number greater 5 than 2 can be written as a sum of two prime numbers (for example, 36 = 5 + 31). Pick any 10 even numbers greater than 2 and see if they follow Goldbach’s conjecture. Can you find an even number greater than 2 that does not obey this 7-8 rule • demonstrating a procedure for finding multiples of a number. • developing a procedure for finding common factors and common multiples of two or more numbers with the aid of a calculator. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-49 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 4. explain the meaning 7-8 exploring the result of repeatedly doubling or halving a quantity, and of powers and roots comparing that to the result of repeatedly adding or subtracting two. of numbers and use calculators to 7-8 conducting experiments that illustrate exponential growth or decay, compute powers and such as measuring the heights of successive bounces of a ball. square roots. 7-8 graphing the results of experiments involving exponential growth or (IV.3.M4) Explain and decay. compute powers and square roots 7-8 investigating the relationship between the length of the side and the Tools: scientific area of a square, and determining one of the two values when the calculators other is known. 5. apply their 6 using published data from a report on contamination in local streams understanding of to compare actual conditions to environmental standards for number acceptable water quality. relationships in solving problems. 7-8 conducting their own environmental studies of air, soil or water quality, radon levels in the home, etc., and writing reports or making (IV.3.M5) Apply presentations on their findings. understanding of number relationships 5 monitoring their food and vitamin intake for one day and comparing to solve real-world problems their consumption to recommended daily allowances. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-50 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project V. Numerical and Algebraic Operations and Analytical Thinking 1. Students understand and use various types of operations (e.g., addition, subtraction, multiplication, division) to solve problems. (Operations and their Properties) In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 1. use manipulatives 5-6 extending familiar models and manipulatives used for operations and diagrams to with whole numbers (number line, multibase blocks, chips, model operations rectangular arrays, etc.) to model operations with fractions, decimals and their inverses and integers. with integers and rational numbers and relate the models to their symbolic expressions. (V.1.M1) Use manipulatives and 7-8 writing mathematical diagrams to model expressions to correspond to 3 operations involving physical models, for example algebraic expressions writing (n+2)·3 to describe. n 2 6-8 drawing pictures to represent mathematical operations, for example a diagram that illustrates (1/3 + 1/2) or (2/3 · 1/4). Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-51 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 2. compute with 7-8 developing strategies for estimating computations, for example integers, rational recognizing that (2.47 x 3.93) is approximately 10 (2.5 · 4). numbers and simple algebraic applying mental estimation to consumer situations, for example expressions using 5 reasoning that the bill for 12 assorted candy bars must be less than $6 mental computation, since each one is less than 50¢. estimation, calculators, and 7-8 paper-and-pencil; developing and practicing strategies for mental computation such as explain what they combining addends that sum to 10 ($13.30 + $5.95 + $3.70 = $17 + are doing and know $5.95 = $23 - 5¢) or using related facts (“5+5 =10, so 6+6 would be which operations to 12, and then 6+7 would have to be 13”). perform in a given 6 situation. comparing the meaning of expressions like (5 + 7) x 3 and 5 + (7 x 3) and explaining the need for an order-of-operations rule for (V.1.M2) Perform 7-8 5 + 7 x 3. operations integers rational numbers developing facility with a calculator, including the use of memory simple algebraic and special keys, and using a calculator to carry out complex expressions procedures. percents (x, , using scientific calculator) (V.1.M2) Explain strategies and know which operation to perform in a given situation Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-52 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 3. describe the 5 using a number line to model the associative and commutative properties of properties for addition with rationals and integers. operations with rationals and 5-6 using an area model to demonstrate the associative and commutative integers (e.g., properties for multiplication with rationals and integers. closure; associative, commutative and distributive 7-8 exploring the distributive property using physical models, diagrams properties) and give and mathematical expressions. examples of how they use those 3 people (2 books + 3 toys) 3(2+3) =3(5) properties. (6 books + 9 toys) (6 + 9) (3 people x (n+2) presents) 3(n + 2) (V.1.M3) Describe and apply properties with 7-8 rationals and integers distributive using manipulatives and diagrams to show that n/n = 1 (n≠0) and division by zero using that result to find equivalent fractions by multiplying or All dividing by numeral (n/n). exploring the relationship between multiplication and division, including multiplication by zero, and offering reasons to explain why division by zero is undefined. 4. efficiently and All reading and discussing problem situations and explaining what accurately apply mathematical operations will be needed, and why. operations with integers, rational 5 predicting the approximate result of a computation before actually numbers, and simple carrying it out. algebraic 6 expressions in solving problems. writing examples of problems that would appropriately be solved with mental computation, paper and pencil, or calculator. (V.1.M4) Use 7-8 operations to solve examining the results of several computations and, without real-world problems calculating, determining which answers are reasonable and which are not. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-53 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project V. Numerical and Algebraic Operations and Analytical Thinking 2. Students analyze problems to determine an appropriate process for solution and use algebraic notations to model or represent problems. (Algebraic and Analytic Thinking) In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… All writing verbal descriptions of situations that could be represented by 1. read and write given algebraic expressions (e.g., 14x + 37y could represent the total algebraic cost of 14 cards and 37 stamps). expressions; develop original examples 7-8 writing algebraic expressions and equations that describe selected expressed verbally situations (e.g., How could you express the total cost of tickets for and algebraically; the field trip if adult tickets are $5 and children’s tickets are $1.50?). simplify expressions 6-8 and translate between translating verbal and algebraic expressions that were written by verbal and algebraic other students. expressions; and solve linear equations 7-8 determining solutions for linear equations and inequalities using and inequalities. informal means including physical models and guess-and-check strategies. (V.2.M1) Translate between verbal and algebraic expressions (V.2.M1) Read, write, and simplify algebraic expressions (V.2.M1) Write and solve linear equations and inequalities Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-54 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-55 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 6-8 engaging in hands-on explorations with manipulatives (e.g., algebra 2. represent algebraic tiles, balance beams) and using them to represent algebraic concepts with expressions, equations and inequalities. geometric models (e.g., algebra tiles), physical models (e.g., balance beam), tables and graphs; and write algebraic expressions to correspond to the multiple representations. 7-8 constructing tables of values, including tables generated with computer spreadsheets, to represent algebraic expressions, such as a (V.2.M2) Use models, table of ordered pairs that satisfy the equation m = 3n - 1. tables, and graphs to represent algebraic concepts 7-8 identifying a pattern in a table of values and using that pattern to construct an algebraic expression to represent the given relationship. x 1 2 3 4 5 y =? y 2 5 10 17 26 7-8 graphing values from a table and explaining how the graph represents the table and the equation derived from that table (e.g., explaining why the graph of entries in a table of circumference vs. diameter of round objects yields a straight line and how the slope of that line relates to the value of pi). 7-8 demonstrating how variables in an algebraic expression are related by showing what happens to one variable when another is changed, for example demonstrating that when the diameter of a circle is tripled, the circumference is also tripled. Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-56 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 3. solve linear 7-8 using number lines to model solutions to linear equations and equalities and inequalities in one variable. inequalities using algebraic and geometric methods, and use the context of the problem to interpret and explain 5 x < 35 their solutions. 7-8 using geometric representations, including graphs in the Cartesian (V.2.M3) Graph and plane, to model solutions of linear equations in two variables. explain solutions for linear equations and inequalities Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-57 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 4. analyze problems 7-8 discussing how individual students understand and represent modeled by linear selected problems and comparing such multiple representations; functions, determine strategies for All devising and explaining their own strategies for solving problems. solving the problems, and 7-8 discussing whether a solution is acceptable in the context of the evaluate the adequacy of the problem (for example, 11.69 is an appropriate solution to 456/39 solutions in the when computing the average cost of 39 items ordered from a context of the catalog—but when computing the number of 39-passenger busses problems. needed to transport 456 students, the answer is 12, while the answer is 11 when determining the number of boxes that can be filled when (V.2.M4) Choose a 456 cookies are packaged in boxes of 39). problem-solving strategy and decide if the solution is appropriate 5. explore problems 5-6 comparing long-distance calling plans from several different carriers that reflect the and deciding on the most economical plan to buy. contemporary uses of mathematics in using a spread sheet to explore the way a savings account grows significant contexts 5-6 given different initial investments and rates of interest, and using the and use the power of 7-8 information to plan a personal savings strategy. technology and algebraic and analytic reasoning to experience the ways mathematics is used in society. (V.2.M5) Use technology, algebraic concepts, and reasoning to solve real-world problems Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-58 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project VI. Probability and Discrete Mathematics 1. Students develop an understanding of the notion of certainty and of probability as a measure of the degree of likelihood that can be assigned to a given event based on the knowledge available; and, make critical judgments about claims that are made in probabilistic situations. (Probability) In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 3. conduct experiments 7-8 conducting experiments where items are replaced after they are and give examples selected compared to experiments where items are not replaced after to illustrate the they are selected. difference between dependent and 7-8 investigating whether the occurrence of the first event affects the independent events. occurrence of the second event. (VI.1.M3) Conduct experiments involving dependent and independent events mutually exclusive compound events 4. explain the 7-8 conducting experiments to generate data that lead to empirical difference between probabilities (such as the probability of rolling “5” on a die, calculated probabilities experimentally by rolling a die 20 times) and comparing those results determined from to the corresponding mathematical or theoretical probability. experiments or chance events 7-8 combining experimental data and using computer simulations to (empirical) and produce large numbers of trials, and comparing the probabilities probabilities derived derived from such large samples with those derived from small mathematically (theoretical), and samples and with the theoretical values. explain how the empirical probability changes for a large number of trials. (VI.1.M4) Compare experimental and theoretical probabilities Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-59 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as ... students have the Level opportunity to… 5. conduct probability All identifying situations in which probability is used to solve real experiments and problems, such as in actuarial applications or product testing. simulations to model and solve 7-8 applying probabilities to model and solve real problems such as using problems. capture-recapture simulations to estimate wildlife populations. (VI.1.M5) Conduct 6-8 exploring the concept of “fair games” by playing and analyzing simple probability experiments and games and by designing original games. simulations to solve real-world problems Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-60 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project VI. Probability and Discrete Mathematics 2. Students investigate practical situations such as scheduling, routing,sequencing, networking, organizing and classifying; and analyze ideas like recurrence relations, induction, iteration and algorithm design. (Discrete Mathematics) In the middle years, Grade By engaging in activities such as … students have the Level opportunity to… 1. use manipulatives, 6-8 using physical manipulatives to model simple permutation and diagrams, and the combination problems. fundamental theorem of counting 7-8 exploring the fundamental theorem of counting (i.e., if one task can to count be performed in m ways and the second task can be performed in n permutations and ways, then the number of ways of performing the two tasks is mn combinations. ways). (VI.2.M1) Use 7-8 observing patterns in the number of permutations of 2, 3, 4, … manipulatives, objects and deriving an expression for the number of possible diagrams, and counting principle to permutations of any number of objects. 7-8 count permutations and combinations giving examples of situations that illustrate the difference between permutations, where the order of objects is important, and combinations, where order is not important. 2. use sets and set All constructing Venn diagrams to represent mathematical situations relationships to (for example, various partitions on the set of integers, or diagrams explore and solve of the relationships among different sets of quadrilaterals). simple algebraic and geometric problems. All describing attributes of given attribute blocks (VI.2.M2) Use sets and set relationships to solve geometric and algebraic problems Venn diagrams Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-61 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as … students have the Level opportunity to… 3. solve problems 7-8 tracing paths through a network of vertices and edges. involving networks, for example planning delivery routes or counting paths between points. (VI.2.M3) Solve 7-8 using networks to solve problems such as laying out the most problems involving efficient route for delivering newspapers. networks 4. explore recurrence All analyzing and extending arithmetic and geometric sequences. relations and iterations 7-8 investigating Pascal’s triangle and the Fibonacci sequence. (VI.2.M4) Explore recurrence relations and iterations 5. continue to use 7-8 designing an algorithm to accomplish a task: manipulatives and - design a flow diagram to “assemble” a sandwich. drawings to model - use a map to write directions to get from point A to point B. the concepts and procedures for the All creating and solving logic problems. standard arithmetic algorithms, and develop and analyze 7-8 investigating the Euclidean algorithm to find the greatest common their own and other factor of two numbers. students' algorithms to accomplish a task or solve a mathematical problem. (VI.2.M5) Develop and analyze algorithms to solve problems Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-62 Suggested Teaching and Learning Activities taken from the Michigan Curriculum Framework Project In the middle years, Grade By engaging in activities such as … students have the Level opportunity to… 6. use discrete All using discrete mathematics concepts to model situations and solve mathematics problems, such as: concepts as - developing schedules for school, buses, television, etc. described above to - creating maps and design delivery routes or rubbish pick-up model situations and routes. solve problems; and look for whether or - using tree diagrams to investigate genetic characteristics. not there is a - investigating the concept of “pyramid schemes” or telephone solution (existence relays. problems), - investigating graphs like the Konigsberg bridge problem. determine how - determining possible combinations of clothing, food, friends, many solutions there activities, etc. and describe results of adding/subtracting items. are (counting problems), and decide upon a best solution (optimization problems). (VI.2.M6) Use discrete math to solve existence, counting, and optimization real- world problems Huron County Mathematics Curriculum Seventh-Eighth Grade 1/01 DRAFT 7-63

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