6th Grade - Gateway 1

The materials reviewed for Math Nation Grade 6 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

The curriculum consists of nine units, including one optional unit. Assessments include Cool-down Tasks, Mid-Unit Assessments, and End-of-Unit Assessments. Examples of assessment items aligned to grade-level standards include:

• Unit 4, Mid-Unit Assessment (B), Question 2, “Kiran has used \frac{4}{5} of the pieces in his jigsaw puzzle. He has used 120 pieces. How many pieces are in the whole puzzle? A. 96; B. 125; C. 150; D. 216” (6.NS.1)

• Unit 7, End-of-Unit Assessment (A), Question 2, “Diego’s dog weighs more than 10 kilograms and less than 15 kilograms. Select all the inequalities that must be true if w is the weight of Diego’s dog in kilograms. A. w>10; B. w<10; C. w>11; D. w<11 ; E. w>15; F. w<15.” (6.EE.5) 

• Unit 8, Lesson 7, Cool-down, Questions 1 and 2, “The two histograms show the points scored per game by a college basketball player in 2008 and 2016. 1. What is a typical number of points per game scored by this player in 2008? What about in 2016? Explain your reasoning.  2. Write 2–3 sentences that describe the spreads of the two distributions, including what spreads might tell us in this context.” (6.SP.5)

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

The materials present opportunities for students to engage in extensive work and the full intent of most Grade 6 standards. Each lesson contains a Warm-Up, a minimum of one Exploration Activity, a Lesson Summary, Practice Problems, and three Check Your Understanding Questions. Each unit provides a Readiness Check and a Test Yourself! practice tool. Examples of full intent include:

• Unit 2, Lesson 8, 2.8.6 Practice Problems, Question 1, engages students with the full intent of 6.RP.3b (Solve unit rate problems including those involving unit pricing and constant speed). Students use double number lines to solve unit pricing problems. “In 2016, the cost of 2  ounces of pure gold was $2,640. Complete the double number line to show the cost for 1, 3,  and 4 ounces of gold.” Students are provided with a double number line with one labeled “cost in dollars” and the other labeled “ounces of gold.”  

• Unit 3, Lesson 6, 3.6.2 Exploration Activity, Question 1, engages students with the full intent of 6.RP.2 (Understand the concept of a unit rate a/b associated with a:b with b≠0, and use rate language in the context of a ratio relationship). Students solve real-world problems using ratios. “Priya, Han, Lin, and Diego are all on a camping trip with their families. The first morning, Priya and Han make oatmeal for the group. The instructions for a large batch say, ‘Bring 15 cups of water to a boil, and then add 6 cups of oats.’ Priya says, ‘The ratio of the cups of oats to the cups of water is 6:15.  That’s 0.4 cups of oats per cup of water.’ Han says, ‘The ratio of the cups of water to the cups of oats is 15:6. That’s 2.5 cups of water per cup of oats.’ 1. Who is correct?  Explain your reasoning. If you get stuck, consider using the table.” Students are provided with a table with one column labeled “Water (cups)” with rows 15, 1, and and the other labeled “Oats (cups)” with rows 6, , and 1.

• Unit 6, Lesson 6, 6.6.6 Practice Problems, Question 3, engages students with the full intent of 6.EE.2a (Write expressions that record operations with numbers and with letters standing for numbers). Students solve problems involving variables. “A bottle holds 24 ounces of water. It has x ounces of water in it. a. What does 24-x represent in this situation? b. Write a question about this situation that has 24-x for the answer.” 

The materials present opportunities for students to engage with extensive work with grade-level problems. Examples of extensive work include:

• Unit 3, Lesson 3, Lesson 4, and Lesson 9 engage students in extensive work with 6.RP.3d (Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities). Lesson 3, 3.3.5 Practice Problems, Question 5, students convert units of measurement. “Clare wants to mail a package that weighs 4\frac{1}{2} pounds. What could this weight be in kilograms?” Answer choices: 2.04, 4.5, 9.92, 4500. Lesson 4, 3.4.2 Exploration Activity, students use information provided in word problems in context to solve ratio reasoning application problems. “Elena and her mom are on a road trip outside the United States. Elena sees this road sign. (Image given stating, “Maximum 80.”) Elena’s mom is driving 75 miles per hour when she gets pulled over for speeding. 1. The police officer explains that 8 kilometers is approximately 5 miles. a. How many kilometers are in 1 mile? b. How many miles are in 1 kilometer? 2. If the speed limit is 80 kilometers per hour, and Elena’s mom was driving 75 miles per hour, was she speeding? By how much?” Unit 3, Lesson 9, 3.9.6 Practice Problems, Question 4, students use ratio reasoning to compare three jobs to find out which paid better. “Andre sometimes mows lawns on the weekend to make extra money. Two weeks ago, he mowed his neighbor’s lawn for \frac{1}{2} hour and earned $10. Last week, he mowed his uncle’s lawn for \frac{3}{2} hours and earned $30. This week, he mowed the lawn of a community center for 2 hours and earned $30. Which jobs paid better than others? Explain your reasoning.”   

• Unit 6, Lesson 16, Exploration Activity, Practice Problem, and Check Your Understanding engage students in extensive work with 6.EE.9 (Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation). In 6.16.2 Exploration Activity, Question 2, students solve problems identifying independent and dependent variables. “Lin notices that the number of cups of red paint is always \frac{2}{5} of the total number of cups. She writes the equation r=\frac{2}{5}t to describe the relationship. Which is the independent variable? Which is the dependent variable? Explain how you know.” In 6.16.5 Practice Problems, Question 1, students analyze the relationship between two variables using graphs and tables and relate these to an equation. “Here is a graph that shows some values for the number of cups of sugar, s, required to make x batches of brownies. A. Complete the table so that the pair of numbers in each column represents the coordinates of a point on the graph. B. What does the point (8, 4) mean in terms of the amount of sugar and number of batches of brownies? C. Write an equation that shows the amount of sugar in terms of the number of batches.” Students are provided a graph that shows the “cups of sugar” on the y-axis and the “batches of brownies” on the x-axis. In 6.16.7 Check Your Understanding, Question 1, students write an equation to represent one quantity in terms of the other quantity. “At a local farm, you can buy 2 boxes of strawberries for $9.00. Which equation represents the relationship between the boxes of strawberries, x, and the cost in dollars, y ?” Answer choices: (A) y=4.5x; (B) y=9x; (C) x=4.5y; (D) x=9y.” 

• Unit 7, Lessons 6 and 7, engage students in extensive work with 6.NS.7 (Understand ordering and absolute value of rational numbers). Lesson 6, 7.6.3 Exploration Activity, Questions 1 and 2, students use information from a word problem to describe what numbers mean in the context of the problem. “A part of the city of New Orleans is 6 feet below sea level. We can use ‘-6 feet’ to describe its elevation, and |-6| ‘feet’ to describe its vertical distance from sea level. In the context of elevation, what would each of the following numbers describe? A. 25 feet; B. |25| feet; C. -8 feet; D. |-8| feet 2. The elevation of a city is different from sea level by 10 feet. Name the two elevations that the city could have.” Lesson 7, 7.7.2 Exploration Activity, Questions 1 and 2, use information from a word problem to place people in order based on a description. “A submarine is at an elevation of −100 feet (100 feet below sea level). Let's compare the elevations of these four people to that of the submarine: Clare's elevation is greater than the elevation of the submarine. Clare is farther from sea level than the submarine. Andre's elevation is less than the elevation of the submarine. Andre is farther away from sea level than the submarine. Han's elevation is greater than the elevation of the submarine. Han is closer to sea level than is the submarine. Lin's elevation is the same distance away from sea level as the submarine's. 1. Complete the table as follows. A. Write a possible elevation for each person. B. Use <, >, or = to compare the elevation of that person to the submarine. C. Use absolute value to tell how far away that person is from sea level (elevation 0). 2. Priya says her elevation is less than the submarine’s and she is closer to sea level. Is this possible? Explain your reasoning.” 

The materials do not provide opportunities for students to meet the full intent of the following standard:

• While students engage with 6.RP.3a (Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.), students have no opportunities to work on plotting the pairs of values on the coordinate plane to meet the full intent of the grade-level standards.

The materials provide limited opportunities for all students to engage in extensive work with the following standard:

• Unit 7, Lesson 11, 7.11.2 Exploration Activity, there are limited opportunities to engage with extensive work on standard 6.NS.6b (Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes). In this example students label, make observations about, and plot coordinates. This is the only opportunity for students to engage with this standard. “1. Label each point on the coordinate plane with an ordered pair. 2. What do you notice about the locations and ordered pairs of 𝐵, 𝐶, and 𝐷? How are they different from those for point A? 3. Plot a point at (-2,5). Label it 𝐸. Plot another point at (3, -4.5). Label it 𝐹. 4. The coordinate plane is divided into four quadrants, I, II, III, and IV, as shown here. A. In which quadrant is G located? H? I? B.  A point has a positive y-coordinate. In which quadrant could it be?”

The materials reviewed for Math Nation Grade 6 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade.

To determine the amount of time spent on major work, the number of units, the number of lessons, and the number of days were examined. Assessment days are included. Any lesson, assessment, or unit marked optional was excluded.

• The approximate number of units devoted to major work of the grade (including supporting work connected to the major work) is 5 out of 8, which is approximately 63%.

• The approximate number of lessons devoted to major work (including assessments and supporting work connected to the major work) is 95 out of 142, which is approximately 67%. 

• The approximate number of days devoted to major work of the grade (including supporting work connected to the major work) is 95 out of 142, which is approximately 67%.

A lesson-level analysis is most representative of the materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 67% of the materials focus on major work of the grade.

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

Materials are designed to connect supporting standards/clusters to the grade’s major standards/clusters. Examples of connections include:

• Unit 2, Lesson 15, 2.15.7 Practice Problems, Question 5, 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.3 (Use ratio and rate reasoning to solve real-world and mathematical problems) and 6.RP.3b (Solve unit rate). Students interpret data from a number line to solve problems involving unit rates students use unit rates to solve real-world problems. “A cashier worked an 8-hour day, and earned $58.00.  The double number line shows the amount she earned for working different numbers of hours. For each question, explain your reasoning. a. How much does the cashier earn per hour? b. How much does the cashier earn if she works 3 hours?”

• Unit 4, Lesson 13, 4.13.5 Exploration Activity, connects the supporting work of 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) to the major work of 6.NS.1 (Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions). Students find the length of a tray given the width and area, calculate how many titles would be needed to cover the tray completely, and draw a diagram to represent their answer. “Noah would like to cover a rectangular tray with rectangular tiles.  The tray has a width of 11\frac{1}{4} inches and an area of 50\frac{5}{8} square inches. 1. Find the length of the tray in inches. 2. If the tiles are \frac{3}{4}  inch by \frac{9}{16}  inch, how many would Noah need to cover the tray completely, without gaps or overlaps? Explain or show your reasoning. 3. Draw a diagram to show how Noah could lay the tiles. Your diagram should show how many tiles would be needed to cover the length and width of the tray, but does not need to show every tile.”

• Unit 5, Lesson 8, 5.8.7 Check Your Understanding, Question 1, 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.EE.A (Apply and extend previous understandings of arithmetic to algebraic expressions). Students multiply decimals to solve real-world problems. “Victor decided to take a taxi after watching a football game at Raymond James Stadium. The taxi charges a flat fee of $3.50 for pickup, $0.90 for each of the first 4 miles, and $0.75 for each additional mile. Victor’s apartment is 12 miles from the stadium. Complete the statement by typing a value into the blank space.  The total fare for Victor’s taxi ride is $_________.”

• Unit 6, Lesson 4, 6.4.2 Exploration Activity, 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.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). Students solve equations using the four operations. “Solve the equations in one column. Your partner will work on the other column. Check in with your partner after you finish each row. Your answers in each row should be the same. If your answers aren't the same, work together to find the error and correct it. [Column A 18 =2x; Column B 36 = 4x] [Column A 17 = x +9; Column B 13 = x + 5] [Column A 8x = 56; Column B 3x = 21].”

The materials reviewed for Math Nation 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 major work to major work throughout the grade-level materials, when appropriate. Examples include:

• Unit 4, Lesson 11, Cool-down, Question 2, connects the major work of 6.NS.A (Apply and extend previous understandings of multiplication and division to divide fractions by fractions) to the major work of 6.EE.B (Reason about and solve one-variable equations and inequalities). Students use fractions to solve word problems. “If \frac{4}{3} liters of water are enough to water \frac{2}{5} of the plants in the house, how much water is necessary to water all the plants in the house? Write a multiplication equation and a division equation for the situation, then answer the question. Show your reasoning.” 

• Unit 5, Lesson 6, 5.6.3 Exploration Activity, 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). Students use area diagrams to compute the products of decimals. “1. In the diagram, the side length of each square is 0.1 unit. a. Explain why the area of each square is not 0.1 square unit. b. How can you use the area of each square to find the area of the rectangle? Explain or show your reasoning. c. Explain how the diagram shows that the equation (0.4)\cdot(0.2)=0.08 is true.  2. Label the squares with their side lengths so the area of this rectangle represents 40\cdot20. a. What is the area of each square? b. Use the squares to help you find 40\cdot20. Explain or show your reasoning. 3. Label the squares with their side lengths so the area of this rectangle represents (0.4)\cdot(0.2). Next, use the diagram to help you find (0.4)\cdot(0.2). Explain or show your reasoning.” Students are provided a rectangle divided into eight square boxes for each set of problems.

• Unit 6, Lesson 7, 6.7.2 Exploration Activity, connects the major work of 6.RP.A (Understand ratio concepts and use ratio reasoning to solve problems) to the major work of 6.EE.B (Reason about and solve one-variable equations and inequalities). Students perform repeated calculations involving percentages and generalize to write algebraic expressions.   “1. Answer each question and show your reasoning. A. Is 60% of 400 equal to 87? B. Is 60% of 200 equal to 87?  C. Is 60% of 120 equal to 87?  2. 60% of x is equal to 87. Write an equation that expresses the relationship between 60%, x, and 87. Solve your equation. 3. Write an equation to help you find the value of each variable. Solve the equation. A. 60% of c is 43.2 B. 38% of e is 190.”

• Unit 8, Lesson 4, 8.4.2 Exploration Activity, connects the supporting work of 6.SP.A (Develop understanding of statistical variability) to the supporting work of 6.SP.B (Summarize and describe distributions). Students construct a dot plot from a set of data, make observations about the dot plot, and summarize their observations. “1. Use the tables from the warm-up to display the number of toppings as a dot plot. Label your drawing clearly. 2. Use your dot plot to study the distribution for number of toppings. What do you notice about the number of toppings that this group of customers ordered? Write 2-3 sentences summarizing your observations.”

The materials reviewed for Math Nation 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.

Grade-level concepts related explicitly to prior knowledge from earlier grades along with content from future grades is identified and related to grade-level work in the Teacher Edition. Generally, explicit connections are found in the Course Guide or the Full Lesson Plan. 

Examples of connections to future grades include:

• Unit 3, Lesson 15, Full Lesson Plan, Lesson Narrative, connects 6.RP.3 to 7.RPA. “This lesson focuses on finding ‘A% of B’ as efficiently as possible…The third activity hints at work students will do in grade 7, namely finding a constant of proportionality and writing an equation to represent a proportional relationship.”

• Unit 6, Lesson 13, Full Lesson Plan, 6.13.2 Classroom activity, connects 6.EE.4 to 7.EE.A, A-SSE.A and A-SSE.B. “The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure (MP7) to compare exponential expressions. To this end, encourage students to rewrite expressions in a different form rather than evaluate them to a single number. For students who are accustomed to viewing the equal sign as a directive that means ‘perform an operation,’ tasks like these are essential to shifting their understanding of the meaning of the equal sign to one that supports work in algebra, namely, ‘The expressions on either side have the same value.’”

• Unit 7, Lesson 14, Full Lesson Plan, Lesson Narrative, connects 6.NS.6 to 7.NS.1 and 8.EE.B, “In this lesson, students explore ways to find vertical and horizontal distances in the coordinate plane. [...] Students will use these skills in Grade 7 to find distances on maps. In Grade 8, they will use these skills to draw slope triangles in the coordinate plane and find the lengths of their sides when considering graphs of proportional and nonproportional relationships.”

Examples of connections to prior knowledge include:

• Unit 2, Lesson 2, Full Lesson Plan, 2.2.1 Warm-Up, builds on 4.NF.4 and 5.NF.3 and connects to 6.RP.1. “This number talk helps students recall that dividing by a number is the same as multiplying by its reciprocal. Four problems are given, however, they do not all require the same amount of time. Consider spending 6 minutes on the first three questions and 4 on the fourth question. In grade 4, students multiplied a fraction by a whole number, using their understanding of multiplication as groups of a number as the basis for their reasoning. In grade 5, students multiply fractions by whole numbers, reasoning in terms of taking a part of a part, either by using division or partitioning a whole…Two important ideas that follow from this work and that will be relevant to future work should be emphasized during discussions: Dividing by a number is the same as multiplying by its reciprocal. We can multiply numbers in any order if it makes it easier to find the answer.”

• Unit 4, Lesson 2, Full Lesson Plan, Lesson Narrative and 4.2.1 Warm-Up, builds on 3.OA.2 and connects to 6.NS.A. Lesson Narrative, “In this lesson, students revisit the relationship between multiplication and division that they learned in prior grades. Specifically, students recall that we can think of multiplication as expressing the number of equal-size groups, and that we can find a product if we know the number of groups and the size of each group. They interpret division as a way of finding a missing factor, which can either be the number of groups, or the size of one group. They do so in the context of concrete situations and by using diagrams and equations to support their reasoning.”... 4.2.1 Warm-up, “The purpose of this warm-up is to review students' prior understanding of division and elicit the ways in which they interpret a division expression. This review prepares them to explore the meanings of division in the lesson.”

• Unit 7, Lesson 4, Full Lesson Plan, 7.4.1 Warm-Up, builds on 4.NBT.2 and 5.NBT.3b and connects to 6.NS.C. “The purpose of this warm-up is for students to review strategies for comparing whole numbers, decimal numbers, and fractions as well as the use of inequality symbols. The numbers in each pair have been purposefully chosen based on misunderstandings students typically have when comparing.”