7th Grade - Gateway 2

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

According to the MathLinks Program Information, in Focus, Coherence and Rigor, “Conceptual understanding, the bedrock of a MathLinks course, frequently drives the other two components of rigor. It is a MathLinks philosophy to make sure all students have the opportunity to make meaning for every concept presented, and we focus on the conceptual development of Big Ideas in depth and make them plausible through investigations, activities, and practice. This is commonly done throughout lessons in all units, oftentimes with the help of teacher Lesson Notes and Slide Decks. Opportunities for independent work within a Student Packet appear as Practice pages within lessons, in the Review section as activities, and as Spiral Reviews in subsequent units. Unit Resources on the Teacher Portal also contain problems, tasks, and projects to support conceptual development.” A table is provided that identifies multiple ‘concept development activities’ throughout the lessons. 

Materials develop conceptual understanding throughout the grade level. (Note - Lesson Notes come after the workbook page in the Teacher Edition.) For example:

• Unit 3, Lesson 3.1, Teacher Edition, Lesson Notes S3.1a: Proportional Relationships, students use tables, unit rates, and graphs to develop understanding about elements that determine proportional relationships (7.RP.2). As students work through the slides, teachers are given prompts such as, “How do the unit rates relate? Does it make sense to connect the graphed points? How are they related to the unit rate? How is the graph in problem 4 different than the other graphs? Which two tables contain all equivalent unit rates?” The work and discussion lead to discoveries such as the constant of proportionality is the same as a unit rate and the graphs fall on a line through the origin.

• Unit 5, Lesson 5.1, Teacher Edition, Lesson Notes S5.1a: Multiplying Integers with Counters 1, students work with counters to develop understanding of multiplying integers (7.NS.2). With the guidance of the teacher, students first look at 2\cdot3=6 and use positive counters to show their work. They then use negative counters to show 2\cdot(-3)= -6. Teachers are given prompts such as, “Use the ‘think aloud’ sentence frames; What must happen if we put only groups of positive counters on our work space?; What is a factor? What is a product?” Students then answer several more questions to practice this new concept. With each computation, they record drawings using positive and negative symbols. Later in the lesson, students summarize the rules, Practice 1, Problem 7, “The product of two positive numbers is; The product of two negative numbers is; The product of one positive and one negative number is…”

• Unit 7, Lesson 7.3, Teacher Edition, Slide Deck Alternative S7.3a: Graphing Inequalities, students develop understanding of inequalities (7.EE.4). As students work through the slides, teachers are provided with prompts such as, “What do variables stand for? What is a solution to an equation or inequality? How many numbers are greater than 3? Is 3.00001 included? Is 3\frac{3}{4}? Is -3? What do these symbols mean? (given inequality symbols)” Students compare the meaning of n=3, n>3, n\geq3. They also copy each equation or inequality and describe it using words and a graph. For example: 6) x\geq-2; 9) This graph shows that everyone in gym class ran, but no more than 10 laps. Explain: a) Why is there a closed circle at 10? b) Why is the shaded portion to the left of 10? c) Why is there an open circle at 0?”

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. For example:

• Unit 1, Lesson 1.3, The Cereal Box Simulations, students demonstrate understanding of the probability of chance processes and probability models (7.SP.6-8). “There are six different animal prizes in Krispi Krunchy Cereal, and you want to collect all six. You have an equally likely chance of getting any of the prizes when buying a box. How many boxes do you think you need to buy to get all six? Create a simulation and carry it out. 1) First make a prediction. What is your ‘gut feeling?’ 2) What tools or materials will you use to generate a simulation for collecting 6 objects? How many times will you perform the experiment? 3) Perform your experiment to collect and then organize your data. 4) Write a few sentences to analyze the data using statistics. 5) Write a few concluding sentences about the process. Did your prediction agree with your actual results?” 

• Unit 6, Lesson 6.3, Practice 9, Problem 7, students demonstrate understanding of equivalent expressions by analyzing an explanation (7.EE.A). “Aretha looked at the expressions 2n and n+2. She substituted the value of 2 for n in both expressions, and said, ‘They’re both equal to 4, so they must be equivalent expressions.’ Critique Aretha’s reasoning.” 

• Unit 7, Lesson 7.2, Balanced and Unbalanced Scales, Problem 7, students demonstrate conceptual understanding of equations by showing balance on a scale to solve equations. (7.EE.4). “Iggy built the balanced scale to the right. a) Write the equation it represents. b) Why can Iggy remove 3 units from both sides? c) Draw the new balanced scale and write the equation it represents. d) Does the equation in part c represent the solution to the equation in part a?”

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

According to the MathLinks Program Information, in Focus, Coherence and Rigor, “In MathLinks, we thoughtfully develop new procedural skills and provide opportunities for students to gain fluency throughout the year. Skills practice in each unit is found in the Student Packets in the following ways: Practice pages – These pages support concept development. Review activities – These pages often include skills practice. Spiral Review – These pages have distributed practice of prior skills. Math Path Fluency Challenges - This Activity Routine, which is in the Spiral Review section, utilizes mental math skills and supports fluency development in a puzzle format.”

In addition to what is in the student packets, teachers have access to additional support for developing procedural skill and fluency. “Grade-level skills practice is in each unit, as well as practice to fill in gaps. Both can be found on the Teacher Portal in Other Resources in the following ways: Essential Skills – This entire section reviews skills and concepts important for success in a given unit. Activity Routines such as Big Square Puzzles, Open Middle Problems, and Four-in-a-Row games are also in these sections for some units. They provide a practice alternative to “drill and kill.” Extra Problems – Skills practice by lesson is available for all units. Non routine Problems – In addition to skills practice that is embedded in nonroutine problems, Big Square Puzzles, Open Middle Problems, and Four-in-a-Row games are located in this section for some units.” A table is provided that identifies multiple “examples of fluency work” throughout the lessons.

Materials develop procedural skills and fluency throughout the grade level. (Note: Lesson notes come after the workbook page in the Teacher Edition.) For example:

• Unit 1, Lesson 1.3, Getting Started, Problems 1-13, students are introduced to a probability game to practice converting fractions into decimals (7.NS.2d).  The activity starts with “Find the decimal equivalent for each fraction below. Use a repeat bar when necessary. 1) \frac{1}{1}; 2) \frac{1}{2}; 3) \frac{1}{3} …..13) Box all the fractions that are equivalent to repeating decimals. Circle all the fractions that are equivalent to terminating decimals.”

• Unit 2, Lesson 2.3, Matching Scale Drawings Of Triangles And Rectangles, Problem 2, students practice problems involving scale drawings of geometric figures (7.G.1). “Your teacher will give you some geometric shapes. Cut them out. Determine which figures are scale drawings of the others. Then complete this table.” The first column is the original figure. In the second column, students identify the “Scale drawing of figure”. The third column is “Scale factor”; the fourth is “Enlargement or reduction?”; fifth is “Measures of angles in the actual figure”; and the last column is “Measures of angles in the corresponding figure”. Students have six original figures to work through. 

• Unit 7, Lesson 7.4, Lesson Notes S7.4: Equations with Rational Numbers, Problems 1-6, teacher guides discussion about multiple solution strategies for equations with prompts such as, “Why did Anita multiply both sides of the equation by 10? Why did Kim choose to multiply each term by 8? What did Gerry do first?” Students practice solving equations (7.EE.4a), for example: “1) -4.2=-n-0.4n-2.8;  2) 2.2-4w+4.4=-0.4”

Materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. For example:

• Unit 4, Lesson 4.1, Practice 2, Problems 10-18, students add and subtract rational numbers (7.NS.1d). “10) 7+(-2); 11) -9+9; 12) -1+(-3); 13) 11+12; 14) 3+(-8);  15) -5+6; 16) 2+(-2); 17) -3+(-6); 18) -13+3.”

• Unit 6, Lesson 6.3, Practice 9, Problem 5, students expand linear expressions with rational coefficients (7.EE.1). “Apply the distributive property to each expression below. Use cups and counters or a picture as needed. Then match each expression in Row I to an equivalent expression in Row II as a check.” Row I includes: “2(x+1); 2(x-1); 2(-x+1); 2(-x-1)” and Row II: “-2(x+1); -2(x-1); -2(-x+1); -2(-x-1).”

• Unit 8, Lesson 8.1, Problem 2, students use facts about angles to write and solve simple equations for an unknown angle (7.G.5). “Refer to the diagram above. Write two different equations that could be used to find the value of (2n+13). Solve for n in both equations, and write the value of (2n+13).” The diagram shows intersecting lines with 4 angles labeled.     

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

According to the MathLinks Program Information, in Focus, Coherence and Rigor, “Problem solving is an important driver of instruction within MathLinks courses. In MathLinks, we include engaging mathematical problems and applications with accessible entry points for all students, multiple approaches or solutions, and extensions to challenge and enrich. All units begin with an Opening Problem, which introduces a concept or establishes a ‘need to know.’ In many cases, students require more instruction throughout the unit before they are fully prepared to bring the problem to its conclusion. Substantial problems exist throughout the units as well.” A table is provided that identifies multiple “examples of problem-solving lessons” throughout the lessons and additional resources such as tasks and projects. 

Materials include multiple routine and non-routine applications of mathematics throughout the grade level. Though, in many cases, problems labeled “non-routine” in the materials are actually routine problems since there is only one solution path to an expected answer, even though the context of the problem may be novel. For example:

• Unit 2, Teacher Portal, Other Resources, Technology Reproducibles, Dueling Discounts, Problems 1-5, students apply understanding of percent in a non-routine real-world problem (7.RP.3). Given images from a Dan Meyer website, “(Discussion) Which do you think is better, $20 off or 20% off? Explain. Perform calculations and compare prices. Highlight the better discount for each item. What tools are appropriate to make these computations? Explain how you calculated $20 off. Explain how you calculated 20% off. Under what conditions do you think that 20% off is better than $20 off? Explain.” 

• Unit 3, Lesson 3.3, Jenna’s Cornbread Recipe, students apply their skills in operations with rational numbers in a routine real-world problem (7.NS.3). “Granny and Auntie both love the cornbread Jenna brought to the family dinner, so Jenna says, ‘Here’s what I did. I started by using 1\frac{1}{2} cups of milk, 2\frac{1}{2} cups of cornmeal, 1\frac{1}{4} cups of flour, and…” “Wait!” Granny says. “I just want to make it for myself, not for a party!” Auntie agrees. Jenna says, “You both know a lot about ratios. I’ll give you the rest and you figure it out!” Granny and Auntie want their cornbread to taste the same as Jenna’s. Analyze the cornbread recipe representations below. Let M and C represent parts milk and cornmeal, respectively. 1) Finish the tape diagram below using some of Jenna’s initial quantities.” “3) Auntie plans to use 1 cup of cornmeal. Finish the tape diagram below to represent the quantities Auntie will need.” “5) How many cups of milk are needed for \frac{3}{4} cups of cornmeal?”

• MathLinks Portal, Puzzles and Games, Shape-Up, students apply understanding of congruence in geometrical figures by drawing dividing lines to create new shapes in a non- routine application (7.G.A). “Description: Each problem presents three pictures of the same equilateral shape: triangle, square, pentagon, hexagon, heptagon, or octagon. Directions are to draw one line in each of the shapes to create two other shapes that meet the specified criteria. The three drawings must differ. Shapes created by the one line may be congruent or non-congruent. If there is a finite number of ways to draw one line to make the shapes, the directions will read “Show all ways.” If there is an infinite number of solutions, the directions will read “Show three of the ways.” In Shape-Up #29, the instructions is for Heptagon Shape- Ups, “Draw one line to make 1 triangle and 1 heptagon. Show three of the ways. Draw one line to make 2 non-congruent pentagons. Show three of the ways.” 

Materials provide opportunities for students to independently demonstrate multiple routine and non- routine applications of the mathematics throughout the grade level. For example:

• Unit 4, Lesson 4.3, Extending Your Thinking, students apply their knowledge of adding and subtracting rational numbers in a routine real-world context (7.NS.1). “1) A rancher is digging a well. Ground level has an elevation of zero. First write an expression to describe his actions. Then solve the problem. From ground level he digs down 13 feet, and then stops for the day. Overnight wind blew 2 feet of dirt back into the hole. The second day he digs another 9 feet. The third day he decides the hole is now too deep, and fills in 6 feet of dirt. What is the elevation at the bottom of the well after his work is complete?”

• Unit 6, Teacher Portal, Other Resources, Task Reproducibles, Rectangle Reasoning, Problem 3, students apply understanding of perimeter to write variable expressions and explain their equivalence in a non-routine problem (7.EE.4). “William finds the perimeter of a rectangle by adding the length and the width and then doubling this sum. Matthew finds the perimeter of a rectangle by doubling the length, doubling the width and then adding the doubled amounts. Write a variable expression that shows how William finds the perimeter. Write a variable expression that shows how Matthew finds the perimeter. Explain why their expressions are equivalent.”

• Unit 9, Lesson 9.2, Penny Drop Probabilities, Problems 1 and 2, students apply knowledge of area to predict probabilities (7.G.4, 7.SP.2). “In the Penny Drop Game, a player drops a penny on a board on the floor. If the penny does not land on the board, the player drops it again. If the penny lands on the board and is at least half way in a white space, the player wins. If not, the player loses. Figures A, B, and C above represent boards for the Penny Drop Game. All three are squares that have side lengths equal to 1 foot. All the circles within board B have the same diameter length. All the circles within board C have the same diameter length. 1) Predict which board you think provides the greatest chance of winning. 2) Test your prediction by calculating the probabilities of winning and losing for each board. What is your conclusion?”

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout each grade level. For example:

• Unit 2, Teacher Portal, Nonroutine Problems Reproducibles, “Calculate”-ing Percent, Problem 3, students apply knowledge of percent in a real-life situation (7.RP.3). “Allie and Evan spend $45 on lunch. They want to leave a 20% tip. How much will each person spend on lunch?”

• Unit 7, Lesson 7.2, Balanced and Unbalanced Scales, Problems 1-7, students develop conceptual understanding of solving equations (7.EE.4). “For each problem, start with this original balanced scale, 4=4. Draw a sketch to illustrate the action described. Write the resulting equation or inequality.” Examples include: “2) One unknown (x) is added to both sides of the original balanced scale. 4) “Two x’s are added to the right side, and one x to the left side of the original scale.”

• Unit 7, Spiral Review, Problem 1, students practice fluently multiplying and dividing rational numbers (7.NS.2). Students work through a maze of 16 problems. “Follow the math path to computational fluency.” Example problems include: “-3(-5); -1.2\div0.4; \frac{1}{2}(\frac{3}{4}); \frac{1}{2}\div-\frac{1}{8}.”

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout each grade level. For example:

• Unit 2, Teacher Portal, Non-Routine Problems Reproducibles, Buying a Tablet, students develop procedural skills with percentages and apply their knowledge in a real-world situation (7.RP.3). “1) A company buys a tablet for $129.00. Sales tax varies from state to state, so the total amount paid for the tablet varies from state to state. a) Jordan lives in Hawaii and pays 4% sales tax when he buys the tablet. What is the total amount he pays for the tablet? b) Kendra paid $138.03 for the same tablet. How much did she pay in sales tax? c) What percentage sales tax did Kendra pay? 2) Cheyenne lives in California and bought a similar tablet to Jordan and Kendra. She paid a total price of $161.25 with 7.5% sales tax. How much was the tablet before tax?”

• Unit 5, Lesson 5.3, Practice 8, Problems 1-4, students develop conceptual understanding about negative numbers, procedural skills for computing with negative numbers and apply those in equations to solve problems involving rational numbers in real-world situations (7.EE.4). “Here are two equivalent equations for converting between the Celsius and Fahrenheit scales. Let C = degrees Celsius and F = degrees Fahrenheit. F=\frac{9}{5}+32; C=\frac{5}{9}(F-32). 1) The NFL Championship game on December 31,1967 between the Green Bay Packers and the Dallas Cowboys in Green Bay, Wisconsin is known as the “Ice Bowl.” The low temperature for that game was 13 degrees below zero (F). a) Write this temperature as an integer. b) Choose one of the equations above. Substitute this value to solve for C. 4) In Sochi, Russia, the historical average high temperature for January is about 50\degreeF. When they hosted the XXII Olympic Winter Games in 2014, temperatures reached 20\degreeC. Is this temperature higher or lower than the historical average high,and by how much?”

• Unit 7, Teacher Portal, Nonroutine Problems Reproducibles, MIxed Problems, Problem 5, students develop conceptual understanding and practice fluency when recognizing real-world situations that lead to 0 (7.NS.1). “Which of these questions have an answer equal to 0? Select ALL that apply. a) Landry jumped into a pool from a diving board 5 meters above the water. He sank 5 meters and then swam straight up to the surface of the water. How many meters did Landry swim? b) Annie left her house and walked 1.2 miles directly east. Then she walked 1.2 miles directly west. At this point, how many miles did Annie walk? c) A trail begins at an elevation of -40 feet. The trail ends at an elevation of 40 feet. By how many feet does the elevation of the trail change from beginning to end? d) Jesse walked 0.75 miles directly north from home to get to school. He walked 0.75 miles directly south after school. How many meters is Jesse from home?”

• Unit 7, Lesson 7.2, Practice 4, Problems 5-7, students use conceptual understanding to recognize mistakes in solutions and develop procedural fluency to correct the mistakes (7.EE.3). “Circle the part of each equation-solving process that contains a mistake. Correct it and continue the solution process underneath the problem. 5) 12=-3x+15-6x // -12=3x+15 // -27=3x // -9=x. Correction(s):___.” Students are expected to describe what needs to be corrected.

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the Mathematical Practice standards. 

Mathematical Practice Standards are identified in the Teacher Edition and Student Packets. The Teacher Edition has Teaching Tips that include abbreviated descriptions of the Standards for Mathematical Practice and specific examples and explanations that describe how the practices are developed within the unit. 

There is intentional development of MP1 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 1.3, Teacher Edition, Lesson Notes S1.3: Spinner Puzzles, students work to understand information in analogous problems (MP1). “Students work in groups to create a circular spinner from a set of clues. The clues describe the probabilities of each spinner section in words and numbers. A table accompanies each spinner as a more complete recording mechanism. Note that one set of clues is purposely flawed and must be corrected.” Students use clues to determine probabilities. The probabilities are converted into fractions, decimals, and percents. “Change 1 clue to make Spinner C work.”

• Unit 2, Lesson 2.2, Getting Started, Problem 2, students interpret information given to make sense of a problem about percent (MP1). “It is common for a clothing store to buy merchandise from a manufacturer and then mark up the price by about 100% when selling the item. a) What does it mean to mark up the price of a pair of jeans by 100%?  b) If a clothing store buys jeans for $25 each, what will be the selling price of the jeans after a 100% markup? c) When these jeans are purchased, a 9% sales tax is required. What is the total cost of purchasing these jeans?” 

• Unit 7, Lesson 7.4, Practice 9, Problem 5, students analyze givens, constraints, relationships, and check correctness to make sense of inequalities (MP1). “Solve each problem using algebra (an equation or inequality). Define variables, answer the question and check the solution(s). Gerardo is a salesperson. He is paid $300 per week plus $15 per sale. This week he wants his pay to be more than $900. How many sales does he have to make this week?”

There is intentional development of MP2 to meet its full intent in connection to grade-level content. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 2, Lesson 2.0, Opening Problem, Using Coupons, students reason quantitatively by decontextualizing a situation and using operations flexibly to figure out the best way to save money (MP2).”Bridget has four coupons for the CAMY’s department store. Coupon A offers 25% off any item. Coupon B offers $20 off any item. Coupon C offers 10% off any item. Coupon D offers $10 off any item. She needs to buy the following items. One set of sheets for $45. One mattress for $400. One set of 4 pillows for $60. One bed frame for $120. If she is allowed to use only one coupon per item, how should she use her coupons to save the most money?”

• Unit 6, Lesson 6.2, Practice 5, Problem 1, students attend to the meaning of quantities to create multiple representations of the problem (MP2). Students use tables, graphs, and input-output rules to describe visual patterns. “Build steps 1-3 for tile patterns A and B. Then build and draw step 4 for each pattern. Complete the tables and draw the graphs with titles and labels.”

• Unit 9, Teacher Portal, Quiz B, Problem 4, students reason abstractly and quantitatively by considering the units involved with the area of circles (MP2). “Weston was making a wheel out of wood for a train decoration. (See diagram.) a) He first cut out a large circle with a radius of 3.5 feet. Find the area of this large circle. Use 3.14 for \pi. b) He then cut a smaller circle from the middle of the larger circle with a diameter of 4 feet. What is the area of the remaining part?  (The shaded part of the diagram).”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Mathematical Practice Standards are identified in the Teacher Edition and Student Packets. The Teacher Edition has Teaching Tips that include abbreviated descriptions of the Standards for Mathematical Practice and specific examples and explanations that describe how the practices are developed within the unit. 

There is intentional development of MP3 to meet its full intent in connection to grade-level content. Students construct viable arguments and critique the reasoning of others as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 1, Lesson 1.1, Practice 1, Problem 8, students construct viable arguments as they explain their strategies and thinking orally or in writing, using concrete models, drawings, or numbers (MP3). “Ryder wants to play a game with the marbles above. She will choose a marble out of the bag 20 times (and replace it after each turn). If Ryder chooses a blue or yellow marble, she gets a point. If she chooses a red or green marble, you get a point. Is this game fair? Explain using words, diagrams, and/or numbers.” There are 10 blue, 5 yellow, 4 green, and 1 red marbles.

• Unit 2, Lesson 2.2, Practice 8, Problem 2, students critique the reasoning as they perform error analysis with percents (MP3). “Rosando said to Carlos, ‘You’re taking 25% off for your discount, and then adding 6% sales tax. Since 25–6=19, just take off 19%.’ Critique Rosando’s reasoning.”

• Unit 3, Lesson 3.3, Practice 4, Problem 1, students critique the reasoning of others as they perform error analysis with equivalent fractions (MP3). “1) Some students explored the equation \frac{3}{5}=\frac{6}{10} and rewrote it in a few different ways. a) Circle the three true equations. b) For the equation that is not true, explain to that student why it is not true and a way to revise the work.” 

• Unit 7, Teacher Portal, Task Reproducibles, Reasoning About Solutions, Problem 3, students construct viable arguments about solutions to equations (MP3). “Wendy tried different strategies to solve the equation 2(3+x)=2x+6. After a while, she became exasperated!  ‘It doesn’t seem to matter what x is!  I can put any number in there and the equation will be true.’ Explain why Wendy is correct.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Mathematical Practice Standards are identified in the Teacher Edition and Student Packets. The Teacher Edition has Teaching Tips that include abbreviated descriptions of the Standards for Mathematical Practice and specific examples and explanations that describe how the practices are developed within the unit. From the Applying Standards for Mathematical Practice (SMP) section in the Teacher Edition, “[All Lessons] Students record mathematics vocabulary as it is introduced in lessons. They use precise language in writing and exercises. Precise definitions are located in the Student Resources section in the back of the unit.” This is true for all units.

There is intentional development of MP6 to meet its full intent in connection to grade-level content and the instructional materials attend to the specialized language of mathematics. Students attend to precision as they work with the support of the teacher and independently throughout the units. Examples include:

• Unit 2, Lesson 2.1, Practice 4, Problem 2, students calculate accurately and efficiently to express numerical answers with a degree of precision appropriate for the problem context. “Use a calculator as needed and round appropriately. 2) Steven buys one video game for $20 and another for $30. His total at the register is $53.50. What is the tax rate that Steven paid?”

• Unit 3, Review, Match and Compare Sort: Solving Equations, Problems 1-3, students differentiate between pairs of words that might be closely related, and write at least one detailed set of reasons of similarities and differences. Students “connect concepts to vocabulary words and phrases, 1) Individually, match words with descriptions. Record results into a table. 2) Partners, choose a pair of numbered matched cards and record the attributes that are the same and those that are different.” A Venn diagram is provided for students to place vocabulary words in boxes and explanations in circles that list similarities and differences. “3) Partners, choose another pair of numbered matched cards and discuss the attributes that are the same and those that are different.” The vocabulary for this unit is independent variable, dependent variable, unit rate, unit price, proportional relationship, constant of proportionality, input-output rule, equation. Match and Compare Sorts are included in many units.

• Unit 6, Lesson 6.4, Expression Card Sort… and More, Problem 3, students attend to precision when identifying equivalence. “Circle all expressions that are equivalent to m-3(4-m): m-12-m; m-12-m; m-12+3m; 4m-12; 2m-12; m-2(6-m); m-(12-3m); m+[-3(4-m)]”

• Unit 7, Review, Vocabulary Review crossword, each puzzle clue focuses on specialized language, “2) set of numbers that includes natural numbers and zero, 9) a number less than zero.” Each unit review includes a vocabulary crossword.

The materials reviewed for MathLinks: Core 2nd Edition Grade 7 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

Mathematical Practice Standards are identified in the Teacher Edition and Student Packets. The Teacher Edition has Teaching Tips that include abbreviated descriptions of the Standards for Mathematical Practice and specific examples and explanations that describe how the practices are developed within the unit.

There is intentional development of MP7 to meet its full intent in connection to grade-level content. Students look for and make use of structure as they work with the support of the teacher and independently throughout the units. Examples include: 

• Unit 1, Lesson 1.3, Probability Experiments: Games and Puzzles, The Terminator: Theoretical Probability, Problems 1-3, students search for and decompose a complicated scenario into a simpler scenario to make connections among mathematical concepts and apply general mathematical rules to complex situations. “1) Make an outcome grid to determine the theoretical probabilities of winning and losing the Terminator game. Using two different colored cubes helps to keep track of outcomes. 2) Determine the theoretical probabilities of winning and losing as a fraction, decimal and percent. 3) Based on the theoretical probabilities, out of 3,000 rolls, about how many times is winning expected?”

• Unit 3, Lesson 3.2, Practice 3, Problems 1-5, students find patterns and look for structure to simplify multiple related problems. “Fruity-Fizzy-Water (FFW) is made using 5 cups of soda water for every 2 cups of fruit juice. 1) Fill in the table for different mixtures of FFW. 2) Complete the paragraph: To keep the same flavor, a 1 cup increase in soda water requires an increase of ___cups of juice. The unit rate of cups of juice per 1 cup soda water is ___. An equation that relates the amounts of juice to soda water is ___. One ordered pair is (1, ___). Within the context of FFW, this represents ___. Another ordered pair is (0, ___). Within the context of FFW, this represents ___.  3) How many cups of juice are needed to make the exact same flavor of FFW if 40 cups of soda water are used? 4) How many cups of soda water are needed to make the exact same flavor of FFW if 40 cups of juice are used? 5) How many cups of FFW can be made using 10 cups of juice?”

• Unit 7, Lesson 7.2, Getting Started, Problems 1-4, students look for and explain structure within mathematical representations. “In each problem below, all the shapes have some weight, the same shapes have the same weight, and different shapes have different weights. All problems are independent of one another. Use what you know about balance to answer each question.” Students use this understanding to equate shapes to numbers and to solving equations. 

There is intentional development of MP8 to meet its full intent in connection to grade-level content. Students Look for and express regularity in repeated reasoning as they work with the support of the teacher and independently throughout the units. Examples include: 

• Unit 4, Lesson 4.3, Getting Started, Number Line Addition, Problem 13, students generalize rules for integer addition. “Look at problems 1 – 12 above. Do the addition rules we learned in a previous lesson hold for these problems? Do you think that these rules hold for all rational number addition?” 

• Unit 5, Lesson 5.2, Determining the Sign of a Product, Problem 7, students notice repeated calculations to create a shortcut to determine the sign for the solution when multiplying integers. “Make conjectures about multiplying nonzero numbers. a) If there are an odd number of negative factors, the product is ___. b) If there are an even number of negative factors, the product is ___.”

• Unit 7, Lesson 7.3, Exploring Inequalities, Problems 3-4, students generalize the conditions under which the sign of an inequality is preserved or reversed. “3) In the table above, look closely at the last column and circle every result where the inequality changed direction compared to the original inequality. 4) Under what circumstances did the direction of the inequality symbol change?”