6th Grade - Gateway 1

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Examples of supporting standards/clusters connected to the major standards/clusters of the grade include but are not limited to:

• In Lesson 2-6, Teacher’s Lesson Guide, reason about and solve one-variable equations (6.EE.7) by using common factors to rewrite expressions using the distributive property (6.NS.4). The teacher displays different representations. The students demonstrate how to combine like terms and justify their steps. The teacher asks, “How do the representations show that the problems are all similar?”
• In Lesson 4-6, Teacher’s Lesson Guide, students apply the properties of operations to generate equivalent expressions (6.EE.3) to find the area of special quadrilaterals (6.G.1). The teacher displays rectangles as students record the matching expression of each rectangle area by looking at patterns. The students create a general rule for each pattern based on how they think the Distributive Property works. The teacher prompt states, “Ask a volunteer to describe how each expression represents the area of the corresponding rectangle. What factor is being distributed? How do you know the equations are true even though the expressions on either side of the equal sign look different?”  
• In Lesson 5-3, Teacher’s Lesson Guide, students find the area of right triangles (6.G.1)  to write, read, and evaluate expressions in which letters stand for numbers (6.EE.2). Students investigate finding the area of triangles. The teacher asks, “How is the area formula for a parallelogram similar to or different from the area of a triangle? How would you write a formula for the area of a triangle?”  
• In Lesson 8-9, Teacher’s Lesson Guide, students fluently divide multi-digit numbers using the standard algorithm (6.NS.2) to understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship (6.RP.2). Mental Math and Fluency teacher prompt states, “On their slates, have students record a unit rate that describes each situation.” Some examples provided for the teacher include, “96 students in 6 classes, 300 miles per 15 gallons, and 75 miles over 6 hours.”

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations that the amount of content designated for one grade level is viable for one year. 

Recommended pacing information is found on page xxii of the Teacher’s Lesson Guide and online in the Instructional Pacing Recommendations. As designed, the instructional materials can be completed in 170 days:

• There are 8 instructional units with 107 lessons. Open Response/Reengagement lessons require 2 days of instruction adding 8 additional lesson days.
• There are 43 Flex Days that can be used for lesson extension, journal fix-up, differentiation, or games; however, explicit teacher instructions are not provided.
• There are 20 days for assessment which include Progress Checks, Open Response Lessons, Beginning-of-the-Year Assessment, Mid-Year Assessment, and End-of-Year Assessment.  

The materials note lessons are 60-75 minutes and consist of 3 components: Warm-Up: 5-10 minutes; Core Activity: Focus: 35-40 minutes; and Core Activity: Practice: 20-25 minutes.

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations for being consistent with the progressions in the Standards. The instructional materials relate grade-level concepts explicitly to prior knowledge from earlier grades and present extensive work with grade-level problems. The instructional materials relate grade-level concepts with work in future grades, but there are a few lessons that contain content from future grades that is not clearly identified as such.

The instructional materials relate grade-level concepts to prior knowledge from earlier grades. Each Unit Organizer contains a Coherence section with “Links to the Past”. This section describes “how standards addressed in the Focus parts of the lessons link to the mathematics that children have done in the past.” Examples include:

• Unit 1, Teacher’s Lesson Guide, Links to the Past, “6.NS.5: In Grade 4, students identified lines of symmetry and recognized that when a figure is folded along its line of symmetry, the two parts match.”  
• Unit 4, Teacher’s Lesson Guide, Links to the Past, “6.EE.1: In Grade 4, students informally explored situations that involve whole-number exponents by solving problems that involve multiplying the same factor repeatedly. In Grade 5, students read, wrote, and compared numbers in standard and exponential notations.”  
• Unit 6, Teacher’s Lesson Guide, Links to the Past, “6.EE.4: In Grade 5, students both identified and generated equivalent expressions, including in the context of working with measurements with different units. In Unit 4, students compared equivalent expressions when writing numbers using four 4s. They also wrote and compared equivalent expressions when modeling and solving growing pattern problems.”  

The instructional materials relate grade-level concepts with work in future grades. Each Unit Organizer contains a Coherence section with “Links to the Future”. This section identifies what students “will do in the future.” Examples include:

• Unit 2, Teacher’s Lesson Guide, Links to the Future, “6.RP.2: Throughout Grade 6, students will continue to explore ratio situations and solve ratio problems. In Grade 7, students will extend their work with ratios to represent proportional relationships with equations. In addition, they will begin a formal exploration of ratios in the context of working with linear equations and slope.”
• Unit 6, Teacher’s Lesson Guide, Links to the Future, “6.EE.5: Throughout Grade 6, students will continue to practice writing equations and inequalities to model and solve problems. In Unit 7, students will write and interpret inequalities to help them identify mystery numbers, to determine the ingredients for a healthy salad, and when using spreadsheets to solve problems. In Unit 8, students will write equations to model and solve various real-world situations.”
• Unit 8, Teacher’s Lesson Guide, Links to the Future, “6.RP.3: In Grade 7, students will continue to practice using proportional relationships to solve multistep ratio and percent problems.”

In some lessons, the instructional materials contain content from future grades that is not clearly identified as such. Examples include:

• Lesson 6-3 “introduces the bar model, which can be an effective tool for solving equations with variables on one or both sides of an equation.” Throughout the lesson, students use bar models to solve given equations or equations that arise from word problems. However, the majority of the equations that students solve in the lesson do not align to 6.EE.7. For example, in Student Math Journal, Mathe Message, students solve “3d + 12 = 20 + d”; in Teacher Lesson Guide, Focus, Solving Equations with Bar Models, students solve 2p + 21 = 39 and 3e + 17 = 29 + 2e”; and in Teacher Lesson Guide, Math Masters, Home Link, students solve “4a + 12 = 96 and 6d + 7 = d + 22.”
• In Lesson 6-4, students solve multiple pan-balance problems, but the problems involve comparing the weights of objects and do not align to 6.EE.7. For example, in the Teacher Lesson Guide, Math Message, Problem 1, “Let m be the number of cylinders and n be the number of spheres. Write an equation that shows that the weights of the two sides are equal. 2m = m + 2n”; in Teacher Lesson Guide, Student Math Journal, Problem 3, “One cube weighs as much as ____ marbles.”; and in Math Masters, Home Link, Problem 2, “One ball weighs as much as ____ coin(s).” 
• In Lesson 6-5, students solve multiple pan-balance problems, but the problems involve comparing the weights of objects or equations in forms that do not align to 6.EE.7. For example, in the Teacher Lesson Guide, Math Message, Problem 1, “These two pan balances are each in perfect balance. a. Use the relationships in the pan balances shown above to determine which of the pan balances below are balanced. Circle the ones that are in balance. b. For any pan balance above that you did not circle, add or cross out objects to balance the pans.”, and in Math Masters, Home Link 6-5, students, “Find the value of the missing number that will balance each set of pans below. The same number is missing from both sides of a pan balance.”
• In Lesson 6-8, Student Math Journal, T-Shirt Cost Estimates, Focus, Comparing Models and Strategies, “Students compare and analyze models and strategies they used to solve real-world problems, (6.EE.7).” For example, in Problem 1, “Travis has 64 baseball cards and buys 3 new cards every week. When will Travis have 73 baseball cards? Define a variable and write an equation for Travis’s situation. Let g be the number of weeks. Equation: 64 + 3g = 73.” The equations that students write and solve in this lesson do not align to 6.EE.7.
• In Lesson 7-8, Student Math Journal, Problem 2a, “Complete the table, and write the equation to represent the rule. Rule: 2 * x + 2 = y”, and in Problem 6c, “Record an equation that represents the rule for the number of rhombuses in each step. Rule: 3(x) + 1 = y.” The form of these rules do not align to 6.EE.7.
• In Lesson 8-6, “Students explore how a mobile balances and use the balance formula to solve problems (6.EE.7).” In Student Math Journal, Solving Mobile Problems, Problem 2, “What is the distance from the fulcrum to each of the objects?” Students solve 20 * x = 15 * (x + 3).” In Math Masters, Home Links, Solving Mobile Problems, Problem 2, “What is the distance of each object from the fulcrum?” Students solve 8(x + 4) = 16(x - 4). Solving linear equations with variables on both sides of the equation does not align to 6.EE.7.
• In Lesson 8-8, Anthropometry, Focus, Using the Prediction Line, Teacher’s Lesson Guide, “Explain that these points fall on what is called a prediction line. The prediction line shows the exact values that result from using the formula representing the relationship between height and tibia (6.EE.9, 6.SP.5, 6.SP.5c).” In the Student Math Journal, Problems 2 and 3, “The following rule is sometimes used to predict the height (H) of an adult from the length of the adults tibia (t). Measurements are in inches. H = 2.6t + 25.5. Why do you think this rule might not predict the relationship for everyone? Use the rule above to complete the table. Tibia Length (in.) 11, 14, 19, 17 $$\frac{1}{2}$$: Height Predicted (in.) ?, ?, ?, ?.” Knowing that straight lines are widely used to model relationships between two quantitative variables is aligned to 8.SP.2.

Examples of the materials giving all students extensive work with grade-level problems include:

• In Lesson 3-2, Math Journal 1, Zooming in on Number Lines, “In Lesson 1-12, you zoomed in to find fractions between fractions. This process is even easier when you want to find decimals between decimals. a. Look at each number line separately and estimate where Point A is located on each. b. How far apart are the tick marks on each number line?” (6.NS.6)
• In Lesson 4-10, Math Masters, Student Reference Book, students solve inequalities by being the first player in their groups to discard all of their cards. “When it is your turn: Discard any of your number cards that is a solution to the inequality on the Solution Search Card.” (6.EE.5, 6.EE.8) 
• In Lesson 7-5, Math Journal 2, Problem 4, “Ten teaspoons of sugar is 40 grams of sugar. a. Compare the ratio/rate table to find the number of teaspoons of sugar in 1 fluid ounce of each drink type. b. Find the number of teaspoons of sugar in 1 fluid ounce of each drink. Cola: ___ Tsp sugar per fluid ounce, Fruit punch: ___  Tsp sugar per fluid ounce, Sports drink: ___ Tsp sugar per fluid ounce.” (6.RP.3b)

The instructional materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations for fostering coherence through connections at a single grade, where appropriate and required by the Standards.

Materials include learning objectives that are visibly shaped by CCSSM cluster headings. Focus and Supporting Clusters addressed in each section are found in the Table of Contents, the Focus portion of each Section Organizer, and in the Focus portion of each lesson. Examples include:

• The Lesson Overview for Lesson 2-1, “Students find factors and the greatest common factor (GCF) of two or more numbers,” is shaped by 6.NS.B, “Compute fluently with multi-digit numbers and find common factors and multiples.”
• The Lesson Overview for Lesson 4-3, “Students write numerical expressions for special cases of a pattern and learn to generalize a pattern using an algebraic expression,” is shaped by 6.EE.A, “Apply and extend previous understandings of arithmetic to algebraic expressions.”
• The Lesson Overview for Lesson 7-8, “Students represent a growing pattern using numbers, symbols, words, and graphs, and make connections between the representations,” is shaped by 6.EE.A, “Apply and extend previous understandings of arithmetic to algebraic expressions” and 6.EE.B, “Reason about and solve one-variable equations and inequalities.”
• The Lesson Overview for Lesson 8-3, “On a coordinate grid, students enlarge the dimensions of scale drawings for representations of artwork and use the scale to reproduce the artwork at its original size.” (Teacher’s Lesson Guide, page 732). This is shaped by the cluster heading, 6.RP.A, “Understand ratio concepts and use ratio reasoning to solve problems.” 

The materials include problems and activities connecting two or more clusters in a domain, or two or more domains in a grade, in cases where these connections are natural and important. Examples include:

• Lesson 2-5 connects 6.NS.A with 6.EE.A as students compare models and analyze strategies for fraction multiplication. In the Student Math Journal, Problems 1 and 2, “Use an area or number-line model to show how to find the solution for each problem. 1. $$\frac{2}{3}$$ * $$\frac{6}{8}$$ =      2. $$\frac{6}{8}$$ * $$\frac{2}{3}$$ =    .” In the Teacher’s Lesson Guide, “Have volunteers display their models, and prompt students to describe their models by asking questions like the following: How does your model represent the factors? Where is the answer in your representation?”
• Lesson 4-9 connects 6.EE.A with 6.EE.B as students match number sentence statements to descriptions of inequalities. In the Student Math Journal, Problem 1, “Any number greater than or equal to 5.” Students choose the appropriate inequality statement from an answer bank.
• Lesson 7-9 connects 6.EE.C with 6.RP.A as students use ratio tables to show relationships between dependent and independent variables. In the Student Math Journal, students compare rates in an Ironman Triathlon. They calculate rates in minutes per mile, complete 3 ratio tables relating time and distance, and graph the information on a coordinate grid.