6th Grade - Gateway 1

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Within the MathLinks: Core 2nd Edition materials, the quizzes and cumulative tests are found online in the Teacher Portal in PDF and editable Microsoft Word versions. Cumulative tests are primarily multiple-choice, while quizzes are typically short answer. Materials assess grade-level standards and do not include above-grade assessment items. Examples include:

• Unit 1, Quiz A, Problem 1, students use a numerical data set to determine measures of center and variability in questions a-j. “The list below represents the ages of professional soccer players who retired from the sport last year: 35, 35, 38, 37, 45, 31, 30, 30, 45, 27, 30, 30. j) The Mean Absolute Deviation (MAD) of the data set is ___. (rounded to the nearest tenth) Recall that the MAD is the average of the distance of the data points to the mean.” (6.SP.5c)

• Unit 4, Task - A Triple Celebration, Problem 1, students use visual fraction models to represent cakes shared in class. “Andy, Brandy and Candy are triplets, each in a different sixth grade class. Their mom wants to celebrate their birthday by bringing brownies to their 3 classes. She makes 2 sheet pans of brownies to share and wants to share the same amount with each class. 1) Draw a picture to show how the 2 pans of brownies can be equally shared among the 3 classes. How much of a pan does each class get?” (6.NS.1)

• Unit 5, Quiz A, Problem 2, students determine percent. ”Devon got 16 out of 20 problems correct on a test. She wanted to get at least 80% of them correct on this test. Did she reach her goal? Explain.” (6.RP.3c)

• Cumulative Tests, Test 6, Problem 5, students demonstrate understanding about parts of an expression. “Choose all of the statements that are true about the expression 7m+3n+2+6p. A) The expression contains 3 terms. B) 7, 3, and 6 are the coefficients of the variables. C) The constant term is 2. D) 7, 3, 2 and 6 are all constant terms.” (6.EE.2b)

• Cumulative Tests, Test 10, Problem 9, students reason about inequalities. “Choose all numbers below that are solutions to x>6.  A) -6.02,  B) -4.5, C) -10.7,  D) 3.93.” (6.EE.5)

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. 

Materials present all students with extensive work with grade-level problems. Examples include:

• Unit 1, Lesson 1.2, Practice 3, Problems 1-2, students describe a set of data based on its center and spread. “This data shows the number of hours of online games played in one week by 13 teens: 13, 28, 15, 10, 10, 17, 4, 15, 17, 10, 8, 11, 6.  1) Rewrite the data set in order from least to greatest in the table below. 2) Calculate measures of center and spread associated with the median: range, median, five-number summary, interquartile range (IQR).” (6.SP.2)

• Unit 2, Lesson 2.2, Practice 3, Problem 10, students find the least common multiple. “The drama club meets in the school auditorium every 6 days and the choir meets there every 9 days. If the groups are both meeting in the auditorium today, how many days from now will be the next day that they have to share the auditorium?” (6.NS.4)

• Unit 6, Lesson 6.1, Practice 3, Problems 1 and 2, students perform arithmetic operations in the conventional order. “1) Evaluate 2\cdot8+12\div4 .2) Copy the expression in problem 1. Then insert exactly one set of parentheses to make an expression whose value is 10. Show work to justify.” (6.EE.2c)

Materials present opportunities for all students to meet the full intent of the standard. 

• In both the student and teacher editions, grade-level standards for each unit are listed. If the standard is only partially addressed during the unit, the remainder of the text is struck through then identified in a different unit, making it clear when the full intent has been met. For example: 6.RP.3 - “Using ratio and rate reasoning to solve real-world and mathematical problems” is first addressed in Unit 3. The standard has strike throughs on “equations”, “plot the pairs of values on the coordinate plane”, and “c” is not addressed. However, this standard is addressed again in Units 4, 5, 7, 8, and 9 with various elements struck through, until all parts of the standard are addressed in multiple problems. Example problems for 6.RP.3 include: 

• Unit 3, Lesson 3.1, Practice 3, Problem 2, students use ratio and rate reasoning to solve problems by creating tape diagrams and tables. “Sam makes tie-dyed shirts. Her most frequently used colors are orange and green. a) For the orange dye, she uses red and yellow in a ratio of 3:2. How many ounces of red and yellow dye will she need if she wants to make 80 ounces of orange dye? Use a tape diagram. b) For the green dye, she uses blue and yellow in a ratio of 5:2.  How many ounces of yellow dye will she need if she is using 40 ounces of blue dye? Use a table.” 

• Unit 4, Lesson 4.2, Practice 7, Problem 2, students use ratio and rate reasoning to solve problems by finding unit rates. “Show which is the best buy: 6 burgers for $22.50; 4 burgers for $18; 5 burgers for $21.”

• Unit 5, Lesson 5.2, Practice 7, Problem 8, students use ratio and rate reasoning to solve problems by finding percent. “Mr. Gold’s 6th grade class earned $1,290 from the fundraiser. They are setting aside \frac{1}{4} of the money for an end of the school year dance, 30% for the buddy program, and the remaining money is for new technology. How much money do they have for: a) the dance?  b) the buddy program?  c) new technology?”

• Unit 7, Lesson 7.2, Practice 6, Problem 7, students use ratio and rate reasoning to solve problems by plotting pairs of values on the coordinate plane to compare unit rate. After graphing data in the prior problem, students answer, “Which graph illustrates a greater cost increase per each additional keychain? How can you see this when comparing graphs?”

• Unit 8, Lesson 8.2, Getting Started, Problem 11, students use ratio and rate reasoning to solve problems by writing equations to determine a unit rate. “The weight of one bag of apples, x, is unknown. There are 5 bags of apples that are all of this weight. The total weight is 40 pounds. For this situation: a) Draw a tape diagram. b) Write an equation and its solution. c) Use the value for x from part b above. Write an equation to express that the sum of the weights of the bag of apples plus the weight of a bag of oranges, y, is 12 pounds. Solve for y.”

• Unit 9, Lesson 9.3, The Food Drive, Problem 2, students use ratio and rate reasoning to solve problems by converting measurement units. “Convert all measurements in the table before making calculation decisions.” In the table, students must convert inches to feet to yards for both a box and a truck.

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

To determine the approximate amount of time spent on major work of the grade, materials were analyzed from three different perspectives; units, lessons, and hours. Lesson reviews, unit reviews, and assessment days are included. In addition, supporting work that connects to major work is included.

• The approximate number of units devoted to major work of the grade is 7.4 out of 10, which is approximately 74%.

• The approximate number of lessons devoted to major work is 25 out of 33, which is approximately 76%. 

• The approximate number of hours devoted to major work of the grade is 115 out of 140, which is approximately 82%. One hundred forty hours includes all lessons, reviews, and assessments, but it does not include time indicated for intervention, enrichment, and school obligations as those needs vary. 

A lesson-level analysis is most representative of the instructional materials, because the lessons include major work, supporting work connected to major work, and have the review and assessment embedded. Based on this analysis, approximately 76% of the instructional materials for MathLinks: Core 2nd Edition Grade 6 focus on major work of the grade.

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Connections between supporting and major work enhance focus on major work of the grade. Examples include:

• Unit 4, Lesson 4.2, Practice 7, Problem 4, connects the supporting work of 6.NS.3 (Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation) to the major work of 6.RP.3b (Solve unit rate problems including those involving unit pricing and constant speed). Students multiply and divide decimals to find a rate and use their solution in related questions. “On Saturday Angela babysat for 5 hours and earned $62.50. a) How much did she get per hour? b) At this rate, how much would she earn in 9 hours? c) On Sunday she babysat again, getting the same pay rate, and earned $43.75. How many hours did she work? d) How much more did she earn Saturday compared to Sunday?”

• Unit 5, Lesson 5.3, Practice 8, Problems 1 and 2, connect the supporting work of 6.SP.4 (Display numerical data in plots on a number line, including dot plots, histograms, and box plots) to the major work of 6.RP.3c (Find a percent of a quantity as a rate per 100; solve problems involving finding the whole, given a part and the percent). Students create double number lines and use the number lines to find percent. A double number line is provided with Quantity and Percent. “1) Complete the double number lines below with the information given. 2) Use the double number lines above to help you answer questions and write equivalent fractions. What is 20% of $60?”

• Unit 6, Lesson 6.1, Practice 1, Problems 6-9, connect the supporting work 6.NS.4 (Find the greatest common factor of two whole numbers less than or equal to 100…) to the major work of 6.EE.3 (Apply the properties of operations to generate equivalent expressions). Students factor out the GCF to generate equivalent expressions. “Rewrite each sum as a product by factoring out the GCF and applying the Distributive property. Check that expressions are equivalent. 6) 14+21; 7) 24-9; 8) 5(3)+5(5); 9) 15-3.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

There are connections from supporting work to supporting work and/or from major work to major work throughout the grade-level materials, when appropriate. Examples include:

• Unit 1, Lesson 1.3, Getting Started, Problems 1-4, connect supporting work of 6.SP.A (Develop understanding of statistical variability) to supporting work of 6.NS.B (Compute fluently with multi-digit numbers and find common factors and multiples.) Students display, analyze, and describe data sets. Given a Line Plot, Histogram, and Box Plot that all display the same data set, “1) Make a list, in order, of the data values in the displays; 2) Write the five-number summary for the data. Circle the median on each display if possible; 3) Find the mean; 4) Which has greater value, the median or the mean? Why?”

• Unit 7, Lesson 7.2, Practice 6, Problems 1-7, connects major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems.) and major work of 6.EE.C (Represent). Students compare and analyze data about two keychains using different representations. “Here are two more keychain packages. LOCKS- 3 for $6 and CUBES- 2 for $5. 1) Complete each table below; 2)  Explain how you know which is the cheaper purchase based on unit price; 3) Explain how you know which is cheaper based on the entries with x=3; 4) Explain how you know which is cheaper based on the entries with y=10; 5) Write a rule for each; 6) Complete a graph for each; 7) Which graph illustrates a greater cost increase per each additional keychain? How can you see this when comparing the graphs?” 

• Unit 8, Lesson 8.3, Practice 5, Problem 12, connects the major work in 6.NS.A (Apply and extend previous understanding of multiplication and division to divide fractions by fractions.) to major work in 6.EE.B (Reason about and solve one-variable equations and inequalities.) as students use their understanding of fraction operations to solve one step equations. “\frac{1}{6}=1\frac{2}{3}x”.

• Unit 10, Lesson 10.4, Practice 7, Problem 2, connects supporting work of 6.G.A (Solve real-world and mathematical problems involving area, surface area, and volume.) and supporting work of 6.NS.B (Compute fluently with multi-digit numbers and find common factors and multiples.) Students have plotted coordinate points to draw a house and yard, identified each polygon by name, and found the side lengths. In this problem, students “Find the area of each section. Show formulas, work, and solution. Driveway, House, Patio, BBQ Area, Grass, Front Yard.”

The materials reviewed for MathLinks: Core 2nd Edition Grade 6 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.

Within the Teacher Edition, General Information, each unit provides information about relevant aspects of the content which involve the progression of mathematics. Additionally, Teacher Notes within some lessons identify when current content is building on prior learning and/or connecting to future concepts. Connections to future content and prior knowledge include:

• Unit 1, 6th Grade Fluency Requirements, “In previous grades, students solve whole number and decimal problems using a progression of sense-making methods that build towards the standard algorithm in grade 6.” Connections to prior work include: “Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. [5.NBT.6]” And, “Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. [5.NBT.7]”

• Unit 4, Fluency, describes how computational fluency is a culmination of learning spanning several years. “Grade 3: objects, drawings, strategies; Grade 4: drawing, strategies, illustrations, explanations; Grade 5: strategies, illustrations, explanations; Grade 6: standard algorithm.” 

• Unit 7, The Algebra Progression in MathLinks: Grade 6, makes connections between this unit and 8th grade content. In this unit, “Inputs and Outputs, students use visuals and contexts to analyze and solve problems with multiple representations. Concepts related to proportional reasoning are reviewed and emphasized. Students' knowledge of expressions enables them to generate equations for relationships relating two variables, called "input-output rules. Without explicitly defining "function" (this is done in grade 8), students begin to develop flexibility when working with variables, expressions, and equations. The problems introduced set the stage for solving a linear equation in one variable since these equations are of the form x+p=q and px=q (ie., "one-step" equations") for cases in which p and q, and are nonnegative rational numbers.”

• Unit 10, Lesson 1, Lesson Notes S10.4b: A Basketball Court, the Teacher Notes connect current grade-level work to future work. “What is the ordered pair for the top of the key? Since circles are not studied in detail until 7th grade, help with reasoning as needed through the fact that if the free throw line is 10 ft, a diameter perpendicular to it is also 10 ft, and the radius of the circle is 5 ft. Therefore, the top of the key is at (0, 10), 25 ft from the baseline and 10 ft from the half court line.”

• Unit 10, Lesson 4, Lesson Notes S10.4a: House Plans, the Teacher Notes connect current grade-level work to future work. “On a coordinate grid, students create a footprint for a house and its adjacent areas, and find lengths of the sides of these figures. Because addition and subtraction of signed numbers is a 7th grade topic, we look at three cases where coordinates lie on a horizontal or vertical line: (1) at least one of the two points falls on an axis, (2) both points lie in the same quadrant, and (3) the points lie in different quadrants. Students use absolute value notation to represent the distance from zero for the coordinate and (distance must not be negative), and add or subtract appropriately.”

Teacher Edition, Big Ideas and Connections in each unit identifies the focus concepts of the grade level and draws connections among the content specific to the current unit. “Grade 6 is organized around seven big ideas. This graphic provides a snapshot of the ideas in Unit 3 and their connections to each other.” Below the graphic, a chart listing “Prior Work” and “What’s Ahead”, and “These ideas build on past work and prepare students for the future.” Examples include: 

• Unit 3, Teacher Edition, Big Ideas and Connections, Prior Work, “Represent and solve problems involving multiplication and division. (Grades 3, 4, 5); Build fluency with concepts and operations of with fractions. (Grades 3, 4, 5); Convert within measurement systems. (Grades 4, 5); Gain familiarity with factors and multiples. (Grade 4); Generate and analyze patterns. (Grades 4, 5)”

• Unit 3, Teacher Edition, Big Ideas and Connections, What’s Ahead, “Apply proportional reasoning to other contexts, including markups, discounts, interest rates, and percents. (Grades 6, 7); Use proportional reasoning to make sense of input/output situations. (Grades 6, 7); Build upon proportional reasoning when studying the broader world of functions. (Grade 8, HS); Use proportional reasoning to solve problems involving similarity and scale. (Grades 7, 8, HS)”