6th Grade - Gateway 2

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 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 3.3, Teacher Edition, Lesson Notes S3.3: Double Number Lines, students develop understanding and reason about equivalent ratios (6.RP.1, 6.RP.3). “Slide 1: For (1), discuss elements of a double number line. Assist students as they create one on their workspace. Why do you think this is called a double number line? Why does the double number line start at 0 and show only positive values? How is it marked and labeled? How many hours correspond to $32? What dollar value goes between $0 and $32? $16. What hour value falls on the tick mark half way between 0 and 4? Is 32:4 equivalent to 16:2? What other values go on the dollar line? The hours line? For (3), ask partners to discuss and share features that are the same or different in the Getting Started table and the double number line. Then, for (4), ask students to share patterns they notice on the double number line and to record the ones that make the most sense to them.” 

• Unit 9, Lesson 9.1, Teacher Edition, Slide Deck Alternative S9.1b: Area of A Parallelogram, students develop conceptual understanding of the similarities between rectangles and parallelograms to find area (6.G.1). In the Teacher Notes, “Slide 1: Demonstrate a “cut up” strategy to show how pieces of a parallelogram can be rearranged to form a rectangle. This process is referred to as composing and decomposing.” “Slide 2: How did the base (b) change? How did the height (h) change? How did the area change? What can we say about the area of these two figures? Is this always true?” 

• Unit 10, Lesson 10.1, Getting Started, Problems 8-12, students develop an understanding of rational numbers as a point on the number line (6.NS.6). “8) Which numbers are greater on a horizontal number line, the numbers further to the right or the numbers further to the left? To the right 9) Which numbers are greater on a vertical number line, the numbers higher on the line or lower on the line? Higher 10) Write the correct integers for each tick mark on the two number lines. 11) Where are positive numbers located on the number lines? To the right of zero (horizontal line) or above zero (vertical line). 12) Where are negative numbers located on the number line?”

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

• Unit 2, Lesson 2.1, Factor Game, students demonstrate understanding of factors while playing a game with another student (6.NS.4). The game, along with the analysis questions, lead to conceptual understanding of prime and composite numbers and finding the Greatest Common Factor. “Player A circles a number on a game board and receives that amount of points. On the same game board, player B circles all of the unique factors of player A’s number except the number itself, and receives that amount of points. Switch roles, continuing play until there are no more legal moves on the board left or time runs out. The winner is the player with the most points when the game is over. 2e) What is the worst first move on your game board? Explain. 2f) What is the best first move on your game board? Explain.”

• Unit 7, Lesson 7.1, Input-Output Rules, Problem 6, students demonstrate conceptual understanding of the quantitative relationship between variables (6.EE.9). “Fill in the missing numbers and blanks (of the table) based on the suggested numerical patterns. In the tables below, the x-value is considered the input value and the y-value is the output value. a) Rate of change: for every increase of x by 1, y increases by; b) Input-output rule (words) add ___ to an x-value (independent variable) to get its corresponding y-value (dependent variable); c) Input-output rule (equation): y=x+___.”

• Unit 8, Lesson 8.2, Getting Started, Problem 4, students demonstrate understanding of equivalent expressions by using tape diagrams (6.EE.3). “For this tape: (given 6 on top and x on bottom) a) Write an equation the diagram represents; b) Increase the amount of tape so that the top and bottom parts are multiplied by 3 (show a total of 3 copies); c) Write the new equation the diagram represents; d) Did this change to the tape diagram change the value of x?”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 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. Computational Fluency Challenge – This Activity Routine, which is in the Spiral Review section, supports fluency development with whole number and decimal operations, which are expectations for 6th grade.” 

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 computational 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 4, Teacher Portal, Other Resources, Math Talks, Slide Alternative, students develop procedural skill with the standard division algorithm (6.NS.2). “Post the first expression and give students think time. Write all values without validation. Then discuss solutions and strategies. Repeat for the next expression(s) in the set, keeping track of student work. Use one set of related division expressions per day. What is the quotient for the first expression? How do you know? How is this next expression similar to / different from the one before? How can we use the previous expression to help us with this expression? How is ___’s strategy the same as / different from ___’s strategy?” Four problem sets are provided for practice.  One example is as follows: “Column A: 50\div5, 35\div5$4, and 85\div5.”

• Unit 6, Lesson 6.1, Lesson Notes S6.1b: Exponential Notation, teacher activates prior knowledge about exponents then students practice writing and evaluating expressions with whole number exponents (6.EE.1). “Evaluate by writing the factors and product. 2) 5^3; 3) 3^5; 4) 1^4; 5) 4^1.”  

• Unit 10, Lesson 10.1, Lesson Notes S10.1c: Distance and Absolute Value, students use number lines and absolute value equations to compare distance and location (6.NS.C). For the teacher, “Slide 1: Show pictures of animals and their locations above and below sea level. Ask students to record locations relative to sea level in the table.” Next students determine an appropriate scale for the number line and graph the points of each animal. Students complete the table with “Distance from zero” and “Absolute value equation for the distance from sea level”. There are 6 animals, which gives students several opportunities to reinforce the skill. 

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

• Unit 4, Lesson 4.4, Exploring Multiply by the Reciprocal, Problem 3, students divide fractions by fractions (6.NS.A). Students solve a fraction division problem in Column I where they use the “divide across rule”; in Column II, they solve the related multiplication problem using the “multiply across rule” and compare their results. “Compute a) \frac{10}{21}\div\frac{2}{7}; b) \frac{7}{8}\div\frac{1}{4};  c) \frac{2}{3}\div\frac{1}{6}; d) \frac{1}{6}\div{2}{3}.” 

• Unit 6, Spiral Review, Problem 1, students fluently multiply and divide multi-digit decimals (6.NS.3). “Computational Fluency Challenge: This paper and pencil exercise will help you gain fluency with multiplication and division. Try to complete this challenge without any errors. No calculators! a) Start with 4.5. Multiply by 4. Multiply the result by 0.7. Multiply the result by 8. Multiply the result by 10. Now you have a “big number”. My big number is ___. b) Start with your big number. Divide it by 14. Divide the result by 0.2. Divide the result by 1.8. Divide the result by 4. What is the final result? ___.” 

• In Unit 9, Lesson 9.1, Practice 2, Problems 1-4, students find the area of polygons by substituting values into the formula (6.EE.2c). “For each problem: Identify the polygon and the corresponding area formula. Measure and label the relevant dimensions to the nearest tenth of a cm (mm). Substitute values into the formula and evaluate to find the area. Use appropriate units in answers. Polygon name; Area formula; Substitute; A= ___. What do the arrows mean on the sides of the polygon?” Students are given a trapezoid, parallelogram, isosceles triangle and scalene triangle.

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 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 the 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 8, Lesson 8.3, Translating Problems into Equations, Problems 1-4 (one set for each operation), students apply their understanding of one-step equations in routine problems (6.EE.7). “Write and solve the equations below as directed. 1) The total number of puppies and kittens is 150. a) Write a numerical equation if there are 80 puppies and 70 kittens; b) Write a variable equation if there are p puppies and k kittens; c) Write an equation and solve for p if there are 57 kittens. 2) The number of trading cards KC has after giving some away is 49. a) Write a numerical equation if KC started with 77 cards and gave away 28; b) Write a variable equation if KC had x trading cards and gave away y of them; c) Write an equation and solve for x if KC gave away 33 cards.”

• Unit 9, Lesson 9.3, The Food Drive, students apply understanding of volume to solve a non- routine real-world problem (6.G.2). “At Maynard Middle School, the student council led a food drive effort to feed needy families. Enough food was donated for 200 families, so they will fill boxes at school and transport them to their local regional food bank. They will purchase 200 boxes at $1.75 each (taxes included). These boxes are in the shape of cubes, 18 inches on each edge. They will rent a truck from U-Move for $19.95, plus mileage and taxes. The distance from school to the food bank is about 10 miles. The truck has inside dimensions that are 10’ long (or deep) \times 6’ wide \times 8’ high for storage space. What additional information do you need to determine the cost to pack and deliver the boxes? If possible, either research the unknowns, or agree as a class, and record reasonable estimates here. Convert all measurements in the table before making calculation decisions.” Students calculate the dimensions and volume for the box and the truck, then record them in a chart given.

• Unit 10, Leson 10.0, Opening Problem, Extreme Temperatures, students apply knowledge of negative values to represent quantities in a non-routine real-world context (6.NS.5). “1) What is the hottest temperature you remember experiencing? Where was this? 2) What is the coldest temperature you remember experiencing? Where was this? 3) What is the difference between the highest and lowest temperatures you experienced in your lifetime? 4) Ask an artificial intelligence device (e.g., Siri, Alexa, internet) for the coldest and hottest temperatures recorded on earth. Find the difference between them. 5) What was notable about the temperatures in International Falls, Minnesota and Key West Florida on January 2, 2014?” 

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, Practice 8, Problem 1, students apply their understanding of dividing fractions by fractions in a routine problem (6.NS.1). “1. Ryan has 1\frac{1}{2} sandwiches leftover from yesterday’s party. A serving size is \frac{3}{4} of a sandwich. How many servings does he have? Represent this situation with a picture and a division expression. Then perform the divide across procedure. Clearly show your work, and the result.”

• Unit 8, Teacher Portal, Quiz A, Problem 6, students apply understanding of how two quantities change in relation to each other in a non-routine real-world context (6.EE.9). “Lori earns money at a constant hourly rate as a math tutor. The table shows the various times she tutored and the amount of money she made. Time in hours (t): 1, 2, 3, ___, 0.5, 1.5; and Money in dollars (m): ___,  28, 42, 84, ___, ___.  a) Complete the table; b) Write an equation that relates t and m; c) Lori made $35 tutoring last Wednesday. Write and solve an equation for this situation based on the equation in part b.” 

• Unit 9, Teacher Portal, Nonroutine Problems, Same or Different, Problem 3, students reason about area and volume of quadrilaterals in a non-routine application (6.G.1 and 6.G.2). “Problems 1 and 2 have many similarities and differences. Create a Venn diagram to support this claim.” Problem 1 involves a proportional relationship between quarts of paint and area of wall; Problem 2 is about how much sand is needed for the pit under the swings.

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 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 4, Lesson 4.4, Getting Started, Problems 2-4, students develop conceptual understanding of a reciprocal as a strategy to compute division of fractions (6.NS.1). “2) Write the reciprocal of the following numbers: a) 3; b) \frac{1}{6}; c) \frac{4}{5}. 3) The following pairs of numbers are reciprocals of one another. Multiply each pair of reciprocals: a) 5, \frac{1}{5}; b) \frac{5}{7},\frac{7}{5}; c) What is the result when a number is multiplied by its reciprocal? 4) Describe an easy way to find the reciprocal of a fraction.”

• Unit 6, Lesson 6.2, Practice 6, Problem 2, students develop procedural skills in evaluating expressions by substitution (6.EE.2c). “2) Evaluate each expression below for m=9. a) 5m+8–2m+7+3m+10–6m–13  b) 4(m+2)+m+7+3(m–1)–8m  c) 3(m+5)+4m+6+3(m+1)–10m–12.”

• Unit 9, Teacher Portal, Nonroutine Problems Reproducibles, From the Math Olympiad, Problem 1, students apply understanding of volume in a real-world problem (6.G.A). “A rectangular box is 2 cm high, 4 cm wide, and 6 cm deep. Mikael packs the box with cubes, each 2 cm by 2 cm, with no spaces left over. How many cubes does Mikael fit into the box?” 

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, Task Reproducibles, Where Do They Fit, Problems 1-5, students apply their understanding and develop their fluency in working with the greatest common factor to place numbers into a Venn diagram (6.NS.4). Given three Venn diagrams labeled Factors of 12, Factors of 20, and Factors of 30, students determine where numbers should fit. “Venn diagrams are helpful to show visual relationships among sets of items. In this case we will use Venn diagrams to show when numbers have common factors or multiples. Choose the best region for each of the following (A-H). Find a number NOT in the table above that fits in each of the following regions.” 

• Unit 3, Teacher Portal, Non-Routine Problems Reproducibles, Ratio Puzzles, Problem 3, students apply their conceptual understanding of ratio reasoning to solve a real-world problem (6.RP.3). “The Paint Mistake: Jim’s daughter says she wants to paint her room pink. He thinks the color can be made by using 2 scoops of white paint for every 3 scoops of red paint. When Jim makes a sample using exactly 5 scoops, his daughter says, ‘No, you were supposed to use 3 scoops of white paint for every 2 scoops of red paint.’ Jim does not want to waste any paint. Without throwing out the 5-scoop mixture he already made, describe how Jim can correct the mistake when making a larger batch of paint the way his daughter wants it.”

• Unit 6, Lesson 6.2, Practice 4, Nonna’s Pizza Menu, students develop procedural skill in writing and solving variable expressions and apply it in a real-world application (6.EE.6). “A group of friends decide to go to Nonna’s Pizza for lunch. Miguel orders a slice of cheese pizza, a slice of pepperoni pizza, and a medium drink. Barry orders two slices of pepperoni pizza and a large drink. Susie orders a slice of pepperoni pizza and a medium drink. Ronni orders two slices of cheese pizza and a large drink. In the table below, record the variable expressions representing the costs of each order separately, and then total the order. 8) The pizza shop owner decides to take $0.10 off the cost of each slice of pizza. Write a numerical expression for the total cost of the order in problem 5, including this discount. Then find the cost.”

• Unit 10, Teacher Portal, Tasks Reproducibles, Sea Diving, students develop conceptual understanding and procedural skill with negative numbers, then apply their knowledge to real-world situations (6.NS.B). “Some scuba divers are exploring a coral reef. The surface of the ocean (sea-level) is considered to be at an altitude of zero feet. The bottom of the ocean is 15 feet below the surface. A diver is currently at 8 feet below the surface. The captain is at an altitude of 5 feet on the deck of the boat. 1) Graph and label the information described on this vertical number line. Which locations are best described using negative numbers? 2) Determine if the following statements are true or false. Support answers with words and at least one relevant equation or inequality. a) The distance between the bottom of the ocean and the diver is greater than the distance between the diver and the surface of the ocean. b) The diver is closer to the bottom of the ocean than he is to the captain. c) If the diver went down two feet more, she would be 15 feet from the captain. 3) Explain the following with words, numbers, and equations. a) What is the distance between the captain and the bottom of the ocean? b) How far below the surface of the ocean should the diver be so that she is equally distant (equidistant) from the captain and the bottom of the ocean? 4) Write one more true-false statements (like problem 2 above) or one more question (like problem 3 above) that could challenge a classmate, and explain the answer.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 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 3, Lesson 3.1, Practice 1, students analyze and make sense of problems, actively engaging to understand equivalent ratios (MP1). “Zachary likes to make fruit soda when he has friends over to his house. He uses 4 parts juice for every 3 parts sparkling water. 1) Make a tape diagram to illustrate this mixture. 2) How much juice and how much sparkling water will Zachary need if he wants to make 14 cups of fruit soda? 3) How much sparkling water should Zachary use if he has 12 cups of juice? 4) How much juice should Zachary use if he wants to make 70 cups of fruit soda? 5) Jane likes to make fruit soda too. Her recipe uses 2 parts juice and 1 part sparkling water. Who makes a fruitier soda, Zachary or Jane? Explain how you know.“  

• Unit 6, Opening Problem, Lesson Notes S6.0:The Problem of 4’s, students plan, reflect, and revise a solution pathway using order of operations (MP1). “The order in which we perform calculations is determined by agreed-upon rules (mathematical conventions). Here we review order of operations conventions from previous grades.” Slides are presented with expressions written on them. “Copy each expression and evaluate it. Think about the order in which you perform the operation. What’s the same? What’s different? What calculator mistake did he make in getting 9? What keystrokes are needed for Student X to get the correct solution? Using exactly four 4’s, write expressions that have the values 1 through 10.”   

• Unit 10, Lesson 10.2, Lesson Notes S10.2b: Graphing Inequalities, Problems 1-6, students use a variety of strategies that make sense to answer questions and solve problems (MP1). Teacher prompts include, “What do variables stand for? What is a solution to an equation or inequality? Why is there an open circle at 3 rather than a closed circle? (1) For n=3, what can n be? How is n=3 graphed? (2) For m>3, what can m be? How is m>3 graphed? (3) For p<3, what can p be? How is p<3 graphed?” Students complete a table that has “Equation or Inequality, Written description of all solutions, Graph of all solutions.”

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 1, Lesson 1.1, Practice 1, students attend to the meaning of quantities to find measures of center (MP2). “2. Rewrite the data in order from least to greatest in the table below. 3. Write the mode(s) of the data set. What does this tell us about Bobbie’s card playing habits? 4. The median of the data set is ___. What does this tell us about her card playing habits? 5. Which of these two measures of center best represents her card playing habits? Explain.”

• Unit 5, Lesson 5.3, Lesson Notes S5.3a: Percent and Double Number Lines, students work to understand relationships between mathematical representations (MP2). “Previously, students found the percent of some amount. Now students use a double number line, connect percent to ratios and equivalent fractions, and find any missing part of a percent problem.” Slide 1,“(1) Copy and solve: What is 40% of $60? (2) Create a double number line to represent the problem.”

• Unit 6, Lesson 6.1, Practice 1, Problem 12, students use quantitative reasoning to decontextualize, consider units involved in a problem, and represent the situation symbolically (MP2). “A store bought 278 shirts for $7 each and sold them for $15 each. How much profit did the store make? How can you use the distributive property to make your computations easier?”

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 3, Lesson 3.1, Teacher Edition, Lesson Notes: S3.1a: Paint Mixtures, Problems 2-3, students construct viable arguments and critique the reasoning of others by determining if their statement is correct and describing what is wrong (MP3). “Critique the reasoning of each student. 2) Anita: ‘Mixture B and Mixture C will be the same because they both have…the same number of parts of red. 3) Drew: Mixture A and Mixture D will be the same because Mixture A has…one more cup of white than red, and Mixture D has…one more cup of white than red.’” 

• Unit 6, Lesson 6.3, Practice 9, Problem 2, students critique the reasoning of others (MP3). This is a reference to a problem where students calculated area and volume for various side lengths, “Sondra thinks that the measures above for A and V are the same when x=1 ft. What is correct about her statement and what is incorrect about her statement?” 

• Unit 9, Lesson 9.2, Teacher Edition, Lesson Notes S9.2b: Who Needs More Paint?, students construct viable arguments as they create their own conjectures (MP3). “Who needs more paint? Answers will vary based on assumptions. If the square footage to be painted in each room is close, then a reasonable answer is that they will need to buy the same amount of paint. Slide 3: Engage students in discussions where they explore the scenarios in which one student will need more paint than the other. All justified explanations should be accepted.”

• Unit 10, Reflection, Problem 3, students construct mathematical arguments as they make  connections between two concepts (MP3).“Mathematical Practice. How are absolute value (a mathematical symbol) and distance (a geometric idea) related? [SMP3] Then circle one more SMP on the back of this packet that you think was addressed in this unit and be prepared to share an example.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 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, Review, Vocabulary Review crossword, each puzzle clue focuses on specialized language. “6) form of a fraction whose numerator and denominator are relatively prime 11) a number whose only factors are itself and 12) product of the length and width of a rectangle.” Each unit review includes a vocabulary crossword.

• Unit 3, Lesson 3.4, Practice 7, Problems 2-9, students attend to precision when converting customary units. “2) Complete this double number line that relates inches to feet. Use the double number line above to complete these conversion statements. You may want to insert values of extended lines to help you. 3) 4 in = ft; 5) 3\frac{1}{2} ft = in; 8) 22 in = __ft.  9) Explain how you found the number of inches in 3\frac{1}{2} feet.”

• Unit 5, Lesson 5.1, Using Division to Change Fractions to Decimals and Percents, Problems 1-2, students calculate accurately and express numerical answers with a degree of precision appropriate for the problem context. “Change each fraction to a decimal and a percent. Recall in Unit 4, division was used to find decimal numbers. 1) Ronni wanted to rename \frac{3}{8} as a decimal and a percent. She divided as shown to the right. Use Ronni’s work to complete each equation below.” A long division problem is written to the right and student complete the chart with \frac{3}{8} as a decimal, fraction to the thousandths, hundredths, and as a percent. “2) Jay input 3 divided by 8 on his calculator and got 0.38. Why do you think Jay’s result is different than Ronni’s?”

• Unit 8, 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 expression, simplify, substitution, solution to an equation, evaluate, equation, variable, and solve an equation. Match and Compare Sorts are included in many units.

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 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 2, Lesson 2.1, Lesson Notes S2.1a: Building Rectangles, Slides 1-4, students look for structures and patterns to make generalizations to solve problems involving factors, composite numbers, and prime numbers. “Slide 1: Discuss the relationship between the given rectangle’s area and its dimensions. What are the dimensions of this rectangle? How are the dimensions and area of each rectangle related? Slide 2: Reveal the factor pairs of 24, and then the numbers that are not factors of 24. What does it mean for a number to be a factor of another number? How are factor pairs related to area? Slide 3: Check for understanding of the meaning of factors and factor pairs. Then use the number 12 to illustrate the meaning of composite number. Slide 4: Continue to build understanding about factors. Use the number 5 to illustrate the meaning of a prime number.”

• Unit 6, Lesson 6.1, Practice 2, Problems 3-6, students make use of structure as they apply Order of Operations to solve problems. “Evaluate each expression. 3) 4\dic2\cdot4$; 4) 8-2\cot3; 5) 16\div8\cdot2^3; 6) (12+8)\div4-2.”

• Unit 7, Lesson 7.3, Running, Problems 5-9, students look for and explain structures and patterns to make connections among mathematical representations of unit rates when using tables, graphs, and equations. “5) Complete the table below. 6) Write an equation for distance in terms of time. 7) What is the meaning of the coefficient of t in the equation? Circle it in the table and the graph. 8) At this rate, how far did Martino run in 2.5 hours? 9) At this rate, how many hours would it take Martino to run 17 miles?”

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 2, Lesson 2.3, Lesson Notes S2.3: Using Factor Ladders for GCF and LCM, students explain a general process/method as they factor. “Slide 1: Demonstrate how to use a factor ladder to find the GCF and LCM of 24 and 36. What number divides both 24 and 36? The slide begins using a divisor of 6, which produces quotients of 4 and 6, respectively. What number divides both 4 and 6? 2, which produces quotients of 2 and 3, which are relatively prime. Therefore, the GCF of 24 and 36 is 6\cdot2=12. What factors remain? 2 and 3. Therefore, the LCM of 24 and 36 is the GCF multiplied by these remaining factors, 12\cdot2\cdot3=72. Try other start divisors (i.e., 2, 3, or 12) if desired to confirm that results are the same. Why do you think this shortcut works? Answers will vary. Vertical numbers on the left are common factors, regardless of their order or composition. Horizontal numbers at the bottom of the ladder are always the remaining factors needed for the LCM.”

• Unit 4, Lesson 4.1, Slide Deck Alternative S1.4a students notice repeated calculations to understand algorithms and make generalizations related to division and repeated subtraction. “Mrs. Stern has 105 pencils to give to her class. If she has 35 students and each student gets the same amount, how many pencils will each student get? How many groups of 35 were subtracted in all? What if Mrs. Stern had 106 pencils? (1) Compute 144\div24 using repeated subtraction. 2) Write the quotient of 148\div24?”

• Unit 8, Lesson 8.1, Inequalities: Extend Your Thinking, Problems 4-7, students describe a general process/method as they solve inequalities. “Under each inequality below are four potential solutions. Circle the solutions that make the inequality true. Then write a description of ALL of the numbers that could be solutions to the inequality. 4) v+5>12; v=9, v=7, v=7.1, v=6.9. Description:____.”