6th Grade - Gateway 1

The materials reviewed for Everyday Mathematics 4 Grade 6 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Materials engage all students in extensive work with grade-level problems. Each lesson provides opportunities during Warm Up, Focus Activities, and Practice. Examples include:

• Lesson 2-11, Equivalent Ratios, Focus: Solving Unit Conversion Number Stories, Math Journal 1, students use ratios to answer questions about width and height, ”Sort the rectangles into groups of rectangles that look similar. Measure the sides of rectangles to the nearest centimeter. Record the measurements below.” Students find the width and height and then write the ratio of width to height. “1. Describe patterns that might help you decide whether rectangles are similar- that is, they have the same ratio for width to height. 2. How could you use this pattern to make another similar rectangle for one set of rectangles? 3a. Which rectangle group that is NOT composed of squares has rectangles that are the ‘most square’? b. What makes them ‘most square’?” Students engage in extensive work with grade-level problems for 6.RP.3, “Use ratio and rate reasoning to solve real-world and mathematical problems.”

• Lesson 4-1, Parenthesis, Exponents, and Calculators, Focus: Reviewing Exponents, Math Journal 1, students evaluate expressions using exponents, “4. Record each exponential expression as a multiplication expression. a. 6^4 b. 7^3. 5. Find the value of each exponential expression. a. 10^6 b. 4^3 c. 2^5 d. 1^6 e. 0^9 f. 25^2. 6. Write each expression using an exponent a. 4\star4. b. 8\star8\star8\star8\star8\star8. c. 72\star72\star72\star72. d. 1,349\star1,349\star1,349.” Lesson 4-2, Solving Problems with Order of Operations, Practice, Math Masters, Problem 4, students evaluate expressions using order of operations and exponents operations, “Evaluate. a. 45-(1+4^)^2+3. b. (2+4)^2\star(1+ 2)^4.” Students engage in extensive work with grade-level problems for 6.EE.1, “Write and evaluate numerical expressions involving whole-number exponents.”

• Lesson 5-3, Focus, Exploring Triangle Areas, Practice: Math Journal 2, students find the area of triangles, “2. Tape Triangles A and B together to form a parallelogram. Triangle A: Base = ___. Height = ___ Area of Triangle = ___. Tape your parallelogram in the space below. Base = ___. Height = ___. Area of parallelogram ___.” Students complete this activity with 4 other triangles including a right, equilateral, isosceles, and scalene triangle. Lesson 4, Composing and Decomposing Polygons to find Areas, Focus, Find the Area of Nevada, Math Journal 2, Problem 3, students decompose Nevada into triangles and parallelograms to find the state’s total area, “Estimate the area of the state of Nevada. a. Use a ruler. Draw line segments to show how to decompose the state into polygons to calculate the approximate area. Measure line segments that you will use to the nearest of a centimeter. Use the map scale to figure out the distances. b. Approximate area of Nevada: ___. c. Explain how you decomposed Nevada into polygons, and describe how you found its total area.” Students engage in extensive work with grade-level problems for 6.G.1, “Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.”

The materials provide opportunities for all students to engage with the full intent of Grade 6 standards through a consistent lesson structure. According to the Teacher’s Lesson Guide, Problem-based Instruction “Everyday Mathematics builds problem-solving into every lesson. Problem-solving is in everything they do. Warm-up Activity- Lessons begin with a quick, scaffolded Mental Math and Fluency exercise. Daily Routines - Reinforce and apply concepts and skills with daily activities. Math Message - Engage in high cognitive demand problem-solving activities that encourage productive struggle. Focus Activities - Introduce new content with group problem-solving activities and classroom discussion. Summarize - Discuss and make connections to themes of the focus activity. Practice Activities - Lessons end with a spiraled review of content from past lessons.” Examples of meeting the full intent include:

• Lesson 1-10, Introducing Integers, Focus: Introducing Negative Numbers, Math Journal 1, Problems 2 and 3, students use negative numbers in real-world situations, “2. Mr. Pima’s class planned a raffle. Three students sold raffle tickets. The goal for each student was to sell $50 in tickets. The table below shows how well each of the three students did. Complete the table. Student A, $5.50 short of the goal, -$5.50 Dollars Above or Below Goal, Student B, Met the goal exactly, $0 Dollars Above or Below Goal, Student C, Passed goal by $1.75, Dollars Above or Below Goal. 3. a. Describe a situation in which you can use positive and negative numbers. Use examples to explain positive, negative, and 0 in your situation. b. On the number line below, plot, and label some points for your situation.” Lesson 13, Locating Negative Rational Numbers on the Number Line, Focus, Introducing Opposites, Math Journal 1, students use integers to locate opposite numbers, “1. What number is the same distance from 0 as 1\frac{1}{2}? 2. What number is the same distance from 0 as -4? 4. Describe how to find the opposite of a number. 5. Why is 0 a special case? 6. Give the opposite of each number. Place the opposites on the number line. 3 ___, -4\frac{1}{2} ___, -5 ___, \frac{1}{3} ___, \frac{3}{4} ___, -2 ___.” Students engage in the full intent of 6.NS.5, “Understand that positive and negative numbers are used together to describe quantities having opposite directions or values and use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.”

• Lesson 5-8, Comparing Areas, Practice: Math Journal 2, Problem 5, students calculate unit rates for better gas mileage, “One car used 15.5 gallons of gasoline to travel 372 miles. A second car traveled 198 miles using 7.2 gallons. Which car got better gas mileage? Use rates to justify your answer.” Unit 6, Lesson 8, Practice, Math Journal 2, Problem 2, “The price for 8 tickets to the school Fun Fair is $42. Find a unit rate to help you complete the table. (Each ticket costs the same amount.)” The table shows Number of Tickets 1, 2, 10, x and Cost ___, ___, ___, ___.” Students engage in the full intent of 6.RP.3b, “Solve unit rate problems including those involving unit pricing and constant speed.”

• Lesson 6-6, Combining Like Terms, Practice: Math Journal 2, students solve real-world problems using an equation, “Katya runs at a rate 6.25 meters per second. Her younger cousin, Liova, runs 2.5 meters per second. Katya runs faster than Liova, she gives Liova a 100-meter head start in a 200-meter race. 1. Using the variable t to represent the number of seconds, write two expressions - one for Katya and one for Liova - that model how far from the start line they will be after t seconds. 2. Use your expressions from Problem 1 to figure out who will win the race. Show your work and explain your answer.” Lesson 8-6, Mobiles and Mathematics, Practice, Math Journal 2, Problem 4, students solve equations in a real-world problem, “Write an equation for each statement. Then solve the equation. a. 125% of x is 625. Equation: ___. Solution: ___. b. \frac{1}{4}of y is 9. Equation: ___. Solution: ___.” engages students in full intent of 6.EE.7, “Solve real-world and mathematical problems by writing and solving equations of the form x+p=q and px=q for cases in which p, q, and x are all nonnegative rational numbers.”

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

Digital materials’ Main Menu links to the “Spiral Tracker” which provides a view of how the standards spiral throughout the curriculum. The Lesson Landing Page contains a Standards section noting standards covered by the lesson. Teacher Edition contains “Correlation to the Standards for Mathematics” listing all grade-level standards and correlating lessons. Examples include:

• Lesson 2-6, Dividing Fractions with Common Denominators, Focus: Visualizing Fraction Division, students 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?”

• Lesson 4-6, The Distributive Property and Equivalent Fractions, Focus: Representing the Distributive Property, 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?” 

• Lesson 5-3, Area of Triangles, Focus: Exploring Triangle Areas, 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?” 

• Lesson 8-9, Planning a Trip, Warm Up: Mental Math and Fluency, 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). 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 materials reviewed for Everyday Mathematics 4 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.

The Teacher Edition contains a Focus section in each Section Organizer identifying major and supporting clusters covered. There are connections from supporting work to supporting work and major work to major work throughout the grade-level materials, when appropriate. Examples include:

• Lesson 1-3, Introducing the Mean, Math Journal 1, Problem 6, students use different strategies to find the mean of a data set, “Find the average number of raisins per box for the groups at right. Group A: 27, 30, 32, 33, 28. Group B: 29, 28, 32, 31, 30.” This connects the supporting work of 6.NS.B, “Compute fluently with multi-digit numbers and find common factors and multiples” to the supporting work of 6.SP.A, “Develop understanding of statistical variability”, and 6.SP.B, “Summarize and describe distributions.”

• Lesson 2-5, Comparing Strategies for Multiplying Fractions, Math Journal 1, Problems 1 and 2, students compare models and analyze strategies for fraction multiplication, “Use an area or number-line model to show how to find the solution for each problem. 1. \frac{2}{3}\star\frac{6}{8}=  2. \frac{6}{8}\star\frac{2}{3}=.” Teacher’s Lesson Guide states, “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?” This connects the major work of 6.NS.A, “Apply and extend previous understandings of multiplication and division to divide by fractions” to the major work of 6.EE.A, “Apply and extend previous understandings of arithmetic to algebraic expressions.”

• Lesson 4-9, Introduction to Inequalities, Math Journal 1, Problem 1, students match number sentence statements to descriptions of inequalities. Students choose the appropriate inequality statement from an answer bank, “Any number greater than or equal to 5.” This connects the major work of 6.EE.A, “Apply and extend previous understandings of arithmetic to algebraic expressions” to the major work of 6.EE.B, “Reason about and solve one-variable equations and inequalities.” 

• Lesson 5-2, Area of Parallelograms, Math Journal 2, Problem 3, students find the area of parallelograms, “Draw a line segment outside Parallelogram C to show its height. Base = ____. Height = ____. Area of parallelogram = ____.” This connects the supporting work of 6.NS.B, “Compute fluently with multi-digit numbers and find common factors and multiples” to the supporting work of 6.G.A, “Solve real-world and mathematical problems involving area, surface area, and volume.”

• Lesson 7-9, Independent and Dependent Variables, Math Journal 2, students use ratio tables to show relationships between dependent and independent variables. Students compare rates in an Ironman Triathlon to calculate rates in minutes per mile, complete 3 ratio tables relating time and distance, and graph the information on a coordinate grid. This connects the major work of 6.EE.C, “Represent and analyze quantitative relationships between dependent and independent variables” to the major work of 6.RP.A, “Understand ratio concepts and use ratio reasoning to solve problems.” 

• Lesson 8-9, Planning a Trip, Math Journal 2, Problem 9, students plan a trip using a budget, “Use your spreadsheet to design a trip that costs less than $1,000 but includes at least two nights in a hotel and three meals in a restaurant. Copy your solution into the spreadsheet on journal page 380.” This connects the major work of 6.EE.B, “Reason about and solve one-variable equations and inequalities” to the major work of 6.RP.A, “Understand ratio concepts and use ratio reasoning to solve problems.”

The materials reviewed for Everyday Mathematics 4 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.

Materials relate grade-level concepts to prior knowledge from earlier grades. Each Section Organizer contains a Coherence section with “Links to the Past” containing information about how focus standards developed in prior units and grades. Examples include:

• Unit 1, Data Displays and Number Systems, 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, Algebraic Expressions and Equations, 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, Equivalent Expressions and Solving Equations, 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.” 

Materials relate grade-level concepts to future work. Each Section Organizer contains a Coherence section with “Links to the Future” containing information about how focus standards lay the foundation for future lessons. Examples include:

• Unit 2, Fraction Operations and Ratios, 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, Equivalent Expressions and Solving Equations, 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, using spreadsheets to solve problems. In Unit 8, students will write equations to model and solve various real-world situations.”

• Unit 8, Applications: Ratios, Expressions, and Equations, 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.”

Materials contain content from future grades in some lessons that is not clearly identified. Examples include:

• Lesson 6-8, T-Shirt Cost Estimates, Focus: Comparing Models and Strategies, Math Journal 2, “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.” Solving real-world two-step equations is aligned to  7.EE.4.

• Lesson 7-8, Connecting Equations, Tables, and Graphs, Math Journal 2, Problem 2a, “Complete the table, and write the equation to represent the rule. Rule: 2\star 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.” Writing linear equations is aligned to 8.F.3.

• Lesson 8-8, Anthropometry, Focus: Using the Prediction Line, “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 adult's 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.