Invention Greeley-Evans School District 6- 6 Grade: 2016-2017

Greeley-Evans School District 6- 6th Grade: 2016-2017

Revised August 9, 2016

Content Area Mathematics Grade Level 6th Grade

Standard Grade Level Expectations (GLE) GLE Code

1. Number Sense, Properties, and

Operations

1. Quantities can be expressed and compared using ratios and rates MA10-GR.6-S.1-GLE.1

2. Formulate, represent, and use algorithms with positive rational numbers with flexibility, accuracy, and

efficiency

MA10-GR.6-S.1-GLE.2

3. In the real number system, rational numbers have a unique location on the number line and in space MA10-GR.6-S.1-GLE.3

2. Patterns, Functions, and

Algebraic Structures

1. Algebraic expressions can be used to generalize properties of arithmetic MA10-GR.6-S.2-GLE.1

2. Variables are used to represent unknown quantities within equations and inequalities MA10-GR.6-S.2-GLE.2

3. Data Analysis, Statistics, and

Probability

1. Visual displays and summary statistics of one-variable data condense the information in data sets into

usable knowledge

MA10-GR.6-S.3-GLE.1

4. Shape, Dimension, and

Geometric Relationships

1. Objects in space and their parts and attributes can be measured and analyzed MA10-GR.6-S.4-GLE.1

Colorado 21st Century Skills

Critical Thinking and Reasoning: Thinking

Deeply, Thinking Differently

Information Literacy: Untangling the Web

Collaboration: Working Together, Learning

Together

Self-Direction: Own Your Learning

Invention: Creating Solutions

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.

Module Titles Length of Unit Dates

Ratios and Unit Rates (Module 1) 36 8/29 - 10/20

Arithmetic Operations Including Division of Fractions (Module 2) 20 10/21 – 11/17

Rational Numbers (Module 3) 21 11/28 – 12/20

Invention



Expressions and Equations (Module 4 ) 40 1/4 – 3/3

Area, Surface Area and Volume Problems (Module 5) 26 3/6 – 4/17

Statistics (Module 6) 25 4/18 – 5/23

MATHEMATICAL PRACTICES

MATHEMATICAL PRACTICE 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.

MATHEMATICAL PRACTICE 2: Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary 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 to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

MATHEMATICAL PRACTICE 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.



MATHEMATICAL PRACTICE 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.

MATHEMATICAL PRACTICE 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.

MATHEMATICAL PRACTICE 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.

MATHEMATICAL PRACTICE 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.



MATHEMATICAL PRACTICE 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.



Name of Assessment

Purpose of Assessment

Location of Assessment

Suggestions

Performance Assessments

OPTIONAL

Pre and Post of each topic with in each module. These are useful for data cycles.

Schoology:

follow these clicks: Groups, Secondary Teachers, Resources, Math, 6th Grade, Assessments, Pre/Post

If you want to condense and create one comprehensive pre and post assessment for each module, the suggestion would be to use only question 1 from each pre and post-test within the desired module.

Accelerated: ( Apply to any class you feel would benefit)

When using the performance assessments, give only question 2 for the pre-test to check for understanding. If question 2 has no access for them, they can do question 1 to identify misconceptions. For the post-test, only give question 2. If you only want to give one pre and one post-test for each module concerning all Topics, you may condense each Topic pre and post-test using all of the question 2s.

End of Module Common Assessments

REQUIRED

Post assessment used to assess student understanding of priority standards for modules.

School City:

District Tab, 2016-2017, Math Grade 6 Collections

Testing window will be open for all assessments at the beginning of the school year. Testing windows will close 10 days after the module completion date on the cover of the curriculum guide.

General classes and accelerated classes will take the same assessment.

Prerequisite Assessments

OPTIONAL

Used to assess student understanding of the foundational standards for the priority standards in each module.

School City:

District Tab, 2016-2017, Math Grade 6 Collections, Prerequisite Assessments

These assessments are useful for data cycles in general 6th Grade math classes. They help to identify holes in learning from prior grades.

The assessments are not necessary in accelerated classes.



Suggested Big Idea Module 1: Ratios and Unit Rates

Content Emphasis Cluster Understand ratio concepts and use ratio reasoning to solve problems.

Mathematical Practices MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.4 Model with mathematics

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

Common Assessment End of Module Assessment

Graduate Competency Prepared graduates make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative

thinking underlies proportional reasoning

CCSS Priority Standards Cross-Content

Connections Writing Focus Language/Vocabulary Misconceptions

CCSS.MATH.CONTENT.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.

CCSS.MATH.CONTENT.6.RP.3.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.

CCSS.MATH.CONTENT.6.RP.3.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?

CCSS.MATH.CONTENT.6.RP.3.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.

Literacy Connections

RST.6-8.4

Determine the meaning

of symbols, key terms,

and other domain-

specific words and

phrases as they are used

in a specific scientific

or technical context

relevant to grades 6-8

texts and topics.

RST.6-8.5

Analyze the structure an

author uses to organize

a text, including how

the major sections

contribute to the whole

and to an understanding

of the topic.

RST.6-8.7

Integrate quantitative or

technical information

expressed in words in a

text with a version of

Writing Connection

WHST.6-8.2

Write

informative/explanatory

texts, including the

narration of historical

events, scientific

procedures/

experiments, or

technical processes.

a. Introduce a topic

clearly, previewing

what is to follow;

organize ideas,

concepts, and

information into

broader categories as

appropriate to achieving

purpose; include

formatting (e.g.,

headings), graphics

(e.g., charts, tables),

and multimedia when

useful to aiding

comprehension.

Academic Vocabulary-

compare, unit,

constant, outcome,

fair, percent,

compare, multiply,

divide

Technical Vocabulary-

ratio, rate, unit rate,

equivalent ratios,

ratio table, percent,

convert, tape

diagram, coordinate

plane, equation

L.6-8.6

Acquire and use

accurately grade-

appropriate general

academic and

domain-specific

words and phrases;

gather vocabulary

knowledge when

considering a word or

Fractions and ratios may represent different

comparisons.

Fractions can express a part-to- whole

comparison, but ratios can express a part-to-

whole comparison or a part-to-part

comparison which can be written as: a to b,

or a:b.

Even though ratios and fractions express a

part-to-whole comparison, the addition of

ratios and the addition of fractions are

distinctly different procedures.

When adding ratios, the parts are added, the

wholes are added and then the total part is

compared to the total whole.

For example, (2 out of 3 parts) + (4 out of 5

parts) is equal to 6 parts out of 8 total parts

(6 out of 8) if the parts are equal.

When dealing with fractions, the procedure

for addition is based on a common

denominator: ( 2 3 ) + ( 4 5 ) = ( 10 15) + (

12 15) which is equal to ( 22 15).

Therefore, the addition process for ratios and

for fractions is distinctly different.

Often there is a misunderstanding that a

percent is always a natural number less than

or equal to 100.

http://www.corestandards.org/ELA-Literacy/L/6/6/



CCSS.MATH.CONTENT.6.RP.3.D

Use ratio reasoning to convert measurement

units; manipulate and transform units

appropriately when multiplying or dividing

quantities.

that information

expressed visually (e.g.,

in a flowchart, diagram,

model, graph, or table).

RST.6-8.8

Distinguish among facts,

reasoned judgment

based on research

findings, and

speculation in a text.

b. Develop the topic with

relevant, well-chosen

facts, definitions,

concrete details,

quotations, or other

information and

examples.

c. Use appropriate and

varied transitions to

create cohesion and

clarify the relationships

among ideas and

concepts.

d. Use precise language

and domain-specific

vocabulary to inform

about or explain the

topic.

e. Establish and maintain

a formal style and

objective tone.

f. Provide a concluding

statement or section that

follows from and

supports the

information or

explanation presented.

WHST.6-8.4

Produce clear and

coherent writing in

which the development,

organization, and style

are appropriate to task,

purpose, and audience.

phrase important to

comprehension or

expression.

L.6-8.4

Determine or clarify

the meaning of

unknown and

multiple-meaning

words and phrases

choosing flexibly

from a range of

strategies.

Provide examples of percent amounts that are

greater than 100%, and percent amounts that

are less 1%.

Students may confuse mathematical terms

such as ratio, rate, unit rate and percent.

Students may not understand the difference

between an additive relationship and a

multiplicative relationship.

http://www.corestandards.org/ELA-Literacy/L/7/4/



Priority Standard:

Strand(s): RP Topic A: Representing and Reasoning About Ratios (6.RP.1, 6.RP.3a) (8 days)

Previous Grade Standard Standard(s) for Grade/Course: Next Grade Standard

CCSS.Math.Content.5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

CCSS.Math.Content.5.OA.B.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

CCSS.Math.Content.6.RP.A.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.

CCSS.Math.Content.7.RP.A.2 Recognize and represent proportional relationships between quantities. 7.RP.A.2.a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 7.RP.A.2.b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and

http://www.corestandards.org/Math/Content/5/OA/B/3/


http://www.corestandards.org/Math/Content/6/RP/A/3/


http://www.corestandards.org/Math/Content/7/RP/A/2/a/

http://www.corestandards.org/Math/Content/7/RP/A/2/b/



ordered pairs on a coordinate plane.

CCSS.Math.Content.5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

CCSS.Math.Content.6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

CCSS.Math.Content.6.RP.A.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.

verbal descriptions of proportional relationships. 7.RP.A.2.c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

7.RP.A.2.d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate

Changes Changes

Previously 6th graders learned to make equivalent fractions in order to add and subtract fractions. Now students are taught fractions are a subset of ratios and how then to find equivalent ratios. Students also previously learned to reduce fractions to simplest form. In 6th they learn to find unit rate. Previously students use rules given in words to generate patterns; In 6th they identify numerical patterns to create and extend ratio tables of equivalent ratios. They use this understanding to solve real problems involving rates or ratios.

6th grade students are introduced to graphing in terms of plotting points and graphing lines given points. In 7th students refine their understanding of unit rate to slope and begin to understand slope as vertical change divided by horizontal change. Instead of finding the unit rate, 7th graders find the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions. Instead of creating and completing ratio tables of equivalent ratios students determine whether quantities are in proportional relationship from tables or graphing. In 6th grade rules are expressed in words, in 7th they begin representing proportional relationships by equations (y=kx).

Anchor Problem: This anchor problem can be extended over the entire module. Most television shows are paid for with commercial advertising. One popular television show wants to change the ratio of commercial time to show time. What information will you need to make this possible? Do you think that all TV shows have the same amount of commercials? Why/Why not?

http://www.corestandards.org/Math/Content/5/MD/A/1/






http://www.corestandards.org/Math/Content/7/RP/A/2/c/

http://www.corestandards.org/Math/Content/7/RP/A/2/d/



Most television shows use 13 minutes of every hour for commercials, leaving the remaining 47 minutes for the actual show. One popular television show wants to change the ratio of commercial time to show time to be 3:7. Do you support the proposal of this popular television show?

Have students think about and discuss their favorite TV show. Things like what they like about it, what time it is on, what day of the week it is on, etc. Then get into some ideas about the math in their show…How long is their program? How many minutes of commercials do you think happen during your show? If you were to take two TV shows, what are some things that could be compared? Some things that the students might come up with that they are interested in are: Compare ½ hour shows to hour long shows, Prime time vs non-prime time, Types of commercials (content), Year comparison (1985 info. vs 2015), Time of the year (seasons)

Write and describe two different ratios for the following description: An average hour of monitored prime-time US network TV programming contained seven minutes, 59 seconds (8 min.) of in-show brand appearances and 13:52 (14 min.) of network commercial messages, for a combined total of 21:51(22 min.) of marketing content, according to TNS Media Intelligence.

One popular television show wants to change the ratio of commercial time to show time to be 3:7. Determine the number of minutes of commercials in a 30 minute sitcom, 60 minute drama, and 120 minute movie. ** Teacher note- this is where it is important that the students (and teacher) note that you have to compare not part to part, but part to whole. Watch for students that try to keep the 3:7 ratio as they solve the problem. Do not tell them up front that they really need to be looking at 3:7 ratio in terms of 10 minute segments.

Unscripted reality programming has an average of 14 minutes per hour of brand appearances, compared with just 6 minutes per hour for scripted programs such as sitcoms and dramas, reported TNS. Lonnie says that means the ratio of unscripted reality programming to scripted programming is 2:7. Is Lonnie correct? Show why or why not.

In 2015, the ratio of commercials to live football action jumped to 20:1. The average primetime television show has a 2:1 programming ratio. If there are approximately 40 minutes of programming in an average prime time hour, how many minutes of live football were there?

- In the Lost, season finale, there was a 2:1 show to commercial ratio that lasted 72 minutes. How many minutes of commercials were there? During a half hour News broadcast, you watch 9 minutes of commercials. Create a ratio table to show how many minutes of commercials you

will watch after 1, 3, 5, and 8 hours of News Broadcasting. During a half hour News broadcast, you watch 9 minutes of commercials. Create a ratio table to show how many minutes of commercials you

will watch after 1, 3, 5, and 8 hours of News Broadcasting. The following prices given for each television show are the cost of advertising for a 30 second commercial segment. Estimate the value of the

ratio for one of the following shows, then determine the exact value of the ratio. Discuss with the students if their estimates were reasonable or not.



Preparing the learner: It is expected that each teacher has not only read the topic they are preparing to teach but have done all of the problems in the unit so as to understand the conceptual development. This is necessary. For those who have Eureka videos to view, it is also expected that you would have viewed that video at least once prior to teaching the content. Be sure to give the both the school city pretest and the formative pretest prior to the beginning of the unit so you have time to analyze them. Make sure to test student knowledge around simplifying fractions and finding equivalent fractions and all operations of positive rational numbers. Be sure to teach them how to use the multiplication chart to do this. It is also okay to provide a calculator. Just know what you are working with. Don’t be afraid to give up a week prior to each module shoring up previous skills that are lacking. These should be identified in the pre-assessment. It will make a difference. You have three days to assign students the stations they need and get them ready for the module. Big ideas for this topic:

Equivalent Ratios Modeling with tape diagrams



Recall/Skills- I can use models (tape diagrams, blocks, drawings) to find

equivalent ratios.

I can find unit ratio.

On a bicycle you can travel 20 miles in 4 hours. What are the unit rates

in this situation, (the distance you can travel in 1 hour and the amount of time required to travel 1 mile)?

Solution: You can travel 5 miles in 1 hour written as and it takes of an

hour to travel each mile written as 5 𝑚𝑖𝑙𝑒𝑠

1 ℎ𝑜𝑢𝑟 or

1

5ℎ𝑜𝑢𝑟

1 𝑚𝑖𝑙𝑒. Students can represent

the relationship between 20 miles and 4 hours.



I can rewrite ratios from part:part to part:whole.

I can translate between ratios (A:B) and fractions ( 𝐴

𝐵) .

A food company that produces peanut butter decides to try out a new version of its peanut butter that is extra crunchy, using twice the number of peanut chunks as normal. The company hosts a sampling of its new product at grocery stores and finds that 5 out of every 9 customers prefer the new extra crunchy version.

a. Let’s make a list of ratios that might be relevant for this situation.

i. The ratio of number preferring new extra crunchy to total number surveyed is _________.

ii. The ratio of number preferring regular crunchy to the total number surveyed is _________.

iii. The ratio of number preferring regular crunchy to number preferring new extra crunchy is _________.

iv. The ratio of number preferring new extra crunchy to number preferring regular crunchy is _________.

Write the ratio 2 blues to every 3 yellows as a fraction and describe it. Making Connections

I can make connections in the ratio table and models and understand that A:B and C:D are equivalent ratios if there is such a positive number c such that C=cA and D=cB.

I can solve problems involving ratios with models (tape diagrams)

Jasmine has taken an online boating safety course and is now

completing her end of course exam. As she answers each question, the progress bar at the bottom of the screen shows what portion of the test



she has finished. She has just completed question 16 and the progress bar shows she is 20% complete. How many total questions are on the test? Use a table, diagram, or equation to justify your answer.

Teacher Ideas for Interaction Eureka/EngageNY Curriculum Guide In this module, students are introduced to the concepts of ratio and rate. Their previous experience solving problems involving multiplicative comparisons, such as “Max has three times as many toy cars as Jack,” (4.OA.2) serves as the conceptual foundation for understanding ratios as a multiplicative comparison of two or more numbers used in quantities or measurements (6.RP.1). Students develop fluidity in using multiple forms of ratio language and ratio notation. They construct viable arguments and communicate reasoning about ratio equivalence as they solve ratio problems in real world contexts (6.RP.3).

L1-2 sets up the modeling with tape diagrams, blocks, drawings, etc. Do not rush this portion. It is okay to spend 2 days just have students find equivalent ratios and display them in ratio tables. Make sure to explicitly teach the ratio language, ex. “for every, out of every, to, for each,” etc. They should only be writing ratios in A:B and A to B format. They also need to know that the order of the ratio is important and has meaning (this is why the labeling is important).

L3-L5 focus on the connection piece that in that ratio table there is a single constant number that if you multiply the given ratio by to create an equivalent ratio. Go slow to go fast. They need this to make sense all year long. Lots of practice and lots of challenging dialogue having students express the patterns they are seeing. Do a formative assessment before heading into L6.

L6 is heavy application using modeling (tape diagrams) Pay attention to whether the ratios are given to them in ratio notation, or if the students have to identify them in context before they can solve.

And L7 is skill based where students get fluent translating between ratios in part:part and ratios that are part:whole. There is a new piece in here that involves a change to the quantities that will change the ratio. Since they have not had any fraction experience, don’t assume that this will be a simple concept. This would be the time to talk about the differences and similarities of fractions and ratios.

Blended resources/Personalized learning resources/differentiated learning resources: CCSS Math Resources

6.RP.1 6.RP.2 6.RP.3 Thinking Blocks outstanding resource Common Core stations 6th grade

http://ccssmath.org/?s=6.RP.1



http://www.mathplayground.com/NewThinkingBlocks/thinking_blocks_ratios.html

http://www.debbiewaggoner.com/uploads/1/2/9/9/12998469/station_activites_for_6th_grade_math.pdf



Pre-Assessment Module 1 Mars Shell Centre

Traveling to School Sharing Costs Traveling to School

Math is Fun

Ratios

Flashcards for equivalent fractions, decimals and percents Vocabulary:

Ratio (A pair of nonnegative numbers, A:B, where both are not zero, and that are used to indicate that there is a relationship between two quantities such that when there are A units of one quantity, there are B units of the second quantity.)

Rate (A rate indicates, for a proportional relationship between two quantities, how many units of one quantity there are for every 1 unit of the second quantity. For a ratio of A:B between two quantities, the rate is A/B units of the first quantity per unit of the second quantity.)

Unit Rate (The numeric value of the rate, e.g., in the rate 2.5 mph, the unit rate is 2.5.)

Value of a Ratio (For the ratio A:B, the value of the ratio is the quotient A/B.)

Equivalent Ratios (Ratios that have the same value.)

Percent (Percent of a quantity is a rate per 100.)

Associated Ratios (e.g., if a popular shade of purple is made by mixing 2 cups of blue paint for every 3 cups of red paint, not only can we say that the ratio of blue paint to red paint in the mixture is 2:3, but we can discuss associated ratios such as the ratio of cups of red paint to cups of blue paint, the ratio of cups of blue paint to total cups of purple paint, the ratio of cups of red paint to total cups of purple paint, etc.)

Representations:

https://drive.google.com/open?id=0B4O2qe_xVD1aMlc3TDN1cHVzOGM

http://map.mathshell.org/lessons.php?collection=8&unit=6200

http://map.mathshell.org/lessons.php?collection=8&unit=6200

http://www.mathsisfun.com/numbers/ratio.html

https://quizlet.com/24993874/flashcards



Probing questions:

What is a ratio? Can you verbally describe a ratio in your own words using this description? What are two quantities that you would love to have in a ratio of 5:2 but hate to have in a ratio of 2:5? Are they the same? Why or why not? What do these three ratios 7:3, 14:6, 21:9, all have in common? Have the students use math language in their responses. How can you tell if ratios are equivalent or not? Give examples of each. “Ratios are equivalent if there is a positive number that can be multiplied by both quantities in one ratio to equal the corresponding quantities

in the second ratio.” Explain what this means in your own words and provide examples or counter examples. What do the numbers in the boxes of the tape diagram represent in terms of the ratios? How did the words, “more than,” change your tape diagram? How did you decide what numbers to use in the ratios (or the tape diagram)? Choose a problem and write down how you would describe to a friend how to solve the problem. Compare and contrast ratios and fractions. Use an example to illustrate your comparison and contrast.



What strategies did you use to determine how to begin the problem? How did the language complicate or help you in the problems? What is the difference between a ratio and a fraction? Make up a ratio and represent it five different ways. Explain your thinking. Enrique says that 1:2 is not the same ratio as 5:10. Kiara says that ratios are not the same, but their values are equal. Present an argument for

Kiara’s case.



Focus/Priority Standard:

Strand(s): RP Topic B: Collections of Equivalent Ratios (6.RP.3a) (7 days)


CCSS.Math.Content.5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. CCSS.Math.Content.5.OA.B.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. CCSS.Math.Content.5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. CCSS.Math.Content.6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. CCSS.Math.Content.6.RP.A.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.


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?

d) Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

CCSS.Math.Content.7.RP.A.2 Recognize and represent proportional relationships between quantities.

a) Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate

http://www.corestandards.org/Math/Content/5/NF/B/7/













Changes Changes

6th grade students use the previous knowledge of dividing wholes and fractions to determine rates and unit rates.

Value of the ratio (rate) is now called the constant of proportionality in 7th Grade. Constant of proportionality then begins to transform into slope.

Big ideas for this topic:

Ratio tables!!! Fill in a ratio table, create an equation from the ratio table and graph the points on the coordinate plane *Note: this WHOLE topic defines the "unit rate" as just the numerical value, this is NOT true, a "unit rate" is the numerical value per one something (ex: 42 miles per hour or 32 oz/$1) **Note: when writing a rate you may have a fraction (1/2 mile per hour) but when writing the ratio it should be written in whole numbers (1 mile to 2 hours) Recall/Skills-

I can find missing values in the tables, and plot the pairs of values on the coordinate plane.

I can solve unit rate problems including those involving unit pricing and constant speed.

If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be

mowed in 35 hours? What is the unit rate?



I can solve percent problems using double number

lines and tape diagrams.

Making Connections

I can solve problems using a variety of models: ratio table, tape diagrams, double number lines, graphing.

A credit card company charges 17% interest on any charges not paid at the end of

the month. If your bill totals $450 for this month, how much interest would you have to pay if you let the balance carry to the next month? Predict the interest charges for a $300 balance.

Teacher Ideas for Interaction Eureka/EngageNY Curriculum Guide

With the concept of ratio equivalence formally defined, students explore collections of equivalent ratios in real world contexts in Topic B.

L9 they build ratio tables (modeling) and study their additive and multiplicative structure (6.RP.3a). In L10 they make connections in the ratio table both vertically and horizontally and begin talking about value of a ratio. Remember, go slow to go fast. Students continue to apply reasoning to solve ratio problems while they explore representations of collections of equivalent ratios and relate those representations to the ratio table (6.RP.3). In L11 students begin to compare to different ratio tables to compare rates. Formative assessment recommended.

Building on their experience with number lines, in L12 students represent collections of equivalent ratios with a double number line model. This helps them interpolate data in between the ratios. This is critically fluency piece and contains a nice class activity during the lesson. They still need structure though – roles assigned. They relate ratio tables to equations using the value of a ratio defined in Topic A. Formative assessment recommended.

Finally in L13 Students define value of a ratio and use this to write a dependent independent equation. Another critical lesson. Remember, we are going slowly to go quickly later. Formative assessment recommended.

L14-15 (Additional practice) have students plot points on the coordinate plane to show that they are proportional or not based on whether the points form a linear function and pass through the origin. Students may or may not have background with the coordinate plane so prepare to go slow through this. It is revisited again at the end of module 3. Formative assessment recommended.

The Mid-Module Assessment follows Topic B (optional)



Additional Teacher Notes

More Information on Soda and Sugar:

Video: http://www.cnn.com/2013/01/14/health/coke-obesity

Video: http://www.teachertube.com/viewVideo.php?video_id=13788

http://www.sugarstacks.com/beverages.htm

http://www.cdc.gov/features/healthybeverages/

Blended resources/Personalized learning resources/Differentiated Learning Resources CCSS Math Resources Thinking blocks Common Core stations 6th grade Probing questions:

Is it possible to write more than one ratio to represent a given rate? Compare and contrast ratios and unit rates.

In a vertically oriented ratio table, how are the values across the rows related?

In a vertically oriented ratio table, how are the values down a column related?

Is there a way to use addition to figure out the next row in a ratio table?

Is there a way to use multiplication to figure out the next row in a ratio table?

When given two ratio tables, how can we compare the rates?

In the coordinate plane, in most cases, what is placed on the horizontal axis?

How do you create the intervals on the 𝑥-axis? Y-axis?

http://www.cnn.com/2013/01/14/health/coke-obesity

http://www.teachertube.com/viewVideo.php?video_id=13788

http://www.sugarstacks.com/beverages.htm

http://www.cdc.gov/features/healthybeverages/


http://www.mathplayground.com/NewThinkingBlocks/thinking_blocks_ratios.html




Focus/Priority Standard:

Strand(s): RP Topic C: Unit Rates (6.RP.2, 6.RP.3b, 6.RP.3d) (10 days)


CCSS.Math.Content.5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. CCSS.Math.Content.5.OA.B.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. CCSS.Math.Content.5.MD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. CCSS.Math.Content.6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. CCSS.Math.Content.6.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use


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? d) Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

CCSS.Math.Content.7.RP.A.2 Recognize and represent proportional relationships between quantities. CCSS.Math.Content.7.RP.A.2.a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. CCSS.Math.Content.7.RP.A.2.b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. CCSS.Math.Content.7.RP.A.2.c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. CCSS.Math.Content.7.RP.A.2.d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with



special attention to the points (0, 0) and (1, r) where r is the unit rate

Changes Changes

Students use the knowledge of dividing wholes numbers and fractions in order to connect ratios to rate and unit rate. Students build on what they have done converting within measurement systems and also begin to convert between measurement systems. New: students use equivalent ratios to make their conversions, understanding that when they divide as a way of finding a value per unit that they are finding the unit rate.

Students move from the idea of value of the ratio (rate) is called the constant of proportionality in 7th Grade. Also, students move to using rational numbers in the ratios rather than just whole number ratios. New: students understand that the unit rate can be modeled in a graph as the slope

Anchor Problem: Our school wants to change the way they use the blacktop space. The dimensions of the space we currently have on the blacktop is 35 yards by 50 yards. The kids at our school want to build a playground that would take up 1600 square feet of the blacktop. In order to make this remodel, what are all of the other things we'd need to consider (space, budget..)? What percent of the blacktop will be left for the other activities or games?

If we were to think about redoing the space on our blacktop and we raised enough money through student council to buy a new playground set, what do you think would make the best blacktop space?

think about space needed for desired blacktop activities dimensions extra cost items needed for construction ratios or comparisons of how the space is shared Should there be more space for basketball, tetherball, 4 square, monkey bars...



Let's think about our blacktop... One of the things we are going to need is bags of concrete mix to install and secure the playground. We need to think about the most affordable option and here's what Home Depot offers: 80 lbs. of concrete mix for $240 or a 30 lb. bag of concrete mix for $102.

Now that we have decided on the better buy for concrete, we are going to need 100 pounds of concrete mix to finish the job. How much will we spend on concrete mix to secure and install the playground?

Your friend’s father is going to do the concrete work for the new playground. The rate he mixes the concrete mix is 8 gallons of water to 2 lbs of concrete mix. Make a table, graph and equation to show how many gallons of water will be used for 1 to 100 lbs. of concrete mix.

The square footage of our new playground is 1600 square feet. What can we do to determine how that square footage relates to the dimensions of our blacktop that is given in yards? Big ideas for this topic

Rate – $𝟏𝟎 for every 4 packs of diet cola is a rate because it is a ratio of two quantities

Unit Rate – The unit rate is 𝟐. 𝟓 because it is the value of the ratio.

Rate Unit – The rate unit is dollars/packs of cola because it costs 𝟐. 𝟓 dollars for every 1 pack of cola

Unit Rate can be found in table, graph, and equation.

Recall/Skills- I can use the unit rate to find equivalent

ratios.

I can explain the difference between ratios, rates, and unit rates.

Most television shows use 13 minutes of every hour for commercials, leaving the remaining 47 minutes for the actual show. One popular television show wants to change the ratio of commercial time to show time to be 3:7. Create two ratio tables, one for the normal ratio of commercials to programming and another for the proposed ratio of commercials to programming. Use the ratio tables to make a statement about which ratio would mean fewer commercials for viewers watching 2 hours of television.



I can 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.

I can find unit rate in tables, graphs and equations.

Find the rate of change

represented in each equation, table or graph.

𝑦 = 2𝑥 + 4

Making Connections



I can determine among products sold in different quantities, which is the better buy.

Teacher Ideas for Interaction Eureka/EngageNY

In Topic C, students build further on their understanding of ratios and the value of a ratio as they come to understand that a ratio of 5 miles to 2 hours corresponds to a rate of 2.5 miles per hour, where the unit rate is the numerical part of the rate, 2.5, and miles per hour is the newly formed unit of measurement of the rate (6.RP.2). Students solve unit rate problems involving unit pricing (6.RP.3b). They apply their understanding of rates to situations in the real world. Students determine unit prices and use measurement conversions to comparison shop.

Lessons 16 and 17 are critical. There are additional problems to solve in L18 you can use. Ultimately students have to understand the difference between ratios, rates, unit rate and rate unit. As always, the lower the student, or the more language challenged a child is the more important the modeling. Be sure to have students always support their answer with a model as well. This needs to be crystal clear so formative assessment will be required. They must also be able to compare and contrast ratios and rates.

Lesson 19 is where students must also be able to find the unit rate in a table, a graph, and an equation. This lesson has a nice matching activity where students



match the equation to the table to the graph. Shell also has one of these.

L20 is very important for connection to real life. Every single person who goes to the grocery store has to determine which the better buy is. This is an opportunity to connect the math to their world. Even better yet though is if you go to the grocery store or any store that has a brand sold at varying quantities and take some pictures on your cell phone and have students determine the better buy. Or you could have students bring in their own ads or pics of some item they would like to purchase that is sold in different quantities.

The most important feature of L21 is conversion using dimensional analysis. They center them on work problems but the focus is the change in measurement. L22 specifically focuses on D=R*T problems still using the dimensional analysis.

**L23 are work problems. This should be considered an extension lesson for those students who need an additional challenge. This is an optional lesson for general population.

Blended Learning Resources/Personalized learning resources/Differentiation Resources CCSS Math Resources Common Core stations 6th grade Common Core Math Tasks Quia (google Quia percents for jeopardy, rags to riches, matching, concentration, or quizzes) Probing questions:

Is it possible to write more than one ratio to represent a given rate?

Will everyone have the same exact ratio to represent the given rate? Why or why not?

Compare and contrast unit rate and ratios.

Will everyone have the same exact unit rate to represent the given rate? Why or why not?

How can a double number line help you solve ratio problems?

How do you identify the unit rate in a table, graph, and equation?



https://commoncoremathtasks.wikispaces.com/6-RP



Why was the unit rate instrumental when comparing rates?

In Example 2, what if it was set up this way: 10 cups × (4 cups / 1 quart) = 40 quarts. What is wrong with that set up?

Describe the relationship between distance, rate, and time. State this relationship as many different ways as you can. How does each of these representations differ? How are they alike?



Priority Standard:

Strand(s): RP Topic D: Percent (6.RP.3c) (6 days)


CCSS.MATH.CONTENT.5.NBT.A.3 Read, write, and compare decimals to thousandths.

CCSS.Math.Content.6.RP.A.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,

CCSS.MATH.CONTENT.7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Changes Changes

In 5th grade students learn how to translate between fraction, decimal and percent and this is hit again in 6th grade. Very little has been done with percents prior to 6th grade. The focus here is helping students see percentages as ratios and using a double number line, learn how to solve basic percent problems. It is NOT however proportional at this stage in the game. Rather students understand 10% of a number and 1% of a number and use these to solve problems.

In 7th grade they continue to use double number line only introduce the proportional model and the algebraic model. Students move into applications of percents such as mark up, mark down, taxes, gratuities, commissions, etc.

Big ideas for this topic:

1. Using a double number line 2. Solving basic percent problems

http://www.corestandards.org/Math/Content/5/NBT/A/3/




Recall/Skills- I can translate between fractions and percents using

tape diagrams and double number lines I can solve percent problems using tape diagrams

and double number lines

What percent is 4

5? Do not have students divide. They should use tape

diagram or double number line. Division comes later!

Find 62% 𝑜𝑓 80? Do not have students multiply just now. Have them use a tape diagram or double number line and find 10% of 80 and 1% of 80.

Making Connections

I can solve real world problems involving percents using tape diagrams and number lines

A sweater is regularly $𝟑𝟐. It is 𝟐𝟓% off the original price this week.



Teacher Ideas for Interaction Eureka/EngageNY Resources

In the final topic of the module, students are introduced to percent and find percent of a quantity as a rate per 100. Students understand that N percent of a quantity has the same value as N/100 of that quantity. Students express a fraction as a percent, and find a percent of a quantity in real-world contexts. Students learn to express a ratio using the language of percent and to solve percent problems by selecting from familiar representations, such as tape diagrams and double number lines, or a com1bination of both (6.RP.3c). An End-of-Module Assessment follows Topic D.

L24-25 are CRITICAL. Make sure you break out the base ten blocks and that students can translate between fractions, decimals and percents. This is important enough to do a formative assessment after these two lessons. You can combine if you like. L26-27 it is essential that you teach students to use tape diagrams when changing from fraction to percent (not dividing!) This helps solidify the concept that percent is ALWAYS out of 100. L28-29 They also must learn to use a double number line to solve percent problems NON-PROPORTIONALLY at this point as ENGAGEny is showing.. In 7th grade they learn to use the double number line to set up a proportion and solve and they learn to solve them algebraically. In 6th grade we want students to intuitively understand percent by thinking of it in terms of 10% and 1%. Two examples have been provided below. Combine these lessons and pick and choose several problems.



Blended Resources, Personal Learning Resources, Differentiated Learning Resources CCSS Math Resources Common Core stations 6th grade Shell Translating between fractions, decimals and percents Module 1 Post Assessment Module 1 Common Assessment Quia (google Quia percents or Quia translating between decimals, fractions, and percents for jeopardy, rags to riches, matching, concentration, or quizzes)

Probing questions:



http://map.mathshell.org/download.php?fileid=1594

https://drive.google.com/open?id=0B4O2qe_xVD1abnZ3ZjZPUUllREU

https://drive.google.com/open?id=0B4O2qe_xVD1abV9QVmQ0ZHBGVFE



Share two ways that you can write 2%. Compare and contrast fractions, decimals, and percents? Is 70% always the same amount? Is 70% of 40 the same as 70% of 60? Why or Why not. Use an example or counterexample to support to

support your claim Explain two different ways to calculate 20% of 70. Explain two different ways to calculate 62% of 180.

Invention Greeley-Evans School District 6- 6 Grade: 2016-2017

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Transcript of Invention Greeley-Evans School District 6- 6 Grade: 2016-2017