Math Performance Descriptors with Lesson and Activity Links

Notes about this page:  A few years ago I created this page for teachers at Providence St. Mel School in Chicago in my role as a volunteer mathematics specialist. I started with the State of Illinois Math Performance Descriptors, assigning each descriptor to what I thought was the appropriate grade level. Then I linked the content of each descriptor to the National Council of Teachers of Mathematics (NCTM) Standards site. Finally, I found good lessons, activities, and/or resources that I judged to be worth viewing while preparing to teach each objective. My rationale was that the teachers would be overwhelmed by a web search on a particular descriptor, finding thousands, if not millions, of results. To simplify the process, I prescreened and identified excellent links. 


 I am in the process of updating the page to match the Common Core Standards.




Mathematics | Kindergarten
In Kindergarten, instructional time should focus on two critical areas: (1)
representing, relating, and operating on whole numbers, initially with
sets of objects; (2) describing shapes and space. More learning time in
Kindergarten should be devoted to number than to other topics.
(1) Students use numbers, including written numerals, to represent
quantities and to solve quantitative problems, such as counting objects in
a set; counting out a given number of objects; comparing sets or numerals;
and modeling simple joining and separating situations with sets of objects,
or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten
students should see addition and subtraction equations, and student
writing of equations in kindergarten is encouraged, but it is not required.)
Students choose, combine, and apply effective strategies for answering
quantitative questions, including quickly recognizing the cardinalities of
small sets of objects, counting and producing sets of given sizes, counting
the number of objects in combined sets, or counting the number of objects
that remain in a set after some are taken away.
(2) Students describe their physical world using geometric ideas (e.g.,
shape, orientation, spatial relations) and vocabulary. They identify, name,
and describe basic two-dimensional shapes, such as squares, triangles,
circles, rectangles, and hexagons, presented in a variety of ways (e.g., with
different sizes and orientations), as well as three-dimensional shapes such
as cubes, cones, cylinders, and spheres. They use basic shapes and spatial
reasoning to model objects in their environment and to construct more
complex shapes.
Counting and Cardinality
• Know number names and the count sequence.
• Count to tell the number of objects.
• Compare numbers.
Operations and Algebraic Thinking
• Understand addition as putting together and
adding to, and understand subtraction as
taking apart and taking from.
Number and Operations in Base Ten
• Work with numbers 11–19 to gain foundations
for place value.
Measurement and Data
• Describe and compare measurable attributes.
• Classify objects and count the number of
objects in categories.
Geometry
• Identify and describe shapes.
• Analyze, compare, create, and compose
shapes.
Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique
the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning.

Counting and Cardinality K.CC
Know number names and the count sequence.
1. Count to 100 by ones and by tens.

2. Count forward beginning from a given number within the known
sequence (instead of having to begin at 1).

3. Write numbers from 0 to 20. Represent a number of objects with a
written numeral 0-20 (with 0 representing a count of no objects).

Count to tell the number of objects.
4. Understand the relationship between numbers and quantities; connect
counting to cardinality.


a. When counting objects, say the number names in the standard
order, pairing each object with one and only one number name
and each number name with one and only one object.
b. Understand that the last number name said tells the number of
objects counted. The number of objects is the same regardless of
their arrangement or the order in which they were counted.
c. Understand that each successive number name refers to a quantity
that is one larger.
5. Count to answer “how many?” questions about as many as 20 things
arranged in a line, a rectangular array, or a circle, or as many as 10
things in a scattered configuration; given a number from 1–20, count
out that many objects.

Compare numbers.
6. Identify whether the number of objects in one group is greater than,
less than, or equal to the number of objects in another group, e.g., by
using matching and counting strategies.1

7. Compare two numbers between 1 and 10 presented as written
numerals.
Operations and Algebraic Thinking K.OA
Understand addition as putting together and adding to, and understand
subtraction as taking apart and taking from.
1. Represent addition and subtraction with objects, fingers, mental
images, drawings2, sounds (e.g., claps), acting out situations, verbal
explanations, expressions, or equations.

2. Solve addition and subtraction word problems, and add and subtract
within 10, e.g., by using objects or drawings to represent the problem.

3. Decompose numbers less than or equal to 10 into pairs in more
than one way, e.g., by using objects or drawings, and record each
decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

4. For any number from 1 to 9, find the number that makes 10 when
added to the given number, e.g., by using objects or drawings, and
record the answer with a drawing or equation.

5. Fluently add and subtract within 5.

1Include groups with up to ten objects.
2Drawings need not show details, but should show the mathematics in the problem.
(This applies wherever drawings are mentioned in the Standards.)
Number and Operations in Base Ten K.NBT
Work with numbers 11–19 to gain foundations for place value.
1. Compose and decompose numbers from 11 to 19 into ten ones and
some further ones, e.g., by using objects or drawings, and record each
composition or decomposition by a drawing or equation (e.g., 18 = 10 +
8); understand that these numbers are composed of ten ones and one,
two, three, four, five, six, seven, eight, or nine ones.
Measurement and Data K.MD
Describe and compare measurable attributes.
1. Describe measurable attributes of objects, such as length or weight.
Describe several measurable attributes of a single object.

2. Directly compare two objects with a measurable attribute in common,
to see which object has “more of”/“less of” the attribute, and describe
the difference. For example, directly compare the heights of two
children and describe one child as taller/shorter.
Classify objects and count the number of objects in each category.
3. Classify objects into given categories; count the numbers of objects in
each category and sort the categories by count.3
Geometry K.G
Identify and describe shapes (squares, circles, triangles, rectangles,
hexagons, cubes, cones, cylinders, and spheres).
1. Describe objects in the environment using names of shapes, and
describe the relative positions of these objects using terms such as
above, below, beside, in front of, behind, and next to.
  • Literature Connection: My Very First Book of Shapes Eric Carle (1974) This is a wordless, split, board book, where the lower pages have a colorful picture and the upper ones have corresponding shadows but in a different order. Children are to flip the pages until they match the picture to the shadow. This book does not fit into the pattern that encourages children, parents, and teachers to think that "learning your shapes" means naming just circle, square, triangle, and rectangle and learning these imperfectly! (Scroll down to the set of activities suggested for this book around the middle of the web page.)
  • Shaping Patterns and Dancing Shapes
  • Barrier Game Grid (3x3) and Positional Words with Pattern Block Barrier Game
  • Activity: Take a GEOMETRY WALK through the school and/or outdoors. Use a digital camera to record the geometric objects you see. For a sample of results, watch this presentation from a class of 6th graders: TeacherTube Videos - Geometry Walk Geometry in the Real World
2. Correctly name shapes regardless of their orientations or overall size.

3. Identify shapes as two-dimensional (lying in a plane, “flat”) or threedimensional
(“solid”).

Analyze, compare, create, and compose shapes.
4. Analyze and compare two- and three-dimensional shapes, in
different sizes and orientations, using informal language to describe
their similarities, differences, parts (e.g., number of sides and
vertices/“corners”) and other attributes (e.g., having sides of equal
length).
5. Model shapes in the world by building shapes from components (e.g.,
sticks and clay balls) and drawing shapes.
6. Compose simple shapes to form larger shapes. For example, “Can you
join these two triangles with full sides touching to make a rectangle?”
3Limit category counts to be less than or equal to 10.




Grade 1


6A Stage A
• Count with understanding, including skip counting by 2's, 5’s, and 10’s from zero. **
• Demonstrate the concept of odd and even using manipulatives.
6A Stage B
• Extend initial understanding of place value and the base ten number system using multiple models. **
  • Professional Development Resource: article comparing base ten blocks with computer for teaching place value
  • Activity: Race to 100.  The students will work in pairs and have a place value mat, base ten blocks, and a die (six sided). The students will roll to see who goes first. The student who is first will roll the die and then the number rolled is the number to represent on the mat. Players take turns. The first player who gets 100 or more wins. The students will need to remember to trade their units into rods when needed. When they reach 100 they will have a flat on their mat. You can return to this activity in a format involving addition later in the school year; see pages 11-12 of this Math Games handout. 
• Use cardinal and ordinal numbers appropriately.
  • Lesson:  Ten Teddy Bears All in a Row. For teddy bear templates, see pages 103-104 of this family literacy document (which contains a poem about teddy bears for more language arts connections). 
  • Literature connection: On the Stairs is about two mice climbing the first through twelfth steps in rhymes.
6B Stage A
• Solve one-step addition and subtraction number sentences and word problems using concrete materials.

• Construct number sentences to match word problems.


• Demonstrate and describe the effects of adding and subtracting whole numbers using appropriate mathematical notation and vocabulary. **


  • Lesson:  Comparing Sets
    Students create subtraction problems and find differences by comparing sets. They present results in the form of a table and illustrate subtraction situations.


• Explore and apply properties of addition and subtraction.
6C Stage A
• Develop and use strategies for whole number computations, with a focus on addition and subtraction. *
  • Lesson: Mystery Trains Children follow a set of clues to make one or more Cuisenaire Rod trains.
• Use mental math counting strategies.


• Describe reasonable and unreasonable sums and differences.
• Utilize a calculator for counting patterns.


6D Stage A
• Compare two or more sets, using manipulatives, to solve problems.


7A Stage A
• Determine the attributes of an object that are measurable (e.g., length and weight are measurable; color and texture are not).
• Measure objects using nonstandard units.
  • Lesson with literature connection:  How Big Is a Foot?    In this lesson children read  the book, How Big Is A Foot? by Rolf Myller, about the confusion that results from not using a standard measurement tool and then do a measuring activity using a non-standard tool. They then discuss the importance of having a standard tool for measuring. 
  • Literature Connection: For Teacher/Actresses Read Miss Nelson is Missing! by Harry Allard. The next day, dress as Viola Swamp, the substitute teacher in the story. Announce to students that you cannot find anything to measure things with, but you do have some candy bars. Students will work with partners to measure several items in the classroom and chart their measurements for comparison. After some discussion, students should discover that nonstandard units are unreliable, since there are many candy bar sizes. The next day, return to the class as yourself and ask students to write about what happened the day before.
• Identify units of money and the value of each.


7A Stage B
• Identify the type of measure (e.g., weight, height, volume, temperature) for each measurable attribute.
• Order events chronologically.






7B Stage A
• Estimate nonstandard measurements of length, weight, and capacity.
  • (length--See ladybug lesson below for 9A Stage B)
7B Stage B
• Estimate elapsed time for a given task.


7C Stage A
• Choose appropriate nonstandard measurement units to measure length, weight, and capacity (e.g., number of handfuls of cubes to fill a container).
8A Stage A
• Check computation using fact families.
8A Stage B
• Sort, classify, and order objects by multiple properties. **
     by Margarette Reid. Students will organize data into categories by sorting and classifying objects. They will also represent the data collected by using concrete objects, pictures, and/or graphs. Students will sort buttons according to their attributes, classify buttons in a set, organize data collected about their buttons, and use pictures or graphs to represent this information. A great resource.
• Create rules for multiple sortings in a single set.


• Recognize, describe, and extend geometric and numeric patterns.


• Create patterns concretely and numerically to match a given letter description (e.g., AAB) and make predictions.


• Change patterns by manipulation of concrete materials.
  • Lessons: Creating, Describing, and Analyzing Patterns The first part, Making Patterns, includes an interactive figure for creating, comparing, and viewing multiple repetitions of patterns. The interactive figure illustrates how students can create pattern units of squares then predict how patterns with different numbers of squares, will appear when repeated in a grid, and check their predictions. In the second part, Describing Patterns, examples of various ways students might interpret the same sequence of cubes are given. This illustrates the importance of discussing and analyzing patterns in the classroom. The third part, Extending Pattern Understandings, demonstrates ways in which students begin to create a "unit of units," or a grouping that can be repeated, and begin to relate two patterns in a functional relationship.
• Describe missing units in a pattern.


• Analyze growing patterns.
8C Stage A
• Solve real life word problems using patterns.
8C Stage B
• Solve problems and justify solutions using patterns.
  • Investigation: Tables and Chairs  (Don't expect first graders to be able to write a formula!)


8D Stage A
• Solve simple number sentences with variables (e.g., missing addend problems).
9A Stage A
• Recognize and describe shapes that have line symmetry. **


  • Lesson: Symmetry and African Artwork -- a lesson from ArtsEdge at the Kennedy Center; also has a literature connection to Anansi the Spider by Gerald McDermott.
9A Stage B
• Describe, name, and interpret direction and distance in navigating space and apply ideas about direction and distance (e.g., nearer/farther). **
  • Lesson: Lesson Plan Series: Ladybug Adventures - Part 1 - In this activity, students will use their knowledge of number, measurement and geometry to design a "virtual path" which enables a ladybug to hide under a leaf. They also develop navigational skills by testing to see if their path is accurate and revising their solutions. Part 2 - Making Triangles - Students will use 45 and 90-degree angles to create triangles, and develop an understanding of the relationship between angles and the shape of triangle. Students use their knowledge of number, measurement and geometry to design a "virtual path" using two different angles to help a ladybug reach its hiding place under a leaf. Part 3: Making Rectangles - Students use their knowledge of number, measurement and geometry to plan the steps necessary for a ladybug to draw rectangles of different sizes. As they experiment, students begin to understand the relationship between the shape of a rectangle and the lengths of its sides. They will also develop a sense of the amount of turn in a right angle. Part 4: Ladybug Mazes - In this activity, students will plan a series of moves that will navigate a ladybug through a maze. Their plans will turn the ladybug at the appropriate corners and keep it on a path without crossing the walls. This activity helps students gain experience in estimating length and angle measures.
• Create and complete shapes that have line symmetry.


9B Stage B
• Identify objects that are the same shape and size.
• Compare and contrast attributes of two- and three-dimensional objects using appropriate vocabulary.
  • Unit: It's a Perfect Fit Comprehensive unit on two- and three-dimensional shapes, complete with lesson plans, activity sheets, etc. Uses pattern blocks and interlocking cubes.


10A Stage A
• Organize, describe, and label simple data such as pictographs, tallies, tables, and bar graphs.
• Compare numerical information derived from tables and graphs.


  • Lesson: Comparing Columns on a Bar Graph
    Students make bar graphs and use comparison subtraction to process the data on the graphs. They also play a subtraction game in pairs.
10B Stage A
• Gather data and pose questions about pictographs, tallies, tables, and bar graphs.
  • Lesson: A Shoe In Note: literature connection -- The Elves and the Shoemaker.
10C Stage A
• Identify possible and impossible results of probability events using concrete materials.




Grade 2


6A Stage B
• Count with understanding, including skip counting from any number by 2’s and 10’s.
• Extend initial understanding of place value and the base ten number system using multiple models.**
  • Lesson: Base Ten Blocks lesson - in particular follow the link for Counting the Rice, where the students themselves model numbers up to 3 digits.
• Describe numeric relationships using comparison notation.
• Recognize and explain the concept of odd and even numbers.
6A Stage C
• Represent, order, and compare whole numbers to demonstrate an understanding of the base ten number system.


6B Stage B
• Solve two-step addition and subtraction number sentences and word problems.
























































  •  Lesson: Addition and subtraction using base 10 blocks  



































































































































































  •  Lesson: Base 10 blocks--understand trading and renaming











































































































    • • Demonstrate the relationship between addition and subtraction.


    • Explore multiplication and division through equal grouping of objects and sharing. **
    • Activity: Give Me Some Sharing "money" represented using base ten blocks.


    • Connect repeated addition to multiplication.
    • Demonstrate fluency with basic addition and subtraction facts. **


    6C Stage B
    • Explain and use mental math strategies to solve simple addition and subtraction problems.
    • Estimate sums and differences of one- or two-digit numbers.
    • Analyze situations to determine whether exact numbers or estimates are appropriate.
    • Utilize a calculator to solve addition and subtraction problems.
    7A Stage B
    • Explore and describe perimeter and area of real objects.
    • Measure objects using standard units.
    • Tell time using an analog clock.
    • Lesson:  Take a Walk Through My Day  Students will tell and record time on an analog clock using five-minute intervals and create a book that uses analog time to help relate a story.
    • Describe relationships within units of time, money, and length (e.g., 12 inches in a foot).
    • Unit plan: Number Cents  Students explore the relationships among pennies, nickels, dimes, and quarters. They count sets of mixed coins, write story problems that involve money, and use coins to make patterns.


    • Count, compare, and order sets of unlike coins.
    • Show equivalent amounts of money.
    • Explore and explain making change using manipulatives.


    7B Stage B
    • Estimate standard measurements of length, weight, and capacity.
    • Estimate the amount of money needed to make purchases.


    7C Stage B
    • Select an appropriate unit and tool for measurement. **


    8A Stage B
    • Extend numeric patterns involving addition and/or subtraction (e.g., 1, 3, 5, … what are the next two terms?).
    8A Stage C
    • Extend geometric and simple numeric patterns using concrete objects or paper and pencil.
    • Demonstrate how to create a pattern given a set of directions.
    • Identify errors in a given pattern.8B Stage B
    • Describe and compare quantitative change (e.g., student grows two inches in one year). **


    8C Stage C
    • Apply the relationship of fact families to solve for an unknown quantity.
    • Lesson:  Fact Families
      Students make connecting cube trains in two colors, then search for the related addition and subtraction facts for a given number and use a calculator to find differences. They also investigate fact families where one addend is 0.
    8D Stage B
    • Solve word problems involving unknown quantities.


    9A Stage B
    • Compare and contrast the attributes of two- and three-dimensional shapes using appropriate vocabulary.
    • Investigate and predict the results of putting together and taking apart two- and three-dimensional shapes (e.g., put two triangles together to make a square). **
    • Perform translations (slides), reflections (flips), and rotations (turns) with concrete objects.
    • Recognize and represent shapes from different perspectives.9B Stage C
    • Apply geometric ideas and relationships to problems that arise in the classroom or in everyday life. **


    10A Stage B
    • Organize and interpret simple data such as pictographs, tallies, tables, and bar graphs.
    • Lesson:  Eyelet Graphs Students will practice estimation and graphing skills by estimating the number of shoe eyelets their classroom contains, gathering and tabulating the data, and presenting the data in the form of a bar graph.
    • Make predictions from data.


    10B Stage B
    • Create and use interview questions to gather data.


    10C Stage B
    • Identify and discuss likely, unlikely, and impossible probability events.
    • Communicate and display results of probability events in order to make predictions of future events.




    Grade 3


    6A Stage B
    • Describe parts of a whole using 1/2, 1/3, and 1/4.
    • Order concrete representations of unit fractions.
    • Describe parts of a set using 1/2, 1/3, and 1/4.
    • Represent, order, label, and compare unit fractions using concrete materials.
    6A Stage C
    • Represent, order, and compare whole numbers to demonstrate an understanding of the base ten number system.
    • Recognize equivalent representations of whole numbers and generate them by composing and decomposing numbers (e.g., 123 = 100 + 20 + 3). **




    • Judge the size of fractions using models, benchmarks, and equivalent forms. **
    • Represent, order, label, and compare familiar fractions.
    • Recognize and generate equivalent forms of familiar fractions. **
    • Explore and discuss uses of decimals.6B Stage C
    • Show and use the relationship between multiplication and division.
    • Demonstrate and describe the effects of multiplying and dividing whole numbers using appropriate mathematical notation and vocabulary.
    • Explore, identify, and use relationships between and among properties of operations (e.g., commutativity applies to addition but not to subtraction).
    • Demonstrate fluency with basic multiplication and division facts.
      • Fact Drill Idea (from PBS ) "Seven-Minute Boogie Playing basic-fact games can be both fun and effective, as long as they are either non-competitive or "self-competitive." In Seven-Minute Boogie, students roll two dice, make a multiplication problem out of the numbers, write it down (with the answer), and repeat the process as many times as they can in seven minutes. Their score for the day is the number of multiplication sentences they got right in the seven minutes. The next day, they try to improve their scores. With regular dice, they practice low numbers, which are the most useful facts anyway. But you can alter dice to cover more facts: try "2-3-4-6-7-8" and "3-5-6-7-8-9," which miss only a few of the non-trivial facts." http://www.learner.org/channel/workshops/math/work_5.html
      • Lesson: Click on Bricks -- Multiplication The relationship between addition and multiplication is outlined, a process for duplicating the activities in the classroom using "Bricks" is included, and sample problems are given. If the answer is incorrect, a visual clue and the chance to try again is offered. A list of links to related math sites is included to offer visitors more fun and interesting sites.
      • Solve multiplication and division number sentences and word problems.


      • Apply knowledge of basic multiplication facts (factors 0-10) to related facts (e.g., 3 x 4 = 12, 30 x 4 = 120, 300 x 4 = 1200).
      • Select and use one of various algorithms to add and subtract.


      6C Stage C
      • Develop and use strategies (i.e. rounding) to estimate the results of whole-number computations and to judge the reasonableness of such results. **
      • Lesson: Sand Babies. Students will measure weight to the nearest pound and construct and interpret a bar graph. Students will measure length using non-standard units and determine area using square tiles. This lesson also relates to 7A Stage C below.
      • Select appropriate methods and tools for computing with whole numbers from mental computation, estimation, calculators, and paper/pencil according to the context and nature of the computation and use the selected method or tool. *
      • Determine whether exact answers or estimates are appropriate for solutions to problems.6D Stage B
      • Compare unit fractions, using manipulatives, to solve problems.


      6D Stage C
      • Describe the relationship between two sets using ">", "<", and "=", "1".


      7A Stage C
      • Explain the need for using standard units for measuring. **
      • Measure objects using standard units in the U.S. customary and metric systems. **
      • Perform simple unit conversions within a system of measurement (e.g., three feet is the same as a yard). **


      • Describe multiple measurable attributes (e.g., length, mass/weight, time, temperature, area, volume, capacity) of a single object.
      • Make change from a given amount using bills and coins.
      • Show and explain perimeter of an object by measuring and adding its linear units.


        • Show and explain the area of an object by counting square units.
        7B Stage C
        • Develop and use common referents for linear measures to make comparisons and estimates.
        • Lesson:   How Much is a Million?   After listening to the story, How Much is a Million? , students work in groups to determine the following: how long one million dollars would be laid out end to end and how tall a stack of one million pennies would be.                                             
        • Estimate perimeter of simple polygons.


        7C Stage C
        • Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, and temperature. *
        • Determine elapsed time between events.
        • Solve problems using perimeter and area of simple polygons.
        • Lesson: Area of Tangram Pieces (find areas of polygons without using formulas). This lesson is part of the same web unit on tangrams suggested in 9B Stage C for congruence and similarity.
        • Discuss temperature using common vocabulary and notation.
        • Lesson: How Cold Is It? (You'll probably need to make modifications to suit your geographical location.) The lesson includes data collection, analysis, and display--appropriate to 10A Stage C below.


        8A Stage C
        • Represent the idea of a variable as an unknown quantity using a letter or a symbol in a numerical sentence. **
        • Literature connection idea from PBS : Read Pat Hutchins' The Doorbell Rang (or a similar story) to your students, and then help them discover the relationship between the number of children and number of cookies each child gets. Encourage your students to use any of a variety of forms of representation to express this relationship (e.g., concrete, verbal, numerical, tabular, pictorial, graphical, or symbolic). Note this activity applies also to the next two performance descriptors below.
        • Express mathematical relationships using equations.


        8B Stage C
        • Represent and analyze simple patterns and operations using words, tables, and graphs. **
        • Describe situations with constant rates of change using words, tables, and graphs (e.g., walking at a constant rate of speed).


        8D Stage C
        • Demonstrate how to select and use an appropriate operation to solve problems involving patterns (e.g., save one penny on day 1, double that amount each day for 10 days).
        • Solve one-step linear equations using concrete materials.


        9A Stage B
        • Recognize and represent shapes from different perspectives.
        9A Stage C
        • Specify locations using a coordinate system. **
        • Predict and describe the results of translations, rotations, and reflections of two-dimensional shapes.
        • Identify, draw, and build polygons.


        9B Stage C
        • Decompose a three-dimensional object into two-dimensional components.
        • Describe the difference between congruence and similarity. **
        • Describe a motion or a series of motions that will show that two shapes are congruent. *
        • Identify and build a three-dimensional object from two-dimensional representations of that object. *


        • Apply geometric ideas and relationships to other disciplines. **


        9C Stage C
        • Make and test conjectures about mathematical properties and relationships and justify the conclusions. **


        10A Stage C
        • Organize, describe, and make predictions from existing data. *
        • Represent data using tables and graphs such as tallies and bar graphs.
        • Describe the important features of a set of data represented by a graph.
        • Determine the median of data on a graph.


        10B Stage C
        • Create and administer a survey that answers real life questions considering how many and what kind of questions will be asked and how the answers will be recorded.
        • Propose a follow-up survey to investigate questions that arise from the initial survey.


        10C Stage C
        • Unit Plan: Chances Are in three parts; from PBS TeacherSource.
        • Describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, equally likely, and impossible. *
        • Explain probability as a fractional part of a group to the whole group (e.g., A tossed coin can land on heads or tails. Therefore, it should land on heads 1/2 of the time.)
        • Make predictions based on the results received from a probability experiment.
        • Create and perform a probability experiment (e.g., a penny is flipped 100 times) and record the results.
        • Lesson: Spin a Graph  Students will predict the results of spinning a Spinner several times; then they will spin the Spinner, record the results, and compare them to their predictions.
        • Lesson:  Roll On
        • Understand that the measure of the likelihood of an event can be represented by a number from zero to one, inclusive. **


        Grade 4


        6A Stage C
        • Recognize and generate equivalent forms of familiar fractions. **
        • Explore and discuss uses of decimals.


        6A Stage D
        • Represent, order, and compare decimals to demonstrate understanding of the place-value structure in the base-ten number system. **
        • Represent fractions as parts of unit wholes, as parts of a set, as locations on a number line, and as divisions of whole numbers. **


        6B Stage D
        • Describe classes of numbers according to characteristics such as factors and multiples. **
        • Solve addition or subtraction number sentences and word problems using fractions with like denominators.


        • Solve multi-step number sentences and word problems using whole numbers and the four basic operations.
        • Select and use one of various algorithms to multiply and divide.
        • Lesson:  "Long Multiplication" is an interactive tutorial that shows several ways to multiply two-digit numbers.
           
        6C Stage D
        • Develop and use strategies (e.g., compatible numbers, front-end estimation) to estimate the results of whole-number computations and to judge the reasonableness of such results. **


        7A Stage D
        • Measure angles using a protractor or angle ruler.
        • Activity: What's My Angle? Do larger hands have larger angles between the fingers?
        • Lesson: Angles and Hinged Mirrors (Pages 9-10). This lesson focuses on understanding what a degree is and how it is used to describe the size of an angle. Students also relate a full-circle rotation to an angle of 360°.
          • Measure with a greater degree of accuracy.




          • Convert U.S. customary measurements into larger or smaller units with the help of conversion charts.
          • Convert linear metric measurements into larger or smaller units with the help of a conversion chart.
          • Create an accurate representation of a polygon with a given perimeter or area.


          7B Stage D
          • Develop and discuss strategies for estimating the perimeters, areas, and volumes of regular and nonregular shapes. **
          • Develop and use common referents for volume, weight/mass, capacity, area, and angle measures to make comparisons and estimates.
          • Lesson: Cool Estimation Students will be divided into two groups. One group will use estimation to make a presweetened drink. The other group will use exact directions. Students will record information and discuss whether estimation was appropriate or not.
          • Lesson: What is a Million? How big a jar would you need to hold 1 million punched holes from a paper punch?


          7C Stage D
          • Select and apply appropriate standard units and tools to measure the size of angles. **
          • Solve problems using money and time.
          • Lesson: Profit or Loss? Variation on a classic problem solving situation--this one involving a Beanie Baby™.
          • Lesson: Shopping for Toys You have just won a $100 gift certificate to buy some toys! You must try to spend as much of it as you can without going over.
          • Determine the volume of a cube or rectangular prism using concrete materials.8A Stage D
          • Identify a number pattern, both increasing and decreasing, and extend the number sequence.
          • Show the missing number(s) in a complex repeating pattern.
          • Construct and solve simple number sentences using a symbol for a variable.


          • Make generalizations given a specific pattern.
          • Create, describe, and extend patterns.
          • Describe a pattern with one operation verbally and symbolically given a table of input/output numbers.


          8B Stage D
          • Create a table that describes a function rule for a single operation.
          • Activity: Guess My Rule (scroll down to suggested classroom activities in the middle of the page)
          • Demonstrate, in simple situations, how a change in one quantity results in a change in another quantity (e.g., increase the measure of the side of a square and the perimeter increases).
          • Identify situations with varying rates of change using words, tables, and graphs (e.g., growth of a plant). **


          8C Stage D
          • Solve problems with whole numbers using appropriate field properties.


          8D Stage D
          • Solve one-step linear equations with one missing value in isolation and in problem solving situations.


          9A Stage D
          • Identify, draw, and label lines, line segments, rays, parallel lines, intersecting lines, perpendicular lines, acute angles, obtuse angles, right angles, and acute, obtuse, right, scalene, isosceles, and equilateral triangles.
          • Identify, draw, and build regular, irregular, convex, and concave polygons.
          • Read and plot ordered pairs of numbers in the positive quadrant of the Cartesian plane.
          • Describe paths and movement using coordinate systems.
          • Differentiate between polygons and nonpolygons.
          • Explore and describe rotational symmetry of two- and three-dimensional shapes. **
          • Game: Squeeze Play In this game for two players, children take turns placing Cuisenaire Rods within a given outline in an attempt to be the last player to place a rod. The winning strategy in this game for each of the first three game boards is based on the idea of rotational symmetry.
          9B Stage D
          • Determine congruence and similarity of given shapes. **
          • Explore polyhedra using concrete models.


          9C Stage D
          • Make and test conjectures about mathematical properties and relationships and justify the conclusions. **


          10A Stage D
          • Represent data using tables and graphs such as line plots and line graphs. **
          • Describe the shape and important features of a set of data and compare related data sets. **
          • Lesson collection: (from NCTM) Accessing and Investigating Data Using the World Wide Web: Part 1- National Population Projections. In this activity, students will examine the United States Census Bureau web site to investigate population projections from 1990-2100. Using the five provided pyramids, students will analyze the data to determine how the population is distributed over time, and explain what factors might contribute to these trends. Part 2- State Population Projections. In this activity, students will examine the United States Census Bureau web site to investigate projections of the total population of states from 1995-2025. Using the provided data, students will analyze statistics from five states of their choice, develop specific research questions using the data, and create three graphs to compare and contrast the information.
          • Arrange given data in order, least to greatest or greatest to least, and determine minimum value, maximum value, range, mode, and median for an odd number of data points.
          • Compare different representations of the same data and evaluate how well each representation shows important aspects of the data. *


          • Propose and justify conclusions and predictions that are based on data. **


          • Resource: Numbers in Search of a Problem The Internet offers a rich collection of data sources. In the hands of a creative teacher (or student) these numbers can be crafted into meaningful, real life mathematics problems.
          10B Stage D
          • Collect data using observations and experiments. **
          • Lesson: Spin to Win Students use spinners to predict and test the fairness of a game.
          • Propose a further investigation to verify or refute a prediction.


          **10C Stage D
          • List all possible outcomes of a single event and tell whether an outcome is certain, impossible, likely, or unlikely.
          • Describe the probability of an event using terminology such as “5 chances out of 8.”




          Grade 5


          6A Stage D
          • Represent, order, and compare decimals to demonstrate understanding of the place-value structure in the base-ten number system. **
          • Identify prime numbers through 100.
          • Recognize equivalent representations for decimals and generate them by composing and decomposing numbers (e.g., 0.15 = 0.1 + 0.05).


          • Represent fractions as parts of unit wholes, as parts of a set, as locations on a number line, and as divisions of whole numbers. **


          6A Stage E
          • Place mixed numbers and decimals on a number line.
          • Explore numbers less than zero by extending a number line and through familiar applications. *


          6A Stage F
          • Represent place values from units through billions using powers of ten.
            Game: Biggest (or Smallest) Number Wins Using prediction strategies, the students will show the ability to order digits to create the highest or lowest possible number.
          • Identify fractional pieces that have the same value but different shapes.
          6B Stage E
          • Determine whether a number is prime or composite.


          • Identify all the whole number factors of a composite number.


          • Explore and identify properties of square numbers.
          • Compute with 10, 100, 1000, and other powers of 10.
          • Explore and use divisibility rules.
          • Solve number sentences and word problems using addition and subtraction of fractions with unlike denominators.
          • Solve number sentences and word problems using addition and subtraction of decimals.
          6C Stage D
          • Estimate the sum or difference of a number sentence containing decimals using a variety of strategies.


          6C Stage E
          • Develop and use strategies to estimate computations involving familiar fractions and decimals in situations relevant to students’ experience * (e.g., double a recipe with 3/8 cup sugar, will more than a cup of sugar be needed).
          • Evaluate estimates to judge their reasonableness and degree of accuracy.
          • Select and use appropriate operation(s) and tool(s) (mental math, pencil-andpaper, estimation, calculator, computer) to perform calculations on whole numbers, fractions, and decimals according to the context and nature of the computation. **
          • Lesson: Beating Heart How long will it take for your heart to beat one million times?
          • Determine and justify whether exact answers or estimates are appropriate.6D Stage D
          • Determine 50% and 100% of a given group in context.


          6D Stage E
          • Identify and express ratios using appropriate notation (i.e., a/b, a to b, a:b).
          • Model the concept of percent using manipulatives or drawings.


          7A Stage E
          • Convert U.S. customary and metric measurements into larger or smaller units.
          • Develop and use formulas to determine the area of squares, rectangles, and right triangles.
          • Draw an angle of any given measure using a protractor or angle ruler.
          • Read and interpret a scale on a map or a scale drawing using the idea of a constant ratio (e.g., 1” represents 1 mile), and use it to answer questions about actual measurement.
          • Lesson: Adventures in Statistics -- a mathematics project involving fifth grade students and the area of classrooms, which incorporates measurement, graphing, computation, data analysis, and presentation of results.
          7B Stage E
          • Explain that all measurements are approximations.


          • Describe how precision is affected by choice of units.


          • Estimate the perimeter, area, and/or volume of regular and irregular shapes and objects.7C Stage E
          • Select appropriate tools to measure, draw, or construct figures.
          • Develop and discuss strategies for determining area and perimeter of irregular shapes.
          • Lessons: Covering and Surrounding (sample lesson from "Connected Mathematics" materials)
                    at An Example of Effective Sequencing of Problems


          8A Stage E
          • Describe, extend, and make generalizations about given geometric and numeric patterns. **
          • Describe a pattern, with at least two operations, verbally and symbolically, given a table of input/output numbers.
          • Illustrate equality of two expressions with variables (e.g., 28 + 35 = 35 + *).
          • Describe situations involving inverse relationships (e.g., the more people, the fewer cookies per person).


          8B Stage E
          • Model problem situations with objects and equations to draw conclusions. **
          • Represent and analyze patterns and functions using words, tables, and graphs. *
          • Demonstrate how the change in one quantity affects the other in a functional relationship involving whole numbers and unit fractions.
          • Identify, describe, and compare situations with constant and varying rates of change using words, tables, and graphs (e.g., two quantities that vary together are the length of the side of a square and its area). **


          8C Stage E
          • Solve problems with whole numbers using order of operations, equality properties, and appropriate field properties.


          8D Stage E
          • Create and solve linear equations involving whole numbers using a variety of methods (e.g., guess and check, bean stick counters).


          9A Stage D
          • Identify and label radius, diameter, chord, and circumference of a circle.
          • Construct a circle with a specified radius or diameter using a compass.


          9A Stage E
          • Investigate and describe the results of subdividing and combining shapes. **
          • Specify locations to describe paths using coordinate systems. **


          • Determine the distance between points along horizontal and vertical lines of a coordinate system. **
          • Identify and describe how geometric figures are used in practical settings (e.g., construction, art, advertising, architecture).
          • Identify and define a tessellation.
          • Create regular and semiregular tessellations using pattern blocks, other manipulatives, or technology to tile a plane.
          • Lesson: Bricks Activity Students explore different possibilities of making brick walls with and without fault lines, using diagram, process, and solution in their problem solving.


          9C Stage E
          • Make and test conjectures about the results of subdividing and combining shapes. **
          • Make and test conjectures about mathematical properties and relationships.
          • Develop logical arguments to justify conclusions. **


          10A Stage E
          • Represent given data using double bar graphs, double line graphs, and stem and leaf plots with and without technology.
          • Select an appropriate graph format to display given data.
          • Read, interpret, infer, predict, draw conclusions, and evaluate data from any graph.
          • Suggestion: USA Today is a good source of published graphs and tables. Have students tell in their own words what the graph or table is saying. Students should then examine the context and summarize the point the author is trying to make. Next they can provide alternative explanations and alternative representations using the same data.
          • Unit Resource: Project SkyMath -- introduces students to the measurement of temperature, to its representation on a map (either by color, contours, or numbers) and to the analysis of temperature change as shown graphically.
          • Determine mean, median, mode, minimum value, maximum value, and range and discuss what each does to help interpret a given set of data.


          10B Stage E
          • Design investigations to address a question and consider how data-collection methods affect the nature of a data set.
          • from NCTM: Lesson Plan Series: Data Collection: Numerical and Categorical Data. Part 1: Categorical Data - Students will formulate and refine questions which can be addressed with categorical data. They will consider aspects of data collection such as how to word questions and how to record the data they collect. Finally they will represent and analyze the data in order to answer the question posed. Part 2 : Numerical Data. Students will pose and refine questions that can be addressed with numerical data. They will consider aspects of data collection such as how to obtain measurements and record the data they collect. They will represent, then analyze the ordered numerical data by describing the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed. In collecting data, students will measure with standard units and carry out simple unit conversions, such as from centimeters to meters or feet to inches. Part 3: Comparing Categorical and Numerical Data.
          • Propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions or predictions. *
          • Resource: Numbers in Search of a Problem The Internet offers a rich collection of data sources. In the hands of a creative teacher (or student) these numbers can be crafted into meaningful, real life mathematics problems.


            10C Stage E
            • List all possible outcomes of compound events (e.g., toss a coin and spin a spinner).
            • Assign a value of zero to probabilities that are impossible and a value of one to probabilities that are certain.
            • Express simple probabilities as a fraction between zero and one.
            • Predict the probability of outcomes of simple experiments and test the predictions. *


             Grade 6


            6A Stage E
            • Show equivalent representations of a number by changing from one form to another form (e.g., standard form to expanded form, fraction to decimal, decimal to percent, improper fraction to mixed number).


            • Differentiate how fractions are used (part of a whole, part of a set, location on a number line, and division of a whole number).
            • Analyze how the size of the whole affects the size of the fraction (e.g., 1/2 of a large pizza is not the same as 1/2 of a small pizza).
            • Describe integers using familiar applications (e.g., a thermometer, above/below sea level).


            6A Stage F
            • Represent place values from units through billions using powers of ten.
            • Represent, order, compare, and graph integers.
            • Compare and order fractions and decimals efficiently and find their approximate position on a number line. **
            • Represent repeated factors using exponents.


            6B Stage F
            • Write prime factorizations of numbers.
            • Lesson: Factor Game (from NCTM). Help students determine whether a given number has many or only a few factors and to show how this property of numbers is useful for problem solving. Part I, Playing the Factor Game (using an interactive applet online), engages students in a friendly contest in which winning strategies involve recognizing the difference between prime numbers and composite numbers. Part II, Playing to Win the Factor Game, guides students through an analysis of Factor Game strategies and introduces the definitions of prime and composite numbers. Part III provides questions that are rich in connections to situations in which factors, multiples, divisors, products, and prime numbers are significant.
            • Determine the least common multiple and the greatest common factor of a set of numbers.
            • Lesson: Analyzing Numeric and Geometric Patterns of Paper Pool - The interactive paper pool game in this investigation provides an opportunity for students to further develop their understanding of ratio, proportion, factors, multiples, rectangles, the relation of being relatively prime. Before seeing how to apply these concepts, students must gather and organize data, then search for patterns.
            • Demonstrate the meaning of multiplication of fractions (e.g.,1/2 x 3 is 1/2 of a group of three objects).


            • Simplify simple arithmetic expressions with rational numbers using the field properties and the order of operations.




            • Recognize and use the inverse relationships of addition and subtraction, multiplication and division to simplify computations and solve problems. **
            • Lesson:  Using Your Melon for Math   Applying Fraction Multiplication and Division to Recipes
              In this lesson, students use recipes to practice their fraction multiplication and division skills by calculating the amounts of ingredients needed to make specific recipes, given varied numbers for the 'yield' quantity. Students will also use these fraction skills to convert small units of cooking measures to larger units.
            • Solve multiplication number sentences and word problems with whole numbers and familiar fractions.


            6B Stage G
            • Write prime factorizations using exponents.
            • Describe relationships between prime factorizations and properties of squares, primes, and composites.
            • Classify numbers according to the number of whole number factors (e.g., square numbers have an odd number of factors).
            • Lesson: The Locker Problem One of my very favorite problems, RICH with mathematics--factors and multiples, primes and composites, odds and evens, perfect squares--and problem solving strategies--solve a similar simpler problem, act it out, make a table, look for a pattern. This web site provides a lot of support for the teacher.
            • Demonstrate and describe the effects of multiplying or dividing by a fraction less than or greater than one.
            • Simplify arithmetic expressions containing exponents using the field properties and the order of
            operations.




            • Justify rules of divisibility for 2, 5, and 10.
            • Solve multi-step number sentences and word problems with rational numbers using the four basic operations.


            • Select and use appropriate operations, methods, and tools to compute or estimate using whole numbers with natural number exponents. **
            • Analyze algorithms for computing with whole numbers, familiar fractions, and decimals and develop fluency in their use. **


            6D Stage F
            • Solve number sentences and word problems using percents.


            • Demonstrate and explain the meaning of percents, including greater than 100 and less than 1. **
            • Create and explain a pattern that shows a constant ratio.
            • Analyze situations to determine whether ratios are appropriate to solve problems.
            • Determine equivalent ratios.
            6D Stage G
            • Work flexibly with fractions, decimals, and percents to solve number sentences and word problems (e.g., 50% of 10 is the same as 1/2 of 10 is the same as 0.5 x 10). **
            • Create and explain ratios and proportions that represent quantitative relationships.
            • Lesson:  How Many Noses are in Your Arm?     Apply the concept of ratio and proportion to determine the length of the Statue of Liberty's torch-bearing arm.
            • Create and explain a variety of equivalent ratios to represent a given situation.
            • Develop, use, analyze, and explain methods for solving numeric or word problems involving proportions. **


            7A Stage F
            • Investigate the history of the U.S. customary and metric systems of measurement.
            • Measure, with a greater degree of accuracy, any angle using a protractor or angle ruler.
            • Develop and use formulas for determining the area of triangles, parallelograms, and trapezoids.


            • Develop and use the formula for determining the volume of a rectangular and triangular prism.
            • Calculate the surface area of a cube, rectangular prism, and triangular prism.
            • Develop and use formulas for determining the circumference and area of circles.
            7B Stage F
            • Estimate distance, weight, temperature, and elapsed time using reasonable units and with acceptable levels of accuracy.


            7C Stage F
            • Select and justify an appropriate formula to find the area of triangles, parallelograms, and trapezoids. **
            • Select an appropriate formula or strategy to find the surface area and volume of rectangular and triangular prisms. **
            • Lesson: In " Jumping Out Of Windows, " students use measuring, multiplication, division, and remainders to understand situations involving movie stunts. Problems deal with area and volume.


            8A Stage F
            • Evaluate algebraic expressions for given values.
            • Express properties of numbers and operations using variables (e.g., the commutative property is m + n = n + m).
            • Simplify algebraic expressions involving like terms.
            8B Stage F
            • Graph simple inequalities on a number line.
            • Create a table of values that satisfy a simple linear equation and plot the points on the Cartesian plane.
            • Describe verbally, symbolically, and graphically, a simple relationship presented by a set of ordered pairs of numbers.
            • Lesson: Making Connections Formalize the input/output model for function and connect multiple representations: tables, function rules, equations, and graphs.


            8C Stage F
            Resource: The Formal Rules of Algebra


            • Identify and explain incorrect uses of the commutative, associative, and distributive properties.
            • Identify and provide examples of the identity property of addition and multiplication.
            • Identify and provide examples of inverse operations.
            • Explain why division by zero is undefined.


            8D Stage F
            • Create, model, and solve algebraic equations using concrete materials.
            9A Stage E
            • Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes. *
            • Classify two- and three-dimensional shapes according to their properties (e.g., regular and irregular, concave and convex, types of quadrilaterals, pyramids, and prisms). **
            • Lesson: Exploring Properties of Rectangles and Parallelograms Using Dynamic Software. In this Internet Mathematics Excursion students examine the properties of rectangles and parallelograms, and identify what distinguishes a rectangle from a more general parallelogram. Using spatial relationships, they examine the properties of two-and three-dimensional shapes.
            • Identify and justify rotational symmetry in two- and three-dimensional shapes. **


            • Identify, sketch, and build two- and three-dimensional shapes given attribute clues.
            • Lesson: Building Viewpoints Using cubes and grid paper, children interpret shapes in both two and three dimensions. This activity helps develop children's skills in geometry, spatial perception, and connecting mathematics to the physical world.
            • Lesson: Cube-n-ometry
            • Copy a line segment or an angle using a straightedge and a compass.
            • Construct a perpendicular bisector of a line segment.


            9A Stage F
            • Plot and read ordered pairs of numbers in all four quadrants.
            • Describe sizes, positions, and orientations of shapes under transformations, including dilations.
            • Perform simple constructions (e.g., equal segments, angle and segment bisectors, or perpendicular lines, inscribing a hexagon in a circle) with a compass and straightedge or a mira.
            • Determine and describe the relationship between pi, the diameter, the radius, and the circumference of a circle.


            • Solve problems for unknown angle measures using angle relationships and properties of triangles and quadrilaterals.


            9B Stage E
            • Demonstrate congruence of plane figures using transformations (translation, rotation, reflection).
            • Determine if two polygons are congruent using measures of angles and sides.
            • Match a front, right side, and top view drawing with a three-dimensional model built with cubes.
            • Activity: Plot Plans and Silhouettes In this activity you figure out a three-dimensional structure based on two-dimensional silhouettes, or shadows.
            • Activity: Shadows Can you judge an object by its shadow? In this activity you will be asked to determine if a shadow can be produced by a particular shape.
            • Worksheet: Match views Match a stack of cubes to its top, front, or side view.
            • Identify and describe the five regular polyhedra.


            9B Stage F
            • Determine the relationships between the number of vertices or sides in a polygon, the number of diagonals, and the sum of its angles.
            • Solve problems that involve vertical, complementary, and supplementary angles.
            • Analyze quadrilaterals for defining characteristics.
            • Create a three-dimensional object from any two-dimensional representation of the object, including multiple views, nets, or technological representations.
            9C Stage F
            • Make and test conjectures about various quadrilateral and triangle relationships, including the triangle inequality.


            • Justify the relationship between vertical angles.
            • Justify that the sum of the angles of a triangle is 180 degrees.


            10A Stage F
            • Construct, read, interpret, infer, predict, draw conclusions, and evaluate data from various displays, including circle graphs. **
            • Recognize and explain misleading displays of data due to inappropriate intervals on a scale.


            10B Stage F
            • Conduct simple simulations to gather data.
            • Collect data over time with or without technology.10C Stage F
            • Record probabilities as fractions, decimals, or percents.
            • Demonstrate that the sum of all probabilities equals one.
            • Determine empirical probabilities from a set of data provided.
            • Set up a simulation to model the probability of a single event.
            • Lesson:  Rock Around the Clock    Apply the Monte Carlo method of simulation to determine a reasonable estimate. 
            • Discuss the effect of sample size on the empirical probability compared to the theoretical probability.


            • List outcomes by a variety of methods (e.g., tree diagram).
            • Determine theoretical probabilities of simple events.
            • Lesson: The Smithville Families   Students will create the number sequences of Pascal's triangle and discover a relationship that this triangle has to theoretical probability.


             Grade 7


            Resource:  Mathematics Enhancement Program Year 7 materials, including detailed lesson plans, overhead slides, activities, etc., from the CENTRE for INNOVATION in MATHEMATICS TEACHING.


            6A Stage G
            • Represent any large number using scientific notation.
            • Show relationships between sets of numbers, including rational numbers, whole numbers, natural numbers, and integers.
            6B Stage G
            • Write prime factorizations using exponents.
            • Describe relationships between prime factorizations and properties of squares, primes, and composites.


            6B Stage H


            • Determine the least common multiple and greatest common factor of a set of numbers using prime factorization containing exponents.
            • Simplify arithmetic expressions containing integers using the field properties and order of operations.


                • Describe and use the inverse relationships of squaring and finding square roots to simplify
                computations and solve problems. **


                • Justify divisibility rules for 3, 4, 6, 8, and 9.
                6C Stage G
                • Select, use, and justify appropriate operations, methods, and tools to compute or estimate with integers and familiar rational numbers. **
                • Develop, use, and explain strategies to compute exact answers mentally with integers and simple rational numbers using a variety of techniques (e.g., estimate and compensate, halve and double, compatible numbers, decomposition and recomposition using the distributive property).


                • Resource: BEATCALC Strategies for rapid mental computation for squaring, multiplication, division, addition, subtraction, and percents.
                • Analyze algorithms for computing with rational numbers and develop fluency in their use. **


                7A Stage G
                • Select and justify the choice of either U.S. customary or metric systems of measurement according to the situation (e.g., measure fabric in yards, measure dry chemicals in grams).
                • Lesson: Metrics Made Easy     The students will convert standard units of measurement to metric units using tape measures, meter sticks, and rulers.  In particular, click on the link to the last handout at the bottom of the page.
                • Make simple measurements using indirect techniques (e.g., determining the height of a flagpole using its shadow and similar right triangles).


                • Explore and explain derived measurements (e.g., velocity and density).
                • Develop and discuss strategies to find the area of combined shapes. **


                7B Stage G
                • Estimate angle measure, area, and volume using reasonable units and with acceptable levels of accuracy.
                • Activity: Popcorn Compare the volume of two cylindrical containers of popcorn.
                • Lesson: Let's Go Home NCTM Student Math Notes. Apply knowledge of area to practical problems about painting and carpeting a house that needs remodeling. Also addresses working within a budget.
                • Determine and describe acceptable levels of accuracy in estimation situations.
                7C Stage G
                • Select and use appropriate units and tools to measure volume, surface area, and mass/weight accurately for a given situation. **


                • Select an appropriate formula to determine the circumference and the area of circles. **
                • Select and explain an appropriate formula or strategy to find the surface area and volume of rectangular and triangular pyramids, cylinders and cones. **


                • Lesson: Volume Experiment (links to a page with further discussion of the lesson and videos)
                • Solve simple problems involving rate, time, and distance.
                • Solve problems involving mixed units of the same attribute, including time, money, length, and area.
                8A Stage F
                • Investigate, extend, and describe arithmetic and geometric sequences of numbers whether presented in numeric or pictorial form. **
                8A Stage G
                • Investigate, describe, and generalize a variety of patterns using variable or recursive techniques. **
                • Represent situations using variables.
                • Recognize and generate equivalent forms of simple algebraic expressions. **
                8A Stage H
                • Determine the given term of a pattern of numbers or drawings.8B Stage G
                • Create a table of values that satisfy a power or exponential relationship and plot the points on the Cartesian plane.
                • Graph two inequalities with a single variable, including the intersection or union of these inequalities, on a number line.


                8C Stage G
                • Solve arithmetic and linear equations using the properties of equality and inequality.


                • Identify and provide examples or counter examples as appropriate for the reflexive, symmetric and transitive properties of inequality.8D Stage F
                • Solve linear equations, including direct variation, with whole number coefficients and solutions using algebraic or graphical representations.


                8D Stage G
                • Solve simple linear equations, including direct variation, with integral coefficients using algebraic or graphical representations.
                • Solve simple problems involving quadratic relationships using technology for graphing.


                9A Stage G
                • Draw geometric shapes with specified properties, such as side lengths or angle measures. **
                • Perform constructions of congruent angles and parallel lines using a compass and straightedge, paper folding, or a mira.


                • Lesson: Putt-Putt Can you make a hole-in-one on this course?


                9B Stage G
                • Describe, classify, and justify relationships among types of two- and three-dimensional objects using their defining properties.
                • Solve problems using properties of polygons and circles.
                • Determine the relationship among the number of edges, faces, and vertices in a three-dimensional object.
                • Lesson: Let's Be Discrete NCTM STUDENT MATH NOTES. Introduces the Euler relationship through graph theory.
                • Classify and order quadrilaterals according to their properties.


                9B Stage H
                • Create and analyze scale models using proportional reasoning.
                • Solve problems involving similar figures.
                9C Stage G
                • Create and critique arguments concerning geometric ideas and relationships, such as the number of diagonals in a polygon, or the formula for the sum of the interior angles of any polygon. **
                • Justify the area formulas for triangles, parallelograms, and trapezoids based on the formula for the area of a rectangle.
                • Make and test conjectures about the relationships between side length and angle measure in various triangles and quadrilaterals.
                • Justify the properties of angles formed by parallel lines cut by a transversal using appropriate terminology.
                10A Stage G
                • Construct, read, interpret, infer, predict, draw conclusions, and evaluate data from various displays, including box and whiskers plots. **
                • Lesson: Graphing Probabilities (pages 8-10). Choose, create and use various graphical representations of data (line plots, bar graphs, stem-and-leaf plots, histograms, scatter plots, circle graphs, and box-and-whisker plots) appropriately and effectively. Note: the data used in this lesson come from the lesson "Mystery Bags" suggested for 10C Stage H below, which should precede this lesson.
                • Find, use, and interpret measures of center and spread, including interquartile range. **
                • Construct an equivalent data representation given data in a different form.
                • Recognize potential bias in data collection methods or data presentation.
                10B Stage G
                • Choose and use appropriate techniques to gather data.
                • Formulate new questions using conjectures and plan new studies to answer them. **
                10B Stage H
                • Formulate questions, design studies, and collect data. **
                • Analyze potential methods of collecting information and decide which methods would produce the most reliable and accurate data.
                • Analyze instruments used for surveys for errors and bias.


                • Analyze potential experiments or simulations for errors and bias.
                • Resource: Numbers in Search of a Problem The Internet offers a rich collection of data sources. In the hands of a creative teacher (or student) these numbers can be crafted into meaningful, real life mathematics problems.
                10C Stage G
                • Discuss odds versus probability.
                • Make and test conjectures about the results of experiments and simulations using proportionality and basic understanding of probability. **
                • Lesson: The Next Billion. In 1999 the world population passed the 6 billion mark. In this lesson, students predict when it will reach 7 billion using an on-line counter that simulates the changing world population. They time the counter to find how long it takes for the population to increase by, say, 50 or 100 people. They use that measurement to predict how long it would take for the population to increase by 1 billion. Students discuss the reliability of their predictions, compare them to past trends, and discuss social factors that can affect population growth.
                • Compute probabilities for simple compound events using such methods as organized lists and tree diagrams. *


                10C Stage H
                • Describe and explain complementary and mutually exclusive events using appropriate terminology. **
                  • Design and conduct experiments or simulations for probability, including the possible use of technology to simulate events.
                  • Discuss the difference in empirical and theoretical probability.
                  • Lesson: Mystery Bags (pages 4-7) Use probability to make predictions about an unknown population. Find the experimental probability and compare it to theoretical probability for the same event. Understand that probability models are used to predict what will happen in the long run over many trials.
                  • Compute probabilities for simple compound events using a variety of methods, including area models.
                  • Identify situations where dependent and independent events occur.


                  • Determine probabilities using simple counting techniques.
                  • Lesson: Pigeonhole How many people would have to be in a school before it contained at least two people with the same first and last initials?
                  • Discuss situations where permutations and combinations should be used in counting outcomes.


                   Grade 8


                  Resource:  Mathematics Enhancement Program Year 8 materials, including detailed lesson plans, overhead slides, activities, etc., from the CENTRE for INNOVATION in MATHEMATICS TEACHING.


                  6A Stage H
                  • Recognize and use exponential, scientific, and calculator notation. **
                  • Represent, order, and compare rational numbers.
                  • Place rational numbers on a number line.


                  6B Stage H
                  • Determine and describe the effects of arithmetic operations with decimals and integers (e.g., multiply by a decimal between zero and one, divide by a negative integer).


                  6B Stage I
                  • Determine an appropriate numerical representation of a problem situation, including roots and powers, if applicable.


                  • Judge the effects of such operations as multiplication, division, and computing powers and roots on the magnitudes of quantities. *


                  • Judge the reasonableness of numerical computations and their results. *
                  6C Stage H
                  • Select, use, and justify appropriate operations, methods, and tools to compute or estimate with real numbers. **
                  • Analyze algorithms for computing with real numbers and develop fluency in their use. **


                  6D Stage H
                  • Develop, use, analyze, and explain methods for solving number sentences or word problems involving proportions with rational numbers. **
                  • Solve problems that involve percents, including percent increase and decrease, regardless of the piece of information that is missing.




                  7A Stage H
                  • Determine derived measurements.
                  • Lesson: Five's a Crowd (From NCTM) Estimate population density in five selected countries. Good social studies connections.
                  • Determine the surface area of three-dimensional figures.
                  • Lesson:   Volume and Surface Area      This lesson is designed to help students give meaning to volume and surface area by solving problems using a meaningful situation rather than formulas.
                  • Determine the volume of a sphere.
                  • Lesson: Valley Springs Snow Cream  Students will explore the relationship between the volume of cones, spheres, and cylinders.
                  7B Stage H
                  Resource: Accuracy and Precision
                  • Measure any quantity to the greatest degree of accuracy determined by the tool.
                  • Determine the maximum error in measurements.


                  7C Stage H
                  Relevant Unit Plan:  World Travels
                  • Solve simple problems involving rates and other derived measurements such as velocity and density. **
                  • Solve problems involving angle measurement in polygons and circles.


                  • Develop and describe surface area and volume formulas for cones and cylinders by relating pyramids to cones and prisms to cylinders.


                  8A Stage H
                  • Investigate and describe linear, quadratic, and exponential patterns recursively. **
                  • Investigate and write algebraic expressions to describe the nth term of a simple linear, power, or exponential sequence.
                  • Create arithmetic and geometric sequences to fit a given set of conditions.
                  • Lesson: Oh! To Work for a King Who Doesn't Know Math! This activity takes a problem-solving approach to introduce geometric progressions. This activity is a good jumping-off point for the study of arithmetic and geometric progressions.
                  • Recognize and generate equivalent forms for linear equations, including transforming linear equations into standard and slope-intercept form. **
                  8B Stage H 
                  • Graph linear equations and inequalities on the Cartesian plane.
                  • Graph a set of points and describe the relationship as linear or nonlinear.
                  • Describe the relationships between symbolic expressions and graphs of lines using the appropriate vocabulary for the intercepts and slope of the line. **
                  • Graph absolute values on a number line.
                  • Determine the slope of a line from a graph.


                  8C Stage H
                  • Solve arithmetic and simple algebraic equations using properties of real numbers, equality, and inequality, and justify the procedures.
                  • Solve simple algebraic equations for a given variable using inverse operations.


                  8D Stage H
                  • Solve algebraic equations or word problems that involve linear equations or inequalities using algebraic or graphical representations. **
                  • Lesson: Cats and Canaries Algebraic language; simultaneous equations; proportional reasoning.
                  • Solve absolute value equations or inequalities in one variable using algebraic or graphical representations.
                  • Create word problems that meet given conditions and represent linear relationships.9A Stage G
                  • Examine and describe geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides, using coordinate geometry. **
                  • Examine and describe line or rotational symmetry of objects in terms of transformations.


                  • Draw transformations of figures in a plane to match specified criteria.
                  9A Stage H
                  • Represent and analyze the properties of geometric shapes using coordinate geometry. **


                  • Examine the congruence or similarity of objects using transformations. **
                  • Draw the image of an object after a combination of transformations.
                  • Identify possible types of two- or three-dimensional figures that would match a set of given conditions.
                  • Determine if a triangle is possible using side lengths and the triangle inequality.


                  • Solve pictorial or word problems that involve geometric relationships within a single geometric shape or figure, including the Pythagorean theorem.


                    • Analyze the results of a combination of reflections, rotations, and translations of a figure, and determine alternate motions that could produce the same results.
                    • Combine simple construction techniques to construct squares, equilateral triangles, or other simple combinations of equal segments, angles, etc.


                    • Analyze properties of a shape or a combination of shapes that enable it to tessellate the plane.


                    9B Stage I
                    • Solve problems using triangle congruence and similarity of figures.
                    • Extend knowledge of plane figure relationships to relationships within and between geometric solids.
                    • Lesson: Tetralope. NCTM STUDENT MATH NOTES. In this activity, students cut and fold an envelope to create and interesting geometric solid.
                    • Identify relationships between circles, arcs, chords, tangents, and secants.


                    • Solve problems in and gain insights into other disciplines and other areas of interest such as art and architecture using geometric ideas. **


                    9C Stage H
                    • Create and critique arguments concerning geometric ideas and relationships, such as congruence, similarity, the Pythagorean relationship, or formulas for surface areas or volume of simple three dimensional objects. **
                      • Justify the simple construction methods used to produce angle bisectors, perpendicular lines, and equilateral triangles.
                      • Represent, solve, and explain numerical and algebraic relationships using geometric concepts.


                      • Provide examples or counter-examples to either illustrate or disprove conjectures about geometric characteristics.


                      9C Stage I
                      • Find a counter-example to disprove a conjecture.


                      • Develop conjectures about geometric situations with and without technology.
                        Lesson: A Sweet Dilemma NCTM STUDENT MATH NOTES. Problems abut packing fudge in rectangular boxes involve functions, graphing, optimization, parametric equations, geometry.
                      • Justify constructions using geometric properties.
                      9D Stage G
                      • Analyze the relationship between sides of right triangles using the Pythagorean theorem.
                      • Solve problems that involve the use of proportions and the Pythagorean theorem in similar right triangles with whole number side lengths.


                      10A Stage H
                      • Construct, read, interpret, infer, predict, draw conclusions, and evaluate data from various displays, including histograms and scatter plots. **


                      • Determine the best measure of central tendency from mean, median, or mode
                      • Discuss how data can be manipulated to represent different points of view based on the use of different measures of central tendency and based on different graphical displays.
                      • Discuss biased reporting of data and questions that should be asked when data is viewed.
                      • Analyze graphical displays of data for possible misleading characteristics.
                      • Make conjectures about the possible correlation between two characteristics of a sample on the basis of scatter plots of the data and approximate lines of fit. *
                      • Lesson: Shedding Light on the Subject: Function Models of Light Decay. This activity explores the development of a mathematical model for the decay of light passing through water. The goal of this investigation is a rich exploration of exponential models in context. This NCTM web activity has a rich collection of ancillary teaching materials, including an interactive grapher and videotape clips.
                      10B Stage I
                      • Describe the characteristics of well-designed studies, including the role of randomization in surveys and experiments. **
                      • Lesson: Predicting M&Ms (pages 11-20) What percent of each color of plain M&Ms does the candy company manufacture? This lesson helps students understand the concept of random sampling. They learn how to make accurate and fair predictions — based on representative samples of data.
                      • Discuss informally different populations and sampling techniques.
                      • Create a question and design a set of questions that might get the data desired.
                      • Decide if the survey was “successful” in gathering the intended data.


                      10C Stage I
                      • Determine geometric probability based on area.


                      • Calculate probability using Venn diagrams.
                      • Determine simple probabilities using frequency tables.


                      • Construct empirical probability distributions using simulations. **
                      • Describe the concepts of conditional probability.
                      • Develop an understanding of permutations and combinations as counting techniques. *


                      Algebra
                      Resources: Algebra for All Students and The Teachers and Algebra Project
                        Resource:  Practical Algebra Lessons
                        Resource:  Mathematics Enhancement Program Year 9 materials, including detailed lesson plans, overhead slides, activities, etc., from the CENTRE for INNOVATION in MATHEMATICS TEACHING.


                        6A Stage I
                        • Illustrate the relationship between second and third roots and powers of a number.
                        • Organize problem situations using matrices.
                        • Represent, order, and compare real numbers.
                        • Place real numbers on a number line.


                        6B Stage I
                        • Compare and contrast the properties of numbers and number systems, including the rational and the real numbers. **


                        • Solve problems using simple matrix operations (addition, subtraction, scalar multiplication).
                        • Develop fluency in operations with real numbers using mental computation or paper-and-pencil calculations for simple cases and technology for more complicated cases. **


                        6C Stage I
                        • Develop fluency in operations with real numbers and matrices using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases. **
                        • Determine and explain whether exact values or approximations are needed in a variety of situations.
                        • Determine an appropriate number of digits to represent an outcome.


                        6D Stage I
                        • Explain how ratios and proportions can be used to solve problems of percent, growth, and error tolerance.
                        • Set up and solve proportions for direct and inverse variation of simple quantities.


                        7A Stage I
                        • Choose units and scales that are appropriate for problem situations involving measurement. **
                        • Solve simple scale conversions, contractions, and dilations in maps and diagrams.
                        • Determine linear measures, perimeters, areas, surface areas, and volumes of similar figures using the ratio of similitude.
                        • Determine the ratio of similar figure perimeters, areas, and volumes using the ratio of similitude.


                        • Calculate by an appropriate method the length, width, height, perimeter, area, volume, surface area, angle measures or sums of angle measures of common geometric figures, or combinations of common geometric figures.
                        7B Stage I
                        • Estimate the magnitude and directions of physical quantities (e.g., velocity, force, slope).


                        • Determine answers to an appropriate degree of accuracy using significant digits.


                        7C Stage I
                        • Solve problems using indirect measurement by choosing appropriate technology, instruments, and/or formulas.
                        • Check measurement computations using unit analysis. **
                        • Lesson: Faster During the 100 meter dash in the 1988 Olympic Games in Seoul, Florence Griffith-Joyner was timed at 0.91 seconds for 10 meters. At that speed, could she pass a car traveling 15 miles per hour in a school zone?
                        • Convert between the U.S. customary and metric systems given the conversion factor.
                        • Solve problems involving time, temperature, mass, speed, distance, density, and monetary values.
                        • Lesson (time): Time Zones (go to challenge # 46 on page 45)
                        • Solve problems involving scale drawings, models, maps, or blueprints.
                        • Explain the meaning of a measurement answer in context.


                        • Solve problems using derived measurements.


                        • Describe the general trends of how the change in one measure affects other measures in the same figure (e.g., length, area, volume).
                          • Lesson: Patios Does bigger perimeter mean bigger area?
                          • Lesson: Taking Up Space/Get A Clue (pages 75 ff. including worksheets) explores perimeter and area using color tiles and counters.


                            8A Stage I
                            • Write equivalent forms of equations, inequalities, and systems of equations. **
                            • Represent and explain mathematical relationships using symbolic algebra. **
                            • Lesson from NCTM: Will the Best Candidate Win? Activities allow students to explore alternative voting methods. They discover what advantages and disadvantages each method offers and also see that each fails, in some way, to satisfy some desirable properties. Connections with social studies.
                              • Model and describe slope as a constant rate of change.
                              • Explain the difference between constant and nonconstant rate of change.


                              • Create an equation of a line of best fit from a set of ordered pairs or set of data points.
                                • Simplify algebraic expressions using a variety of methods, including factoring.
                                  • Justify the results of symbol manipulations, including those carried out by technology. **
                                  • Activity: Mystery Operation The computer produces an output given two numbers as input. Your job is to figure out the pattern--what is the computer doing with the numbers to get the result? Interactive and fun--could be used for enrichment.
                                  • Identify essential relationships in a situation and determine the class or classes of functions (e.g., linear, quadratic) that might model the relationships. **
                                  • Represent relationships arising from various contexts using algebraic expressions.


                                  • Rewrite absolute value inequalities in terms of two separate equivalent inequalities with the appropriate connecting phrase of "AND" or "OR".
                                  8B Stage I
                                  • Describe the relationships of the independent and dependent variables from a graph.
                                  • Interpret the role of the coefficients and constants on the graph of linear and quadratic functions given a set of equations.


                                  • Determine the effect of translations on linear relations.


                                  • Create and connect representations that are tabular, graphical, numeric, and algebraic from a set of data.


                                  • Recognize and describe the general shape and properties of the graphs of linear, absolute value, and quadratic functions.


                                  • Approximate and interpret rates of change from graphical and numerical data. *


                                  • Identify slope in an equation and from a table of values.


                                  • Graph absolute values of linear functions on the Cartesian plane.


                                  • Recognize direct variation, inverse variation, linear, and exponential curves from their graphs, a table of values, or equations. **
                                  • Lesson:  Growing, Growing, Graphing  In this statistics lesson, students focus on China's population growth. They graph data on graph paper using a graphing calculator or spreadsheet software. Students predict future population numbers and decide if the population growth is linear or exponential.
                                  8C Stage I
                                  • Describe and compare the properties of linear and quadratic functions. **
                                  • Solve problems by recognizing how an equation changes when parameters change.
                                  • Interpolate and extrapolate to solve problems using systems of numbers.
                                  • Solve problems using translations and dilations on basic functions.


                                  8D Stage I
                                  • Solve equivalent forms of equations, inequalities, and systems of equations with fluency—mentally or with paper-and-pencil in simple cases and using technology in all cases. **
                                  • Lesson: Buying T-Shirts (pages 19-21) In this investigation, students use their knowledge of functions to decide which t-shirt company the Student Council should hire to make t-shirts for the school. Various representations are used to try to find a common solution to two different equations. Students find their answer graphically and then interpret it in the context of the problem. Finally, students provide their classmates with convincing arguments that support their company choice.
                                  • Interpret and use functions as a geometric representation of linear and non-linear relationships.


                                  • Create word problems that meet given conditions and represent simple power or exponential relationships, or direct or inverse variation situations.
                                  • Solve simple quadratic equations using algebraic or graphical representations.
                                  • Solve problems of direct variation situations using a variety of methods.
                                  9A Stage I
                                  • Describe and apply properties of polygons and circles in a problem-solving situation.


                                  • Classify angle relationships for two or more parallel lines crossed by a transversal.


                                  • Analyze and describe the transformations that lead to successful tessellations of one or more figures.
                                  • Analyze geometric situations using Cartesian coordinates. **


                                  • Represent transformations of objects in the plane using sketches, coordinates, and vectors.
                                  • Design a net that will create a given figure when folded.


                                  • Solve problems using constructions.


                                  • Gain insights into and answer questions in other areas of mathematics using geometric models.**


                                  • Calculate distance, midpoint coordinates, and slope using coordinate geometry.
                                  • Visualize three-dimensional objects from different perspectives and describe their cross sections. **
                                  9C Stage I
                                  • Develop a formal proof for a given geometric situation on the plane.
                                  • Lesson: Prize Numbers In this lesson, students explore what a proof is, how and why mathematicians create them and finally they compose essays on how reason and logic are employed in the workplace.
                                  • Describe the difference between an inductive argument and a deductive argument.


                                  9D Stage H
                                  • Recognize Pythagorean Triples.


                                  • Identify the basic trigonometric ratios in terms of lengths of the sides of a right triangle and an acute angle.


                                  • Solve for missing side lengths using the trigonometric ratios in right triangles.
                                  • Determine and justify the side length relationships present in 45-45-90 triangles and 30-60-90 triangles.
                                  • Determine the ratio of lengths of sides of a right triangle with given measures for its acute angles using appropriate technologies.


                                  9D Stage I
                                  • Determine distances and angle measures using indirect measurement and properties of right triangles.
                                  • Solve problems using 45-45-90 and 30-60-90 triangles.


                                  10A Stage I
                                  Resource: Course guide - Chance Guide


                                  • Describe the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable. **


                                  • Display a scatter plot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients for bivariate measurement data using technological tools.
                                  • Evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions. *


                                  • Analyze two-variable data for linear or quadratic fit.


                                  • Make decisions based on data, including the relationships of correlation and causation.
                                  • Lesson: Cookie Experiment  An activity that has been found quite successful is to have students design an experiment to determine if there is a significant correlation between their rating of the cookies and the price. 


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                                    4/10/2010




                                  Copyright © 2010 Marsha Landau



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