6th to 8th Grade - Gateway 1

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 8 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. 

The assessments are aligned to grade-level standards and do not include content from future grades. Each unit includes Sub-Unit Quizzes and an End-of-Unit Assessment, both available in Forms A and B, and offered in print and digital formats.

Examples include:

• Grade 6, Unit 4: Dividing Fractions, End-of-Unit Assessment: Form A, Screen 12, Problem 7b, students interpret and compute quotients of fractions and solve word problems involving the division of fractions by fractions. The materials state, “Amir and his grandma are making roti, an Indian bread. Amir’s grandma uses a 3443​ - cup scoop. They need 512521​ cups of flour. How many scoops do they need?” (6.NS.1)

• Grade 7, Unit 1: Scale Drawings, Sub-Unit Quiz, Screen 9, Problem 5, students solve problems involving scale drawings of geometric figures, including determining actual lengths and areas and reproducing drawings at different scales. The materials state, “Rectangle S is 3 units by 5 units. a. Sketch a scaled copy of rectangle S with an area of 60 square units. Label each side length of the copy. b. What is the scale factor between rectangle S and your copy?” An interactive graph is included that allows students to sketch a rectangle. (7.G.1)

• Grade 8, Unit 3: Proportional and Linear Relationships, End-of-Unit Assessment: Form A, Screen 9, Problem 5, students compare proportional relationships represented in multiple formats. THe materials state, “One day, three runners ran 10 miles, each at their own constant speed. Which runner ran the fastest? Runner 1, Runner 2, Runner 3.” The problem includes distinct visual representations for each runner. Runner 1 is represented by a graph showing a line with distance (miles) on the vertical axis and time (minutes) on the horizontal axis. Runner 2 is represented by a table with two columns—time (minutes) and distance (miles)—containing five data entries. The representation for Runner 3 shows an equation d=18td=81​t where t=time(minutes) and d=distance(miles). (8.EE.5)

The materials reviewed for Amplify Desmos Math Grades 6 through 8 meet expectations for having assessment information included in the materials to indicate which standards are assessed. 

Formal assessments, including Sub-Unit Quizzes and End-of-Unit Assessments, are consistently aligned to grade-level content standards and Mathematical Practice Standards. This alignment is clearly identified in the program’s Assess and Respond section.

Examples include:

• Grade 6, Unit 8: Describing Data, End-of-Unit Assessment: Form A, Screen 16, Problem 7, “Create a dot plot with: At least five points. A median of 6. A mean that is less than the median.” The Assess and Respond Item Analysis denotes the standards assessed as 6.SP.4 ,6.SP.5c, and MP6.

• Grade 7, Unit 7: Angles, Triangles, and Prisms, Sub-Unit Quiz, Screen 8, Problem 5c, “Here are three lines that intersect at one point. Laila wrote the equation x+18=90x+18=90. Describe the error that Laila might have made.” The Assess and Respond denotes the standards assessed as 7.EE.4, 7.G.5, MP3 and MP7.

• Grade 8, Unit 8: The Pythagorean Theorem and Irrational Numbers, Sub-Unit Quiz, Screen 5, Problem 4, “Drag the movable points to plot these values on the number line.” Students are given the points 643364

, 2727​, 99​, 7337​. The Assess and Respond denotes the standards assessed as 8.NS.2 and MP7.

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 8 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Summative assessments include Sub-Unit Quizzes and End-of Unit Assessments. The assessments measure students’ progress toward key concepts addressed in individual lessons as well as broader concepts that span an entire unit. Across all grades, assessments consistently reflect the full intent of grade-level content and practice standards. They utilize a variety of item types, including: multiple choice, select all, short answer/fill in the blank, extended response prompts, graphing, mistake analysis, matching, constructed response and technology-enhanced items. 

Examples include:

• Grade 6, Unit 7: Positive and Negative Numbers, Sub-Unit Quiz and End-of-Unit Assessment: Form A, develops the full intent of 6.NS.6c as students identify, plot, and label both positive and negative rational numbers on a horizontal number line and on a coordinate plane. Sub-Unit Quiz , Screen 3, Problem 2 states, “1. Drag each number to its approximate location on the number line. 2. Plot and label the opposite of each number on the number line.” Students are given a number line with 0 and 1 labeled, and given two numbers −4−4 and 8338​ to place on the number line. End-of-Unit Assessment: Form A states, “Problem 3. Plot and label these numbers on the number line: −43−34​, 33, −2−2, −45−54​, 4334​. Problem 6a. Here are four points. A (2, 4) B (2, ‐3) C (‐3, 0) D (‐3, 2). a. Plot and label each point.” Students are provided a coordinate plane ranging from 0 to 5 both horizontally and vertically.

• Grade 7, Unit 3: Measuring Circles, End-of-Unit Assessment: Form B, Problem 7, develops the full intent of MP7 as students look for and make use of structure while solving a real-world problem finding the area of a circle. Problem 7 states, “Brielle needs to paint the bottom of two pools. One can of paint covers 125 square feet. a. Pool A is a circular pool with a diameter of 24 feet. How many square feet does Brielle need to cover with paint? b. What is the fewest number of cans of paint Brielle must buy to cover the bottom of Pool A? Explain your thinking. c. Pool B is a circular pool with a diameter of 48 feet. Brielle says that twice the amount of paint is needed to cover the bottom of Pool B. Is Brielle’s claim correct? Yes No Not enough information. Explain your thinking.”

• Grade 8, Unit 6: Associations in Data, End-of-Unit Assessment: Form B, Screen 9, Problem 6b, and Screen 10, Problem 6c, develops the full intent of 8.SP.4 as students use a two-way table to generate a relative frequency table. The materials state, “6b. This two-way frequency table shows the number of adults and children who prefer pizza or hot dogs. Complete the relative frequency table by row. Round to the nearest percent. Use paper or a calculator if that helps with your thinking. 6c. Here are the frequency and relative frequency tables from the previous screen. Is there evidence of an association between age and preference for pizza or hot dogs? Explain your thinking.” A table is provided showing the preference of hot dogs and pizza for both adults and children. Students use this data to complete the relative frequency table.

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 8 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials provide students with consistent opportunities to engage in the full intent of all Grade 6–8 standards. Each lesson includes a Warm-Up, one or more Activities, Lesson Practice, a Lesson Synthesis, and a Show What You Know task. Units also include Pre-Unit Checks and Practice Days. Pre-Unit Checks help identify students’ prior knowledge and readiness for the unit content. Practice Days offer time for students to apply and reinforce the knowledge and skills developed throughout the unit.

Examples include:

• Grade 6, Unit 4: Dividing Fractions, Lesson 4, Lesson Practice (Screen 4, Problem 4 and Screen 7, Problem 7) engages students with 6.NS.1(Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem). Students interpret quotients of fractions and create word problems involving division of fractions. The materials state, “4. Abena wrote the expression 6÷346÷43​ to represent how many potatoes fill 1 planter. Describe a situation that represents 8÷458÷54​. 7. A painter is making a mural. They use 3 gallons of paint for 3883​ of the mural. How many gallons of paint would they need to paint the whole mural?” 

• Grade 7, Unit 6: Expressions, Equations, and Inequalities, Lesson 16, (Screen 7, Problem 7 and Screen 8, Problem 8) engages students with the full intent of 7.EE.4b (Solve word problems leading to inequalities of the form px+q>rpx+q>r or px+q, where pp, qq, and rr are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem). Students work with inequalities to convince someone that their response is not correct. The problem states, “7a. Solve 25−4x<125−4x<1. 7b. Graph the solutions to the inequality. 8. Chloe made a mistake solving the inequality 25−4x<125−4x<1. Chloe says the solutions to the inequality are x<6x<6. Explain what you think is incorrect about Chloe’s work.” A graph modeling Chloe’s incorrect solution is shown.

• Grade 8, Unit 5: Functions and Volume, Lesson 3, Screen 8, engages students with the full intent of 8.F.1 (Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output). Students complete a card sort. An example is as follows: “Sort the graphs based on whether they represent a function or not.”

The materials reviewed for Amplify Desmos Math Grades 6 through 8 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade:

In Grade 6:

• 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 supporting work connected to the major work) is 75 out of 112, 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 96 out of 142, which is approximately 68%.

In Grade 7:

• 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 supporting work connected to the major work) is 62 out of 94, which is approximately 66%.

• The approximate number of days devoted to major work of the grade (including supporting work connected to the major work) is 83 out of 123, which is approximately 67%.

In Grade 8:

• The approximate number of units devoted to major work of the grade (including supporting work connected to the major work) is 6 out of 8 which is approximately 75%.

• The approximate number of lessons devoted to major work (including supporting work connected to the major work) is 72 out of 98, which is approximately 73%.

• The approximate number of days devoted to major work of the grade (including supporting work connected to the major work) is 96 out of 130, which is approximately 74%.

An instructional day analysis across Grades 6 through Grade 8 is most representative of the instructional materials as the days include major work, supporting work connected to major work, and the assessments embedded within each unit. Any day marked optional was excluded. As a result, approximately 68% of the materials in Grade 6, 67% of the materials in Grade 7, and 74% of the materials in Grade 8 focus on major work of the grade.

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 8 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Materials are designed so that supporting standards/clusters are connected to the major standards/ clusters of the grade. These connections are listed for teachers within the Teacher’s Edition. 

Examples include:

• Grade 6, Unit 7: Positive and Negative Numbers, Lesson 12, Lesson Practice 6.7.12, connects the supporting work of 6.G.3 (Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems) to the major work of 6.NS.8 (Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate). Problem 8 states, “Gabriela drew a quadrilateral with these descriptions: It has exactly one pair of parallel sides. Two of the vertices are (‐4, 5) and (2, ‐3). At least one side has a length of 5 units. Create a quadrilateral that Gabriela could have made.” A blank four-quadrant coordinate plane, ranging from –5 to 5 on both axes, is shown.

• Grade 7, Unit 8: Probability and Sampling, Lesson 3, Screen 11, Show What You Know, connects the supporting work of 7.SP.6 (Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability) to the major work of 7.RP.A (Analyze proportional relationships and use them to solve real-world and mathematical problems). The problem states, “A new bag has 5 blocks. Some are red and some are blue. Based on these results, how many blocks are likely to be red?” An image shows Total Picks: 100, Red 63 and Blue 37. 

• Grade 8, Unit 8: The Pythagorean Theorem and Irrational Numbers, Lesson 14, Lesson Practice 8.8.14, connects the supporting work of 8.NS.1 (Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number) to the major work of 8.EE.2 (Use square root and cube root symbols to represent solutions to equations of the form x2=px2=p and x3=px3=p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 22

​ is irrational). Problem 3 states, “Determine whether each number is rational or irrational. 11616​1​, 1010​, −213−231​, 0.5320.532.”

The materials reviewed for Amplify Desmos Math Grades 6 through Grade 8 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.

Connections among the major work of the grade are present throughout the materials where appropriate. These connections are listed for teachers in the Teacher’s Edition within each Unit Overview and may appear in one or more phases of a typical lesson: Warm-up, Activity, Synthesis, or Show What You Know. 

Examples include:

• Grade 6, Unit 7: Positive and Negative Numbers, Lesson 7, Lesson Practice 6.7.07 connects the major work of 6.NS.C (Apply and extend previous understandings of numbers to the system of rational numbers) to the major work of 6.EE.B (Reason about and solve one-variable equations and inequalities). Problems 8 & 9 state, “One day in Boston, the temperature was above 52°52°Fand below 60°60°F. 8. Make two graphs, one to represent temperatures above 52°F and another to represent temperatures below 60°F. 9. Write two inequalities to represent the possible temperatures, T, on that day.” Students are given two separate number lines both ranging from 30 to 80.

• Grade 7, Unit 5: Operations With Positive and Negative Numbers, Lesson 5, Screen 3, Puzzle #2, connects the major work of 7.NS.A (Apply and extend previous understandings of operations with fractions) to the major work of 7.EE.B (Solve real-life and mathematical problems using numerical and algebraic expressions and equations). The materials state, “Make true equations by dragging and flipping the cards. Try to use as few flips as possible.” Students are provided an equation tool with the following equations: an unknown plus 5 is equal to an unknown, and an unknown subtracted from an unknown is equal to 9. The original card choices are: 1, 2, 3, 4, -5, -6, -7 and -8. All cards can be flipped from positive to negative and vice versa.

• Grade 8, Unit 8: The Pythagorean Theorem and Irrational Numbers, Practice Day 2, Student Edition, connects the major work of 8.G.B (Understand and apply the Pythagorean Theorem) to the major work of 8.EE.A (Expressions and Equations Work with radicals and integer exponents). Problem 6 states, “A 17-foot ladder leans against a wall. The ladder reaches a window 15 feet up the wall. How far from the wall is the base of the ladder? Show your thinking.” An image of a ladder is provided.

The instructional materials reviewed for Amplify Desmos Math Grades 6 through Grade 8 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. 

The Teacher Edition includes Unit and Lesson Overviews that identify content standard connections. Each Unit features a Math of the Unit section and a Connections to Future Learning component, both of which illustrate how current concepts relate to prior and future standards within the course and across grade levels. At the lesson level, materials specify the standards addressed and indicate how each lesson builds on prior learning, addresses current content, and/or prepares for future learning, categorized as Building On, Addressing, or Building Toward.

An example of a connection to future grades in Grade 6 includes:

• Unit 3: Unit Rates and Percentages, Connections to Future Learning connects 6.RP.2 (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) to the work related to proportional relationships (7.RP.2). Connections to Future Learning states, “In this unit, students continue their work with ratios and unit rates. In Grade 7, students will apply these skills to proportional relationships with fractions. Example: A 4-by-6-inch photograph can be scaled and printed to be many different sizes. In the table, each value in the second column is 3223​ times the length of the value in the first column.”

An example of a connection to prior knowledge in Grade 6 includes:

• Unit 6: Expressions and Equations, Lesson 1, Overview connects 6.EE.6 (Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set) and 6.EE.7 (Solve real-world and mathematical problems by writing and solving equations of the form x+p=qx+p=q and px=qpx=q for cases in which p, q and x are all nonnegative rational numbers) to the work related to solving multi-step word problems using the four operations (4.OA.3). Prior Learning states, “In Grade 4, students used letters to represent unknown amounts. In Unit 3, they used tape diagrams to reason about percentages and determine unknown values.”

An example of a connection to future grades in Grade 7 includes:

• Unit 1: Scale Drawings, Connections to Future Learning connects 7.G.1 (Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale) to the work related to understanding congruence and similarity (8.G.3 and 8.G.4). Connections to Future Learning states, “In this unit, students use scale factors to create and compare scaled copies. In Grade 8, Unit 2, students will apply this to understand similar figures, which are scaled copies of one another. Example: Figures are similar if one can fit exactly over the other after rigid transformations (translations, rotations, and reflections) and dilations. Dilations are a transformation in which each point on a figure moves along a line and changes its distance from a fixed point (called the center of dilations). Each distance is multiplied by the same scale factor. In this example, each point in AB′C′D′ is twice as far from the center of dilation (A) as it is in ABCD. The scale factor from figure ABCD to figure AB′C′D′ is 2, and ABCD is similar to AB′C′D′.”

An example of a connection to prior knowledge in Grade 7 includes:

• Unit 7: Angles, Triangles, and Prisms, Lesson 1, Overview connects 7.G.5 (Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure) to the work related to understanding concepts of angle and their measurements (4.MD.5). Prior Learning states, “In Grade 4, students described and estimated angle measures including right, acute, obtuse, and straight angles. In Unit 6, students learned how to write and solve equations for unknown values.”

An example of a connection to future grades in Grade 8 includes:

• Unit 6: Associations in Data, Connections to Future Learning connects 8.SP.2 (Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line) to the work related to correlation coefficients in high school (HSS.ID.8). Connections to Future Learning states, “In this unit, students fit linear functions to scatter plots of bivariate data. In High School, students will use technology to calculate the correlation coefficient of a linear relationship to assess the strength and direction of the linear correlation. Example: The r-value, also called the correlation coefficient, is a number between –1 and 1 that describes the strength (weak, strong) and direction (negative, positive) of the linear association.” Examples show Strong and Positive r=0.9r=0.9, Strong and Negative r=−0.8r=−0.8, Weak and Positive r=0.3r=0.3, and Weak and Negative r=−.04r=−.04.

An example of a connection to prior knowledge in Grade 8 includes:

• Unit 4: Linear Equations and Linear Systems, Lesson 1, Overview connects 8.EE.7 (Solve linear equations in one variable) and 8.F.1 (Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output) to the work related to solving equations using different representations (7.EE.4). Prior Learning states, “In Grade 7, students worked with different representations to solve equations, including hanger diagrams. In Unit 3, students were introduced to the term linear relationship, but did not yet get the opportunity to practice solving linear equations algebraically.”