invention - greeleyschools.org math...greeley-evans school district 6- 7th grade: 2016-2017 revised...

Greeley-Evans School District 6- 7th Grade: 2016-2017 revised 8/9/16

Content Area Mathematics Grade Level 7th Grade

Standard Grade Level Expectations (GLE) GLE Code

1. Number Sense, Properties, and Operations

1. Proportional reasoning involves comparisons and multiplicative relationships among ratios

MA10-GR.7-S.1-GLE.1

2. Formulate, represent, and use algorithms with rational numbers flexibly, accurately, and efficiently

MA10-GR.7-S.1-GLE.2

2. Patterns, Functions, and Algebraic Structures

1. Properties of arithmetic can be used to generate equivalent expressions MA10-GR.7-S.2-GLE.1

2. Equations and expressions model quantitative relationships and phenomena MA10-GR.7-S.2-GLE.2

3. Data Analysis, Statistics, and Probability

1. Statistics can be used to gain information about populations by examining samples MA10-GR.7-S.3-GLE.1

2. Mathematical models are used to determine probability MA10-GR.7-S.3-GLE.2

4. Shape, Dimension, and Geometric Relationships

1. Modeling geometric figures and relationships leads to informal spatial reasoning and proof

MA10-GR.7-S.4-GLE.1

2. Linear measure, angle measure, area, and volume are fundamentally different and require different units of measure

MA10-GR.7-S.4-GLE.2

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 Proportional Relationships 29 Days 8/22 – 9/30 Acc: 8/29 - 9/21 (17 days)

Rational Numbers 24 Days 10/1 – 11/4 Acc: 9/22 – 10/10 (13 days)

Expressions and Equations 29 Days 11/7 – 12/20 Acc: 10/11 – 11/8 (20 days)

Percent and Proportional Relationships 20 Days 1/4 – 2/1 Acc: 11/9 – 11/22 (12 days)

Statistics and Probability 15 Days 2/2 – 2/24 Acc: 11/28 – 12/13 (11 days)

Geometry 14 Days 2/27 – 3/17 Acc: 1/4 – 1/31 (19 days)

Module 4, 5, and 6 revisited 35 days 3/27 -5/16

Invention


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, 7th 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:

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 7 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 7 Collections, Prerequisite Assessments

These assessments are useful for data cycles in general 7th 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 Proportional Relationships

Content Emphasis Cluster Analyze proportional relationships and use them to solve real-world and mathematical problems.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Draw, constructs, and describes geometrical figures and describes the relationships

between them.

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

MP.2. Reason abstractly and quantitatively.

MP.3. Construct viable arguments and critique the reasoning of others.

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.

MP.8. Look for and express regularity in repeated reasoning.

Common Assessment End of Module Common Assessment

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

quantities. Multiplicative thinking underlies proportional reasoning.

Prepared graduates use critical thinking to recognize problematic aspects of situations, create

mathematical models, and present and defend solutions

Prepared graduates apply transformation to numbers, shapes, functional representations, and data

CCSS Priority Standards Cross-Content

Connections Writing Focus Language/Vocabulary Misconceptions

CCSS.MATH.CONTENT.7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

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.,

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.

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

Academic Vocabulary- Cross discipline language-

Compute, Identify,

Represent, Explain,

Estimate, Solve

Technical Vocabulary- Discipline-specific

language- Proportional To,

Proportional Relationship,

Constant of Proportionality,

One-to One

Correspondence, Scale

Drawing and Scale Factor ,

Ratio, Rate, Unit Rate ,

Equivalent Ratio , Ratio

Table

A common error is to reverse

the position of the variables

when writing equations.

Students may find it useful to

use variables specifically

related to the quantities rather

than using x and y.

Constructing verbal models

can also be helpful. A student

might describe the situation as

“the number of packs of gum

times the cost for each pack is

the total cost in dollars”. They

can use this verbal model to

construct the equation.


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. CCSS.MATH.CONTENT.7.G.1 Solve problems involving scale

drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

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 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.

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.

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

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 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.

Students can check their

equation by substituting values

and comparing their results to

the table. The checking process

helps student revise and

recheck their model as

necessary. The number of

packs of gum times the cost for

each pack is the total cost (g x

2 = d).

Student’s may have

misconceptions about correctly

setting up proportions, how to

read a ruler, doubling side

measures, and does not double

perimeter.

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

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


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.

Priority Standard:

Strand(s): RP Topic A: Proportional Relationships (7.RP.2a) August 26 – September 2 (6 days) ACCELERATED PACE and SUGGESTIONS: Combine lessons 3-4 and 5-6 (4 days)

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

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. CCSS.MATH.CONTENT.6.RP.A.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.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.

CCSS.Math.Content.8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. CCSS.Math.Content.8.F.B.4 Construct a function to model a linear relationship between two quantities.


CCSS.MATH.CONTENT.6.RP.A.3.B Solve unit rate problems including those involving unit pricing and constant speed. CCSS.MATH.CONTENT.6.RP.A.3.C

- Addressed Later 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.

Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Changes Changes

In 6th grade the real focus is on equivalent ratios. Students build ratio tables. They begin to analyze the relationships within the table to find other ratios. They learn to find unit rate to compare two sets of data. They also learn to also write them in fractional form. They also learn to plot these points and discover they are a line if they are indeed equivalent and write the equation of the line in terms of y=kx where k is the constant.

In 7th grade instead of talking about unit rate and refer to it as constant of proportionality which could be change in y over change in x. In 8th grade this becomes slope and is strictly defined as vertical change over horizontal change. Students also start learning to write equations with given slopes and points.

Anchor Problem:

Preparing the learner: The first 5 days of the year are set aside for Jo Boaler’s “Week of Inspiration” on youcubed. If you feel that your students will be better served by having one week of review on Number Sense from 6th grade content, please use this time to get them as caught up as possible. Pre-assessment is really


important here. Students have to know how to use the multiplication chart or a calculator to find equivalent ratios and simplify ratios. They have to be able to find unit rate both as whole and rational values. If students can use a ratio to create a ratio table and plot the points on the coordinate plane in a line they are ready to proceed. Use these three days to shore up the skills and get them ready for topic A in module 1. Almost immediately they change the language and continue on into other forms of proportionality Review with students so that they are able to convert measurement and add and subtract mixed measurements. Also make sure students understand the difference between reductions and enlargements and how to produce them proportionally. Big ideas for this topic:

Proportionality in tables, graphs, and equations!

Recall/Skills- I can determine whether values in a ratio table are

proportional

I can determine the unit rate (constant) in a table

I can plot points in the coordinate plane

I can write an equation using the constant in the form y = kx

How much fruit should you use with 7

cups of nuts? Is this proportional? Describe how you know.

Determine the unit rate in the table at right.

Plot the data from the chart above to determine whether the function is proportional. How did you make this decision?

Write an equation in the form of 𝑦 = 𝑘𝑥 from either the ratio table or the graph.

Making Connections I can solve real life problems involving

proportionality

Angel and Jayden were at track practice. The track is .5 kilometers around. Angel ran 1 lap in 2 minutes. Jayden ran 3 laps in 5 minutes.


How many minutes does it take Angel to run three kilometers? What about Jayden?

How far does Angel run in one minute? What about Jayden?

Who is running faster? Explain your reasoning.

Teacher Ideas for Interaction

Eureka

In Module 1, students build upon their Grade 6 reasoning about ratios, rates, and unit rates (6.RP.1, 6.RP.2, 6.RP.3) to formally define proportional relationships and the constant of proportionality (7.RP.2). In Topic A, students examine situations carefully to determine if they are describing a proportional relationship. Their analysis is applied to relationships given in tables, graphs, and verbal descriptions (7.RP.2a).

I cannot stress enough how important it is to do some station work using 6th grade module 1 materials. This time will pay off. L1 starts off having students determine the better buy. They did quite a bit of this in the 6th grade and it is a great connection piece to use. Remember the focus is on using double number lines and unit rates.

L2 replaces “constant” with constant of proportionality and the next 3 lessons have students practice completing a ratio table, plotting points on the coordinate plane to discover they form a line and pass through the origin, and write the equation in context over and over. Focus on choosing a few quality problems to dig into deeply. Encourage student to student presentations, writing, and student dialogue around these problems. Use all five days. They will seem like they are the same thing over and over but kids need repetition. After the third day, give an exit ticket and differentiate on the last two days.

Blended Resources, Personal Learning Resources, Differentiated Learning Resources CCSS Math Resources Common Core stations 7th grade Thinking blocks (excellent resource) Dan Meyer Blog Quia (google quia “proportional relationships) for jeopardy, rags to riches, matching, concentration, or quizzes)

http://ccssmath.org/?s=7.RP.2

http://www.debbiewaggoner.com/uploads/1/2/9/9/12998469/station_activies_for_7th_grade_math.pdfhttp:/www.debbiewaggoner.com/uploads/1/2/9/9/12998469/station_activites_for_6th_grade_math.pdf


http://www.mathplayground.com/thinking_blocks_ratios_iPad.html

http://blog.mrmeyer.com/


Howard County • Hair and Nails MARS Shell Center • Proportion or Not? Pre-Assessment Module 1 Representations: Equations y=kx Vocabulary:

Ratio- A ratio is a statement of how two numbers compare. It is a comparison of the size of one number to the size of another number. All of the lines below are different ways of stating the same ratio.3:4

Rate- In mathematics, a rate is the ratio between two related quantities.[1] Often it is a rate of change

Unit Rate -When rates are expressed as a quantity of 1, such as 2 feet per secondor 5 miles per hour, they are called unit rates. If you have a multiple-unit rate such as 120 students for every 3 buses, and want to find the single-unit rate, write a ratio equal to the multiple-unit rate with 1 as the second term.

Equivalent Ratio- These ratios are equivalent because they have the same meaning - the amount of water is six times the amount of squash. You can find equivalent ratios by multiplying or dividing both sides by the same number.

Proportional To (Measures of one type of quantity are proportional to measures of a second type of quantity if there is a number k > 0 so that for every measure x of a quantity of the first type the corresponding measure y of a quantity of the second type is given by kx, i.e., y = kx.)

Coordinate Plane

Flour Sugar

2

4

6

3

6

9

Ratio Table

https://grade7commoncoremath.wikispaces.hcpss.org/Unit+2+Ratios+and+Proportional+Relationships

http://map.mathshell.org/materials/lessons.php?taskid=483

https://drive.google.com/open?id=0B4O2qe_xVD1aR1Z5NGNleFBCWmc

https://en.wikipedia.org/wiki/Mathematics

https://en.wikipedia.org/wiki/Ratio

https://en.wikipedia.org/wiki/Quantity

https://en.wikipedia.org/wiki/Rate_(mathematics)#cite_note-1


Proportional Relationship (A one-to-one matching between two types of quantities such that the measures of quantities of the first type are proportional to the measures of quantities of the second type.)

Constant of Proportionality (If a proportional relationship is described by the set of ordered pairs that satisfies the equation y = kx, where k is a positive constant, then k is called the constant of proportionality.; e.g., If the ratio of y to x is 2 to 3, then the constant of proportionality is 2/3 and y = 2/3 x.)

One-to- One Correspondence (Two figures in the plane, S and S', are said to be in one-to-one correspondence if there is a pairing between the points in S and S', so that, each point P of S is paired with one and only one point P' in S' and likewise, each point Q' in S' is paired with one and only one point Q in S.)

Scale Factor

Probing questions:

What does it mean that the “cost is proportional to weight”?

How do we know if two quantities are proportional to each other?

How can we recognize a proportional relationship when looking at a table or a set of ratios?

Explain how the constant was determined. Describe appropriate steps for graphing a set of data to determine proportionality. It may help you to actually make one to think about the

steps. Describe how to determine proportionality in a table and a graph. Give an example to illustrate your point.

What is a common mistake a student might make when deciding whether a graph of two quantities shows that they are proportional to each other?

Compare and contrast the constant of proportionality and the unit rate alike?

How do we compare ratios when we have varying sizes of quantities?

How do we know if two quantities are proportional to each other? How can we recognize a proportional relationship when looking at tables or a set of ratios? Do the 𝑥- and 𝑦-values need to increase at a constant rate? In other words, when the 𝑥- and 𝑦-values both go up at a constant rate, does this

always indicate that the relationship is proportional? When looking at ratios that describe two quantities that are proportional in the same order, do the ratios always have to be equivalent? How can you use a table to determine whether the relationship between two quantities is proportional? Do the following represent a proportional relationship? Prove it two different ways. Which


Priority Standard:

Strand(s): RP Topic B: Unit Rate and the Constant of Proportionality (7.RP.2b, 7.RP.2c, 7.RP.2d, 7.EE.4a) September 6-September 9 (4 days) ACCELERATED PACE and SUGGESTIONS: Combine lessons 8-9 (3 days)

Previous Grade Standard Standard(s) for Grade/Course: Next Grade Standard 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. CCSS.MATH.CONTENT.6.RP.A.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.A.3.B Solve unit rate problems including those involving unit pricing and constant speed. CCSS.MATH.CONTENT.6.RP.A.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. CCSS.MATH.CONTENT.6.RP.A.3.D Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. CCSS.MATH.CONTENT.6.G.A.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. CCSS.MATH.CONTENT.6.EE.B.7

CCSS.Math.Content.7.RP.A.2- ** This will be the tested priority Recognize and represent proportional relationships between quantities.

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

CCSS.Math.Content.8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. CCSS.Math.Content.8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

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

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

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

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

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


http://www.corestandards.org/Math/Content/8/EE/B/5/

http://www.corestandards.org/Math/Content/8/F/B/4/


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.

Changes Changes

6th graders spend a huge amount of time on determining if things in a ratio are proportional by identifying the constant. This is extended into 7th grade as we begin to see that same constant reappear in both the coordinate plane and in linear equations in the form 𝑦 = 𝑘𝑥.

The constant of proportionality gets narrowed specifically to slope in a line. The constant of proportionality or the slope is then used to determine other points that satisfy the equation.

Preparing the learner: Make sure to give both the school city pre-assessment AND the formative pre test as well. At this point students have to be able to recognize how to find constant of proportionality in table, graph and equation. Make sure students can divide rational numbers or have a tool to assist them and that they can plot points on the coordinate plane. Big ideas for this topic:

Constant of proportionality Y = kx

Recall/Skills- I can identify the constant of proportionality in a

table, graph or equation.

Making Connections

I can identify and interpret in context the point (1, 𝑘)

Great Rapids White Watering Company rents rafts for $125 per hour. Explain


on the graph of a proportional relationship where 𝑘, is the unit rate.

I can interpret what points on the graph of a proportional relationship mean in terms of the situation or context of the problem, including the point (0, 0).

why the point (0,0) and (1,125) are on the graph of the relationship, and what these points mean in the context of the problem.


Eureka

With the concept of ratio equivalence formally defined, students explore collections of equivalent ratios in real world contexts in Topic B. They build ratio tables and study their additive and multiplicative structure (6.RP.3a). They relate ratio tables to equations using the value of a ratio defined in Topic A. In Topic B, students learn that the unit rate of a collection of equivalent ratios is called the constant of proportionality and can be used to represent proportional relationships with equations of the form y = kx, where k is the constant of proportionality (7.RP.2b, 7.RP.2c, 7.EE.4a). Students relate the equation of a proportional relationship to ratio tables and to graphs and interpret the points on the graph within the context of the situation (7.RP.2d). Do both L9-L10, L15 with fidelity. Take your time because this is really, really important that students see these connections. 14 are great applications we don’t have time for. In 3 days, you need to make sure they get the connection. Choose one awesome problem to go deeply if you want.


Finally in L15, students expand their experience with the coordinate plane (5.G.1, 5.G.2) as they represent collections of equivalent ratios by plotting the pairs of values on the coordinate plane. The Mid-Module Assessment follows Topic B.

Blended Resources, Personal Learning Resources, Differentiated Learning Resources CCSS Math Resources Common Core stations 7th grade Quia (google “Quia proportional relationships” for jeopardy, rags to riches, matching, concentration, or quizzes) Do math together Probing questions:

How do I find the constant of proportionality?

What type of relationship can be modeled using an equation in the form y = kx , and what do you need to know to write an equation in this form?

Give an example of a real-world relationship that can be modeled using this type of equation and explain why.

How do you determine which value is 𝑥 (independent) and which value is y (dependent)?

Which makes more sense: to use a unit rate of “ears of corn per dollar” or of “dollars/cents per ear of corn”? Which one is independent?

Give an example of a real-world relationship that cannot be modeled using this type of equation and explain why.

What points are always on the graph of two quantities that are proportional to each other?

How can you use the unit rate to create a table, equation, or graph of a relationship of two quantities that are proportional to each other?

How can you identify the unit rate from a table, equation, or graph?

How do you determine the meaning of any point on a graph that represents two quantities that are proportional to each other?

http://ccssmath.org/?s=7.rp.2

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

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

http://domathtogether.com/proportional-relationships/


Focus Standard:

Strand(s): RP Topic C: Unit Rates (6.RP.2, 6.RP.3b, 6.RP.3d) September 12-September 16 (approximately 5 days) ACCELERATED PACE and SUGGESTIONS: Eliminate lesson 12 (4 days)

Previous Grade Standard Standard(s) for Grade/Course: Next Grade Standard 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. CCSS.MATH.CONTENT.6.RP.A.3.B Solve unit rate problems including those involving unit pricing and constant speed. CCSS.MATH.CONTENT.6.G.A.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. CCSS.MATH.CONTENT.6.EE.B.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.

**These two are a focus standard, but not priority CCSS.MATH.CONTENT.7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. 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.

CCSS.Math.Content.8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. CCSS.Math.Content.8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Changes Changes

6th graders are learning the difference between ratio and fractions. They are learning to find equivalent ratios, determine proportionality, and to use various modeling tools in this process. 7th graders take this to an application model. 6th graders learn to solve one step equations and distribute while 7th graders must become fluent at two step equations and distributing.

8th graders must be able to solve any multistep linear equation. They must be able to determine proportionality in a table, graph, or an equation. They use the proportional relationships to solve real applications. The focus really shifts to understanding rate of change in multiple settings.




http://www.corestandards.org/Math/Content/8/EE/B/5/

http://www.corestandards.org/Math/Content/8/F/B/4/


Preparing the learner: Make sure students can convert measurement and add and subtract mixed measurements. This should be on the pretest and probably should include a station prior to the beginning of this unit. Also make sure students understand the difference between reductions and enlargements and how to produce them proportionally. Big ideas for this topic:

Fractional rates, Constant of proportionality and scale factor, application of proportional relationships Recall/Skills-

I can identify proportionality in tables, graphs and equations.

I can find the constant of proportionality.

I can find the unit rate when it is a complex fraction.

I can make a scale drawing with a given scale factor.

Determine if the quantities of nuts and fruit are proportional for each serving size

listed in the table. If the quantities are proportional, what is the constant of proportionality or unit rate that defines the relationship? Explain how you determined the constant of proportionality and how it relates to both the table and graph.

If a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction 1/21/4 miles per hour, equivalently 2 miles per hour.


Make a scale drawing with a scale factor of 1

10 of the rectangle below.

Making Connections I can solve real life problems using proportionality

I understand constant of proportionality and unit rate are the same.

Angel and Jayden were at track practice. The track is .5 kilometers around.

o Angel ran 1 lap in 2 minutes. o Jayden ran 3 laps in 5 minutes.

a. How many minutes does it take Angel to run one kilometer? What about Jayden? b. How far does Angel run in one minute? What about Jaden? c. Who is running faster? Explain your reasoning. Compare constant or proportionality and unit rate. Be sure to use an example to

illustrate your comparison.

Teacher Ideas for Interaction Eureka

In Topic C, students extend their reasoning about ratios and proportional relationships to compute unit rates for ratios and rates specified by rational numbers, such as a speed of ½ mile per ¼ hour (7.RP.1). Students apply their experience in the first two topics and their new understanding of unit rates for ratios and rates involving fractions to solve multistep ratio word problems (7.RP.3, 7.EE.4a).

Lesson 11 is really critical. Don’t be so quick to teach the process or answer. In teams, encourage students to commit to who is faster in L11 and prove it. You want them proving it with a variety of models. Make them each present their ideas and encourage respectful discussion. L12 could be an optional lesson. In this lesson students have to remodel a room (Tile, carpet, baseboard, etc.) If you have the time and your kids need this kind of activity it is very applicable to the real world but will take several days. L13 is almost completely dividing complex fractions so take a look at your kids needs and decide to do a portion of this lesson or skip it. L14 is critical. Heavy application but most importantly make students prove their response with some type of modeling and share out the modeling plans. Don’t be afraid to spend a couple of really quality days on these types of problems. And finally L15 is very important as well as they start


plotting fractional points and determining that when points are proportional they do indeed form a line and pass through the origin. L11, 14, and 15 are the focus over the seven class days.

Blended Resources, Personal Learning Resources, Differentiated Learning Resources CCSS Math Resources

7.RP.A1 7.RP.A3

Common Core stations 7th grade MARS Shell Center • A Golden Crown • Bike Ride • Buses • Journey • Shelves Quia (google quia proportional ratio and percent for jeopardy, rags to riches, matching, concentration, or quizzes)

Probing questions:

What is a complex fraction? How can unit rate be helpful? What is a consumer? Compare and contrast a commission and a discount markdown. What does it mean to be proportional? Explain how you determine proportionality in tables, graphs, and equations.

What is a scale drawing?

What are possible uses for enlarged drawings/pictures? What are the possible purposes of reduced drawings/pictures Describe what corresponding points or parts means. Be sure to use an example to help support your description.

How do scale drawings related to rates and ratios?

Where is the constant of proportionality represented in scale drawings?

What step(s) are used to calculate scale factors?

What operation(s) is (are) used to create scale drawings?





http://map.mathshell.org/tasks.php






Focus

Strand(s):RP Topic D: Ratios of Scale Drawings (7.RP.2b, 7.G.1) September 20 – September 28 (7 days) ACCELERATED PACE and SUGGESTIONS: Combine 16-17, 18-19 and eliminate 20 and 21 (3 days)

Previous Grade Standard Standard(s) for Grade/Course: Next Grade Standard 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. CCSS.MATH.CONTENT.6.RP.A.3.B Solve unit rate problems including those involving unit pricing and constant speed. 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. 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."

CCSS.MATH.CONTENT.7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

CCSS.MATH.CONTENT.8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. CCSS.MATH.CONTENT.8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. CCSS.MATH.CONTENT.8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the




CCSS.MATH.CONTENT.7.RP.A.2 Recognize and represent proportional relationships between quantities.

1. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

equation y = mx + b for a line intercepting the vertical axis at b.

Changes Changes

Much time is spent in 6th grade with equivalent ratios and finding unit rate. This has to be fluent. In 7th grade students apply the constant of proportionality to find scale factor in scale drawings.

The idea of scale drawings moves to similarity in 8th grade. Constant of proportionality moves to slope.

Anchor Problem:


From your computer shows you a map of Island Grove in Greeley, Colorado. The scale of the map tells us that 2 inch on the map is 1 mile. Students find a familiar place using an online map tool (Google earth, mapquest, etc.). *Teachers, try your online tool out before the students to ensure that there is a scale key. Students find the scale that is given to them on the map tool. Discuss what this means in terms of measuring the place that they are observing. What happens when you zoom out, zoom in? Does the scale change? Create a reduction and enlargement of your original place. What information do you need to have to create your scale drawings? What information do you need to have on your scale drawings?

So when we we’re looking at the map of a location of your choice, what happens when you zoom out on the map? What happens when you zoom in?

Clicking a resizing button on the computer screen will result in an

image of Island Grove where the exact same-sized bar now represents 1000 feet.

a.Do you think the size of Island Grove under the 1000 ft scale will appear smaller or larger than it was under the 1 mile scale? b.Draw an accurate picture/map of Island Grove under this new 1000 ft scale. c.Was your guess in part a correct? Can you explain why the size of the map changed as it did?

How would knowing the scale factor for the drawing of the map of your location help in determining if you had a reduction?

If you had the dimensions of the map for your location how could you find the area? If you knew the actual dimensions for the map area in real life, how could you relate the areas as ratios?

Preparing the learner: Students have to be able to identify enlargements and reductions and how to create them with scale factor. This should be in your pre-assessment. Big ideas for this topic: Recall/Skills-

https://www.google.com/maps/place/Island+Grove+Regional+Park/@40.4390425,-104.7015733,16z/data=!4m2!3m1!1s0x0:0xfb52149852709cfe


I can determine whether a replica is a reduction or an enlargement

I can identify scale factor in scale drawings

Is the figure at right a reduction or enlargement? Explain how you know.


I can make a scale drawing given a scale factor

I can find the area of an scale drawing given the scale factor

A 1 –inch length in the scale drawing corresponds to actual length of 12 feet in the room.

o Describe how the scale factor can be used to determine the actual area of the dining room

o Find the actual area of the dining room. o Can a rectangular table that is 7 feet long and 4

feet wide fit into the narrow section of the dining room. You must justify your answer with math.

Making Connections

I understand that constant of proportionality in a scale drawing is the same as scale factor

I understand that enlargements are created with the product of a length and value greater than one and reductions are created with the product of a length and a value between 0 and 1.

I can create my own scale drawing

Compare proportionality in a scale drawing and scale factor. Use an example to support

your comparison. If two figures have a scale factor of 1:3 is it an enlargement or reduction? Explain your

response.

Variable possibilities


Eureka

In the final topic of this module, students bring the sum of their experience with proportional relationships to the context of scale drawings (7.RP.2b, 7.G.1). Given a


scale drawing, students rely on their background in working with side lengths and areas of polygons (6.G.1, 6.G.3) as they identify the scale factor as the constant of proportionality, calculate the actual lengths and areas of objects in the drawing, and create their own scale drawings of a two-dimensional view of a room or building. The topic culminates with a two-day experience of students creating a new scale drawing by changing the scale of an existing drawing.

Later in the year, in Module 4, students will extend the concepts of this module to percent problems.

L16 could totally be turned into a station in a station rotation at the beginning of the module. This is where enlargement and reductions have to become fluent. Students must know an enlargement results in a product of a number and a scale factor greater than one while a reduction results in a product of a number and scale factor between 0 and 1. Negative numbers do not affect scale factor. Remind students these reflect the figures. L17 solidifies that the constant of proportionality and the scale factor are one in the same. I think there might be better resources out there but the objective is critical. L18 is where I would begin teaching once you KNOW from a formative assessment that students understand how to reduce and enlarge. We are back to scale drawings. Lots of practice here and worthwhile time. L19 is optional as it asks students to do the scale drawing and then compare the areas. This is an exposure topic in 7th grade. L20-21 are project oriented. You could differentiate the various lessons for students with the minimum being “I can make a scale drawing given a scale” to scaling projects. Blended Resources, Personal Learning Resources, Differentiated Learning Resources CCSS Math Resources Common Core stations 7th grade Quia (google Quia scale drawings for jeopardy, rags to riches, matching, concentration, or quizzes) Post Assessment Module 1 Common Assessment Module 1

Probing questions: What is the difference between an original picture and a scale drawing? How do you know if a scale drawing is a reduction or an enlargement of the original image? How can we prove that a reduction has a scale factor less than 1 and an enlargement has a scale factor greater than 1? How does the size of the area you are looking at affect your scale? How can we relate area to ratios? What materials and information do you need to create a scale drawing? Why would you need to produce a scale drawing of a different scale? How does your scale drawing change when a new scale factor is presented?

http://ccssmath.org/?s=7.G.1


https://drive.google.com/open?id=0B4O2qe_xVD1aanpOMFA5SWtyckk

https://drive.google.com/open?id=0B4O2qe_xVD1aM2J3ZUNGcUtubFU


If you made a scale drawing of your map location to include in a party invite why would you need to change to a different scale?

invention - greeleyschools.org math...greeley-evans school district 6- 7th grade: 2016-2017 revised...

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