# Problem of the Month Tri Triangles

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 In the first level of the POM, students view a pattern of triangles composed of toothpicks. Their task is to determine the number of toothpicks that make up each pattern. The function is proportional. In level B, students examine a linear pattern that involves a constant. The task involves a set of triangular tables that are arranged adjacently in a row. The task asks students to determine the relationship between the number of tables and the number of people who can sit around the tables. Students need to extend the pattern. They also find the inverse relationship, i.e., find the pattern number when given the total number of people seated. Level C requires students to determine how a pattern grows. Students need to see that the pattern grows by square numbers. They identify the relationship and then explain a valid process for finding these values. In level D, students are asked to generalize a rule for finding a value in the triangular number sequence. They are also asked to explain the process for finding an inverse value for the triangular number sequence by finding the term given the total. In the final level E, students generate a closed expression for a sequence that grows as the sum of two exponential functions. In addition, the students must justify their findings.

3 Problem of the Month Tri - Triangles Level A: Lisa is making triangle patterns out of toothpicks all the same length. A triangle is made from three toothpicks. Her first pattern is a single triangle. Her second pattern is shown below: How many toothpicks are needed for her second pattern? Her third pattern is shown below: How many toothpicks are needed for her third pattern? If she continued the same pattern, how many toothpicks are needed for the fifth pattern? How many toothpicks are needed for the tenth pattern? Problem of the Month Tri - Triangles Page 1 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

4 If you had 81 toothpicks, what pattern number could you make? Explain how you know. Problem of the Month Tri - Triangles Page 2 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

5 Level B: Your classroom has triangular shape tables. Three students can sit around one table. Two tables can be pushed together so that two sides are adjacent. How many students can sit around the tables in this arrangement? Tables can be added to the arrangement by pushing together tables so that each additional table is adjacent to one side of the row of tables. The arrangement may grow to be a long row of tables. How many students can sit around three tables in this arrangement? How many students can sit around with five tables in a row arrangement? Determine how many students can sit around twelve desks in a row without drawing the arrangement? Explain how you figured it out. Problem of the Month Tri - Triangles Page 3 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

6 How many tables in a row are needed to seat 105 students? Explain your answer. Problem of the Month Tri - Triangles Page 4 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

7 Level C: Anne uses triangles to make the following patterns: Pattern 1 Pattern 2 Pattern 3 The pattern continues in the same geometric design. Draw Pattern 4, how many triangles are needed? How many triangles are needed to construct Pattern 7? How many triangles are needed to construct Pattern 16? Explain how you determined your rule. Write a rule to find the number of triangles needed for the nth pattern? Explain your rule. Problem of the Month Tri - Triangles Page 5 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

8 Suppose a pattern had 2,025 triangles, what is the pattern number? Explain. Problem of the Month Tri - Triangles Page 6 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

9 Level D: Jo constructs triangular patterns using dots. Pattern 1 Pattern 2 Pattern 3 The pattern continues in the same geometric design. How many dots are needed to make Pattern 5? How many dots are needed for the nth pattern? Explain your rule. Jo was born in 1953, she was wondering if she could make a triangular pattern out of exactly 1,953 dots. If she could, what would the pattern number be? Explain your answer. Problem of the Month Tri - Triangles Page 7 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

10 Level E: Pattern 1 Pattern 2 Pattern 3 Craig constructs the designs above from equal line segments. The design in Pattern 1 is made up of three line segments. Pattern 2 is made up of nine line segments. Pattern 3 is made up of thirty line segments, and so on. How many line segments are needed to make Pattern 8? How many line segments are needed to make Pattern 16? Determine a function for finding the number of line segments needed to make the pattern for any number n. Justify why your function works. You have 6,294,528 equal line segments. Can you construct a design that belongs in this sequence using just those line segments? If so, what pattern number would that be? If not, how many more line segments might you need to construct a design that fits the sequence? Problem of the Month Tri - Triangles Page 8 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

11 Problem of the Month Tri - Triangles Primary Version Level A Materials: A picture of the three patterns, paper, pencil and toothpicks for students to make the different patterns. Discussion on the rug: (Teacher holds up the pictures of the triangle patterns) Here are different patterns. How many toothpicks would it take to make the first pattern? (Students may build and count) How many toothpicks do you need to build the second pattern? (Students may build and count) (Teacher asks the children to think about how the number of toothpicks changed from first pattern to the second.) In small groups: (Each student has access to toothpicks, paper, pencil and the picture of the first three patterns. The teacher explains that they may either build or draw the pattern to help find the answers. Teacher asks the following questions. Only go on to the next question if students have success.) How many toothpicks do you need to build the first pattern? How many toothpicks do you need to build the second pattern? How many toothpicks do you need to build the third pattern? How many toothpicks do you need to build the fourth pattern? How many toothpicks do you need to build the sixth pattern? (At the end of the investigation have students either discuss or dictate a response to this summary question.) Tell me how you know. Problem of the Month Tri - Triangles Page 9 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

12 Problem of the Month Tri - Triangles First Pattern Second Pattern Third Pattern Problem of the Month Tri - Triangles Page 10 (c) Noyce Foundation To reproduce this document, permission must be granted by the Noyce Foundation:

13 Problem of the Month Tri-Triangles Task Description Level A This task challenges a student to view a pattern of triangles composed of toothpicks. Students must determine the number of toothpicks that make up each pattern. This function is proportional. Common Core State Standards Math - Content Standards Operations and Algebraic Thinking Represent and solve problems involving multiplication and division. 3.OA.1. Interpret products of whole numbers, e.g. interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. 3.OA.2 Interpret whole number quotients of whole numbers, e.g., interpret 56 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares,, or as a number of objects in each share when 56 objects are partitioned into equal shares of 8 objects each. 3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and mathematical quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problems. Understand properties of multiplication and the relationship between multiplication and division. 3.OA.5 Apply properties of operations as strategies to multiply and divide. 3.OA.6 Understand division as an unknown-factor problem. Solve problems involving the four operations, and identify and explain patterns in arithmetic. 3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Ass the reasonableness of answers using mental computation and estimation strategies including rounding. Generate and analyze patterns. 4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Common Core State Standards Math Standards of Mathematical Practice MP.1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

14 MP.3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

15 Problem of the Month Tri-Triangles Task Description Level B This task challenges a student to examine a linear pattern that involves a constant. The task involves a set of triangular tables that are arranged adjacently in a row. Students determine the relationship between the number of tables and the number of people who can sit around the tables. Students need to extend the pattern and find the inverse relationship, i.e., find the pattern number when given the total number of people seated. Common Core State Standards Math - Content Standards Operations and Algebraic Thinking Understand properties of multiplication and the relationship between multiplication and division. 3.OA.5 Apply properties of operations as strategies to multiply and divide. 3.OA.6 Understand division as an unknown-factor problem. Solve problems involving the four operations, and identify and explain patterns in arithmetic. 3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Ass the reasonableness of answers using mental computation and estimation strategies including rounding. Generate and analyze patterns. 4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Write and interpret numerical expressions. 5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluations them. Analyze patterns and relationships. 5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns and graph the ordered pairs on a coordinate plane. Expressions and Equations Represent and analyze quantitative relationships between dependent and independent variables. 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variable using graphs and tables, and relate these to the equations. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Common Core State Standards Math Standards of Mathematical Practice MP.3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze

16 situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. MP.4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

17 Problem of the Month Tri-Triangles Task Description Level C This task challenges a student to determine values that grow in a square area relationship and then to explain a valid process for finding these values and extending the pattern. The also find the inverse relationship for specific values. Common Core State Standards Math - Content Standards Operations and Algebraic Thinking Generate and analyze patterns. 4.OA.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Write and interpret numerical expressions. 5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluations them. Analyze patterns and relationships. 5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns and graph the ordered pairs on a coordinate plane. Expressions and Equations Apply and extend previous understandings of arithmetic to algebraic expressions. 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents. Represent and analyze quantitative relationships between dependent and independent variables. 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variable using graphs and tables, and relate these to the equations. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Common Core State Standards Math Standards of Mathematical Practice MP.3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

18 MP.4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

20 interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

21 Problem of the Month Tri-Triangles Task Description Level E This task challenges a student to generate a closed expression for a sequence that grows as the sum of two exponential functions. In addition, the students must justify their findings. Common Core State Standards Math - Content Standards Expressions and Equations Expressions and equations work with radicals and integer exponents. 8.EE.1 Know and apply the properties of inter exponents to generate equivalent numerical expressions. High School Algebra Creating Equations Create equations that describe numbers or relationships. A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions. High School Functions Building Functions Build a function that models relationship between two quantities. F-BF.1 Write a function that describes a relationship between two quantities. F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations and translate between the two forms. Build new functions from existing functions. F-BF.4 Find inverse functions. High School Functions Linear, Quadratic, and Exponential Models Interpret expressions for functions in terms of the situation they model. F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context. Common Core State Standards Math Standards of Mathematical Practice MP.4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. MP.8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1,2) with slope 3, middle school students might abstract the equation (y -2)/(x- 1) = 3. Noticing the regularity in the way terms cancel when expanding (x-1)(x+1), (x-1)(x 2 + x + 1),

22 and (x-1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

24 MP.4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

### Problem of the Month: Perfect Pair

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

### Problem of the Month The Wheel Shop

Problem of the Month The Wheel Shop The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core

### Problem of the Month: On Balance

Problem of the Month: On Balance The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core

### Problem of the Month: Double Down

Problem of the Month: Double Down The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core

### Problem of the Month: William s Polygons

Problem of the Month: William s Polygons The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common

### Problem of the Month Pick a Pocket

The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems

### Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

### Problem of the Month Through the Grapevine

The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems

### Division with Whole Numbers and Decimals

Grade 5 Mathematics, Quarter 2, Unit 2.1 Division with Whole Numbers and Decimals Overview Number of Instructional Days: 15 (1 day = 45 60 minutes) Content to be Learned Divide multidigit whole numbers

### Problem of the Month: Digging Dinosaurs

: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of

### Problem of the Month: Fair Games

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

### Problem of the Month: Once Upon a Time

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

### PA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis*

Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed

### Problem of the Month: Cutting a Cube

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

### Unit 1: Place value and operations with whole numbers and decimals

Unit 1: Place value and operations with whole numbers and decimals Content Area: Mathematics Course(s): Generic Course Time Period: 1st Marking Period Length: 10 Weeks Status: Published Unit Overview Students

### Standards for Mathematical Practice: Commentary and Elaborations for 6 8

Standards for Mathematical Practice: Commentary and Elaborations for 6 8 c Illustrative Mathematics 6 May 2014 Suggested citation: Illustrative Mathematics. (2014, May 6). Standards for Mathematical Practice:

### Grades K-6. Correlated to the Common Core State Standards

Grades K-6 Correlated to the Common Core State Standards Kindergarten Standards for Mathematical Practice Common Core State Standards Standards for Mathematical Practice Kindergarten The Standards for

### For example, estimate the population of the United States as 3 times 10⁸ and the

CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number

### Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities

Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned

### Describing and Solving for Area and Perimeter

Grade 3 Mathematics, Quarter 2,Unit 2.2 Describing and Solving for Area and Perimeter Overview Number of instruction days: 8-10 (1 day = 90 minutes) Content to Be Learned Distinguish between linear and

### Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

### Performance Assessment Task Gym Grade 6. Common Core State Standards Math - Content Standards

Performance Assessment Task Gym Grade 6 This task challenges a student to use rules to calculate and compare the costs of memberships. Students must be able to work with the idea of break-even point to

### Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. Add and subtract within 20. MP.

Performance Assessment Task Incredible Equations Grade 2 The task challenges a student to demonstrate understanding of concepts involved in addition and subtraction. A student must be able to understand

### Problem of the Month Diminishing Return

The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems

### Problem of the Month: Circular Reasoning

Problem of the Month: Circular Reasoning The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common

### Pearson Algebra 1 Common Core 2015

A Correlation of Pearson Algebra 1 Common Core 2015 To the Common Core State Standards for Mathematics Traditional Pathways, Algebra 1 High School Copyright 2015 Pearson Education, Inc. or its affiliate(s).

### Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### Math at a Glance for April

Audience: School Leaders, Regional Teams Math at a Glance for April The Math at a Glance tool has been developed to support school leaders and region teams as they look for evidence of alignment to Common

### Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series

Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them

### Evaluation Tool for Assessment Instrument Quality

REPRODUCIBLE Figure 4.4: Evaluation Tool for Assessment Instrument Quality Assessment indicators Description of Level 1 of the Indicator Are Not Present Limited of This Indicator Are Present Substantially

### Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %

Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### Georgia Standards of Excellence 2015-2016 Mathematics

Georgia Standards of Excellence 2015-2016 Mathematics Standards GSE Coordinate Algebra K-12 Mathematics Introduction Georgia Mathematics focuses on actively engaging the student in the development of mathematical

### Overview. Essential Questions. Grade 8 Mathematics, Quarter 4, Unit 4.3 Finding Volume of Cones, Cylinders, and Spheres

Cylinders, and Spheres Number of instruction days: 6 8 Overview Content to Be Learned Evaluate the cube root of small perfect cubes. Simplify problems using the formulas for the volumes of cones, cylinders,

### Mathematics Interim Assessment Blocks Blueprint V

6-7 Blueprint V.5.7.6 The Smarter Balanced Interim Assessment Blocks (IABs) are one of two distinct types of interim assessments being made available by the Consortium; the other type is the Interim Comprehensive

### Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)

New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct

### G C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.

### Geometry Solve real life and mathematical problems involving angle measure, area, surface area and volume.

Performance Assessment Task Pizza Crusts Grade 7 This task challenges a student to calculate area and perimeters of squares and rectangles and find circumference and area of a circle. Students must find

### Overview. Essential Questions. Grade 2 Mathematics, Quarter 4, Unit 4.4 Representing and Interpreting Data Using Picture and Bar Graphs

Grade 2 Mathematics, Quarter 4, Unit 4.4 Representing and Interpreting Data Using Picture and Bar Graphs Overview Number of instruction days: 7 9 (1 day = 90 minutes) Content to Be Learned Draw a picture

### INDIANA ACADEMIC STANDARDS. Mathematics: Grade 6 Draft for release: May 1, 2014

INDIANA ACADEMIC STANDARDS Mathematics: Grade 6 Draft for release: May 1, 2014 I. Introduction The Indiana Academic Standards for Mathematics are the result of a process designed to identify, evaluate,

### Performance Level Descriptors Grade 6 Mathematics

Performance Level Descriptors Grade 6 Mathematics Multiplying and Dividing with Fractions 6.NS.1-2 Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

K-12 Louisiana Student Standards for Mathematics: Table of Contents Introduction Development of K-12 Louisiana Student Standards for Mathematics... 2 The Role of Standards in Establishing Key Student Skills

### 5 th Grade Common Core State Standards. Flip Book

5 th Grade Common Core State Standards Flip Book This document is intended to show the connections to the Standards of Mathematical Practices for the content standards and to get detailed information at

### DELAWARE MATHEMATICS CONTENT STANDARDS GRADES 9-10. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s))

Prentice Hall University of Chicago School Mathematics Project: Advanced Algebra 2002 Delaware Mathematics Content Standards (Grades 9-10) STANDARD #1 Students will develop their ability to SOLVE PROBLEMS

### #1 Make sense of problems and persevere in solving them.

#1 Make sense of problems and persevere in solving them. 1. Make sense of problems and persevere in solving them. Interpret and make meaning of the problem looking for starting points. Analyze what is

### 1 ST GRADE COMMON CORE STANDARDS FOR SAXON MATH

1 ST GRADE COMMON CORE STANDARDS FOR SAXON MATH Calendar The following tables show the CCSS focus of The Meeting activities, which appear at the beginning of each numbered lesson and are taught daily,

### Overview. Essential Questions. Precalculus, Quarter 3, Unit 3.4 Arithmetic Operations With Matrices

Arithmetic Operations With Matrices Overview Number of instruction days: 6 8 (1 day = 53 minutes) Content to Be Learned Use matrices to represent and manipulate data. Perform arithmetic operations with

### Glencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 3-3, 5-8 8-4, 8-7 1-6, 4-9

Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 6-8 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,

### Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

### CORE Assessment Module Module Overview

CORE Assessment Module Module Overview Content Area Mathematics Title Speedy Texting Grade Level Grade 7 Problem Type Performance Task Learning Goal Students will solve real-life and mathematical problems

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

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers

### Topic: 1 - Understanding Addition and Subtraction

8 days / September Topic: 1 - Understanding Addition and Subtraction Represent and solve problems involving addition and subtraction. 2.OA.1. Use addition and subtraction within 100 to solve one- and two-step

### Common Core State Standards for Mathematics for California Public Schools Kindergarten Through Grade Twelve

Common Core State Standards for Mathematics for California Public Schools Kindergarten Through Grade Twelve Adopted by the California State Board of Education August 2010 Updated January 2013 Prepublication

### Georgia Standards of Excellence Mathematics

Georgia Standards of Excellence Mathematics Standards GSE Algebra I K-12 Mathematics Introduction Georgia Mathematics focuses on actively engaging the student in the development of mathematical understanding

### Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter A. Elementary

Elementary 111.A. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter A. Elementary Statutory Authority: The provisions of this Subchapter A issued under the Texas Education Code,

### Grade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills

Grade 6 Mathematics Assessment Eligible Texas Essential Knowledge and Skills STAAR Grade 6 Mathematics Assessment Mathematical Process Standards These student expectations will not be listed under a separate

### Indiana Academic Standards Mathematics: Algebra I

Indiana Academic Standards Mathematics: Algebra I 1 I. Introduction The college and career ready Indiana Academic Standards for Mathematics: Algebra I are the result of a process designed to identify,

### Performance Assessment Task Bikes and Trikes Grade 4. Common Core State Standards Math - Content Standards

Performance Assessment Task Bikes and Trikes Grade 4 The task challenges a student to demonstrate understanding of concepts involved in multiplication. A student must make sense of equal sized groups of

### Greater Nanticoke Area School District Math Standards: Grade 6

Greater Nanticoke Area School District Math Standards: Grade 6 Standard 2.1 Numbers, Number Systems and Number Relationships CS2.1.8A. Represent and use numbers in equivalent forms 43. Recognize place

### CCGPS Curriculum Map. Mathematics. 7 th Grade

CCGPS Curriculum Map Mathematics 7 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Unit 1 Operations with Rational Numbers a b

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### Common Core State Standards for Mathematics Accelerated 7th Grade

A Correlation of 2013 To the to the Introduction This document demonstrates how Mathematics Accelerated Grade 7, 2013, meets the. Correlation references are to the pages within the Student Edition. Meeting

September: UNIT 1 Place Value Whole Numbers Fluently multiply multi-digit numbers using the standard algorithm Model long division up to 2 digit divisors Solve real world word problems involving measurement

### Overview. Essential Questions. Grade 4 Mathematics, Quarter 4, Unit 4.1 Dividing Whole Numbers With Remainders

Dividing Whole Numbers With Remainders Overview Number of instruction days: 7 9 (1 day = 90 minutes) Content to Be Learned Solve for whole-number quotients with remainders of up to four-digit dividends

### Mathematics. Mathematical Practices

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

### GRADE 7 SKILL VOCABULARY MATHEMATICAL PRACTICES Add linear expressions with rational coefficients. 7.EE.1

Common Core Math Curriculum Grade 7 ESSENTIAL QUESTIONS DOMAINS AND CLUSTERS Expressions & Equations What are the 7.EE properties of Use properties of operations? operations to generate equivalent expressions.

### Grade 6 Mathematics Performance Level Descriptors

Limited Grade 6 Mathematics Performance Level Descriptors A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Grade 6 Mathematics. A student at this

### 1) Make Sense and Persevere in Solving Problems.

Standards for Mathematical Practice in Second Grade The Common Core State Standards for Mathematical Practice are practices expected to be integrated into every mathematics lesson for all students Grades

### History of Development of CCSS

Implications of the Common Core State Standards for Mathematics Jere Confrey Joseph D. Moore University Distinguished Professor of Mathematics Education Friday Institute for Educational Innovation, College

### Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary

Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Planned Course of Study. Algebra I Part 1 NORTHWESTERN LEHIGH SCHOOL DISTRICT 6493 ROUTE 309 NEW TRIPOLI, PA 18066

Planned Course of Study Algebra I Part 1 NORTHWESTERN LEHIGH SCHOOL DISTRICT 6493 ROUTE 309 NEW TRIPOLI, PA 18066 NORTHWESTERN LEHIGH SCHOOL BOARD 2013 Darryl S. Schafer, President LeRoy Sorenson, Vice

### Grade 6 Mathematics Common Core State Standards

Grade 6 Mathematics Common Core State Standards Standards for Mathematical Practice HOW make sense of problems, persevere in solving them, and check the reasonableness of answers. reason with and flexibly

### Utah Core Curriculum for Mathematics

Core Curriculum for Mathematics correlated to correlated to 2005 Chapter 1 (pp. 2 57) Variables, Expressions, and Integers Lesson 1.1 (pp. 5 9) Expressions and Variables 2.2.1 Evaluate algebraic expressions

### Math-U-See Correlation with the Common Core State Standards for Mathematical Content for Fourth Grade

Math-U-See Correlation with the Common Core State Standards for Mathematical Content for Fourth Grade The fourth-grade standards highlight all four operations, explore fractions in greater detail, and

### Indiana Academic Standards Mathematics: Algebra II

Indiana Academic Standards Mathematics: Algebra II 1 I. Introduction The college and career ready Indiana Academic Standards for Mathematics: Algebra II are the result of a process designed to identify,

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### N Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Performance Assessment Task Swimming Pool Grade 9 The task challenges a student to demonstrate understanding of the concept of quantities. A student must understand the attributes of trapezoids, how to

### Performance Assessment Task Which Shape? Grade 3. Common Core State Standards Math - Content Standards

Performance Assessment Task Which Shape? Grade 3 This task challenges a student to use knowledge of geometrical attributes (such as angle size, number of angles, number of sides, and parallel sides) to

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### Overview of Math Standards

Algebra 2 Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse

### Content Emphases for Grade 7 Major Cluster 70% of time

Critical Area: Geometry Content Emphases for Grade 7 Major Cluster 70% of time Supporting Cluster 20% of Time Additional Cluster 10% of Time 7.RP.A.1,2,3 7.SP.A.1,2 7.G.A.1,2,3 7.NS.A.1,2,3 7.SP.C.5,6,7,8

### Mathematics. Curriculum Content for Elementary School Mathematics. Fulton County Schools Curriculum Guide for Elementary Schools

Mathematics Philosophy Mathematics permeates all sectors of life and occupies a well-established position in curriculum and instruction. Schools must assume responsibility for empowering students with

### Investigations. Investigations and the Common Core State Standards GRADE. in Number, Data, and Space. Infinity Prime Donna Casey

Infinity Prime Donna Casey GRADE K This fractal is a classic spiral, which is my favorite, and I m always amazed at the variations and the endlessly repeating patterns that can be created out of such a

### Analysis of California Mathematics standards to Common Core standards- Kindergarten

Analysis of California Mathematics standards to Common Core standards- Kindergarten Strand CA Math Standard Domain Common Core Standard (CCS) Alignment Comments in reference to CCS Strand Number Sense

### COMMON CORE STATE STANDARDS FOR MATHEMATICS 3-5 DOMAIN PROGRESSIONS

COMMON CORE STATE STANDARDS FOR MATHEMATICS 3-5 DOMAIN PROGRESSIONS Compiled by Dewey Gottlieb, Hawaii Department of Education June 2010 Operations and Algebraic Thinking Represent and solve problems involving

Academic Standards for Grades Pre K High School Pennsylvania Department of Education INTRODUCTION The Pennsylvania Core Standards in in grades PreK 5 lay a solid foundation in whole numbers, addition,

### New York State Mathematics Content Strands, Grade 6, Correlated to Glencoe MathScape, Course 1 and Quick Review Math Handbook Book 1

New York State Mathematics Content Strands, Grade 6, Correlated to Glencoe MathScape, Course 1 and The lessons that address each Performance Indicator are listed, and those in which the Performance Indicator

### Mathematical Practices

The New Illinois Learning Standards for Mathematics Incorporating the Common Core Mathematical Practices Grade Strand Standard # Standard K-12 MP 1 CC.K-12.MP.1 Make sense of problems and persevere in

### Georgia Standards of Excellence Curriculum Map. Mathematics. GSE 8 th Grade

Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade

### Assessment Anchors and Eligible Content

M07.A-N The Number System M07.A-N.1 M07.A-N.1.1 DESCRIPTOR Assessment Anchors and Eligible Content Aligned to the Grade 7 Pennsylvania Core Standards Reporting Category Apply and extend previous understandings

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and