6th Grade Math Curriculum Notebook
2017
Curriculum Notebook Table of Contents
Standards
Standards indicate the broad goals for a student to master in a course. Standards are typically set by a state or district school board.
Mathematical Practice Standards ......................................................................................................... Page 4 Ratios and Proportional Relationships .................................................................................................. Page 9 The Number System .............................................................................................................................. Page 9 Expressions and Equations .................................................................................................................... Page 10 Geometry .............................................................................................................................................. Page 10 Statistics and Probability ....................................................................................................................... Page 11
Essential Learning Standards
Particular standards/objectives/indicators that a school/district defines as critical for student learning. In fact, they are so critical that students will receive intervention if they are not learned. Essentials are chosen because they 1. have endurance, 2. have leverage, and 3. are important for future learning.
Math .............................................................................................................................................................. Page 12
Curriculum Resources
The materials teachers use to plan, prepare, and deliver instruction, including materials students use to learn about the subject. Such materials include texts, textbooks, tasks, tools, and media. Sometimes organized into a comprehensive program format, they often provide the standards, units, pacing guides, assessments, supplemental resources, interventions, and student materials for a course.
Developing Mathematical Thinking............................................................................................................... Page 15 5 Strands of Mathematical Proficiency ......................................................................................................... Page 18 USBE Core Content Guides............................................................................................................................ Page 19
Pacing Guide
The order and timeline of the instruction of standards, objectives, indicators, and Essentials over the span of a course (semester or year).
Math Essentials Pacing Guide ....................................................................................................................... Page 108
Units
A plan for several weeks of instruction, usually based on a theme, that includes individual lesson plans. Units often also include: Standards, learning targets/goals, skills, formative and summative assessment, student materials, essential questions, big ideas, vocabulary, questions, and instructional methods.
Understanding By Design .............................................................................................................................. Page 109 Math .............................................................................................................................................................. Page
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Assessment Standards
A set of criteria to guide the assessment of student learning in a course that is based on Standards/Essentials of the course; this might include formative assessment practices, summative assessments/practices, common assessment plans, feedback practices, and a schedule for testing.
SAGE ...................................................................................................................................................... Page 110 Ethics ..................................................................................................................................................... Page 112
Intervention Standards
A set of criteria to guide teachers to provide additional instruction to students who did not master the content in Tier 1 instruction. This might include: commercial intervention programs, teacher-developed intervention materials, diagnostic testing, RTI/MTSS processes, and a list of essential knowledge/skills that will prompt intervention if the student does not demonstrate mastery.
RTI ......................................................................................................................................................... Page 114 MTSS...................................................................................................................................................... Page 116
Supplemental Resources
Instructional materials, beyond the main curricular materials, used to strategically fill gaps/weaknesses of the core program materials.
Provo Way Instructional Model ............................................................................................................ Page 118 Mathematics ......................................................................................................................................... Page 121
Evidence-based Pedagogical Practices
A list of teaching strategies that are supported by adequate, empirical research as being highly effective.
John Hattie ............................................................................................................................................ Page 122
Glossary
Terms and acronyms used in this document ........................................................................................ Page 123
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Mathematics
Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
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 solutions pathway rather than simply jumping into a solution attempt. The 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 the continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
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2 Reason abstractly and quantitatively
Mathematically proficient students make a sense of the quantities and their relationships in problem situations. Students bring two complimentary abilities to bear on problems involving quantitative relationships: the ability to decontextualize–to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents–and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
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 tow plausible arguments, distinguish correct logic or reasoning form 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.
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4 Model with mathematics
Mathematically proficient students can apply mathematics the 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 on 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, possible improving the model if it has not served its purpose.
5 Use appropriate tools strategically
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
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6 Attend to precision
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
7 Look for and make use of structure
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y )2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.
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)(x2 +x+1),and(x–1)(x3 +x2 +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.
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Grade 6 Mathematics
In Grade 6, instructional time should focus on four critical areas:
1. Use reasoning about multiplication and division to solve ratio and rate
problems about quantities.
2. Use the meaning of fractions, the meanings of multiplication and division,
and the relationship between multiplication and division to understand and
explain why the procedures of dividing fractions make sense.
3. Understandtheuseofvariablesinmathematicalexpressions.
4. Building on and reinforcing understanding of numbers, begin to develop
ability to think statistically.
Ratios and Proportional Relationships
• Understand ratio concepts and use ratio reasoning to solve problems
The Number System
• Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
• Compute fluently with multi-digit numbers and find common factors and multiples
• Apply and extend previous understandings of numbers to the system of rational numbers.
Expressions and Equations
• Apply and extend previous understandings of arithmetic to algebraic expressions.
• Reason about and solve one-variable equations and inequalities.
• Represent and analyze quantitative relationships between dependent and independent variables.
Geometry
• Solve real-world and mathematical problems involving area, surface area, and volume.
Statistics and Probability
• Develop understanding of statistical variability
• Summarize and describe distributions
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
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Ratios and Proportional Relationships RP
Understand ratio concepts and use ratio reasoning to solve problems.
1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
2. Understand the concept of a unit rate a/b associated with a ratio a:b with b≠0, and use rate language in the context of a ratio relationship
3. Use ratio and rate reasoning to solve real-world and mathematical problems by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
The Number System NS
Understand ratio concepts and use ratio reasoning to solve problems.
1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions by using visual fraction models and equations to represent the problem.
Compute fluently with multi-digit numbers and find common factors and multiples
2. Fluently divide multi-digit numbers using the standard algorithm.
3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard
algorithm for each operation.
4. Find the greatest common factor of two whole numbers less than or equal to 100 and the
least common multiple of two whole numbers less than or equal to 12.
5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values
6. Understand a rational number as a point on the number line.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on
the number line.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of
the coordinate plane.
c. Find and position integers and other rational numbers on a horizontal or vertical
number line diagram.
7. Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
b. Write, interpret, and explain statements of order for rational numbers in real-world contexts.
c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane.
Apply and extend previous understanding of numbers to the system of rational
numbers.
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Expressions and Equations EE
Apply and extend previous understandings of arithmetic to algebraic expressions.
1. Write and evaluate numerical expressions involving whole-number exponents
2. Write, read, and evaluate expressions in which letters stand for numbers
a. Write expressions that record operations with numbers and with letters standing for numbers.
b. Identify parts of an expression using mathematical terms.
c. Evaluate expressions at specific values of their variables.
3. Apply the properties of operations to generate equivalent expressions.
4. Identify when two expressions are equivalent.
Reason about and solve one-variable equations and inequalities
5. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true.
6. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px=q for cases in which p, q, and x are all nonnegative rational numbers.
8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem.
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, in terms of the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
Geometry G
1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
2. Fine the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, an show that the volume is the same as would be found by multiplying the edge lengths of the prism.
3. Draw polygons in the coordinate plane to give coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate.
4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures.
Represent and analyze quantitative relationships between dependent and
independent variables.
Solve real-world and mathematical problems involving area, surface area, and
volume.
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Statistics and Probability SP
Develop understanding of statistical variability
1. Recognize a statistical question as one that anticipates variability in the data related to the question and occurs for it in the answers.
2. Understand that a set of data collected to answer a statistical question has a distribution which can be describe by its center, spread, and overall shape.
3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.
Summarize and describe distributions
4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
5. Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations
b. Describing the nature of the attribute under investigation, including how it was
measured and its units of measurement.
c. Giving quantitative measures of center and variability as well as describing any overall
pattern and any striking deviations from the overall pattern with reference to the
context in which the data were gathered.
d. Relating the choice of measures of center and variability to the shape of the data
distribution and the context in which the data were gathered.
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Mathematics Essential Learning Standards
Essential Skills List for Mathematics
All 8 Standards for Mathematical Practice are essential.
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers. 4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
The standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout their education.
Essential Skills from Standards for Mathematical Content
Grade 6
In grade 6 instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.
Ratios and Proportional Relationships (6.RP)
A. Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio
relationship between two quantities.
6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b≠0
and use rate language in the context of a ratio relationship.
6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems,
e.g., by reasoning about tables of equivalent ratios, tape diagrams, double
number line diagrams, or equations
The Number System (6.NS)
A. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
6.NS.1 Interpret and compute quotients of fractions, and solve word problems
involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem
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B. Compute fluently with multi-digit numbers and find common factors and multiples.
Fluently divide multi-digit numbers using the standard algorithm. Fluently add, subtract, multiply and divide multi-digit decimals using the standard algorithm for each operation.
6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1- 100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.2 6.NS.3
6.NS.5
6.NS.6
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real- world contexts, explaining the meaning of 0 in each situation.
Understand a rational number as a point on the number line, Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
Expressions and equations (6.EE)
A. Apply and extend previous understandings of arithmetic to algebraic expressions
Write, read, and evaluate expressions in which letters stand for numbers Apply the properties of operations to generate equivalent expressions Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them.)
B. Reason about and solve one-variable equations and inequalities.
6.EE.5 Understand solving an equation or inequality as a process of answering a
question: which values from a specified set, if any, make the equation or
inequality true?
6.EE.7 Solve real-world and mathematical problems by writing and solving equations
of the form x + p = q and px = q for cases in which p, q, and x are all
nonnegative rational numbers.
Geometry (6.G)
A. Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and
polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6.G.2 Find the volume of a right rectangular prism with appropriate unit fraction edge lengths by packing it with cubes of the appropriate unit fraction edge lengths (e.g., 31⁄2 x 2 x 6) and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and
V = Bh to find the volumes of right rectangular prisms with fractional edge lengths in the context of solving real –world and mathematical problems.
6.EE.2 6.EE.3 6.EE.4
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Statistics and Probability (6.SP)
B. Summarize and describe distributions
6.SP.4 Display numerical data in plots on a number line, including dot plots,
historgrams, and box plots.
6.SP.5 Summarize numerical data sets in relation to their context
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Curriculum Resources
Developing mathematical Thinking for Practice Standards
Summary of Practice Standards
Prompts to develop mathematical thinking
1. Make sense of problems and persevere in solving them.
Interpret and make meaning of the problem to find a starting point.
Analyze what is given in order to explain to themselves the meaning of a problem.
Plan a solution pathway instead of jumping to a solution.
Monitor their progress and change the approach if necessary.
See relationships between various representations.
Relate current situations to concepts or skills previously learned and connect mathematical ideas to one another.
Continually ask themselves, “Does this make sense?” Can understand various approaches to solutions
How would you describe the problem in your own words? How would you describe what you are trying to find? What do you notice about . . .?
Describe the relationship between quantities.
Describe what you have already tried. What might you change? Talk me through the steps in the steps you’ve used to this point. What steps in the process are you most confident about?
What are some other strategies you might try?
What are some other problems that are similar to this one?
How might you use one of your previous problems to help you begin?
How else might you organize . . . represent . . . show . . .?
2. Reason abstractly and quantitatively.
Make sense of quantities and their relationships.
Decontextualize (represent a situation symbolically and manipulate the symbols) and contextualize (make meaning of the symbols in a problem) quantitative relationships.
Understand the meaning of quantities and are flexible in the use of operations and their properties
Create a logical representation of the problem.
Attend to the meaning of quantities, not just how to compute them.
What do the numbers used in the problem represent?
What is the relationship of the quantities?
How is __________ related to ___________?
What is the relationship between ____________ and ____________?
What does ___________ mean to you? (e.g., symbol, quantity, diagram)
What properties might we use to find a solution?
How did you decide in this task that you needed to use . . .?
Could we have used another operation or property to solve this task? Why or why not?
3. Construct viable arguments and critique the reasoning of others.
Analyze problems and use stated mathematical assumptions, definitions, and established results in constructing arguments.
Justify conclusions with mathematical ideas.
Listen to the arguments of others and ask useful questions to determine if an argument makes sense.
Ask clarifying questions or suggest ideas to improve/revise the argument.
Compare two arguments and determine correct or flawed logic.
What mathematical evidence would support your solution? Howcanwebesurethat...? Howcouldyouprovethat...? Will it still work if . . .?
What were you considering when . . .?
How did you decide to try that strategy?
How did you test whether your approach worked?
How did you decide what the problem was asking you to find? (What was unknown?)
Did you try a method that did not work? Why didn’t it work? Would it ever work? Why or why not?
What is the same and what is different about . . .? How could you demonstrate a counter-example?
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4. Model with mathematics.
Understand this is a way to reason quantitatively and abstractly (able to decontextualize and contextualize, see standard 2 above).
Apply the mathematics they know to solve everyday problems.
Are able to simplify a complex problem and identify important quantities to look at relationships.
Represent mathematics to describe a situation either with an equation or a diagram and interpret the results of a mathematical situation.
Reflect on whether the results make sense, possibly improving/ revising the model
What number model could you construct to represent the problem?
What are some ways to represent the quantities?
What is an equation or expression that matches the diagram, number line, chart, table ?
Where did you see one of the quantities in the task in your equation or expression?
How would it help to create a diagram, graph, table? What are some ways to visually represent . . .? What formula might apply in this situation?
How can I represent this mathematically?
Summary of Practice Standards
Prompts to develop mathematical thinking
5. Use appropriate tools for mathematical practice.
Use available tools recognizing the strengths and limitations of each.
Use estimation and other mathematical knowledge to detect possible errors.
Identify relevant external mathematical resources to pose and solve problems.
Use technological tools to deepen their understanding of mathematics
What mathematical tools could we use to visualize and represent the situation?
What information do you have?
What do you know that is not stated in the problem?
What approach are you considering trying first?
What estimate did you make for the solution?
In this situation would it be helpful to use a graph, number line, ruler, diagram, calculator, manipulative?
Why was it helpful to use ______?
What can using a _______ show us that _______ may not?
In what situations might it be more informative or helpful to use ________?
6. Attend to precision.
Communicate precisely with others and try to use clear mathematical language when discussing their reasoning.
Understand the meanings of symbols used in mathematics and can label quantities appropriately.
Express numerical answers with a degree of precision appropriate for the problem context.
Calculate efficiently and accurately.
What mathematical terms apply to this situation? How did you know your solution was reasonable?
Explain how you might show that your solution answers the problem?
What would be a more efficient strategy?
How are you showing the meaning of the quantities?
What symbols or mathematical notations are important in this problem?
What mathematical language, definitions, properties can you use to explain ______?
How can you test your solution to see if it answers the problem?
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7. Look for and make use of structure.
Apply general mathematical rules to specific situations. Look for the overall structure and pattern in mathematics.
See complicated things as single objects or as being composed of several objects.
What observations do you make about _____ ?
What do you notice when ______?
What parts of the problem might you eliminate or simplify?
What patterns do you find in _______ ?
How do you know if something is a pattern?
What ideas that we have learned before were useful in solving this problem?
What are some other problems that are similar to this one?
How does this problem connect to other mathematical concepts?
In what ways does this problem connect to other mathematical concepts?
8. Look for and express regularity in repeated reasoning?
See repeated calculations and look for generalizations and shortcuts.
See the overall process of the problem and still attend to the details.
Understand the broader application of patterns and see the structure in similar situations.
Continually evaluate the reasonableness of immediate results.
Explain how this strategy will work in other situations. Is this always true, sometimes true, or never true? How would you prove that _______?
What do you notice about ________?
What is happening in this situation?
What would happen if ________?
Is there a mathematical rule for _________?
What predictions or generalizations can this pattern support? What mathematical consistencies do you notice?
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5 Strands of Mathematical Proficiency from NRC’s Adding It Up
Conceptual understanding: Comprehension of mathematical concepts, operations, and relations
Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently and appropriately
Strategic competence: ability to formulate, represent, and solve mathematical problems
Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification
Productive disposition: habitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one's own efficacy
Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.
The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.
In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
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USBE Core Content Guides
Click on a standard to go to the USBE Core Content Guide
6RP1 Understand concept of a ratio and use ratio language
6RP2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0
6RP3 Use ratio and rate reasoning to solve real-world and mathematical problems
6RP3b Solve unit rate problems including those involving unit pricing and constant speed
6RP3c Find a percent of a quantity as a rate per 100
6RP3d Use ratio reasoning to convert measurement units; manipulate and transform units
6NS1 Compute quotients of fractions, solve problems of division of fractions by fractions
6NS2 Fluently divide multi-digit numbers using the standard algorithm
6NS3 Fluently add, subtract, multiply, divide multi-digit decimals using the standard algorithm
6NS4 Find GCF of two whole numbers ≤ 100 & LCM of two whole numbers ≤ 12
6NS5 Understand positives & negatives are used to describe quantities of opposite directions
6NS6a Recognize opposite signs of numbers as indicating locations on opposite sides of 0
6NS6b Understand signed numbers in ordered pairs indicate locations in quadrants
6NS6c Find & position rational numbers on a horizontal or vertical number line diagram
6NS7a Interpret inequalities as statements of relative position of 2 numbers on a number line
6NS7b Write, interpret, explain statements of order for rational numbers in real-world contexts
6NS7c Understand absolute value of a rational number as its distance from 0
6NS7d Distinguish comparisons of absolute value from statements about order
6NS8 Solve real-world and mathematical problems by graphing points in all four quadrants
6EE1 Write and evaluate numerical expressions involving whole-number exponents
6EE2a Writeexpressionstorecordoperationswithnumbersandwithletters
6EE2b Identify parts of an expression using mathematical terms
6EE2c Evaluate expressions at specific values of their variables
6EE3 Apply the properties of operations to generate equivalent expressions
6EE4 Identify when two expressions are equivalent
6EE5 Understand solutions of equations or inequalities are values that make them true
6EE6 Use variables to represent numbers & write expressions for real-world problems
6EE7 Solve problems by writing and solving equations of the form x + p = q and px = q
6EE8 Write an inequality of the form x > c or x < c to represent a real-world problem
6EE9 Use variables to represent 2 quantities that change in relationship to one another
6G1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons
6G2 Find the volume of a right rectangular prism with appropriate unit fraction edge lengths
6G3 Draw polygons in the coordinate plane given coordinates for the vertices
6G4 Represent three-dimensional figures using nets made up of rectangles and triangles
6SP1 Recognize statistical questions anticipate and account for variability in the data
6SP2 Understand that a collected data set has a distribution which can be described
6SP3 Recognize measure of center of a data set summarizes all values as a single number
19
6SP4 6SP5a 6SP5b 6SP5c 6SP5d
Display numerical data in number lines, dot plots, histograms, and box plots Summarizenumericaldatasetsby:Reportingthenumberofobservations
Summarize numerical data sets by: Describing nature of attribute under investigation Summarize numerical data sets by: Giving quantitative measures of center & variability Relate measures of center & variability to the shape of the distribution and the context
20
Ratios and Proportional Reasoning 6RP1
Core Content
Cluster Title: Understand ratio concepts and use ratio reasoning to solve problems.
Standard:1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
MASTERY Patterns of Reasoning:
Conceptual:
• Understand the concept of a ratio as a way of expressing relationships between quantities.
Distinguish when a ratio is describing part to part or part to whole comparison
Procedural:
• Describe ratio relationships between two quantities.
Translate relationships between two quantities using the notation of ratio language
(1:3, 1 to 3, 1/3).
Representational:
• Communicate relationships between two quantities using ratio notation and language.
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the relationship between parts and wholes. Procedural:
• Translate “for every” and other meanings of multiplication into terms. Representational:
• Experience working with set and measurement models. Academic Vocabulary
:, /, ratio, terms of ratio (i.e., the numbers used in a ratio are called its terms)
21
Instructional Strategies Used
Using a variety of situations, describe relationships using ratio, for example:
1. Part to Part: Compare the number of girls to boys in the classroom using the different symbols for ratio (girls: boys, girls to boys,girls/boys , girls out of boys)
Resources Used
Illuminations Ratio Applet:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=178
Ratio coloring activity:
http://www.softschools.com/math/ratios/ratio_coloring_game/
express a ratio in the simplest form
http://learnzillion.com/lessons/304-express-a-ratio-in-the- simplest-form
interpret ratios as if then statements
http://learnzillion.com/lessons/305-interpret-ratios-as-ifthen- statements
http://learnzillion.com/lessons/310-translate-ratios-into-
2. Part to Whole:
Compare the number
of girls to the whole
class. Do the same
thing for the boys in thefractions class
translate ratios into fractions
translate ratios into words
http://learnzillion.com/lessons/306-translate-ratios-into-words
translate words into ratios
http://learnzillion.com/lessons/307-translate-words-into-ratios
translate word problems into ratios
http://learnzillion.com/lessons/309-translate-word-problems- into-ratios
use ratio notation to express relationships
http://learnzillion.com/lessons/308-use-ratio-notation-to- express-relationships
Assessment Tasks Used
Skill-based Task Problem Task
There are four dogs and three The newspaper reported, “For every vote candidate A received, cats. What is the ratio of dogs candidate B received three votes”. Describe possible election to cats and cats to dogs? results using at least three different ratios. Explain your
answer.
22
Ratios and Proportional Reasoning 6RP2
Core Content
Cluster Title: Understand ratio concepts and use ratio reasoning to solve problems.
Standard:2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that a rate is a special ratio that compares two quantities with different units of measure.
• Understand that unit rates are the ratio of two measurements in which the second term is one (e.g. x miles per one hour)
• Understand rate language (per, each, or the @ symbol )
• When using rates , “b” cannot be 0 Procedural:
Representational:
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Ratio knowledge from Standard 1(6.RP.1)
• Equivalent ratio
• Simplifying fractions
Procedural:
Representational:
Academic Vocabulary
Equivalent ratio, unit rate, rate, ratio
23
Instructional Strategies Used
1. Show examples of rates: 300 miles on 10 gallons of gas, $15 for 5 ounces, $30 for 6 hours
2. Connect rates from number 1 with their unit rates: 30 miles per gallons, $3 per 1 ounce, $5 per 1 hour
3. Convert rates from fraction form to written form using per, each, or @. Example 300 miles/10 gallons of gas = 30 miles per gallon of gas
4. Quick write: Students brainstorm examples of unit rates in the real world, e.g. 4 candy bars per $1, 55 miles per hour, 6 points per touchdown)
Assessment Tasks Used
Skill-based Task
Identify (given examples) the difference between a ratio and a rate.
Resources Used
UEN- Lesson “Ratio, Rate, and Proportion”
determine whether two ratios are equivalent http://learnzillion.com/lessons/316- determine-whether-two-ratios-are- equivalent
interpret rates as special types of ratios
http://learnzillion.com/lessons/311- interpret-rates-as-special-types-of-ratios
translate rates into ratios
http://learnzillion.com/lessons/313- translate-rates-into-ratios
translate ratios into rates
http://learnzillion.com/lessons/312- translate-ratios-into-rates
Is the following example a ratio or rate?
[60 heartbeats per minute] Explain your answer.
Problem Task
24
Ratios and Proportional Reasoning 6RP3a
Core Content
Cluster Title: Understand ratio concepts and use ratio reasoning to solve problems.
Standard: 3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations
a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios
MASTERY Patterns of Reasoning:
Conceptual:
• Understand how to make, complete, and read a table of equivalent ratios.
• Understand that tools such as tables of equivalent ratios support the development of
ratio and rate reasoning.
Understand that pairs of values from a table can be plotted on the coordinate plane. Understand that establishing connections between tables and plotted points on the
coordinate plane allow for
• extended reasoning and synthesis of the concept of ratios and rates
Procedural:
• Use a table to compare ratios.
Determine missing values using ratio reasoning. Identify relationships in ratio tables
Representational:
• Plot pairs of values from a table to a coordinate plane Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand that coordinate graphs are two-dimensional and rely on two coordinate points to identify a specific
• location on a plane.
Understand equivalent fractions.
Understand equivalent ratio (from 6.RP.2).
Procedural:
• Experience with coordinate plane graphing in quadrant 1. Read equations
Representational:
• Plotting a point on a coordinate plane when given the coordinates Academic Vocabulary
coordinate plane, tables of equivalent ratios (value table)
25
Instructional Strategies Used Resources Used
1.Have students make a table given a
ratio situation. They should plot thosehttp://amstat.org/education/gaise/index.cfm
points on a coordinate plane and draw conclusions about what’s happening in the ratio situation. 2.Give students a table with missing values and have them identify the missing values.
3. Have students study ratio relationships in a table.
Refer to website under “resource used” level A.
http://www.youtube.com/watch?v=d625kdtsUIw
UEN: Price-Earnings ratio
http://www.uen.org/Lessonplan/preview.cgi?LPid=25290
compare ratios using a ratio table
http://learnzillion.com/lessons/321-compare-ratios- using-a-ratio-table
find equivalent ratios using ratio tables
http://learnzillion.com/lessons/317-find-equivalent- ratios-using-ratio-tables
find ratio values using ratio tables
http://learnzillion.com/lessons/318-find-ratio-values- using-ratio-tables
find the missing value of a proportion
http://learnzillion.com/lessons/319-find-the-missing- value-of-a-proportion
plot points on a coordinate plane
http://learnzillion.com/lessons/320-plot-points-on-a- coordinate-plane
Problem Task
Graph the information from the table on the coordinate plane and explain the relationship of swimmers to life guards.
Assessment Tasks Used
Skill-based Task
Analyze the table below to determine the missing values.
Fill in the missing values on the table below
swimmers 20 30 40 60 90 100 life guards 2 3 4 6
26
Ratios and Proportional Reasoning 6RP3b
Core Content
Cluster Title: Understand ratio concepts and use ratio reasoning to solve problems.
Standard:3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that tools such as tables of equivalent ratios, tape diagrams, double number line diagrams, and equations
• support the development of ratio and rate reasoning.
Understand that rate problems compare two different units, such as miles to hours. Recognize that a unit occurs when at least one of the units is one.
Understand that establishing connections between tools allow for extended reasoning
and synthesis of the concept of
• ratios and rates (e.g., How do tape diagrams and double number lines show rate
reasoning given the same context?).
Procedural:
• Solve real world problems using ratio reasoning. Representational:
• Set up the unit rate correctly Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand ratio reasoning and relationships. Understand equivalent fractions.
Procedural:
• Use the four basic operations (+, -, x, /). Representational:
• Represent equivalent ratios with ratio notation.
Academic Vocabulary
equivalent ratios notation (a/b=c/d) or a is to b as c is to d; ratio; unit rate
27
Instructional Resources Used Strategies Used
1. Identify the question being asked based on the context, and determine a method for finding the unit rate (table of equivalent ratios, tape diagrams, double number line diagrams, and equations).
2. Complete the determined tool to find the unit rate (e.g., use tool to find the ratio in which one of the units is one).
Illuminations measuring up activity:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L511
Math Playground tape diagrams:
http://www.mathplayground.com/ThinkingBlocks/thinking_blocks_start.html
NZ Maths double number lines:
http://nzmaths.co.nz/sites/default/files/Animations/double_numberlines.swf
solve rate and ratio word problems
http://learnzillion.com/lessons/314-solve-rate-and-ratio-word-problems
Assessment Tasks Used
Skill-based Task If5CDscost $60,whatis the price of each CD?
Problem Task
Joe’s Gas and Go has drinks for the following prices: 12flozfor$.89
16flozfor$.99
20flozfor$1.09
32flozfor$1.19
Which drink costs the least per ounce. You may round to the nearest cent and use a calculator if you desire.
28
Ratios and Proportional Reasoning 6RP3c
Core Content
Cluster Title: Understand ratio concepts and use ratio reasoning to solve problems.
Standard:3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems
involving finding the whole, given a part and the percent.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that a percent is rate per 100 and can be represented using tools such as tables of
• equivalent ratios, tape diagrams, double number line diagrams, and equations. Understand that percentage-based rate problems compare two different units
where one of the
• units is 100.
Understand that establishing connections between tools allow for extended reasoning and
• synthesis of the concept of ratios and rates (e.g., How do tape diagrams and double number
• lines show rate reasoning given the same context?). Procedural:
• Writing a percent as a rate over 100.
Finding the percent of a number using rate methods developed in 6.RP.3b. Given the parts and a percent, determine the whole using tools identified above
Representational:
• Represent the relationship of part to whole to describe percents using model. Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the concept of rate as detailed in 6.RP.3b. Understand whole and parts in the context of a ratio.
Procedural:
• Use unit pricing and constant speed to solve problems. Use unit rates to solve problems
Representational:
• Represent unit rates with models. Academic Vocabulary
%, percent
29
Instructional Strategies Used Resources Used
1. Model using a hundreds grid. Color Coloring percent activity: http://www.softschools.com/math/percent/games/ in 30 units and have students write it
as a fraction and percent.
2 Use double number lines and tape diagrams in which the whole is 100 to find the rate per hundred.
Assessment Tasks Used
Skill-based Task
What is 25% of 60?
72% of what number is 300?
NLVMpercent virtual manipulative: http://nlvm.usu.edu/en/nav/frames_asid_160_g_2_t_1.html Tape Diagrams: http://mathgpselaboration.blogspot.com/2010/04/mp5-tape-diagrams.html
Double Number Line for Percents:
http://nzmaths.co.nz/sites/default/files/Animations/double_numberlines2.swf
Problem Task
Stop and Shop has pants for $30 with a 10% discount, while
Stay and Shop has pants for $45 with a 20% discount. Which store has the pants for a better price? Use a table of equivalent values, double number line, or tape diagram to solve and explain your reasoning
30
Ratios and Proportional Reasoning 6RP3d
Core Content
Cluster Title: Understand ratio concepts and use ratio reasoning to solve problems.
Standard:3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that measurement units employ ratio reasoning (e.g., If 3 feet is equal to yard, then 6 feet is equal to 2
• yards).
Understand that tools such as tables of equivalent ratios, tape diagrams, double
number line diagrams, and
• equations can help scaffold understanding for converting measurement units.
Understand that establishing connections between tools allow for extended reasoning and synthesis of the concept
• of ratios and rates (e.g., How do tape diagrams and double number lines show rate reasoning given the same context?).
Procedural:
• Convert customary units using ratio tools and methods.
Convert metric units by multiplying or dividing by powers of ten.
Representational:
• Represent relationships between measurement units using tables of equivalent ratios, tape diagrams, double
• number line diagrams, and equations Supports for Teachers
Critical Background Knowledge
Conceptual:
Understand customary and metric units of measurement.
Understand ratios and unit rates.
Procedural:
Use customary and metric units of measurement.
Be able to multiply and divide by powers of 10.
Representational:
Represent multiplication and division with powers of 10 with charts, tables and manipulatives.
Academic Vocabulary
convert, 10n (power of 10 notation)
31
Instructional Strategies Used Resources Used
1.Usedoublenumberline,tapediagrams,NLVM conversion manipulative: tables of equivalent values, or equations tohttp://nlvm.usu.edu/en/na/frames_asid_272_g_2_t_4.html?open=instructions&from=category_g_2_t_4.html convert measurements in customary and
metric units.
2. If 4 cups equals one quart, how many cups in 12 quarts?
4 /1 = /12
Assessment Tasks Used
Skill-based Task
How many inches are in three feet?
How many inches in two miles?
http://learnzillion.com/lessons/322-convert-between-centimeters-and-inches-using-ratios
convert between gallons and liters using ratios
http://learnzillion.com/lessons/326-convert-between-gallons-and-liters-using-ratios
convert between kilometers and miles using ratios
http://learnzillion.com/lessons/324-convert-between-kilometers-and-miles-using-ratios
convert between meters and feet using ratios
http://learnzillion.com/lessons/323-convert-between-meters-and-feet-using-ratios
convert between pounds and kilograms using ratios
http://learnzillion.com/lessons/325-convert-between-pounds-and-kilograms-using-ratios
Problem Task
In the store a package of candy that weighs 150 grams costs $1.00. A package of 200 candies that each weigh 200 milligrams also costs $1.00. Which package is the better deal?
convert between centimeters and inches using ratios
32
The Number System 6NS1
Core Content
Standard 1: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lbs. of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi. and area 1/2 square mi.?
MASTERY Patterns of Reasoning:
Conceptual:
• Understand how to set up a problem based on the context of the problem.
• Be able to interpret what the quotient represents.
• Recognize that what is known or not known is based on the type of division needed
(partitive—Total / # of groups = size of
• groups—or quotative or measurement—Total / size of group = # of groups) model
• Create a story context using division of fractions.
• Understand that multiplication and division are inverse operations regardless of the
class of numbers.
Procedural:
• Compute the division of fractions.
• Solve a story context using division of fractions. Representational:
• Model division of fractions with manipulatives, diagrams (e.g., bar model, number line) and story contexts.
• Write equations representing authentic problems involving fractions. Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know that multiplication and division are inverse operations.
• Know that division is either fair sharing (partitive) or repeated subtraction (quotative). Procedural:
• Convert between improper fractions and mixed numbers.
• Division by whole numbers.
• Division of a whole number by a fraction.
Representational:
• Model division with manipulatives, diagrams and story contexts. Academic Vocabulary and Notation
quotient, reciprocal, inverse operation
Cluster Title: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
33
Instructional Strategies Used
Use this problem: How many servings of popcorn are in 4••• cups if each person receives 3/4 cup of popcorn
The teacher provides 4••• cups of popcorn. Students use a 3/4 cup measuring cup to solve the problem. Record solutions as a group.
1. Think-Pair-Draw-Share: Put students in pairs. Have one solve the problem using a picture/diagram and the other solve using the algorithm. Then they get together and compare.
2. Think-Pair-Share: Students solve the problem on their own, then get together and discuss how their solutions are the same and how they are different.
3. Four Corners: Give students a problem and the quotient. Give each corner in your room a label and have students go to the corner they think would be the correct label for the quotient.
Resources Used
Fraction Bars from NLVM:
http://www.nlvm.usu.edu/en/nav/frames_asid_265 _g_2_t_1.html?open=activities&from=category_g_2 _t_1.html
Divide fractions by fractions, dividing across
http://www.learnzillion.com/lessons/205-divide- fractions-by-fractions-dividing-across
Divide fractions by fractions using models
http://www.learnzillion.com/lessons/204-divide- fractions-by-fractions-using-models
Divide fractions by fractions using the common denominator http://www.learnzillion.com/lessons/206-divide- fractions-by-fractions-using-the-common- denominator
Divide fractions by using fractions, using the reciprocal http://www.learnzillion.com/lessons/356-divide- fractions-by-fractions-using-the-reciprocal
Divide fractions by whole numbers, using models
http://www.learnzillion.com/lessons/203-divide- fractions-by-whole-numbers-using-models
Divide mixed numbers by fractions, using models
http://www.learnzillion.com/lessons/207-divide- mixed-numbers-by-fractions-using-models
Divide mixed numbers multiplying by the reciprocal
http://www.learnzillion.com/lessons/208-divide- mixed-numbers-multiplying-by-the-reciprocal
Divide whole numbers by fractions using the reciprocal http://www.learnzillion.com/lessons/202-divide- whole-numbers-by-fractions-using-the-reciprocal
Divide whole numbers by non-unit fractions
http://www.learnzillion.com/lessons/201-divide- whole-numbers-by-nonunit-fractions
34
Divide whole numbers by unit fractions using bar models http://www.learnzillion.com/lessons/200-divide- whole-numbers-by-unit-fractions-using-bar-models
Divide whole numbers by unit fractions using visual models http://www.learnzillion.com/lessons/199-divide- whole-numbers-by-unit-fractions-using-visual- models
Interpret remainders when dividing using models
http://www.learnzillion.com/lessons/209-interpret- remainders-when-div-using-models
http://www.learnzillion.com/lessons/214-multiply- fractions-by-fractions-by-mult-across
Multiply fractions by fractions by multiplying across
Multiply fractions by fractions using area models
http://www.learnzillion.com/lessons/213-multiply- fractions-by-fractions-using-area-models
Multiply fractions by whole numbers using bar models http://www.learnzillion.com/lessons/212-multiply- fractions-by-whole-numbers-using-bar-models
Multiply mixed numbers by renaming factors
http://www.learnzillion.com/lessons/215-multiply- mixed-numbers-by-renaming-factors
Multiply whole by mixed numbers, break apart
http://www.learnzillion.com/lessons/211-multiply- whole-by-mixed-numbers-breakitapart
Multiply whole numbers by fractions using repeated addition http://www.learnzillion.com/lessons/210-multiply- whole-numbers-by-fractions-using-repeated- addition
35
Assessment Tasks Used
Skill-based Task:
Use representations to show that 1/4 divided by 1/2 is 1/2, that 2/3 divided by 2/5 is 5/3, that 2/3 divided by 3/4 is 8/9, and that 1••• divided by 6/4 is 1.
Multiply whole numbers by mixed numbers using area models http://www.learnzillion.com/lessons/216-multiply- whole-numbers-by-mixed-numbers-using-area- models
Simplify fraction multiplication problems cancelling common factors http://www.learnzillion.com/lessons/373-simplify- fraction-multiplication-problems-cancelling- common-factors
Problem Task:
You have 5/8 pound of Skittles. You want to give your friends 1/4 lb. each. How many friends can you give Skittles to? Explain your answer.
You have a 3/4-acre lot. You want to divide it into 3/8-acre lots. How many lots will you have? Draw a diagram to justify your solution.
You have a 3/4-acre lot. You want to divide it into 2 sections. How many acres in each section will you have? Draw a diagram to justify your solution.
How wide is a rectangular strip of land with length 3/4 mi. and area 1/2 square mi.?
36
The Number System 6NS2
Core Content
Cluster Title: Compute fluently with multi-digit numbers and find common factors and multiples.
Standard:2. Fluently divide multi-digit numbers using the standard algorithm. MASTERY Patterns of Reasoning:
• Conceptual:
Identify when it is appropriate to use the standard algorithm.
Procedural:
Use the standard algorithm to compute multi-digit division problems with procedural fluency.
Note: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficien • National Research Council).
Representational:
Divide multi-digit numbers using the standard algorithm. Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand the meaning of division.
• Understand place value of multi-digit numbers.
• Know that division is the inverse of multiplication.
• Illustrate and explain the relationship between calculations and models for multiplying
and dividing multi-digit numbers.
Procedural:
• Divide with single-digit numbers.
• Use compatible numbers to make an estimation to determine reasonableness of
answers.
• Use the standard algorithm for division.
• Read division notation.
Representational:
• Model division with manipulatives, diagrams and story contexts. Academic Vocabulary
dividend, division notation ÷, /, divisor, quotient, remainder
37
Instructional Strategies Used Resources Used
1.ThinkAloud-Dotheproblemwithapartnerwhile NationalLibraryofVirtualManipulatives
explaining and telling what you are thinking and doing.
Use a mnemonic with understanding to remember the steps (e.g. Does McDonalds Serve Cheese Burgers – divide, multiply, subtract, compare, bring down).
2. Draw a picture to show 144 divided by 12 makes 12 equal groups
3. Connect students’ existing strategies for division with the standard algorithm.
http://nlvm.usu.edu/en/nav
Assessment Tasks Used
Skill-based Task
248 divided by 18. Explain how you know that your answer is correct.
Problem Task
I spent $504 on 28 tickets for a rock concert. How much did I spend on each ticket? Write an explanation of each step of your solution.
38
The Number System 6NS3
Core Content
Standard:3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand role of place value in the operations of addition, subtraction, multiplication, and division
• Identify when it is appropriate to use the standard algorithm. Procedural:
• Add multi-digit decimals.
• Subtract multi-digit decimals.
• Multiply multi-digit decimals.
• Divide multi-digit decimals.
Representational:
• Model the operations of addition, subtraction, multiplication, and division with manipulatives, diagrams and story contexts for multi-digit decimals.
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand decimal place values.
• Know basic facts for addition, subtraction, multiplication and division. Procedural:
• Add single-digit decimals.
• Subtract single-digit decimals.
• Multiply single-digit decimals.
• Divide single-digit decimals.
Representational:
• Model the operations of addition, subtraction, multiplication, and division with manipulatives, diagrams and story contexts for single digit decimals.
Academic Vocabulary
Addend, sum, difference, factor, product, divisor, dividend, quotient, remainder
Cluster Title: Compute fluently with multi-digit numbers and find common factors and multiples.
Instructional Strategies Used
1-4. Connect students’ knowledge of various strategies to the standard algorithm
1-4. Have students look at student work that contains a common misconception and look at errors and discuss how to correct the error.
Resources Used
39
Assessment Tasks Used
Skill-based Task
1. 242.134 + 308.02 2. 38.9 – 14.334
3. 11.82 X 2.81
4. 341.8 ÷ 1.2
Problem Task
The school had a bake sale and raised $75.55. If each cookie cost $0.05, how many cookies were sold? Explain how you got your answer.
40
The Number System 6NS4
Core Content
Standard:4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distribute property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9+2).
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that greatest common factor and least common multiple are ways to discuss number relationships in multiplication and division.
• Understand the distributive property using sums and its use in adding numbers 1-100 with a common factor (e.g., 20 + 24 = 4(5 + 6)).
Procedural:
• Compute fluently using the distributive property of multiplication over addition.
• Find greatest common factor of two whole numbers less than or equal to 100.Find the
least common multiple of two whole numbers less than or equal to 12.
Representational:
• Model factorization of whole numbers 1-100 using a number line and manipulatives. Supports for Teachers
Critical Background Knowledge
Conceptual:
• Understand that a factor is a whole number that divides without a remainder into another number.
• Understand that a multiple is a whole number that is a product of the number and any other factor.
• Know the distributive property. Procedural:
• Compute using the distributive property
• Find factors and multiples of a given number. Representational:
• Model the distributive property of multiplication over addition using manipulatives, diagrams, and story contexts.
Academic Vocabulary
Distributive property, factor, multiple, least common multiple (LCM), greatest common factor (GCF)
Cluster Title: Compute fluently with multi-digit numbers and find common factors and multiples.
41
Instructional Strategies Used
1-2. Solve for LCM and/or GCF using a variety of strategies (e.g. factor towers, Venn diagrams, factor trees) insert samples of each
3 Use a model to show that 4(9 + 2) is four groups of 9 and four groups of 2
Resources Used
Assessment Tasks Used
Skill-based Task
Find GCF of 24 and 60
Find the least common multiple of 6 and 10
Use the distributive property to show 15 + 75
Problem Task
Hot dogs come in packs of 8. Buns come in packs of 12. How many packs of hot dogs and bags of buns would you have to buy to have an equal number of hot dogs and buns?
You need to make gift bags for a party with the same number of balloons and candy in each bag. One package of candy has 24 pieces. One package of balloons has 20 balloons. How many gift bags can be made containing equal items?
42
The Number System 6NS5
Core Content
Standard:5. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that positive and negative numbers (integers) allow us to talk about quantities that have opposite directions or values.
• Understand that a negative integer is less than zero.
• Understand that the meaning of zero is determined by the real world context (e.g.,
freezing point in the Celsius system—anything below freezing is negative, anything
above freezing is positive).
Procedural:
• Use integers to represent situations in real-world contexts. Representational:
• Represent integers using real-world tools such as a thermometer, balance sheet (money), etc.
Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know where positive integers are on a number line.
• Know the set of positive integers.
• Understand that zero represents a position.
• Know that number lines extend to show positive integers right and up and negative
integers left and down (vertical and horizontal number lines)
Procedural:
• Describe quantities having opposite values Representational:
• Plot integer points on a number line. Academic Vocabulary
→,←, ↑,↓,+, –, integer, negative, positive, rational, zero
Cluster Title: Apply and extend previous understandings of numbers to the system of rational numbers.
43
Instructional Strategies Used Resources Used
Give multiple examples of types of contexts using positive and National Library of Virtual negative integers such as: a bank account, hot air balloons, discs to Manipulatives
show positive and negative charges, thermometer, number line
Give students a number and have them write a real life situation for that number and its opposite that would result in an answer of zero. Explain the meaning of zero in that situation and represent it on the number line
http://nlvm.usu.edu/en/nav
Use a comparison matrix
Definition Example Picture
Assessment Tasks Used
Skill-based Task
Fill in missing numbers on a number line
Problem Task
Negative
Zero
Positive
Joe’s football team had a loss of 5 yards on first down. Write an integer to represent the situation.
44
The Number System 6NS6a
Core Content
Standard:6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the
opposite of the opposite of a number is the number itself, e.g.,–(–3) = 3, and that 0 is its own opposite.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand the meaning of the term opposite.
• Recognize that the opposite of the opposite of the number is the number itself (e.g., -
(-3)).
• Recognize that zero is its own opposite.
Procedural:
• Find the opposite of a number. Representational:
• Extend number line diagrams to include negative numbers.
• Plot opposites on a number line. Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know the characteristics of a number line (extends in both directions, origin, importance of unit).
• Know a number line can be used to represent real life situations. Procedural:
• Create a number line with equidistant tick marks determined by the identified unit. Representational:
• Draw a number line.
• Represent real-life contexts on a number line. Academic Vocabulary
+, –, integer, opposite, rational number, ( ), point
Cluster Title: Apply and extend previous understandings of numbers to the system of rational numbers.
45
Instructional Strategies Resources Used Used
Make a number line on the http://mathstar.lacoe.edu/lessonlinks/integers/integers_main.html floor. Have a student
choose an integer to stand
on then call on a student to
come stand on the opposite integer.
Repeat process asking students to stand on the opposite of an opposite. Have students use appropriate notation to record these integers.
Assessment Tasks Used
Skill-based Task
Locate 4 and its opposite on a number line
Problem Task
This is a skill-based standard. Therefore, no problem task is offered.
46
The Number System 6NS6b
Core Content
Standard 6: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
MASTERY Patterns of Reasoning:
Conceptual:
• Understand that the signs of numbers in ordered pairs represent a singular location on the coordinate plane.
• Understand that changing the sign of one or both numbers in the ordered pair will create a reflection of the point.
• Understand that a reflection on the coordinate plane is defined as a transformation of a point or shape across one or both of the axes.
Procedural:
• Find reflection points across axes.
• Recognize the components of the coordinate plane (Quadrant I (+,+), Quadrant II (-
,+) , Quadrant III (-,-) Quadrant IV (+, -),
• x and y axes, origin)
Representational:
• Plot points in all four quadrants for any given ordered pair. Supports for Teachers
Critical Background Knowledge
Conceptual:
• Know locations of points in the first quadrant. Procedural:
• Identify coordinates of given points in the first quadrant Representational:
• Plotting points in Quadrant I.
Academic Vocabulary and Notation
(x, y), coordinate plane, ordered pair, point, quadrant, reflection, x-axis, y-axis
Cluster Title: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
47
Instructional Strategies Used
• Have students draw and label a coordinate plane, including quadrants and axes.
• Make a set of cards with ordered pairs that have a matching card that is a reflection of the point.
• Have students get in groups and pair the cards that are reflections.
• Given an ordered pair, have students identify in which quadrant the ordered pair is located.
Resources Used
Assessment Tasks Used
Skill-based Task:
If you had a point graphed at (5, -3), what would be one ordered pair that is a reflection of the point? Students may use a coordinate plane to find a solution
Problem Task:
A town was laid out using a coordinate plane. On the city plans, the library is at (3, 2). Which of the following locations is a reflection across the x-axis of where the library is located? Prove your answer is correct using two different methods.
School (-3, -2) Gas Station (-3, 2) Post Office (3, -2)
48
The Number System 6NS6c
Core Content
Standard 6: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. MASTERY Patterns of Reasoning:
Conceptual:
• Understand a rational number as a point on a number line. Procedural:
• Place rational numbers on horizontal and vertical number lines.
• Write an ordered pair using rational numbers to represent a point on the coordinate
plane.
Representational:
• Plot points on a coordinate plane given an ordered pair using rational numbers.
• Extend number line diagrams and coordinate axes to represent points with rational
number coordinates Supports for Teachers
Critical Background Knowledge
Conceptual:
• Have experience with the coordinate plane.
• Understand that an ordered pair is composed of two parts: the first coordinate refers
to the x-axis, the second coordinate refers to the y-axis.
Procedural:
• Identify the coordinates of plotted points Representational:
• Plot points for given ordered pairs. Academic Vocabulary and Notation
+, –, coordinate plane, ordered pair, x-axis, y-axis
Cluster Title: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
49
Instructional Resources Used Strategies Used
Play coordinate plane http://www.lessonplanspage.com/MathBattleshipPlotCoordinates79.htm battleship as a whole (Note: Modify the lesson plan to include rational numbers within a
class and then limited range; for example, halves and fourths between -2
with partners. and 2.)
Assessment Tasks Used
Skill-based Task:
Plot the following ordered pairs: (3, 2), (-4, 5), (-8, -3), (4, -6), (21⁄2 , -5), (-9.75, 0) (Note: Approximation is appropriate.) Create a graph of several points and have students write the ordered pair for each point.
Problem Task:
50
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