7th GradeCurriculum Resources

for Common Core State Standards

2013–2014

2013-2014 Wicomico County Public Schools Page 1 of 60 Last Revised May 24, 2013

Grade 7 Common Core IntroductionIn Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.

1. Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish proportional relationships from other relationships.

2. Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. They use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.

3. Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area of three-dimensional objects. In preparation for work on congruence and similarity in Grade 8 they reason about relationships among two-dimensional figures using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between angles formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.

4. Students build on their previous work with single data distributions to compare two data distributions and address questions about differences between populations. They begin informal work with random sampling to generate data sets and learn about the importance of representative samples for drawing inferences.

2013-2014 Wicomico County Public Schools Page 2 of 60 Last Revised May 24, 2013

PARCC KEY ADVANCES In grade 6, students learned about negative numbers and the kinds of quantities they can be used to represent; they also learned

about absolute value and ordering of rational numbers, including in real-world contexts. In grade 7, students will add, subtract, multiply, and divide within the system of rational numbers.

Students grow in their ability to analyze proportional relationships. They decide whether two quantities are in a proportional relationship (7.RP.2a); they work with percents, including simple interest, percent increase and decrease, tax, markups and markdowns, gratuities and commission, and percent error (7.RP.3); they analyze proportional relationships and solve problems

involving unit rates associated with ratios of fractions (e.g., if a person walks 12 mile in each

14 hour, the unit rate is the complex

fraction

1214

miles per hour or 2 miles per hour) (7.RP.1); and they analyze proportional relationships in geometric figures (7.G.1).

Students solve a variety of problems involving angle measure, area, surface area, and volume (7.G.4–6).

PARCC MAJOR WITHIN-GRADE DEPENDENCIES Meeting standard 7.EE.3 in its entirety will involve using rational number arithmetic (7.NS.1–3) and percents (7.RP.3). Work

leading to meeting this standard could be organized as a recurring activity that tracks the students’ ongoing acquisition of new skills in rational number arithmetic and percents.

Because rational number arithmetic (7.NS.1–3) underlies the problem solving detailed in 7.EE.3 as well as the solution of linear expressions and equations (7.EE.1–2, 4), this work should likely begin at or near the start of the year.

The work leading to meeting standards 7.EE.1–4 could be divided into two phases, one centered on addition and subtraction (e.g., solving x + q = r) in relation to rational number addition and subtraction (7.NS.1) and another centered on multiplication and division (e.g., solving px + q = r and p(x + q) = r) in relation to rational number multiplication and division (7.NS.2).

2013-2014 Wicomico County Public Schools Page 3 of 60 Last Revised May 24, 2013

Common Core Standards for Mathematical Practice

The Standards for Mathematical Practice are central to the teaching and learning of mathematics. These practices describe the behaviors and habits of mind that are exhibited by students who are mathematically proficient. Mathematical understanding is the intersection of these practices and mathematics content. It is critical that the Standards for Mathematical Practice are embedded in daily mathematics instruction.

2013-2014 Wicomico County Public Schools Page 4 of 60 Last Revised May 24, 2013

Common Core Math Practice 1: Make sense of problems and persevere in solving them.

Common Core Math Practice 2: Reason abstractly and quantitatively.

Common Core Math Practice 3: Construct viable arguments and critique the reasoning of others.

Common Core Math Practice 4: Model with mathematics.

Common Core Math Practice 5: Use appropriate tools strategically.

Common Core Math Practice 6: Attend to precision.

Common Core Math Practice 7: Look for and make use of structure.

Common Core Math Practice 8: Look for and express regularity in repeated reasoning.

GRADE 7 COMMON CORE MATH AT-A-GLANCE2013-2014

THE BIG QUESTION IMPORTANT DATES

Unit 1 (50 Days)August 28– November 12

Pages 9 – 17

How can we express proportional relationships in

multiple ways to solve real world problems?

September 2 Labor DaySeptember 27 Professional DayOctober 18 MSEA ConventionOctober 29 End term 1

Unit 2 (25 Days)November 13 – December 20

Pages 18 – 25

How do we extend our knowledge of rational numbers to solve real world problems?

November 4 Professional DayNovember 18–22 American Education WeekNovember 27–29 Thanksgiving HolidayDecember 21-January 5 Winter Break

Unit 3 (26 Days)January 6 – February 14

Pages 26 – 35

As we build fluency, how can we use equations and

inequalities to represent and solve multi-step problems?

January 14-15 Common Core Mid-Year Assessment (score by February 5)January 20 Martin Luther King’s birthdayJanuary 22 End term 2January 24 Professional Day

Unit 4 (25 Days) February 18 – March 28

Pages 36 – 43

How can we use portions of a population to make statistical

comparisons?

February 17 Presidents’ DayMarch 4-5 MSA MathMarch 11-12 MSA ReadingMarch 28 – End of term 3

Unit 5 (19 Days)April 1 – April 30

Pages 44 – 51

How do models and experiments help us predict the

likelihood of an outcome?

March 31 – Professional DayApril 17–21 Spring Holiday

Unit 6 (21 days)May 1 – May 30Pages 52 – 59

How can we use geometric properties to solve problems?

May 26 – Memorial Day

2013-2014 Wicomico County Public Schools Page 5 of 60 Last Revised May 24, 2013

2013-2014 Wicomico County Public Schools Page 6 of 60 Last Revised May 24, 2013

7.RP.A Analyze proportional relationships and use them to solve real-world and mathematical problems. (MAJOR CLUSTER)(M)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. For example, if a person walks 12mile in each

14 hour, compute the unit rate as the complex fraction

1214

miles per hour,

equivalently 2 miles per hour.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.

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.

7.G.A Draw, construct, and describe geometrical figures and describe the relationships between them. (ADDITIONAL CLUSTER)(A)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.

2013-2014 Wicomico County Public Schools Page 7 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 1

Standard Connections, Notes, Examples Resources

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. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. (M)

Connections:In 6th grade, students used models such as a tape diagram and a double number line to look at the relationships between equivalent ratios. At this grade, students will also work with ratios specified by rational numbers,

such as 34 cups flour for every

12 stick butter. Students continue to use ratio

tables, extending this use to finding unit rates.

Notes:Students continue to work with unit rates from 6th grade; however, the

comparison now includes fractions compared to fractions. For example, if 12

gallon of paint covers 16 of a wall, then the amount of paint needed for the

entire wall can be computed by using the ratio 12 gal to

16 wall, and doubling to

write equivalent fractions – 1 gallon to 13 wall; 2 gallons to

23 wall; 3 gallons to

1 whole wall. This calculation gives 3 gallons. This standard requires only the use of ratios as fractions. Fractions may be proper or improper.

Examples:Ratio problem specified by rational numbers: Three approaches

To make Perfect Purple paint mix 12 cup blue paint with

13 cup red paint. If you

want to mix blue and red paint in the same ratio to make 20 cups of Perfect Purple paint, how many cups of blue paint and how many cups of red paint will you need?

Method 1

Background Knowledge:Navigating through Number & Operations in grades 6-8:Proportional Reasoning overview pages 75-84

Ratios and Proportions Progression Document 6-7

Lesson Ideas:Navigating through Number & Operations in grades 6-8:Using a Unit Rate to Solve ProblemsChanging Rates

AIMS Proportional Reasoning:Reading Rates by PagesPulse Rates

Illuminations: What’s your Rate?

Illuminations: Feeding Frenzy

Opus Math 7.RP.1

Moodle: Article about Tape Diagrams

Illustrative Mathematics Tasks:Molly's Run - complex fractions

2013-2014 Wicomico County Public Schools Page 8 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 1

Standard Connections, Notes, Examples Resources

“I thought about making 6 batches of purple because that is a whole number of cups of purple. To make 6 batches, I need 6 times as much blue and 6 times as much red too. That was 3 cups blue and 2 cups red and that made 5 cups purple. Then 4 times as much of each makes 20 cups purple.”

Method 2

“I found out what fraction of the paint is blue and what fraction is red. Then I found those fractions of 20 to find the number of cups of blue and red in 20 cups.”

Method 3

Like Method 2, but in tabular form, and viewed as multiplicative comparisons.

Cooking with the Whole Cup

Molly's Run - Assessment Variation

Track Practice

2013-2014 Wicomico County Public Schools Page 9 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 1

Standard Connections, Notes, Examples ResourcesSource: Progressions for the Common Core State Standards; Ratios & Proportions

7.RP.A.2. Recognize and represent proportional relationships between quantities. (M)

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

Connections:In the first section (7.RP.A.1) students use various methods for finding unit rates and examining equivalent fractions. In this section, they make the connection between proportional relationships and graphs, tables, and function rules (in the form y=kx). They should be able to state what a point (x,y) or (1,r) means in the context of a proportional relationship on a graph. The connection is made to measurement when students convert between different units of measure and write that relationship in the form of a table or graph.

Notes: Students’ understanding of the multiplicative reasoning used with

proportions continues from 6th grade. Students determine if two quantities are in a proportional relationship from a table. Ratio (rate) tables are used to build equivalent ratios/rates.

Proportional relationships can be represented symbolically (equation), graphically (coordinate plane), in a table, in diagrams, and verbal descriptions.

Proportional relationships that are written in the form of an equation should be written in the form y=kx. Proportions that are written in the form of two equivalent fractions should be reserved for unit 3.

Students recognize that graphs that are not lines through the origin and tables in which there is not a constant ratio in the entries do not represent proportional relationships.

Special attention should be spent on analyzing the points (0, 0) and (1, r) where r is the unit rate.

Ratio and proportional reasoning strategies can be extended and applied to multi-step ratio and percent problems.

Examples:1. The table below gives the price for different number of books. Do the

numbers in the table represent a proportional relationship? Students can examine the numbers to see that 1 book at 3 dollars is equivalent to 4 books for 12 dollars since both sides of the tables can be multiplied by 4. However,

Background Knowledge:Teaching Student-Centered Mathematics, Van de WalllePages 154-178

Lesson Ideas:AIMS Proportional Reasoning:Proportional Practice (a)Rectangle Ratios (a)Recognizing Proportions (a)Patterns in Equivalent Fractions (a)Mixing Measures (d)The Shadow Knows (c)

AIMS Looking at Lines:The Thick of ThingsShadow LinesReflection Takes a Turn

Thinking blocks for ratios and proportions

Inside Mathematics: Cat Food

Partial Product (Dan Meyer)

Opus Math 7.RP.2aOpus Math 7.RP.2bOpus Math 7.RP.2cOpus Math 7.RP.2d

2013-2014 Wicomico County Public Schools Page 10 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 1

Standard Connections, Notes, Examples Resources

relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

the 7 and 18 are not proportional since 1 book times 7 and 3 dollars times 7 will not give 7 books for 18 dollars. Seven books for $18 is not proportional to the other amounts in the table; therefore, there is not a constant of proportionality.

Books 1 3 4 7Cost 3 9 12 18

Students graph relationships to determine if two quantities are in a proportional relationship and interpret the ordered pairs. If the amounts from the table above are graphed (number of books, price), the pairs (1, 3), (3, 9), and (4, 12) will form a straight line through the origin (0 books cost 0 dollars), indicating that these pairs are in a proportional relationship. The ordered pair (4, 12) means that 4 books cost $12. However, the ordered pair (7, 18) would not be on the line, indicating that it is not proportional to the other pairs.The ordered pair (1, 3) indicates that 1 book is $3, which is the unit rate. The y-coordinate when x = 1 will be the unit rate. The constant of proportionality is the unit rate. Students identify this amount from tables (see example above), graphs, equations and verbal descriptions of proportional relationships.

2.

Navigating through Number & Operations in grades 6-8:Comparing Tables, Rules, and GraphsBuying PizzaExchanging Currency

Illustrative Mathematics Tasks:Art Class (Assessment Variation)(a)Art Class (Variation 1)(a)Art Class (Variation 2)(a)

Music Companies - stock context (a)

Stock Swaps Variation 2 (b,d)

Stock Swaps Variation 3 (b,d)

Robot Races (b,d)Robot Races (Assessment Variation)(b,d)

Sore Throats (Variation 1) (c)

Buying Bananas Assessment Variation (d)

Buying Coffee (d)

2013-2014 Wicomico County Public Schools Page 11 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 1

Standard Connections, Notes, Examples Resources

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. (M)

Connections:Students should be solving percent problems using models at this point. They will use equations to solve problems later, in Unit 3.

Notes:Students should be able to explain or show their work using a representation (numbers, words, pictures, physical objects, or equations) and verify that their answer is reasonable. Models help students to identify the parts of the problem and how the values are related. For percent increase and decrease, students identify the starting value, determine the difference, and compare the difference in the two values to the starting value.

Lesson Ideas:Illuminations: Grid and Percent It

Illuminations: Understanding Distance, Speed, and Time Relationships

Lesson: Increasing and Decreasing Quantities by a Percent

2013-2014 Wicomico County Public Schools Page 12 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 1

Standard Connections, Notes, Examples Resources

Examples: A sweater is marked down 33%. Its original price was $37.50. What is the

price of the sweater before sales tax?

The discount is 33% times 37.50. The sale price of the sweater is the original price minus the discount or 67% of the original price of the sweater, or

Sale Price = 0.67 x Original Price.

Gas prices are projected to increase 124% by April 2015. A gallon of gas currently costs $4.17. What is the projected cost of a gallon of gas for April 2015?A student might say: “The original cost of a gallon of gas is $4.17. An increase of

100% means that the cost will double. I will also need to add another 24% to figure out the final projected cost of a gallon of gas. Since 25% of $4.17 is about

$1.04, the projected cost of a gallon of gas should be around $9.40.”

$4.17 + 4.17 + (0.25 4.17) = 2.25 x 4.17

A shirt is on sale for 40% off. The sale price is $12. What was the original price? What was the amount of the discount?

Dueling Discounts (Dan Meyer)

Mixing Paint s

Opus Math 7.RP.3

Moodle 7.RP.3 Percent Estimates

Illustrative Mathematics Tasks:Sale!

Anna in DC - sales tax and tip

Buying Protein Bars and Magazines - sales tax with a half percent

Comparing Years in a Calendar - percent difference

Friends Meeting on Bikes - multistep ratio problem (not %)

Music Companies Variation 2 (stock context)- multistep ratio problem

Sand Under the Swing Set - area, volume, and ratio

Gotham City Taxis

2013-2014 Wicomico County Public Schools Page 13 of 60 Last Revised May 24, 2013

100%

$4.17

100%

$4.17

24%

?

Discount Sale Price - $12

Original Price (p)

A student might use a tape diagram or a double number line to solve this problem.

37.50Original Price of Sweater

33% of 37.50Discount

67% of 37.50Sale price of sweater

Math Grade 7 Common Core Unit 1

Standard Connections, Notes, Examples Resources

After eating at a restaurant, your bill before tax is $52.60 The sales tax rate is 8%. You decide to leave a 20% tip for the waiter based on the pre-tax amount. How much is the tip you leave for the waiter? How much will the total bill be, including tax and tip? Express your solution as a multiple of the bill.

The amount paid = (0.20 x $52.50) + (0.08 x $52.50) = (0.28 x $52.50)

Source: Arizona Academic Standards

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. (A)

Connections:Students will be using what they learned about proportional relationships to find the relationship between an object and its scale drawing or between drawings at different scales.

Notes:To compute unknown lengths from known lengths, students can set up proportional relationships in tables, or they can reason about how lengths compare multiplicatively. Students should: Determine the dimensions of figures when given a scale and identify the

impact of a scale on actual length (one-dimension) and area (two-dimensions).

Identify the scale factor given two figures. Using a given scale drawing, students reproduce the drawing at a different scale.

Understand scaling up/down as an application of proportional reasoning. Understand the relationship between dimensions of a scale drawing and the

original figure is proportional. Understand that areas do not scale by the same factor that relates

lengths. (Areas scale by the square of the scale factor that relates lengths, if area is measured in the unit of measurement derived from that used for length.)

Understand the difference between “within ratios” and “between ratios” when comparing or determining missing sides of similar figures (see next page).

Examples: Julie showed you the scale drawing of her room. If each 2 cm on the scale

Lesson Ideas:AIMS Proportional Reasoning:Toy Soldiers Take the CourtGrowing DesignsMovie PropsDoin’ DotsOverhead EnlargementsPicturing ProjectionsPlaying at Math

Lesson: Developing a Sense of Scale

Illuminations: Shopping Mall Math

Opus Math 7.G.1

Illustrative Mathematics Tasks:Floor Plan

2013-2014 Wicomico County Public Schools Page 14 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 1

Standard Connections, Notes, Examples Resources

drawing equals 5 ft, what are the actual dimensions of Julie’s room? What is the area of the room?

If the rectangle below is enlarged using a scale factor of 1.5, what will be the perimeter and area of the new rectangle?

Answer: The perimeter will also increase by a scale factor of 1.5. The area will increase by a scale factor that is the square of 1.5, or 2.25 (Original Area = 14; New area = 31.5; 31.5÷14=2.25)

Source: Arizona Academic Standards

Example of within ratios and between ratios:

If the two rectangles are similar, then how wide is the larger rectangle?

Use a scale factor: Find the scale factor from the small rectangle to the larger one:

2013-2014 Wicomico County Public Schools Page 15 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 1

Standard Connections, Notes, Examples Resources

Use an internal comparison: Compare the width to the height in the small rectangle. The ratio of the width to the height is the same in the large rectangle.

Source: Progressions for the Common Core State Standards; Ratios & Proportions

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2013-2014 Wicomico County Public Schools Page 17 of 60 Last Revised May 24, 2013

7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. (MAJOR CLUSTER)(M)7.NS.A.1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition

and subtraction on a horizontal or vertical number line diagram.a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two

constituents are oppositely charged.b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or

negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

7.N.S.A.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the

properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

c. Apply properties of operations as strategies to multiply and divide rational numbers.d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or

eventually repeats.

7.NS.A.3. Solve real-world and mathematical problems involving the four operations with rational numbers. [Computations with rational numbers extend the rules for manipulating fractions to complex fractions.]

7.EE.B Solve real-life and mathematical problems using numerical and algebraic expressions and equations. (MAJOR CLUSTER)(M)7.EE.B.3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole

numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For

example: If a woman making $25 an hour gets a 10% raise, she will make an additional 110 of her salary an hour, or $2.50, for a new salary of

$27.50. If you want to place a towel bar 934 inches long in the center of a door that is 27

12 inches wide, you will need to place the bar about 9

inches from each edge; this estimate can be used as a check on the exact computation.

2013-2014 Wicomico County Public Schools Page 18 of 60 Last Revised May 24, 2013

2013-2014 Wicomico County Public Schools Page 19 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 2

Standard Connections, Notes, Examples Resources

7.NS.A.1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. (M)a. Describe situations in which

opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

Connections:In 6th grade, students used number lines to identify the relationships between rational numbers (both positive and negative). They also learned about the meaning of absolute value. This is their first introduction to operations that include negative rational numbers. It is imperative that they explore these operations and find patterns in the operations before any discussion of the algorithms takes place.

Notes: Visual representations may be helpful as students begin this work; they

become less necessary as students become more fluent with the operations.

Students should start their work with integers using concrete models such as horizontal and vertical number lines and two-color counters. Models should then be tied to conceptualizations of the operations, and finally to automaticity (this likely means algorithms, though some students may visualize the meaning of the numbers and the operations for automaticity).

Examples: Use a number line to illustrate:

p – q p + (–q) (expressed as “p plus the opposite of q”) Is this equation true? p – q = p + (-q)

In what situations does the order in which you subtract two numbers not matter? (i.e., when does a−b = b−a)

On the number line below, the numbers a and b are the same distance from 0. What is a+b? Explain how you know.

Background Knowledge:Teaching Channel: Adding Integers – Large Numbers

Video Example adding integers - Kahn Academy (for teacher use, not for students)

Lesson Ideas:Illuminations: Zip, Zilch, Zero

Illuminations: Elevator Arithmetic

Lesson: Using Positive and Negative Numbers in Context

Opus Math 7.NS.1aOpus Math 7.NS.1bOpus Math 7.NS.1cOpus Math 7.NS.1d

Moodle MTMS Article: What are you Worth?

Moodle AIMS Grab a Charge

Illustrative Mathematics Tasks:Comparing Freezing Points

Distances on a Number Line

Operations on a Number Line2013-2014 Wicomico County Public Schools Page 20 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 2

Standard Connections, Notes, Examples Resources

A number line is shown below. The numbers 0 and 1 are marked on the line, as are two other numbers a and b.

Which of the following numbers is negative? Choose all that apply. Explain your reasoning.

(a)a−1(b)a−2(c )−b(d )a+b(e )a−b( f )ab+1

7.NS.A.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. (M)a. Understand that multiplication is

extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of

Connections:Students have worked in the past with multiplying and dividing whole numbers and fractions. In this section, the operations are extended to include all negative numbers, and the initial work can be done using number lines and expanded to be more abstract.

Notes: Multiplication and division of integers is an extension of multiplication

and division of whole numbers. Students recognize that when division of rational numbers is

represented with a fraction bar, each number can have a negative sign. Using long division from elementary school, students understand the

difference between terminating and repeating decimals. This understanding is foundational for work with rational and irrational numbers in 8th grade.

Students should be able to identify which fractions will terminate (the denominator of the fraction in reduced form only has factors of 2 and/or 5) This happens because only factors with denominators containing factors of 2 and/or 5 can be rewritten in tenths, hundredths, thousandths, etc…

Examples:

Lesson Ideas:Navigating through Number & Operations in grades 6-8:Linden’s AlgorithmErica’s AlgorithmKeonna’s Conjecture

MAP Task: Division

Opus Math 7.NS.2aOpus Math 7.NS.2bOpus Math 7.NS.2cOpus Math 7.NS.2d

Illustrative Mathematics Tasks:Equivalent fractions approach to non-repeating decimals (d)Repeating Decimal as Approximation (d)

2013-2014 Wicomico County Public Schools Page 21 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 2

Standard Connections, Notes, Examples Resources

rational numbers by describing real-world contexts.

c. Apply properties of operations as strategies to multiply and divide rational numbers.

d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0’s or eventually repeats.

1. Examine the family of equations. What patterns do you see? Create a model and context for each of the products. Write and model the family of equations related to 3 × 4 = 12.

Equation Number Line Model

Context

2 × 3 = 6 Selling two packages of apples at $3.00 per pack

2 × -3 = -6 Spending 3 dollars each on 2 packages of apples

-2 × 3 = -6 Owing 2 dollars to each of your three friends

-2 × -3 = 6 Forgiving 3 debts of $2.00 each

2. The distributive property states that (−1)x+1x is the same as (−1+1)x, and this is 0. It follows that (−1)x is the same as −x. Explain why, then use similar reasoning to explain why (−x)y is the same as −(xy). By the way, is it correct to say, −x is a negative number ?

3. A recipe calls for 234 cups of flour. You only have a

14 cup measuring

cup. How many times will you need to fill the 14 measuring cup?

Source: North Carolina Public Schools

7.NS.A.3. Solve real-world and mathematical problems involving the four operations with rational numbers. [Computations with rational numbers extend the rules for manipulating fractions to complex

Connections:Now that students understand all four operations with rational numbers, they should be able to solve real world and mathematical problems involving all four operations with rational numbers.

Notes:

Background Knowledge:Navigating through Number & Operations in grades 6-8:See Chapter 2 for Information

Lesson Ideas:

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Math Grade 7 Common Core Unit 2

Standard Connections, Notes, Examples Resources

fractions.] (M) Be sure to include problems involving complex fractions.

Examples:1. The three seventh grade classes at Sunview Middle School collected the most box tops for a school fundraiser, and so they won a $600 prize to share among them. Mr. Aceves' class collected 3,760 box tops, Mrs. Baca's class collected 2,301, and Mr. Canyon's class collected 1,855. How should they divide the money so that each class gets the same fraction of the prize money as the fraction of the box tops that they collected?

Navigating through Number & Operations in grades 6-8:Fraction Situations

Illuminations: Classic Middle School Fractions Problems

Task: Pizzeria Profits

Lesson: Increasing and Decreasing Quantities by a Percent

School Supplies – PARCC Prototype Task

Anne’s Family Trip – PARCC Prototype Task

Illuminations: Mangoes Problem

Illustrative Mathematics Task:Sharing Prize Money

7.EE.B.3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of

Connections : In elementary school, the properties of mathematics were introduced with whole numbers. In sixth grade, students saw the properties demonstrated with variables included.

Notes : Students start to see whole numbers, integers, and positive and

negative fractions as belonging to a single system of rational numbers, and they solve multi-step problems involving rational numbers presented in various forms.

Lesson Ideas:Performance Task: Toy Trains

7EE3: Solving Problems

7EE3: Fencing

Ann's Family Trip - operations with rational numbers

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Math Grade 7 Common Core Unit 2

Standard Connections, Notes, Examples Resources

answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. (M)(also on page 17 – do we want it in both places?)

Students use mental computation and estimation to assess thereasonableness of their solutions.

Estimation strategies for calculations with fractions and decimals extend from students’ work with whole number operations. Estimation strategies include, but are not limited to: front-end estimation with adjusting (using the highest place value

and estimating from the front end making adjustments to the estimate by taking into account the remaining amounts),

clustering around an average (when the values are close together an average value is selected and multiplied by the number of values to determine an estimate),

rounding and adjusting (students round down or round up and then adjust their estimate depending on how much the rounding affected the original values),

using friendly or compatible numbers such as factors (students seek to fit numbers together  - i.e., rounding to factors and grouping numbers together that have round sums like 100 or 1000), and

using benchmark numbers that are easy to compute (students select close whole numbers for fractions or decimals to determine an estimate)

Source: Arizona Academic Content Standards

Examples : The following statement appeared in an article about the annual

migration of the Bartailed Godwit from Alaska to New Zealand:She had flown for eight days—nonstop—covering approximately

7,250 miles at an average speed of nearly 35 miles per hour.

Students can make the rough mental estimate8 × 24 × 35 = 8 × 12 × 70  < 100 × 70 = 7000

to recognize that although this astonishing statement is in the rightballpark, the average speed is in fact greater than 35 miles per hour, suggesting that one of the numbers in the article must be wrong.

School Supplies - operations with rational numbers

Spicy Veggies – PARCC Prototype Task

TV Sales – PARCC Prototype Task

Opus Math 7.EE.3

Illustrative Mathematics Tasks:ShrinkingDiscounted BooksGuess my Number

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Math Grade 7 Common Core Unit 2

Standard Connections, Notes, Examples ResourcesSource: Progressions for the Common Core State Standards; Expressions & Equations

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7.NS.A Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. (MAJOR CLUSTER)(M)7.NS.A.3. Solve real-world and mathematical problems involving the four operations with rational numbers. [Computations with

rational numbers extend the rules for manipulating fractions to complex fractions.]

7.EE.A Use properties of operations to generate equivalent expressions. (MAJOR CLUSTER)(M)7.EE.A.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.7.EE.A.2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the

quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

7.EE.B Solve real-life and mathematical problems using numerical and algebraic expressions and equations. (MAJOR CLUSTER)(M)7.EE.B.3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole

numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

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.a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers.

Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

7.RP.A Analyze proportional relationships and use them to solve real-world and mathematical problems. (MAJOR CLUSTER)(M)7.RP.A.2. Recognize and represent proportional relationships between quantities.

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.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.

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Math Grade 7 Common Core Unit 3

Standard Connections, Notes, Examples Resources

7.EE.A.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. (M)

Connections:This is a continuation of work from 6th grade using properties of operations and combining like terms. Students apply properties of operations and work with rational numbers (integers and positive/negative fractions and decimals) to write equivalent expressions.

Notes: Students should recognize subtraction as the addition of the opposite, so

that when they are subtracting one expression from another, they know to add the opposite of the second expression.

Students should be able to multiply a monomial expression (even if it’s a constant) by a polynomial expression, recognizing that they must use the distributive property to multiply the term by each term inside the parentheses.

Examples: Write an equivalent expression for 3 ( x+5 )−2. Suzanne thinks the two expressions 2 (3a−2 )+4a and 10a−2 are

equivalent. Is she correct? Explain why or why not. Write equivalent expressions for: 3a+12.

Possible solutions might include factoring as in 3(a+4 ), or other expressions such as a+2a+7+5.

A rectangle is twice as long as it is wide. One way to write an expression to find the perimeter would be w+w+2w+2w. Write the expression in two other ways.

Solution: 6w or 2w + 2(2w).

An equilateral triangle has a perimeter of 6 x+15. What is the length of each

Background Information:Progressions Document: Expressions and Equations 6-8

Lesson Ideas:Examples for teaching 7EE1 – Learn Zillion

Steps to Solving Equations Lesson

Opus Math 7.EE.1

Introduction to Variables – “Expressions and Equations in the Real World” – Teaching Channel

Video: Simplifying Expressions through Think-Pair-Share – Teaching Channel

Moodle: NCTM Navigating through Algebra in Grades 6-8 : Plotting Land

Illustrative Mathematics Tasks:Equivalent Expressions?Miles to KilometersWriting ExpressionsGuess My Number

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Math Grade 7 Common Core Unit 3

Standard Connections, Notes, Examples Resources

of the sides of the triangle?Solution: 3(2 x+5) can represent the whole perimeter, therefore each side is 2 x+5 units long.

7.EE.A.2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” (M)

Notes:Students understand the reason for rewriting an expression in terms of a contextual situation. For example, students understand that a 20% discount is the same as finding 80% of the cost (.80c).

Examples:1. Which expression is not equivalent to the other three?

2(1+2b+3a)2(1+2a)+2(a+2b)6a+2+4b2(3a+1)+4b+1

2. Jamie and Ted both get paid an equal hourly wage of $9 per hour. This week, Ted made an additional $27 dollars in overtime. Write an expression that represents the weekly wages of both if J = the number of hours that Jamie worked this week and T = the number of hours Ted worked this week? Can you write the expression in another way?

One student might say: To find the total wage, I would first multiply the number of hours Jamie worked by 9. Then I would multiply the number of hours Ted worked by 9. I would add these two values with the $27 overtime to find the total wages for the week. The student would write the expression9J + 9T + 27.

Another student might say: To find the total wages, I would add the number of hours that Ted and Jamie worked. I would multiply the total number of hours worked by 9. I would then add the overtime to that value to get the total wages for the week. The student would write the expression 9(J + T) + 27

Background Knowledge:Teaching Student-Centered Mathematics Grades 5-8, Van de Walle, Chapter 9 “Algebraic Reasoning.”

Lesson Ideas:Opus Math 7.EE.2

Activities in Moodle:Apple Tree Problem, Modeling with Cuisenaire Rods,Tiling around the Garden

AIMS Solving Equations“What’s in the Bank?"

AIMS Looking at Lines:Pattern Block FunctionsRising Tower

Navigating through Number & Operations in grades 6-8:Pledge Drive

Illustrative Mathematics Tasks:Miles to Kilometers

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Math Grade 7 Common Core Unit 3

Standard Connections, Notes, Examples Resources

A third student might say: To find the total wages, I would need to figure out how much Jamie made and add that to how much Ted made for the week. To figure out Jamie’s wages, I would multiply the number of hours she worked by 9. To figure out Ted’s wages, I would multiply the number of hours he worked by 9 and then add the $27 he earned in overtime. My final step would be to add Jamie and Ted wages for the week to find their combined total wages. The student would write the expression (9J) + (9T + 27).

3. Given a square pool with sides of length s, as shown in the picture, write as many different expressions as you can, to find the total number of tiles in the border. Explain how each of the expressions relates to the diagram and demonstrate that the expressions are equivalent. Which expression do you think is most useful? Explain your thinking.

Source: Arizona Academic Standards

Moodle: Tiling the Pool

See NCTM Navigating Through Algebra in Grades 6-8, Tiling Tubs activity, for more information about this question.

Resource for Pool Problem with solutions

7.NS.A.3. Solve real-world and mathematical problems involving the four operations with rational numbers. [Computations with rational numbers extend the rules for manipulating fractions to complex fractions.] (M)

Connections:In Unit 2, students learned how to solve real world problems using operations with rational numbers. In this unit, we are using what they learned to solve real world problems and compare the solutions to solutions obtained by writing equations (see: “Compare to Arithmetic Solutions” in web).

Notes:This is an introduction to solving equations, standard 7.EE.B.3

Examples:1. You might take the “tiling the pool” problem from the last section, where

students figured out that the expression for the number of tiles is 4s + 4, and

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Math Grade 7 Common Core Unit 3

Standard Connections, Notes, Examples Resources

ask them a question such as, “If you have 60 tiles, how large a pool could you tile?” Students figure out that they have to subtract 4 from 60 (for the corners), and then divide that number by 4 for the 4 sides of the pool. Relate this to an equation in which students solve 4s + 4 = 60 (in order to solve, subtract 4 from both sides, then divide by 4). This would work with many of the expressions problems you did in the last section.

7.EE.B.3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. (M)

Connections:The focus here is on using equations to solve word problems. Students should see the connection between solving a problem arithmetically and writing and solving an equation to answer the same question.

Notes:Students solve contextual problems using rational numbers. Students convert between fractions, decimals, and percents as needed to solve the problem. Students use estimation to justify the reasonableness of answers.

Examples:1. The youth group is going on a trip to the state fair. The trip costs $52. Included in that price is $11 for a concert ticket and the cost of 2 passes, one for the rides and one for the game booths. Each of the passes cost the same price. What is the price of one pass?

Students may solve arithmetically, finding that if $52 was spent in all, and $11 of that was for a ticket, then the remainder, or $41, was spent on the two passes. When that is divided by two, we see that each pass cost $20.50.

Write an equation representing the cost of the trip and determine the price of one pass.

When student solve this problem using an equation, they see that they are using the same steps they used when they solve the problem arithmetically.

2x + 11 = 52 2x = 41 x = $20.5

Background Knowledge:Teaching Student-Centered Mathematics Grades 5-8, Van de Walle, Chapter 9 “Algebraic Reasoning.”

Lesson Ideas:Opus Math 7.EE.3

Illuminations: Everything Balances Out in the End

Moodle: Installing a Mirror

AIMS Solving Equations“Widgets”“ESP: Extraordinary Solution Prediction”

Teaching Channel: Expression & Equations in the Real World (Video related to ESP activity above) – Long video, but watch at least the first few minutes!

Illustrative Mathematics Tasks:Guess my NumberAnna in D.C.

2013-2014 Wicomico County Public Schools Page 31 of 60 Last Revised May 24, 2013

x x 11

52

Math Grade 7 Common Core Unit 3

Standard Connections, Notes, Examples Resources

Source: Arizona Academic Content StandardsDiscounted BooksShrinking

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. (M)a. Solve word problems leading to

equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least

Notes:Students graph inequalities and make sense of the inequality in context. Inequalities may have negative coefficients. Problems can be used to find a maximum or minimum value when in context.

Examples: Amie had $26 dollars to spend on school supplies. After buying 10 pens,

she had $14.30 left. How much did each pen cost? The sum of three consecutive even numbers is 48. What is the smallest of

these numbers?

Solve:

54n+5=20

Florencia has at most $60 to spend on clothes. She wants to buy a pair of jeans for $22 dollars and spend the rest on t-shirts. Each t-shirt costs $8. Write an inequality for the number of t-shirts she can purchase.

Steven has $25 dollars. He spent $10.81, including tax, to buy a new DVD. He needs to set aside $10.00 to pay for his lunch next week. If peanuts cost $0.38 per package including tax, what is the maximum number of packages that Steven can buy? Write an equation or inequality to model the situation. Explain how you

determined whether to write an equation or inequality and the properties of the real number system that you used to find a solution.

Solve

12x+3>2

and graph your solution on a number line.

Source: Arizona Academic Content Standards

Background Knowledge:Van de Walle, Teaching Student-Centered Mathematics Grades 5-8, Chapter 9 “Algebraic Reasoning.”

Lesson Ideas:Opus Math 7.EE.4aOpus Math 7.EE.4b

AIMS Solving Equations“What’s in a Case?”“Jumping to Solutions”“Backtracking ESP”“Manipulating ESP”

Illustrative Mathematics Tasks:Gotham City Taxis (a)Fishing Adventures 2 (b)Sports Equipment Set (b)

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Math Grade 7 Common Core Unit 3

Standard Connections, Notes, Examples Resources

$100. Write an inequality for the number of sales you need to make, and describe the solutions.

7.RP.A.2. Recognize and represent proportional relationships between quantities. (M)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.

Connections:In Unit 1, students wrote equations to express proportional relationships using the form y=kx. In this unit, students are SOLVING equations that express proportional relationships, and seeing the connections between

equations in the form y=kx and equations in the form ab= cd .

Notes: Students connect their work with equations to their work with tables and

diagrams from Unit 1. To see the rationale for cross-multiplying, note that when the fractions are

given the common denominator 2 × T , then the numerators become 5 × T and 2 × 100, respectively. Once the denominators are equal, the fractions are equal only when their numerators are equal, so 5×T must equal 2 × 100 for the unit rates to be equal. This is why we can solve the equation 5 × T = 2 × 100 to find the amount of time it will take for Seth to run 100 meters.

Source: Progressions for the Common Core State Standards, Ratios & Proportions

Examples:1. If Seth runs 5 meters every 2 seconds, then how long will it take Seth to run 100 meters at that rate? The traditional method is to formulate an equation, 52=100T , cross-multiply, and solve the resulting equation to solve the problem.

If 52 and

100T are viewed as unit rates obtained from the equivalent ratios 5 : 2

and 100 : T , then they must be equivalent fractions because equivalent ratios have the same unit rate.2. Students should recognize nonexamples such as: If Josh is 10 and Reina is 7, how old will Reina be when Josh is 20? We cannot solve this problem with

Background Knowledge:Video Explaining Work Problems (see Example 2)

Khan Academy video on proportions

Lesson Ideas:Illuminations: How Much is a Million?

Inside Mathematics: Proportion and Non-Proportion Situations

Inside Mathematics: Photographs

Inside Mathematics: Cereal

Inside Mathematics: Lawn Mowing

Illustrative Mathematics Tasks:Two-School Dance - multistep ratio problem

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Math Grade 7 Common Core Unit 3

Standard Connections, Notes, Examples Resources

the proportion 107

=20R because it is not the case that for every 10 years that

Josh ages, Reina ages 7 years. Instead, when Josh has aged 10 another years, Reina will as well, and so she will be 17 when Josh is 20.3. Students should recognize that work problems cannot be solved with proportions: If it takes 2 people 5 hours to paint a fence, how long will it take 4 people to paint a fence of the same size (assuming all the people work at the

same steady rate)? We cannot solve this problem with the proportion 25= 4H

because it is not the case that for every 2 people, 5 hours of work are needed to paint the fence. When more people work, it will take fewer hours. With twice as many people working, it will take half as long, so it will take only 2.5 hours for 4 people to paint a fence. Students must understand the structure of the problem, which includes looking for and understand the roles of “for every,” “for each,” and “per.”3. A common error in setting up proportions is placing numbers in incorrect locations. This is especially easy to do when the order in which quantities are stated in the problem is switched within the problem statement. For example, the second of the following two problem statements is more difficult than the first because of the reversal.

“If a factory produces 5 cans of dog food for every 3 cans of cat food, then when the company produces 600 cans of dog

food, how many cans of cat food will it produce?”

“If a factory produces 5 cans of dog food for every 3 cans of cat food, then how many cans of cat food will the company

produce when it produces 600 cans of dog food?”

Source: Progressions for the Common Core Standards, Ratios & Proportions

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,

Connections:In Unit 1, students solved percent problems using models. In this unit, the focus should be on writing and solving equations in order to solve these percent problems.

Notes:Students extend their work to solving multistep ratio and percent problems.

Lesson Ideas:Inside Mathematics: Mixing Paints

Inside Mathematics: Cat Food

Illustrative Mathematics 2013-2014 Wicomico County Public Schools Page 34 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 3

Standard Connections, Notes, Examples Resources

percent error. (M) Problems involving percent increase or percent decrease require careful attention to the referent whole.

Examples:1. Consider the difference in these two percent decrease and percent increase problems:

Skateboard problem 1. After a 20% discount, the price of a SuperSick skateboard is $140. What was the price before the discount?

Skateboard problem 2. A SuperSick skateboard costs $140 now, but its price will go up by 20%. What will the new price be after the

increase?

The solutions to these two problems are different because the 20% refers to different wholes or 100% amounts. In the first problem, the 20% is 20% of the larger pre-discount amount, whereas in the second problem, the 20% is 20% of the smaller pre-increase amount. Notice that the distributive property is implicitly involved in working with percent decrease and increase. For example, in the first problem, if x is the original price of the skateboard (in dollars), then after the 20% discount, the new price is x−20% ∙ x. The distributive property shows that the new price is 80 % ∙ x:

x−20% ∙ x¿100 % ∙ x−20 % ∙ x¿ (100 %−20 % ) ∙ x

¿80% ∙ x

Tasks:Finding a 10% Increase

Chess Club - percent increase and decrease

Selling Computers - % increase

Tax and Tip

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Math Grade 7 Common Core Unit 3

Standard Connections, Notes, Examples Resources

Source: Progressions for the Common Core Standards, Ratios & Proportions

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7.SP.A Use random sampling to draw inferences about a population. (SUPPORTING CLUSTER)(S)7.SP.A.1. Understand that statistics can be used to gain information about a population by examining a sample of the population;

generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

7.SP.A.2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

7.SP.B Draw informal comparative inferences about two populations. (ADDITIONAL CLUSTER)(A)7.SP.B.3. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the

difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

7.SP.B.4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

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Math Grade 7 Common Core Unit 4

Standard Connections, Notes, Examples Resources

7.SP.A.1. Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. (S)

Connections:This section is closely related to the first unit, on ratios & proportions. Because random samples can be expected to be approximately representative of the full population, one can imagine selecting many samples of that same size until the full population is exhausted, each with approximately the same characteristics. Therefore the ratio of the size of a portion having a certain characteristic to the size of the whole should be approximately the same for the sample as for the full population.Source: Progressions for the Common Core State Standards; Ratios & Proportions

Notes: Students recognize that it is difficult to gather statistics on an entire

population. Instead a random sample can be representative of the total population and will generate valid results.

Random sampling tends to produce representative samples. A random sample must be used in conjunction with the population to get accuracy. For example, a random sample of elementary students cannot be used to give a survey about the prom.

Finding a valid, representative sample will enable valid inferences to be made about a population. Students should understand what it means to have a valid, random sample representative of a population(s).

Examples: The school food service wants to increase the number of students who eat

hot lunch in the cafeteria. The student council has been asked to conduct a survey of the student body to determine the students’ preferences for hot lunch. They have determined two ways to do the survey. The two methods are listed below. Identify whether each type of sample is a random sample, and whether it is representative of the population. Which survey option should the student council use and why?1.Write all of the students’ names on cards and pull them out in a drawing to

determine who will complete the survey.2.Survey the first 20 students that enter the lunch room.

Background Information:Progressions document: Statistics and Probability 6-8

Lesson Ideas:MARS Task: Ducklings

MAP Task: Counting Trees

MAP Task: Candy Bars

Opus Math 7.SP.1

Dana Center: Estimating Mean State Area

Cool Places to get Data: Reaction Time Test for Data

Collection NCTM Data Sets Random Sampler from

Census at Schools (Amstat.org)

Illustrative Mathematics Task:Mr. Briggs’ Class Likes MathElection PollElection Poll Variation 2Election Poll Variation 3

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Math Grade 7 Common Core Unit 4

Standard Connections, Notes, Examples ResourcesSource: Arizona Academic Content Standards

Connection of Statistics & Probability to Ratios & Proportions:

Source: Progressions for the Common Core State Standards; Ratios & Proportions

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Math Grade 7 Common Core Unit 4

Standard Connections, Notes, Examples Resources

7.SP.A.2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. (S)

Notes: Students collect and use multiple samples of data to answer question(s)

about a population. Issues of variation in the samples should be addressed. Use proportional reasoning to make estimates relating to a population on

the basis of a sample. Inferences about a population are only valid if the sample is random and

representative. Having multiple samples for the same population allows for gauging the

variation of estimates or predictions.

Examples: Below is the data collected from two random samples of 100 students

regarding student’s school lunch preferences. Make at least two inferences based on the results.

Source: Arizona Academic Content Standards

Lesson Ideas:

MARS Task: Counting Trees

Opus Math 7.SP.2

Reese’s Pieces Activity: Sampling from a Population

Go Fish! (Capture Recapture) Teacher Notes

Go Fish! (Capture Recapture) Student Worksheet

Election PollElection Poll Variation 2Election Poll Variation 3

Estimating Mean State Area

Illustrative Mathematics Task:Valentine Marbles

7.SP.B.3. Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of

Notes: This is the students’ first experience with comparing two data sets. Students

build on their understanding of graphs, mean, median, Mean Absolute Deviation (M.A.D.) and interquartile range from 6th grade. Mean absolute deviation is an element of a data set that is the absolute difference between that element and a given point.

Measures of center can be used to compare data and measure variability between data sets. Use the mean (or other measures of center) to find the difference between the centers as a measure of variability.

Background Information:Video about Mean Absolute DeviationAnother Video about M.A.DVideo about Interquartile Range

Lesson Ideas:MARS Task: Temperatures

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Math Grade 7 Common Core Unit 4

Standard Connections, Notes, Examples Resources

players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. (A)

Data displays are used to visually compare data sets and draw informal comparative inferences. Box plots are way to show measures of variability such as the range (Other data displays may highlight other measures of variability).

Use observations to about mean differences between two or more samples to make conjectures about the populations from which the samples were taken.

Examples:A pharmaceutical company is running a trial of a new medicine designed to relieve headaches. The company recruits 20 participants and randomly divides them into two groups. They give Group A the medicine, and they give Group B a placebo (a pill without any medical contents). The patients record the number of minutes from when they took the pill to when the headache ends. Use measures of center and variability to compare the patients' responses to the two pills.

Opus Math 7.SP.3

Mean Absolute Deviation (Glencoe Lesson Seed)

Illustrative Mathematics Task:College AthleticsOffensive Linemen

7.SP.B.4. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example,

Notes: For random samples, students should understand that medians and means

computed from samples will vary from sample to sample and that making informed decisions based on such sample statistics requires some knowledge of the amount of variation to expect. Just as for proportions, a good way to gain this knowledge is through simulation, beginning with a

Lesson Ideas:Opus Math 7.SP.4

Random Sampler from Census at Schools (Amstat.org)

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Math Grade 7 Common Core Unit 4

Standard Connections, Notes, Examples Resources

decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. (A)

population of known structure. Measures of center include mean, median, and mode. The measures of

variability include range, mean absolute deviation, and interquartile range.

Examples: The two data sets below depict random samples of the housing prices sold in

the King River and Toby Ranch areas of Arizona. Based on the prices below which measure of center will provide the most accurate estimation of housing prices in Arizona? Explain your reasoning.o King River area

{1.2 million, 242000, 265500, 140000, 281000, 265000, 211000}o Toby Ranch homes

{5million, 154000, 250000, 250000, 200000, 160000, 190000}Source: Arizona Academic Content Standards

Ms. G is a college track coach who is curious about the appropriate amount of distance training for her athletes. Ms. G decides to run an experiment. She randomly divides her runners into two groups. The first group does about 20 miles of distance training per week, and the second group does about 30 miles of distance training per week. At the end of the two weeks, all of the runners compete in a mile run, yielding the results below. Based on the center and variability of each distribution, what inferences can you draw about the two populations?

Illustrative Mathematics Tasks:College AthleticsOffensive Linemen

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Math Grade 7 Common Core Unit 4

Standard Connections, Notes, Examples Resources

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2013-2014 Wicomico County Public Schools Page 45 of 60 Last Revised May 24, 2013

7.SP.C Investigate chance processes and develop, use, and evaluate probability models. (SUPPORTING CLUSTER)(S)7.SP.C.5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event

occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

7.SP.C.6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

7.SP.C.7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities

of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

7.SP.C.8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for

which the compound event occurs.b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event

described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to

approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

2013-2014 Wicomico County Public Schools Page 46 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 5

Standard Connections, Notes, Examples Resources

7.SP.C.5. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. (S)

Notes: “What is my chance of getting the correct answer to the next multiple

choice question?” is not a probability question in the relative frequency sense. “What is my chance of getting the correct answer to the next multiple choice question if I make a random guess among the four choices?” is a probability question because the student could set up an experiment of multiple trials to approximate the relative frequency of the outcome. And two students doing the same experiment will get nearly the same approximation.

This is students’ first formal introduction to probability. Students recognize that all probabilities are between 0 and 1, inclusive, and that the sum of all possible outcomes is 1. The probability of any single event can be recognized as a fraction. The closer the fraction is to 1, the greater the probability the event will occur. Larger numbers indicate greater likelihood.

Probability can be expressed in terms such as impossible, unlikely, likely, or certain or as a number between 0 and 1 as illustrated on the number line.

Examples: The container below contains 2 gray, 1 white, and 4 black marbles.

Without looking, if you choose a marble from the container, will the probability be closer to 0 or to 1 that you will select a white marble? A gray marble? A black marble? Justify each of your predictions.

Lesson Ideas:Inside Mathematics: Evaluating Statements about Probability

Inside Mathematics: Will It Happen?

MARS Task: Counters

Marble ManiaRandom Drawing Tool

Opus Math 7.SP.5

2013-2014 Wicomico County Public Schools Page 47 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 5

Standard Connections, Notes, Examples Resources

Source: Arizona Academic Content Standards

7.SP.C.6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. (S)

Notes: Students collect data from a probability experiment, recognizing that

as the number of trials increases, the experimental probability approaches the theoretical probability (the Law of Large Numbers).

The focus of this standard is relative frequency -- The relative frequency is the observed number of successful events for a finite sample of trials. Relative frequency is the observed proportion of successful events.

It must be understood that the connection between relative frequency and probability goes two ways. If you know the structure of the generating mechanism (e.g., a bag with known numbers of red and white chips), you can anticipate the relative frequencies of a series of random selections (with replacement) from the bag. If you do not know the structure (e.g., the bag has unknown numbers of red and white chips), you can approximate it by making a series of random selections and recording the relative frequencies.

Different representations of a sample space:

All the possible outcomes of the toss of two coins can be represented as an organized list, table, or tree diagram. The sample space becomes a probability model when a probability for each simple event is specified.

Background Knowledge:

Lesson Ideas:Yellow Starbursts (Dan Meyer)

Opus Math 7.SP.6

Rock, Paper, Scissors

Rolling for the Big One

The Game of Skunk

Illustrative Mathematics Task:Rolling DiceTossing Cylinders

2013-2014 Wicomico County Public Schools Page 48 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 5

Standard Connections, Notes, Examples Resources

Examples: Each group receives a bag that contains 4 green marbles, 6 red

marbles, and 10 blue marbles. Each group performs 50 pulls, recording the color of marble drawn and replacing the marble into the bag before the next draw. Students compile their data as a group and then as a class. They summarize their data as experimental probabilities and make conjectures about theoretical probabilities (How many green draws would you expect if you were to conduct 1000 pulls? 10,000 pulls?).

Students create another scenario with a different ratio of marbles in the bag and make a conjecture about the outcome of 50 marble pulls with replacement. (An example would be 3 green marbles, 6 blue marbles, 3 blue marbles.)

Source: Arizona Academic Content Standards

2013-2014 Wicomico County Public Schools Page 49 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 5

Standard Connections, Notes, Examples Resources

7.SP.C.7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. (S)a. Develop a uniform probability model

by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

Notes: Probabilities are useful for predicting what will happen over the long

run. Using theoretical probability, students predict frequencies of outcomes. Students recognize an appropriate design to conduct an experiment with simple probability events, understanding that the experimental data give realistic estimates of the probability of an event but are affected by sample size.

The product rule for counting outcomes for chance events should be used in finite situations like tossing two or three coins or rolling two number cubes. There is no need to go to more formal rules for permutations and combinations at this level. Students should gain experience in the use of diagrams, especially trees and tables, as the basis for organized counting of possible outcomes from chance processes.

Examples: If you choose a point in the square, what is the probability that it is

not in the circle?

Source: Arizona Academic Content Standards

When you toss a thumbtack in the air, there are only two possibilities for it landing – it will land point up, or point down. Is the probability of a tossed thumbtack landing point up ½? How could you find out whether it is or not?

Source: Progressions for the Common Core State Standards: Stats & Probability

Background Knowledge:Khan Academy: Probability of Rolling DoublesKhan Academy: Probability Through Counting Outcomes

Lesson Ideas:MARS Task: Fair Game?

Opus Math 7.SP.7aOpus Math 7.SP.7b

Illuminations: What Are My Chances?

Illuminations: Stay or Switch?

Illustrative Mathematics Task:How many buttons? (a)

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Math Grade 7 Common Core Unit 5

Standard Connections, Notes, Examples Resources

7.SP.C.8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. (S)a. Understand that, just as with simple

events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.

c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

Notes: Students use tree diagrams, frequency tables, and organized lists, and

simulations to determine the probability of compound events. Probabilities are useful for predicting what will happen over the long

run. Using theoretical probability, students predict frequencies of outcomes. Students recognize an appropriate design to conduct an experiment with simple probability events, understanding that the experimental data give realistic estimates of the probability of an event but are affected by sample size.

Examples: Students conduct a bag pull experiment. A bag contains 5 marbles.

There is one red marble, two blue marbles and two purple marbles. Students will draw one marble without replacement and then draw another. What is the sample space for this situation? Explain how you determined the sample space and how you will use it to find the probability of drawing one blue marble followed by another blue marble.

Suppose, over many years of records, a river generates a spring flood about 40% of the time. Based on these records, what is the chance that it will flood for at least three years in a row sometime during the next five years?

Show all possible arrangements of the letters in the word FRED using a tree diagram. If each of the letters is on a tile and drawn at random, what is the probability that you will draw the letters F-R-E-D in that order? What is the probability that your “word” will have an F as the first letter?

Background Knowledge:Khan Academy: LeBron Asks: What is the probability of making 10 free throws in a row?

Lesson Ideas:Memory Games

Rolling for the Big One

Opus Math 7.SP.8aOpus Math 7.SP.8bOpus Math 7.SP.8c

Rock, Paper, Scissors

The Game of Skunk

Sticks and Stones

Cereal Box Problem

Illustrative Mathematics Tasks:Rolling TwiceWaiting TimesSitting Across from Each Other (a, b)Sticks and StonesCereal Box Problem

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Math Grade 7 Common Core Unit 5

Standard Connections, Notes, Examples Resources

Source: Arizona Academic Content Standards

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2013-2014 Wicomico County Public Schools Page 53 of 60 Last Revised May 24, 2013

7.G.A Draw, construct, and describe geometrical figures and describe the relationships between them. (ADDITIONAL CLUSTER)(A)7.G.A.2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on

constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

7.G.A.3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

7.G.B Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. (ADDITIONAL CLUSTER)(A)7.G.B.4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of

the relationship between the circumference and area of a circle.7.G.B.5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple

equations for an unknown angle in a figure.7.G.B.6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects

composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

2013-2014 Wicomico County Public Schools Page 54 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 6

Standard Connections, Notes, Examples Resources

7.G.A.2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. (A)

Connections:Students understand the characteristics of angles that create triangles. For example, can a triangle have more than one obtuse angle? Will three sides of any length create a triangle? Students recognize that the sum of the two smaller sides must be equal to or larger than the third side.

Notes: This can be an informal exploration into the relationships of sides and

angles of triangles. This is the exploration phase done prior to more formalized work in high school around the relationships between shapes and their angles and sides. For example, students may explore the Triangle Sum Theorem, SAS, SSS, etc.

Depending on the attributes given, a unique triangle, more than one triangle, or no triangle can be the result.

There are certain given conditions that will produce only one, unique triangle. Some given conditions may produce more than one triangle or no triangle at all.

Examples: Is it possible to draw a triangle with a 90˚ angle and one leg that is 4 inches

long and one leg that is 3 inches long? If so, draw one. Is there more than one such triangle?

Draw a triangle with angles that are 60 degrees. Is this a unique triangle? Why or why not?

Can you draw a triangle with sides that are 13 cm, 5 cm and 6cm?

Draw a quadrilateral with one set of parallel sides and no right angles.

Background Knowledge:Progressions document: Geometry K-6

Lesson Ideas:Problem of the Month: Polly Gone

Opus Math 7.G.2

Van de Walle Triangle sorting activity

2013-2014 Wicomico County Public Schools Page 55 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 6

Standard Connections, Notes, Examples Resources

Draw an isosceles triangle with only one 80 degree angle. Is this the only possibility or can you draw another triangle that will also meet these conditions?

7.G.A.3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. (A)

Notes: Visualize the cross-sections of rectangular prisms and rectangular

pyramids. While in elementary grades we analyze the 2D faces of 3D figures. Now students will investigate the 2D shapes found inside 3D figures.

Investigate and compare 2D cross-sections parallel to the base of a 3D figure. For example the cross-sections parallel to the base of a rectangular prism are all congruent rectangles. The cross-sections parallel to the base of a rectangular prism are all rectangles, but they get smaller as they are sliced closer to the vertex

Examples:Below are three cross sections of a pyramid with a square base:

If the pyramid is cut with a plane (green) parallel to the base, the intersection of the pyramid and the plane is a square cross section (red).

If the pyramid is cut with a plane (green) passing through the top vertex and perpendicular to the base, the intersection of the pyramid and the plane is a triangular cross section (red).

Lesson Ideas:Opus Math 7.G.3

Activity with Clay Molds

Interactive Slicing Activity

Shodor Website on Cross-Sections

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Math Grade 7 Common Core Unit 6

Standard Connections, Notes, Examples Resources

If the pyramid is cut with a plane (green) perpendicular to the base, but not through the top vertex, the intersection of the pyramid and the plane is a trapezoidal cross section (red).

7.G.B.4. Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. (A)

Notes: Students understand the relationship between radius and diameter.

Students also understand the ratio of circumference to diameter can be expressed as Pi. Building on these understandings, students generate the formulas for circumference and area.

Develop and utilize formulas for the area and circumference of circles.

Examples: Students measure the circumference and diameter of several circular

objects in the room (clock, trash can, door knob, wheel, etc.). Students organize their information and discover the relationship between circumference and diameter by noticing the pattern in the ratio of the measures. Students write an expression that could be used to find the circumference of a circle with any diameter and check their expression on other circles.

Explore the relationship between the circumference and area of a circle by cutting the circle into segments and laying them side-by-side in a shape that is close to the rectangle. How can you use the circumference and the radius or diameter to find the area? How does this relate to the area of a rectangle?

Lesson Ideas:AIMS Looking at Lines:Functions in Circles

Inside Mathematics: Pizza Crusts

Inside Mathematics: Which is Bigger?

Circular Reasoning (Circumference vs. Diameter)

MAP Task: Historic Bicycle

Popcorn Picker

Illuminations Lesson: Apple Pi (Relationship between Circumference & Diameter)

Penny Circle (Dan Meyer)Coffee Traveler (Dan Meyer)Pizza Doubler (Dan Meyer)Coin Carpet (Dan Meyer)

2013-2014 Wicomico County Public Schools Page 57 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 6

Standard Connections, Notes, Examples Resources

Source: Delaware Learning Progressions

Opus Math 7.G.4

Illustrative Mathematics Tasks:Eight Circles

Measuring the Area of a Circle

7.G.B.5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. (A)

Notes: Angle relationships that can be explored include but are not limited to:

Supplementary & complementary angles Adjacent angles are angles that share a common ray from the vertex. Same-side (consecutive) interior and same-side (consecutive) exterior

angles are supplementary. The relationships between the angles formed by intersecting lines

(vertical angles). Use these relationships to solve problems.

Examples: Find the measure of angle b. Then, find the measure of angle a.

Note: Not drawn to scale.Solution:Because, the 45°, 50° angles and b form are supplementary angles, the measure of angle b would be 85°. The measures of the angles of a triangle

Background Knowledge:Learn Zillion Videos: Use facts about angles to solve simple equations

Lesson Ideas:Opus Math 7.G.5

IXL: Identify angles

IXL: Find measures of angles

Sketchpad Activity on Complementary & Supplementary Angles

Complementary Angles GSP Sketch

Complementary &

2013-2014 Wicomico County Public Schools Page 58 of 60 Last Revised May 24, 2013

Math Grade 7 Common Core Unit 6

Standard Connections, Notes, Examples Resources

equal 180° so 75° + 85°+ a = 180°. The measure of angle a would be 20°.

Write and solve an equation to find the measure of angle x.

Write and solve an equation to find the measure of angle x.

Supplementary Angles GSP Sketch

Exterior Angles of Polygon GSP Sketch

7.G.B.6. Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. (A)

Connections:Students continue work from 5th and 6th grade to work with area, volume and surface area of two-dimensional and three-dimensional objects. (including composite shapes)

Notes: “Know the formula” does not mean memorization of the formula. To

“know” means to have an understanding of why the formula works and how the formula relates to the measure (area and volume) and the figure. This understanding should be for all students.

Students should be building on their knowledge of area, to see that the volume of a prism is the area of the base times the height. This is true for triangular and rectangular prisms.

Students will not work with cylinders, as circles are not polygons. Surface area formulas are not the expectation with this standard. Building

on work with nets in the 6th grade, students should recognize that finding the area of each face of a three-dimensional figure and adding the areas will give the surface area. No nets will be given at this level.

Lesson Ideas:Inside Mathematics: Parallelograms

Inside Mathematics: Gold Rush (Maximizing Area)

Moodle: Packing Up Popcorn

Illuminations: Tetrahedral Kites

Illuminations: Popcorn, Anyone?

Illuminations: Area Contractor

MARS Task: Winter Hat

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Math Grade 7 Common Core Unit 6

Standard Connections, Notes, Examples Resources

Examples: A cereal box is a rectangular prism. What is the volume of the cereal box?

What is the surface area of the cereal box? (Hint: Create a net of the cereal box and use the net to calculate the surface area.) Make a poster explaining your work to share with the class.

Find the area of a triangle with a base length of 3 units and a height of 4 units.

Find the area of the isosceles trapezoid shown below using the formulas for rectangles and triangles.

Choose one of the figures shown below and write a step by step procedure for determining the area. Find another person that chose the same figure as you did. How are your procedures the same and different? Do they yield the same result?

MAP Task: Designing a Sports Bag

MAP Task: Octagon Tiles

Opus Math 7.G.6

Holes (Dan Meyer)

Illustrative Mathematics Tasks:Sand Under the Swing Set

2013-2014 Wicomico County Public Schools Page 60 of 60 Last Revised May 24, 2013