Grade: 6

Unit Title: From Wholes to Parts

(Last Revision 1-10-06)

Approximate Length: 40 classes

 

Brief Description of unit:

Investigations in this unit are designed to help students refine their number sense. For example, rather than being presented with a procedure determining equivalent fractions, students are drawn into an investigation that allows them to see patterns in pairs and groups of equivalent fractions. They are encouraged to find ways to use these patterns to determine whether two fractions are equivalent and to write another fraction that is equivalent to a given fraction. This approach helps students to trust and rely on their own reasoning skills to determine whether their results are reasonable.

The same emphasis on number sense occurs with fractional operations. Students come up with their own strategies for finding exact answers mentally, estimate answers to more difficult problems, and make general statements concerning what must be true about the answer to a problem before they have calculated it. Students communicate their thinking processes and strategies both verbally and in written form.

Problem solving takes place later in the unit when students develop and use algorithms to add and subtract fractions. These problem-solving skills are challenged when students are asked to use what they know about fractions to develop and communicate strategies for multiplying and dividing fractions.

NH Frameworks/Proficiency Standards to be addressed:

Problem Solving and Reasoning

1b. K-12 Broad Goal: Students will use mathematical reasoning.

PURPOSE: Students need to recognize that memorized facts, rules, and procedures are only a part of mathematics. They need opportunities to use these facts, rules, and procedures to make conjectures, develop and refine their reasoning abilities, gather evidence, and produce valid rules and generalizations. Students need to be able to justify their thinking through examples and explanations and appreciate that how a problem is solved is as important as the answer.

End of Grade 6:

• Continue a pattern involving integers and positive rational numbers.
• Solve problems involving two-and three-dimensional geometric shapes and explain one's reasoning.
• Use elementary deductive reasoning to solve word problems.
• Use models, known facts, properties, and relationships to explain thinking and to justify answers and solution processes.

Communication and Connections

2a. K-12 Broad Goal: Students will communicate their understanding of mathematics.

PURPOSE: Reading, writing, talking, listening, and modeling, provide students with the oppoApril 3, 2006world, and help them to develop understanding. Actively exploring, investigating, describing, and explaining mathematical ideas promote communication which leads to a greater comprehension of mathematical concepts.

End of Grade 6:

• Demonstrate an understanding of mathematical concepts and relationships through a variety of methods (for example: writing, graphing, charts, diagrams, number sentences, or symbols).
• Explain, analyApril 3, 2006ented by others.
• Explain conclusions, thought processes, and strategies in problem-solving situations.
• Make conjectures and defend generalizations.
• Evaluate the validity of a mathematical statement.

Numbers, Numeration, Operations, and Number Theory

3a. K-12 Broad Goal: Students will develop number sense and an understanding of our numeration system.

PURPOSE: Students must understand numbers if they are to make sense of the ways numbers are used in their everyday world. Numbers are used to describe and interpret real-world phenomena. Students need to use numbers to quantify, to identify location, to identify a specific object in a collection, to name, to measure, and to model real-world situations. They need to understand relative magnitude in order to make sense of everyday situations.

End of Grade 6:

• Name and identify a fraction or decimal, given a physical representation.
• Given a decimal representation in tenths or hundredths, write an equivalent fraction.
• Given an integer or a positive rational number, represent the number with the use of physical models or diagrams.
• Explain the use of numbers in various every-day contexts (for example: calendars, clocks, signs, or literature).
• Given a set of fractional models, name and write those that represent equivalent fractions.
• Given a pair of fractions, determine which is larger by using physical models or illustrations.
• Develop and use order relations for integers and positive rational numbers.
• Apply number theory to the factoring of whole numbers and the equivalency of positive rational numbers.

3b. K-12 Broad Goal: Students will understand the concepts of number operations.

PURPOSE: Students need to build an awareness of the properties of an operation, see relationships among operations, and acquire insight into the effects of operations on real numbers. Students need to recognize conditions in real-world situations where the use of these operations is indicated and useful.

End of Grade 6:

• Apply the associative, commutative, and distributive properties in a problem-solving situation.
• Apply the multiplicative and additive properties of zero and the multiplicative property of one.
• Demonstrate an understanding of multiplication as repeated addition and of division as repeated subtraction.
• Demonstrate an understanding that the product of two whole numbers greater than 1 is greater than either of the factors.
• Demonstrate an understanding that when dividing two whole numbers that are greater than one, the quotient will be smaller than the dividend.

3c. K-12 Broad Goal: Students will compute.

PURPOSE: The purpose of computation is to solve problems. While computation remains important in mathematics and in everyday life, advances of technology require us to rethink how computation is done today. Students must recognize that estimation, mental computation, use of calculators, and paper and pencil calculation are all appropriate ways to compute solutions to problems. Basic fact memorization should be incorporated into a rich curriculum rather than be its primary focus.

End of Grade 6:

• Demonstrate mastery of the multiplication facts with factors less than or equal to 10.
• Select an appropriate computational technique in the solution of problems and check the reasonableness of results through mental computation and estimation strategies.
• Use calculators in appropriate problem solving situations.
• Add integers using models or representations.
• Multiply three digit whole numbers by two digit whole numbers.
• Divide three digit whole numbers by two digit whole numbers.
• Multiply and divide two and three digit decimals.
• Using physical models and illustrations, determine the sum or difference of fractions with like and unlike denominators (using only halves, fourths, and eighths).
• Using physical models, illustrations, and calculators, determine the sum or difference of decimals.

3d. K-12 Broad Goal: Students will use mental computation and estimation skills and strategies and know when it is appropriate to do so.

PURPOSE: Students should know what is meant by estimation and mental computation, when they are appropriate, and how close an estimate is required in a given situation. Students should be encouraged to estimate the solution of problems before computation or measurement is done, and to use estimation to determine the reasonableness of answers, and to recognize when an estimate is sufficient as an answer.

End of Grade 6:

• Use estimation and mental computation to determine the reasonableness of answers obtained from the four basic operations on rational numbers.
• Select and use appropriate mental computation and estimation strategies in problem situations when exact answers are not needed.

NECAPGLEs/GSEs to be addressed

Number and Operations

M (N&))-6-1 Demonstrates conceptual understanding of rational numbers with respect to ratios (comparison of two whole numbers by division a/b, a:b and b, where b 0) and rates (e.g, a out of b 25%) using models, explanations, or other representations.

M (N&O)-6-2 Demonstrates understanding of the relative magnitude of numbers by ordering or comparing numbers with whole number bases and whole number exponents (eg. ), integers, or rational numbers within and across number formats (fractions, decimals, or whole number percents from 1-100) using number lines or equality and inequality symbols.

M (N&O) 6-3 Demonstrates conceptual understanding of mathematical operations by describing or illustrating the meaning of power by representing the relationship between the base (whole number) and the exponent whole number) ( eg. ), and the effect on the magnitude of a whole number when multiplying or dividing it by a whole number decimal or fraction.

M (N&O) 6-4 Accurately solves problems involving single or multiple operations on fractions (proper, improper, and mixed), or decimals: and addition or subtraction of integers: percent of a whole; or problems involving greatest common factor or least common multiple. Important: applies the conventions of order of operations with and without parentheses)

Unit Essential Questions:

Phase One: What methods can be used to determine the GCF and LCM

Phase Two: What tools can be used to model fractions and compare fractions

Phase Three: What methods are used to add and subtract fractions and mixed numbers

Phase Four: What strategies may be helpful when multiplying and dividing mixed numbers

Core Unit Content

In this unit, students use the number sense they have developed with whole numbers to begin investigating fractions. The unit opens with an examination of whole numbers, building on prime factorization and multiples to develop equivalent fractions and common denominators. It then extends to the four basic operations.

Core Unit Skills

Phase One: The Whole of It

Lesson 1: Shapes and Factors

Goal: Explore the definitions of prime factor and common factor

Skills students will be able to do:

Understand the concepts of factors and prime factors

Find common factors of whole numbers

Lesson 2: The Great Factor Hunt

Goal: Determine the greatest number of factors they can use to express a number.

Skills students will be able to do:

Find prime factorization of whole numbers

Find greatest common factors of whole number pairs

Lesson 3: Multiple Approaches

Goal: Explore common multiples

Skills students will be able to do:

Identify and generate multiples of whole numbers

Find least common multiples of two or more whole numbers

Lesson 4: First Things First

Goal: To underscore the need for the order of operations

Skills students will be able to do:

Understand and recall order of operations

Follow correct order of operations in solving problems

Lesson 5: Putting it All Together

Goal: Demonstrate understanding of commutative, associative, and distributive properties along with order of operations

Skills students will be able to do:

Correctly apply arithmetic properties (commutative, associative, and distributive to help solve problems

Solve work problems using factors, multiples, and order of operations

Phase Two: Between the Whole Numbers

Lesson 6: Designer Fractions

Goal: Use models to identify missing fractions

Skills students will be able to do:

Use an area model to show fractions

Understand that equal fractions of an area contain equal areas, but not necessarily the same shape

Understand that the sum of the fractional parts of a whole equal 1

Phase 7: Area Models and Equivalent Fractions

Goal: Use models to identify equivalent fractions, developing a sense of the number relationships involved

Skills students will be able to do:

Use an area model of fractions

Find equivalent fractions with like denominators

Compare Fractions

Lesson 8: Fraction Lineup

Goal: Introduce mixed numbers and improper fractions using the number line

Skills students will be able to do:

Use a number line model of fractions

Understand mixed numbers and improper fractions

Understand that fractions extend the whole number system

Find common denominators and equivalent fractions

Lesson 9: Focus on the Denominator

Goal: Use metal computation to order fractions

Skills students will be able to do:

Compare fractions

Use generalized procedures for finding the common denominators

Phase Three: Adding Parts and Taking Them Away

Lesson 10: Sums and Differences on the Line

Goal: Explore sums and differences of fractions on number lines

Skills students will be able to do:

Use the number line to understand addition and subtraction of fractions

Use common denominators

Lesson 11: Numbers Only

Goal: Explore fraction addition and subtraction without models.

Skills students will be able to do:

Develop and use an algorithm for adding and subtracting fractions

Lesson 12: Not Proper but Still Okay

Goal: Convert between mixed numbers and improper fractions

Skills students will be able to do:

• Understand mixed and improper fractions
• Convert between mixed numbers and improper fractions
• Add mixed numbers and improper fractions

Lesson 13: Sorting our Subtraction

Goal: Discover and describe subtraction methods for problems with both mixed numbers and improper fractions

Skills students will be able to do:

• Subtract mixed numbers and improper fractions

Lesson 14: Calc and the Numbers

Goal: Review algorithms for faction addition and subtraction

Skills students will be able to do:

• Solve application problems involving fraction addition and subtraction

Phase Four: Fractions in Groups

Lesson 15: Picturing Fraction Multiplication

Goal: Explore multiplication by using pictures, area models and number lines to find fractions of fractions

Skills students will be able to do:

• Use area models and number lines to find fractional parts of a whole number
• Understand what it means to multiply a whole number by a fraction
• Find and apply methods for multiplying whole numbers by fractions

Lesson 16: Fractions of Fractions

Goal: Use grid models to find fractions of fractions and then analyze the results to develop an algorithm for fraction multiplication

Skills students will be able to do:

• Use area models to find fractions of fractions
• Find and apply algorithm for multiplying fractions

Lesson 17: Estimation and Mixed Numbers

Goal: To practice estimation with fractions

Skills students will be able to do:

• Estimate fraction products
• Develop a method for multiplying mixed numbers

Lesson 18: Fraction Groups within Fractions

Goal: Model a series of problems and use the results to make predictions and develop the algorithm for division

Skills students will be able to do:

• Use an area model to divide a whole number by a fraction
• Divide a proper and improper fraction and mixed number

Lesson 19: Understanding Fraction Division

Goal: Explore the effects of dividing by numbers between 0 and 1, and by numbers greater than 1

Skills students will be able to do:

• Develop understanding of fraction division
• Estimate fraction quotients

Lesson 20: Multiplication vs. Division

Goal: Apply understanding of fraction multiplication and division

Skills students will be able to do:

• Understand the relationship between fraction multiplication and fraction division
• Solve application problems involving fraction multiplication and division

 

Grade: 6

Unit Title: Patterns in Numbers and Shapes

(Last Revision 1-10-06)

Approximate Length: 3 Weeks

Brief description of unit:

In this Unit students will explore the broad theme of patterns and generalizations that form the basis of this unit introducing the ideas and processes of algebra. Student will also explore Functional Relationships in a variety of settings such as growing shapes, boat trips, sneaking up lines, and receiving payments. They will finally explore multiple representations, as it is present in every lesson.

NH Frameworks/Proficiency Standards to be addressed:

1a . K-12 Broad Goal: Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content.

• Solve problems that require the use of strategies (for example: working backwards; looking for patterns and relationships; guess and check; making tables, charts, and graphs; solving a simpler version of a problem; looking for similar problems; drawing a diagram; or creating a model).
• Formulate, solve, and verify problems from every-day and mathematical situations and interpret the results.
• Solve multi-step problems, solve problems with multiple solutions, recognize when a problem has no solution, and recognize problems where more information is needed.

1b. K-12 Broad Goal: Students will use mathematical reasoning.

• Continue a pattern involving integers and positive rational numbers.
• Solve problems involving two-and three-dimensional geometric shapes and explain one's reasoning.
• Use elementary deductive reasoning to solve word problems.
• Use models, known facts, properties, and relationships to explain thinking and to justify answers and solution processes.

6a. K-12 Broad Goal: Students will recognize patterns and describe and represent relations and functions with tables, graphs, equations and rules, and analyze how a change in one element results in a change in another.

• Generalize simple patterns using words.
• Extend a pattern using models.
• Identify properties and relationships related to prime numbers, composite numbers, rational numbers, multiples, factors, and exponents.
• Determine how a change in length or width affects perimeter, area, and volume of two and three-dimensional figures.
• Solve simple linear equations by using concrete materials, tables, or graphs.
• Apply the following properties when appropriate: commutative, associative, distributive, inverse, and identity elements.

6b. K-12 Broad Goal: Students will use algebraic concepts and processes to represent situations that involve variable quantities with expressions, equations, inequalities, matrices and graphs.

• Plot points on a number line or in the plane.
• Use trial and error to find a solution to an equation from among a given replacement set.
• Solve simple linear equations using concrete, informal methods.
• Given a table or graph, select a sentence describing the underlying relationship(s).

7a . K-12 Broad Goal: Students will be able to use concepts about mathematical change in analyzing patterns, graphs, and applied situations.

• Recognize and extend sequences of number and geometric patterns.
• Describe and interpret change from graphs and/or tables of data
• Find averages (for example: batting averages, or grade point averages) and compute rates in familiar contexts (for example: soft drink consumption, distance per unit of time, hourly wages, or paint mixing).

NECAP GLEs/GSEs to be addressed

M (F&A) 6 – 1: Identifies and extends to specific cases a variety of patterns (linear and nonlinear) represented in models, tables, sequences, graphs, or in problem situations; or writes a rule in words or symbols for finding specific cases of a linear relationship; or writes a rule in words or symbols for finding specific cases of nonlinear relationship; and writes an expression or equation using words or symbols to express the generalization of a linear relationship.

M (F&A) 6-2: Demonstrates conceptual understanding of linear relationships (y=kx; y=mx+b) as a constant rate of change by constructing or interpreting graphs of real occurrences and describing the slope of linear relationships (faster, slower, greater, or smaller) in a variety of problem situations; and describes how change in the value of one variable relates to change in the value of a second variable in problem situations with constant rates of change.

M (F&A) 6-3 Demonstrates conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write linear algebraic expressions involving two or more of the four operations; or by evaluating linear algebraic expressions (including those with more than one variable); or by evaluating an expression within an equation (e.g., determine the value of y when x=4 given y=3x-2)

M (F&A) 6 - 4: Demonstrates conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write linear algebraic expressions involving two or more of the four operations; or by evaluating linear algebraic expressions) including those with more than one variable); or by evaluating an expression within an equation.

 

Unit Essential Questions:

Phase 1: What types of patterns can be described from recorded data?

Phase 2: What patterns can be described using Variables and expressions?

Phase 3: What type pf patterns can be described using a Coordinate Plane

and graphs?

Phase 4: What tools can be used to find and extend patterns?

Core Unit Content:

Students will need to find patterns and make sense of situations. They will need to organize data, describe patterns, and then generalize their rules to solve related problems. In phase one, students will practice looking for patterns in different places- drawings and numbers, sometimes a story. Students will make table of data and discover ways of extending patterns. In phase two, students will look for patterns in letters that grow. They will use the langue of algebra by giving the rules for these patterns using variables and expressions. In phase three, students will plot points on a coordinate grid. Students will also compare graphs and make decisions based on their interpretations. Finally in phase four, Students will understand how to create an algebraic rule to describe patterns. They will then use their data to make a choice as to which is a better outcome.

Core Unit Skills:

While exploring the patterns in this unit, students will have acquired the skills to verbally describe mathematical relationships, make tables and graphs, and use algebraic language of variables and expressions.

PHASE 1

Lesson 1: Calendar Tricks

Goals: Find and describe number patterns

Skills students will be able to do:

Explore patterns originating with numbers

Identify, describe, and generalize patterns

Apply computation skills

Lesson 2: Painting Faces

Goals: Explore patterns and organize data

Skills students will be able to do:

Organize data into a table

Visualize three-dimensional patterns

Find linear number patterns with one quantity that varies

Lesson 3: Crossing the River

Goals: Use tables to describe and predict patterns

Skills students will be able to do:

Reason about factors, multiples, and division

Reason about inverse operations

PHASE 2

Lesson 4: Letter Perfect

Goals: Use variables and expressions to describe patterns

Skills students will be able to do:

Identify, describe, and generalize patterns

Relate visual and numeric patterns

Invent patterns

Lesson 5: Tiling Garden Beds

Goals: Describe patters with equivalent expressions

Skills students will be able to do:

Relate pattern rules to a visual situation

Use inverse operations to solve problems

Determine whether two expressions are equivalent

Lesson 6: Chocolate by the Box

Goals: Explore patterns with two variables

Skills students will be able to do:

Explore patterns with two variables

Relate visual and numeric patterns

Reason about equivalent expressions

PHASE 3

Lesson 7: Gridpoint Pictures

Goals: Use a coordinate plane

Skills students will be able to do:

Name points of the coordinate plane

Plot points on the coordinate plane from given coordinates

Plot and identify points in all quadrants of the coordinate plane

Lesson 8: Points, Plots, and Patterns

Goals: Show rules on the coordinate plane

Skills students will be able to do:

Generate a list of ordered pairs based on the same rule

Plot points on a coordinate grid

Relate a number rule to a graphed line

Relate graphed line to a number rule

Lesson 9: Payday at Planet Adventure

Goals: Compare different patterns on a graph

Skills students will be able to do:

Make a table of ordered pairs

Interpret data from a graph

Identify patterns on the coordinate grid and use them to solve problems

PHASE 4

Lesson 10: Sneaking up the Line

Goals: Solve a simple problem to find a pattern

Skills students will be able to do:

Choose appropriate tools for discovering patterns

Identify, describe, and generalize patterns

Make a table and graph the data on a coordinate grid

Lesson 11: Something Fishy

Goals: Explore geometric patterns of growth

Skills students will be able to do:

Describe a pattern or rule using variables and expressions

Create and describe new patterns

Lesson 12: The Will

Goals: Use patterns to make decisions

Skills students will be able to do:

Make and extend tables

Graph data from a table

Use a variety of tools to look for patterns

      Grade: 6 

Unit Title: What Does the Data Say?

(Last Revision 1-10-06)

Approximate Length: 5 weeks

Brief description of unit:

Introduces students to a number of central ideas from statistics, such as measures of central tendency (mean, median, and mode) and measures of variability (the range).   Students use some of the common ways of representing data graphically – frequency graphs, single bar graphs, double bar graphs, points of a coordinate grid, broken-line graphs- throughout the unit.

Some basic concepts of probability are also presented.   These include the concepts of chance and randomness, theoretical and experimental probability, and the idea of probability as a ratio of a given event to the number of total events.

Some of the work students’ encounter in this unit relates to the strands of Measurement.   In phase three, students find ways to measure performance in various tasks in order to analyze change over time.   Various computational procedures, such as those for computing means, computing difference scores, and comparing probabilities expressed as ratios, are introduced as students proceed through the unit.    

NH Frameworks/Proficiency Standards to be addressed:

5a. K-12 Broad Goal: Students will use data analysis, statistics and probability to analyze given situations and the outcomes of experiments.

PURPOSE: Collecting, organizing, displaying, and interpreting data, as well as using the information to make decisions and predictions, have become very important in our society. Statistical instruction should be carried out in a spirit of investigation and exploration so students can answer questions about data. Probability must be studied in familiar contexts encouraging students to model situations. Students need to investigate fairness, chances of winning, and uncertainty. Technology should be used as a tool throughout the investigation process.

End of Grade 6:

• Construct and interpret line plots, stem and leaf plots, frequency distributions, and graphs.
• Use multiple representations to display equivalent data.
• Select appropriate data to solve simulations and real world problems.
• Simulate, display, graph and analyze data in a variety of mediums.
• Determine and explore various uses of mean, median, and mode.
• Use sampling techniques to make predictions.
• Given a sample space find probabilities of events.

NECAP GLEs/GSEs to be addressed:

M(DSP)–6–1 Interprets a given representation (circle graphs, line graphs, or stem-and-leaf plots) to

answer questions related to the data, to analyze the data to formulate or justify conclusions, to make predictions,

or to solve problems. (IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–6–2 .)

M(DSP)–6–2 Analyzes patterns, trends or distributions in data in a variety of contexts by

determining or using measures of central tendency (mean, median, or mode) or dispersion (range) to

analyze situations, or to solve problems.

M(DSP)–6–4 Uses counting techniques to solve problems in context involving combinations or simple

permutations using a variety of strategies (e.g., organized lists, tables, tree diagrams, models,

Fundamental Counting Principle, or sc others).

M(DSP)–6–5 For a probability event in which the sample space may or may not contain equally likely

outcomes, determines the experimental or theoretical probability of an event in a problem-solving situation.

Not included in the mathscape program – needs to be added

 

Unit Essential Questions:

Phase one Essential Question: What are measures of central tendency and how are they analyzed?

Phase two essential question: How do you collect, present, and analyze data?

Phase three essential question: How do you collect, present, and analyze data to measure progress over time?

Phase four essential question: How do you use data to make informed predictions, determine probabilities, and calculate theoretical and experimental probabilities?

Core Unit Content:  

Phase one – Students will be able to conduct surveys, collect data, represent data using a frequency graph and analyze data using the measures of central tendency.   .  

Phase two – Students will explore the effects of different ways of representing data.   They interpret and compare results of two sets of data from a survey.

Phase three – Students measure and represent progress over time.   They look for trends in data on graphs to make predictions.   Understand the implications of graphed representations of data.  

Core Unit Skills:  

Phase One

 Lesson 1 Class Survey  

Goal :Analyze data to find modes and ranges

Students will be able to:

Collect data by conducting a survey

Make and interpret frequency graphs

Determine the mode and rang for a data set

Use the mode and range to analyze data

Lesson 2 Name Exchange

Goal: Exploring means and medians

Students will be able to:

Collect numerical data

Make and interpret frequency graphs

Determine the mean and median for a data set

Compare means, medians, and modes

Lesson 3 TV Shows

Goal: Investigating Ratings and Distributions

Students will be able to:

Understand numerical rating scales

Interpret different distributions of data

Collect numerical data

Make and interpret frequency graphs

Determine means, medians, modes, and ranges

Write an analysis that is clearly supported by the data

Phase Two

Lesson 4 Animal Comparisons

Goal: Investigating Bar Graphs and Scales

Students will be able to:

Represent data with bar graphs

Select appropriate scales to represent data accurately and fairly

Recognize that the choice of scale can affect people’s impressions of the data

Recognize that scales need to have equal intervals

Lesson 5 Double Data

Goal: Creating and Interpreting Double Bar Graphs

Students will be able to:

Represent data by making accurate and complete double bar graphs

Select appropriate scales for double bar graphs

Analyze data from double bar graphs

Make recommendations based on data from double bar graphs

Demonstrate and understanding of when a double bar graph is appropriate to represent data.

Lesson 6 Across the Ages

Goal: Making Comparisons and Recommendations

Students will be able to:

Compare and analyze two set of survey data

Represent data from two populations using double bar graphs

Interpret double bar graphs

Select appropriate scales

Determine the mean, median, mode, and range

Make recommendations that are based in data

Phase Three

Lesson 7 Are you Improving?

Goal: Using Statistics to measure progress

Students will be able to:

Collect data from performance experiments

Organize and represent data in a broken-line graph

Analyze data in graphs to describe progress

Determine means, medians, modes, and ranges

Use data analysis to make predictions

Communicate about results

Lesson 8 How close can you get?

Goal: Graphing and analyzing errors

Students will be able to:

Use data on the sizes of errors to measure progress

Organize and represent data in a broken-line graph and recognize that there are different graphical representations of progress

Select appropriate scales

Determine mean, median, mode, and range

Analyze data to describe progress and make predictions

Communicate about results

Lesson 9 Stories and Graphs

Goal: Interpreting Multiple Representations of Data

Students will be able to:

Relate graphs of progress over time to descriptions of actual events

Organize and represent data in a broken-line graph.

Determine means, median, modes, and ranges

Analyze data in graphs to describe progress

Use data analysis to support conclusions and predictions

Communicate results in mathematical terms

Phase Four

Lesson 10 What are the chances?

Goal: Determining probability

Students will be able to:

Build an informal understanding of chance and randomness

Introduce qualitative and quantitative ways of describing probability

Introduce theoretical and experimental probabilities in an informal way

Apply data collection and recording skills

Applying data analysis skills, particularly the use of measures of central tendency

Lesson 11 Changing the Chances

Goal: Experimenting with probability

Students will be able to:

Build and understanding of probability as a ratio

Determine theoretical and experimental probabilities

Compare and rank-order probabilities

Apply data collection and recording skills

Apply data analysis skills, particularly the use of measures of central tendency

Lesson 12 Which Bag is Which?

Goal: Applying probability and Statistics

Students will be able to:

Apply understanding of probability

Use sampling to make informed predictions

Apply understanding of bar graphs

Apply data collection and recording skills

Use proportional reasoning to solve problems

  

Grade: 7

Unit Title: Chance Encounters 

(Last Revision 1-19-06)            

Approximate Length:  24 – 45 minute classes

Brief description of unit: 

In this unit, students see probability through a mathematical lens by investigating games and simulations.  The investigations are full of opportunities for students to explore the mathematical themes of theoretical and experimental probability, multiple representations of probability, and modeling situations with simulations.  Whether conducting experiments for games or designing their own simulations, students learn how mathematical thinking can provide insights into the world around them.

NH Frameworks/Proficiency Standards to be addressed:

1a. K-12 Broad Goal: Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content

• Formulate problems from everyday and mathematical situations.
• Solve problems with and without using manipulatives and calculators.
• Formulate, solve, and verify problems from every-day and mathematical situations and interpret the results.
• Solve problems using manipulatives, graphs, charts, diagrams, and calculators.
• Determine, collect, and organize the relevant data needed to solve real-world

3a. K-12 Broad Goal: Students will develop number sense and an understanding of our numeration system.

• Name and identify a fraction or decimal, given a physical representation.
• Given a decimal representation in tenths or hundredths, write an equivalent
• Given a set of fractional models, name and write those that represent equivalent fractions.

5a. K-12 Broad Goal: Students will use data analysis, statistics and probability to analyze given situations and the outcomes of experiments.

• Collect data, construct, and interpret picture and bar graphs.
• Interpret circle graphs.
• Write a story problem using information from a graph.
• Given appropriate information, determine which is most likely to happen or whether one event is more likely than another.
• Select appropriate data to solve simulations and real world problems.
• Use sample sets to make appropriate inferences and predictions.
• Predict and find the probability of outcomes of a simple probability experiment.

8a. K-12 Broad Goal: Students will use a variety of tools from discrete mathematics to explore and model real-world situations.

• Use counting techniques to determine the number of outcomes for situations (for example: handshake problems, menu ordering, or clothes matching).

NECAP GLEs/GSEs to be addressed:

Number and Operations

7-1 – Demonstrates conceptual understanding of rational numbers with respect to percents as a means of comparing the same or different parts of the whole when the wholes vary in magnitude and percents as a way of expressing multiples of a number using models, explanations, or other representations.

7-4 – Accurately solves problems involving proportional reasoning; percents involving discounts, tax, or tips; and rates.

Date, Statistics, and Probability

7-1 – Interprets a given representation (circle graphs, scatter plots that represent discrete linear relationships, or histograms) to analyze the data to formulate or justify conclusions, to make predictions, or to solve problems.

7-2 – Analyzes patterns, trends, or distributions in data in a variety of contexts by solving problems using measures of central tendency (mean, median, or mode), dispersion (range or variation), or outliers to analyze situations to determine their effect on mean, median, or mode; and evaluate the sample from which the statistics were developed (bias).

7-3 – Identifies or describes representations or elements of representations that best display a given set of data or situation, consistent with the representations required in DSP 7-1 above.

7-5 – For a probability event in which the sample space may or may not contain equally likely outcomes, determines the experimental or theoretical probability of an event in a problem-solving situation

 Unit Essential Questions:

• What mathematics is involved in testing and analyzing games of chance?
• What is the meaning of chance?
• How can you change situations to improve your chances?
• How can you tell if a game is fair?

    Core Unit Content: 

            Phase One

                        Students will:

·        Explore the concept of chance.

·        Understand how to use strip graphs, frequency graphs, and probability lines to investigate probability.

Phase Two

            Students will:

·        Test games played with a circular spinner that is divided into unequal parts, then experiment to improve their chances of winning.

·        Draw spinners to match clues that describe situations.

Phase Three

            Students will:

·        Analyze and test the fairness of games involving two independent events.

·        Create outcome grids that show the theoretical probabilities of all possible outcomes in games. 

  Core Unit Skills: 

Phase One – students will:

• Refine understanding of the concept of chance.
• Explore the variability of results in games of chance.
• Conduct probability experiments
• Build an intuitive sense of natural variability and the law of large numbers.
• Represent experimental data using strip graphs and frequency graphs.
• Describe probabilities using qualitative terms.
• Describe probabilities quantitatively using fractions, decimals, and percentages.
• Rank the probability of events on a scale of 0 to 1.
• Use a probability line to represent and compare the likelihood of events.

Phase Two – students will:

• Determine the probability of events.
• Distinguish events that have equal probabilities from events that have unequal probabilities.
• Explore area models of probability.
• Understand that the parts of circular spinners with larger areas tend to be spun more often than the parts with smaller ones.
• Use fractions, decimals, and percentages to express probabilities.
• Represent probabilities as parts of a circular spinner.
• Relate verbal, visual, and numeric representations or probability.
• Use logical reasoning.
• Represent probabilities as areas of a circular spinner.
• Use fractions, decimals, and percentages to express probabilities.
• Relate verbal, visual, and numeric representations of probability.

Phase Three – students will:

• Determine all possible combinations of two independent events.
• Determine theoretical probabilities using outcome grids.
• Apply probability to determining fairness.
• Compare theoretical and experimental probabilities.
• Determine all possible combinations of two independent events.
• Use visual models to determine theoretical probabilities and connect them with numerical representations.
• Apply probability in determining fairness.
• Explore numerical representations of probabilities and connect them with visual models.
• Develop a sense of the relative sizes of different probabilities.
• Recognize that the number of favorable outcomes needs to be considered in proportion to the total number of outcomes.
• Apply proportional reasoning to predict outcomes with large number of trials

Grade: 7

Unit Title: Getting in Shape

(Last Revision 1-20-06)

Approximate Length: 24 – 45 minute classes

Brief description of unit:

In this unit, students explore different types of polygons, as well as circles, and discover important relationships among these two-dimensional figures. Using the mathematical ideas they discover in their investigations, students enlarge and create logos and geometric transformations and, as a culminating activity, a geometric design made up of many of the figures they study.  In discussions and written work, students communicate important geometric concepts and exchange findings using the language of geometry.

NH Frameworks/Proficiency Standards to be addressed:

4a. K-12 Broad Goal: Students will name, describe, model, classify, and compare geometric shapes and their properties with an emphasis on their wide applicability in human activity.

• Identify, describe, and name properties of triangles, quadrilaterals, and other polygons.
• Identify point and line symmetry in given polygons.
• Measure and classify angles.
• Identify and draw congruent and similar figures using graph paper.
• Represent and solve problems using the properties of two and three dimensional geometric figures.
• Use technology, manipulatives, and/or coordinate geometry to deduce and explain the properties of and the relationships among geometric figures.
• Translate between synthetic and coordinate representations.

4b. K-12 Broad Goal: Students will develop spatial sense.

• Tessellate (tile) a plane with a given figure and create a figure that will tile the plane.
• Describe the shadow of certain figures.
• Sketch specific two dimensional figures, given definitions and/or properties.
• Demonstrate that the conditions necessary for congruence or the conditions necessary for similarity are met.
• Use technology, manipulatives, and/or coordinate geometry to explain properties of transformations (for example: translations, line reflections, rotations, dilations, and the composition of these transformations).
• Demonstrate an understanding of properties among two and three dimensional figures.

4c. K-12 Broad Goal: Students will develop an understanding of measurement and systems of measurement through experiences which enable them to use a variety of techniques, tools, and units of measurement to describe and analyze quantifiable phenomena.

• Find and/or estimate the perimeter and area of a given quadrilateral or triangle.
• Demonstrate an understanding of the use of maps, scale drawings, and timelines.
• Compare the relationship between similar figures and their areas.
• Identify and use appropriate units of measurement.
• Approximate areas of irregular shapes drawn on a grid.
• Convert commonly used measurements to equivalent ones within a measurement system.
• Apply the formulas for and choose an appropriate unit of measurement to find the linear and area measures associated with two dimensional figures and the volume and surface area of three dimensional figures. .
• Select an appropriate procedure to determine a measure when a direct measurement cannot be made.
• Use ratio and proportion to find the measure of all sides of similar figures.

4d. K-12 Broad Goal: Students will know the basic concepts of trigonometry and apply these concepts to real-world problems.

• Make scale drawings, keeping sides in proportion. (Scale factor to be kept to a small whole number or fraction with denominator less than 6.)

NECAP GLEs/GSEs to be addressed:

Geometry and Measurement

7-1 –Uses properties of angle relationships resulting from two or three intersecting lines(adjacent angles, vertical angles, straight angles or angle relationships formed by two non-parallel lines cut by a transversal) or two parallel lines cut by a transversal to solve problems.

7-2 – Applies theorems or relationships (triangle inequality or sum of the measures of interior angles of regular polygons) to solve problems.

7-4 – Applies the concepts of congruency by solving problems on a coordinate plane involving reflections, translations, or rotations.

7-5 – Applies concepts of similarity by solving problems involving scaling up or down and their impact on angle measures, linear dimensions and areas of polygons, and circles when the linear dimensions are multiplied by a constant factor. Describe effects using models or explanations.

7-6 – Demonstrates conceptual understanding of the area of circles or the area of circles or the area or perimeter of composite figures (quadrilaterals, triangles, or parts of a circle), and the surface area of rectangular prisms, or volume of rectangular prisms, triangular prisms, or cylinders using models, formulas, or by solving related problems. Express all measures using appropriate units.

Unit Essential Questions:

• What are the relationships among two dimensional figures?
• How do we communicate important geometric concepts using the language of geometry?
• How would geometric shapes and designs influenced by function?

Core Unit Content:  

Phase One - Students will:

• Estimate angle measures and draw and measure different types of angles.  
• They will investigate various angle and side relationships in triangles.
• Explore different ways to classify triangles.  
• Check for lines of symmetry and explore similarity in triangles.

Phase Two - Students will:

• Develop a sorting system for classifying polygon tiles.
• Explore how different four sided figures are related.
• Investigate angle relationships in polygons.
• Expand their knowledge of congruency to include polygons.
• Explore how to use transformations to determine whether polygons are congruent.
• Investigate congruence using transformations on the coordinate plane.

Phase Three - Students will:

• Investigate the unique properties of circles
• Discover how the circumference and the diameter of a circle are related.
• Develop formulas for finding the area of regular polygons and the area of circles.

Core Unit Skills:  

Phase One – students will:

• Estimate and measure angles
• Name and classify angles
• Find how many ways three angles can be combined to fit inside a circle.
• Measure and find the sum of the angles of a triangle.
• Find the measure of the angle of a triangle given the other angle measures.
• Explore the relationships among the sides of a triangle.
• Determine the relationships among the angles and sides of triangles.
• Classify triangles in several different ways.
• Investigate possible combinations of triangle types.
• Use symmetry to classify figures and understand their properties.
• Write true triangle statements.
• Create an enlargement using grid paper.
• Investigate similarity in a real world context of logos
• Determine whether two triangles can be classified as similar
• Write explanations for why triangles can be classified as similar

Phase Two – students will:

• Develop a classification method for sorting polygons.
• Match quadrilaterals to definitions of six types.
• Investigate quadrilateral relationships.
• Summarize facts about quadrilaterals.
• Explore angles of quadrilaterals.
• Investigate the relationship between the number of sides of a polygon and the sum of it’s angles.
• Develop and apply the formula to regular polygons.
• Explore congruence using flips, turns and slides.
• Investigate reflections, rotations and translations, on the coordinate plane.
• Write about transformations.
• Explore the symmetry lines of polygons.
• Look for patterns that show relationships between number of symmetry lines and types of polygons.
• Write full descriptions of regular polygons.

Phase Three – students will:

• Identify the radius and diameter of a circle.
• Collect display and analyze data on the relationship between the circumference and diameter of a circle.
• Write an equation to describe the relationship between the circumference and diameter of a circle.
• Apply understanding of circle relationships to solve a problem with a real world context.
• Re-arranges the wedges of a regular polygon to form a rectangle.
• Relate the area of a rectangle and the area of a regular polygon.
• Develop a formula for the area of a regular polygon and apply this formula to find the area of a regular polygon.
• Estimate the area of a circle.
• Re-arrange the sectors of a circle to form a rectangle.
• Develop a formula for area of a circle.
• Use a formula to evaluate estimates.

Grade: 7

Unit Title: The Language of Algebra   (Last Revision 1-10-06)

Approximate Length:  24 – 45 minute classes

Brief description of unit:  

 In this unit, students explore the language and tools of algebra through a series of investigations.  Whether they are writing equations that describe their ideal school or creating a graph of the relationship between Fahrenheit and Celsius, students have many opportunities to use the basic tools of algebra and see the connections between them.  Throughout the unit, students analyze school fund-raising opportunities and field trip transportation to make recommendations to the student government and school board.  This provides a relevant context for putting the language and tools of algebra to work.

NH Frameworks/Proficiency Standards to be addressed: Under Construction

NECAP GLEs/GSEs to be addressed:

Functions and Algebra

• Identifies and extends to specific cases a variety of patterns (linear and non linear) represented in models, tables, sequences, graphs, or in problem situations; and generalizes a linear relationship using words and symbols; generalizes a linear relationship to find a specific case; or writes an expression or equation using words or symbols to express the generalization of a non linear relationship.
• Demonstrates conceptual understanding of linear relationships(y = kx;y = mx + b) as a constant rate of change by solving problems involving the relationship between slope in concrete situations, or informally determining the slope of a line from a table or graph; and distinguishes between constant and varying rates of change in concrete situations represented in tables or graphs; or describes how change in the value of one variable relates to change in the value of a second variable in problem situations with constant rates of change.
• Demonstrates conceptual understanding of algebraic expressions by using letters to represent unknown quantities to write algebraic expressions (including those with whole number exponents or more than one variable); or by evaluating algebraic expressions (including those with whole number exponents or more than one variable); or by evaluating an expression within an equation (e.g., determine the value of y when x =4 given y = 5x 3 – 2). We will not use exponents.
• Demonstrates conceptual understanding of equality by showing equivalence between two expressions (expressions consistent with the parameters of the left- and right-hand sides of the equations being solved at this grade level) using models or different representations of the expressions, solving mult-step linear equations of the form ax ± b = c with a ≠ 0, and ax ± b = cx ± d with a, c ≠ 0, and (x/a) ± b = c with a ≠ 0, where a, b, c, and d are whole numbers; or by translating a problem-solving situation into an equation consistent with the parameters of the type of equations being solved for this grade level.

Unit Essential Questions: